Growth model of the reared sea urchin Paracentrotus ... - SciViews

U L
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bio mar
UNIVERSITE LIBRE DE BRUXELLES
FACULTE DES SCIENCES
LABORATOIRE DE BIOLOGIE MARINE
Growth model of the reared sea urchin
Paracentrotus lividus (Lamarck, 1816)
Committee:
Prof. G. Josens (president)
Prof. Ph. Dubois (secretary)
Prof. M. Jangoux
Prof. M. Russell
Prof. J.-L. Deneubourg
Prof. Ch. Lancelot
Thesis submitted
in fulfillment of
the degree of Doctor
in Agronomic Sciences
and Biological Engineering
Supervisor: Prof. M. JANGOUX
Philippe GROSJEAN – September 2001

To my mother for her patience
To my father for showing me the way
To 'Zazouille' for all she gave during 7 years
1

Acknowledgements
ACKNOWLEDGEMENTS
Well, well, well… this thesis is written in English, fine! But I do not
feel confident enough with Shakespeare's language to express my feelings.
So, let's switch to French for one page or two…
En tout premier lieu, je tiens à exprimer toute ma gratitude au
Professeur Michel JANGOUX pour m'avoir accueilli dans son laboratoire
(ou devrais-je dire, dans l'annexe ô combien humide et salée de nos locaux
à la Station Marine de Luc-sur-mer). Je le remercie de m'avoir témoigné
toute sa confiance et d'avoir tout fait pour que mes conditions de travail
soient aussi optimales que possible.
Je tiens également à remercier les professeurs Claude LARSONNEUR,
Jacques AVOINE et Marie-Paule CHICHERY pour m'avoir accueilli au
Centre Régional d'Etudes Côtières. Merci à Didier BUCAILLE pour s'être
occupé des élevages avec tant de minutie. Les techniciens de la Station
Marine (Jean-Paul LEHODEY, Jean-Pierre DESMASURES et Alain
SAVINELLI) méritent un énorme bravo pour leur travail de qualité et pour
leur aide précieuse dans la construction de matériel échinicole spécialisé.
Je me dois également de signaler combien le travail des animaliers de tout
poil (objecteurs, C.E.S., étudiants) a été vital et je les en remercie, en
particulier Alexis DECTOT. Enfin, je remercie Brigitte GARCIA pour
s'être acquittée de son travail d'intendance –et même plus– avec efficacité
et… humour.
A toute l'équipe "oursin", j'adresse mes remerciements du fond du cœur
tant pour la coopération sur le plan professionnel, que pour les aprèsboulots
mémorables: Christine SPIRLET, Pol GOSSELIN, Devaragen
VAITILINGON, Jean-Marc OUIN, Cristina DE AMARAL, Raphaël
MORGAN, Corentin CAM, Yolaine BEYENS, Hélène RABAHIE, ainsi
que les étudiants et étudiantes qui ont transité de façon plus brève dans
l'équipe, trop nombreux que pour être cités tous, qu'ils m'en excusent.
3

Acknowledgements
Je voudrais également exprimer toute ma reconnaissance à Michel,
Marie-Pierre, Ludo, Véronique, Joël, Alexandra, Patrice, Bernard, Jeloul,
Jeff, Céline, Jean-Paul, Julie, Laurent, François, Laurence, Roseline,
Stéphane, et bien sûr à Isabelle, pour tous les bons moments passés en leur
compagnie. Je tiens aussi à remercier les laboratoires de Biologie Marine
de Bruxelles et de Mons pour leur acceuil.
Je remercie Christian VAN OSSELAER pour ses encouragements, ses
discussions fructueuses et aussi pour LE conseil qui m'a permis de finir
cette thèse: "Saint-John's Wort". A ma famille, j'exprime ma
reconnaissance pour m'avoir soutenu dans mon travail et pour sa présence
dans les moments difficiles.
Enfin, bien que ce ne soit pas usuel, je crois utile de signaler qu'un
certain nombre de personnes ont rendu ce travail possible de manière
indirecte. Ainsi, c'est en parcourant les écrits de Ludwig VON
BERTALANFFY, de Thomas EBERT et de Roger KOENKER que… plaf
(bruit de la main qui frappe le front), bon sang, mais c'est bien sûr…! Des
trois, je n'ai eu l'occasion de rencontrer que Thomas EBERT, et je me
souviens encore de ses yeux écarquillés comme des billes de loto lorsque
j'ai essayé de lui expliquer comment un modèle flou défuzzifié pouvait être
une solution au problème qui nous préoccupait… A la réflexion, j'espère
être plus explicite par écrit et après avoir maturé la question,… sinon, je
risque bien de me retrouver de nouveau face à des billes de loto à la
soutenance! Enfin, je voudrais adresser un très grand merci à tous les
programmeurs qui ont fait de "R" un logiciel statistique aussi fantastique.
Ce travail a été rendu possible par la collaboration entre le laboratoire
de Biologie Marine de l'Université Libre de Bruxelles (Belgique) et le
Centre Régional d'Etudes Côtières de l'Université de Caen (France). Il a pu
être réalisé grâce à l'appui financier de la Commission Européenne
(contrats FAR AQ2.530 BFE "Sea urchins cultivation" et FAIR CT96-
1623 BFN "Biology of sea urchins under intensive cultivation [closed
cycle echiniculture]").
4

Abstract
ABSTRACT
A rearing protocol for the edible European sea urchin Paracentrotus
lividus in a closed cycle (control of the whole life cycle of the echinoid)
and in a recirculating system (control of the environment around the
echinoid) is set up and tested at a pilot scale. This protocol is used to
experiment on growing postmetamorphics whose age and genetic origin
are perfectly known. Among the various measurements of size, we
determined that the test diameter is both rapid and accurate for quantifying
somatic growth. Causes and mechanisms of asymmetrical, or even
sometimes multimodal, size distributions among previously homogeneous
cohorts are studied. Results evidence the existence of a size-based
intraspecific competition, causing a reversible growth inhibition of smaller
individuals. A new growth model (called 'fuzzy-remanent'), including a
component of intraspecific competition, is elaborated by defuzzifying a
fuzzy model. Traditional least-square regression is abandoned in favor of
quantile regression to fit it. Both the model and the regression method are
adapted to include individual variations (we call this an 'envelope model').
This envelope model has functionally interpretable parameters. One of
them quantifies the degree of inhibition caused by intraspecific
competition. Since many similar fuzzy-remanent functions can be designed
and fitted with this method, this approach is promising to model growth of
other organisms in a functional way. This model rehabilitates von
Bertalanffy's theory on individual growth. Moreover, the latter theory is
now verified for Paracentrotus lividus, despite the observation of an initial
lag phase in growth. A functional classification of growth curves is
proposed.
Keywords: sea urchin, growth model, intraspecific competition, quantile
regression, fuzzy logic, aquaculture, Paracentrotus lividus.
5

Abstract
6

Résumé
RESUME
Un protocole d'élevage pour l'oursin comestible européen Paracentrotus
lividus en cycle fermé (contrôle de tout le cycle de vie de l'échinide) et dans un
système à recirculation d'eau (contrôle de l'environnement autour de
l'échinide) est mis au point et testé à l'échelle pilote. Ce protocole est utilisé
pour effectuer des expériences sur des individus postmétamorphiques en
croissance dont l'âge et l'origine génétique sont parfaitement connus. Parmi les
différentes manières de mesurer la taille de l'oursin, nous avons déterminé que
le diamètre de son test est à la fois une mesure rapide et précise pour quantifier
la croissance somatique. Les causes et les mécanismes responsables de
distributions de tailles asymétriques, voire parfois multimodales au sein de
cohortes initialement homogènes sont étudiés. Les résultats démontrent la
présence d'une compétition intraspécifique basée sur la taille. Cette
compétition entraîne une inhibition réversible des plus petits individus. Un
nouveau modèle de croissance (dit 'à rémanence floue'), incluant une
composante de compétition intraspécifique, est élaboré par défuzzification
d'un modèle flou. La traditionnelle régression par les moindres carrés est
abandonnée au profit de la régression quantile pour son ajustement. Tant le
modèle que la méthode de régression sont modifiés pour inclure les variations
individuelles (ce que nous appelons un 'modèle enveloppe'). Ce modèle
enveloppe présente des paramètres que l'on peut interpréter fonctionnellement.
L'un d'eux quantifie le degré d'inhibition occasionnée par la compétition
intraspécifique. Etant donné que beaucoup de modèles à rémanence floue
peuvent être conçus et ajustés de la sorte, cette approche est prometteuse pour
modéliser la croissance d'autres organismes de manière fonctionnelle. Ce
modèle réhabilite la théorie de von Bertalanffy sur la croissance des
organismes. Cette théorie se vérifie par ailleurs dans le cas de Paracentrotus
lividus, malgré l'observation d'une phase de latence initiale dans sa croissance.
Une classification fonctionnelle des courbes de croissance est proposée.
Mots clefs: oursin, modèle de croissance, compétition intraspécifique,
régression quantile, logique floue, aquaculture, Paracentrotus lividus.
7

Résumé
8

Table of contents
TABLE OF CONTENTS
ACKNOWLEDGEMENTS.............................................................................. 3
ABSTRACT.................................................................................................. 5
RESUME ..................................................................................................... 7
TABLE OF CONTENTS................................................................................. 9
LIST OF FIGURES...................................................................................... 13
LIST OF TABLES ....................................................................................... 17
LIST OF EQUATIONS................................................................................. 19
LIST OF SYMBOLS .................................................................................... 23
FOREWORD.............................................................................................. 29
GENERAL INTRODUCTION ....................................................................... 31
Economical interest of sea urchins .........................................................................32
a. Sea urchin markets and fisheries .........................................................................32
b. Aquaculture potentials .........................................................................................34
Overview of the biology of Paracentrotus lividus ..................................................35
Growth models .........................................................................................................42
a. The exponential curve, a simple Malthusian growth model ................................42
b. The logistic function for asymptotic growth ........................................................44
c. The Gompertz model, an asymmetrical sigmoidal curve .....................................45
d. The von Bertalanffy curves ..................................................................................46
e. The Richards model, a flexible curve that contains many others.........................47
f. The Weibull model, a polyvalent and flexible function.........................................48
g. The Jolicoeur curve, another flexible model........................................................49
h. The Johnson model, a heavily asymmetrical sigmoid..........................................50
i. The Preece-Baines 1 model for human growth.....................................................51
j. The Tanaka model for indeterminate growth........................................................51
Modelling sea urchins growth.................................................................................52
a. Choice of the growth model for sea urchins ........................................................53
b. Fitting of growth models on real data for echinoids ...........................................56
AIM OF THE THESIS.................................................................................. 61
PART I: SET UP OF AN EXPERIMENTAL REARING PROCEDURE FOR
ECHINOIDS ............................................................................................... 65
9

Land-based closed-cycle echiniculture of Paracentrotus lividus (Lamarck)
(Echinoidea: Echinodermata): a long-term experiment at a pilot scale .............67
a. Abstract ................................................................................................................67
b. Introduction..........................................................................................................67
c. Material and methods...........................................................................................69
d. Results ..................................................................................................................78
e. Discussion ............................................................................................................83
f. Conclusions...........................................................................................................90
g. Acknowledgements...............................................................................................90
PART II: MEASUREMENT FOR SIZE IN THE SEA URCHIN ........................ 95
Comparison of three body-size measurements for echinoids ..............................97
a. Abstract ................................................................................................................97
b. Introduction..........................................................................................................97
c. Material and methods...........................................................................................98
d. Results and discussion .......................................................................................100
e. Conclusions ........................................................................................................104
f. Acknowledgements..............................................................................................104
Choice of measurement .........................................................................................105
PART III: EXPERIMENTAL STUDIES OF THE INTRASPECIFIC
COMPETITION ........................................................................................ 111
Experimental study of growth in the echinoid Paracentrotus lividus (Lamarck,
1816) (Echinodermata)...........................................................................................113
a. Abstract ..............................................................................................................113
b. Introduction........................................................................................................113
c. Material and methods.........................................................................................115
d. Results ................................................................................................................118
e. Discussion ..........................................................................................................124
f. Acknowledgements..............................................................................................126
Intraspecific competition: an additional experiment .........................................127
PART IV: A GROWTH MODEL WITH INTRASPECIFIC COMPETITION.... 135
A functional growth model with intraspecific competition applied to a sea
urchin, Paracentrotus lividus (Lamarck, 1816)....................................................137
a. Abstract ..............................................................................................................137
b. Introduction........................................................................................................138
c. Material..............................................................................................................141
d. Results ................................................................................................................144
e. Discussion ..........................................................................................................164
f. Conclusions.........................................................................................................177
g. Acknowledgments...............................................................................................178
GENERAL CONCLUSIONS ....................................................................... 181
REFERENCES.......................................................................................... 189
Table of contents
10

ANNEXES................................................................................................ 213
Annex I: R code for fitting growth models ..........................................................215
a. Code for analyzing data and fitting envelope models........................................215
b. The 'nlrq' package for nonlinear quantile regression........................................246
Annex II: dataset of the cohort measured during seven years ..........................255
Annex III: abstracts of publications and symposia ............................................257
a. International journals ........................................................................................257
b. Reports and other publications..........................................................................262
c. International symposia.......................................................................................265
Table of contents
11

Table of contents
12

List of figures
LIST OF FIGURES
Figure 1. Paracentrotus lividus in a tidal pool in Morgat, Brittany,
France. Page 36.
Figure 2. Paracentrotus lividus. A. Echinopluteus. B. Postlava a few
days after metamorphosis. Page 37.
Figure 3. External anatomy of a regular sea urchin. A. Oral view. B.
Aboral view. Page 38.
Figure 4. Internal anatomy of a regular sea urchin, side view. Page 39.
Figure 5. Mature adult Paracentrotus lividus with oral region removed
showing the five gonads. Page 40.
Figure 6. Location of the sampled Paracentrotus lividus population.
Page 41.
Figure 7. Example of an exponential curve. Page 43.
Figure 8. Example of a logistic curve. Page 44.
Figure 9. Example of a Gompertz curve. Page 45.
Figure 10. Both von Bertalanffy 1 and von Bertalanffy 2 curves.
Page 47.
Figure 11. Shape of Richards curves depending on values of m. Page 48.
Figure 12. Examples of Weibull curves with different values for m.
Page 49.
Figure 13. Examples of Jolicoeur curves with different values for m.
Page 50.
Figure 14. Example of a Johnson curve. Page 50.
Figure 15. Example of a Preece-Baines 1 curve. Page 51.
13

List of figures
Figure 16. Example of a Tanaka curve. Page 52.
Figure 17. Overview of the closed-cycle process and devices used to
produce sea urchins on land at a pilot scale. Page 71.
Figure 18. Changes with time in the size distribution and survival rate of
one fertilization issued from a single larval rearing tank and
followed over 7 years. Page 80.
Figure 19. Change with time in the biomass of a reared cohort of sea
urchins (the same batch as shown in Fig. 18). Page 81.
Figure 20. Data acquisition system. Page 99.
Figure 21. Principal components analysis of the 14 measurements.
Page 106.
Figure 22. Evolution of a single cohort of P. lividus (Fb) reared in stable
environmental conditions according to time. Page 120.
Figure 23. Size distribution of Fc juveniles in each batch in the
beginning of the experiment (A) and 4 months later (B).
Page 121.
Figure 24. Size distribution of Fd individuals in each batch at the
beginning of the experiment (A) and 4 months later (B).
Page 122.
Figure 25. Size distributions of the two different fertilizations used in the
additional experiment (Ff and Fg). Page 128.
Figure 26. Change in size distributions of the Ff and Fg batches (large,
mixed and small) with time. Page 130.
Figure 27. Change in size distributions of the Ff and Fg batches with
time after large individuals were removed from the mixed
batches. Page 131.
14

List of figures
Figure 28. A. Histograms of size distributions of a cohort of reared P.
lividus with time. Top of the box: a projection of three
quantiles (0.025, 0.5 and 0.075) issued from those size
distributions and also presented in B. Page 143.
Figure 29. Survival with time of the same reared cohort of P. lividus as
in Fig. 28A. Page 143.
Figure 30. Construction of the fuzzy growth model. Page 150.
Figure 31. First step of constraining parameters of the new growth model
(origin forced to {t0, D0}). Page 157.
Figure 32. A. Variation of l as a function of 1-τ for several quantile
regressions. B. Variation of k1 (black squares) and k2 (white
triangles) as functions of 1-τ. Page 159.
Figure 33. Envelope model fitted (upper surface) to the whole dataset
(lower surface). Page 162.
Figure 34. Diagnostic of the envelope model fitted in Fig. 33. A.
Contour plot of the residuals. B. Three "slices" cut in the 3Dsurfaces
of Fig. 33 at t' = 300 (1), 600 (2) and 1800 (3) days.
Page 163.
Figure 35. A classification of growth models based on their functional
features. Page 184.
15

List of figures
16

List of tables
LIST OF TABLES
Table 1. Models used to fit sea urchins growth data. Page 54.
Table 2. Age, density, number, and survival rate of sea urchins at each
rearing stage. Page 79.
Table 3. Gonadal and maturity indices of wild and reared sea urchins.
Page 82.
Table 4. Comparison of the three selected measurements for body size.
Page 101.
Table 5. Allometric relations between parameters for Paracentrotus
lividus from Morgat. Page 103.
Table 6. Principal components analysis: contribution of the parameters
to the three first axes. Page 107.
Table 7. Statistical analysis of the size frequency distribution of single
cohorts of P. lividus at different ages. Page 119.
Table 8. Statistics on the four batches of Fc in the beginning of the
experiment and after 4 months. Page 121.
Table 9. Statistics on the six batches of Fd in the beginning of the
experiment and after 4 months. Page 122.
Table 10. Size distribution of Fe echinoids after having been reared
individually (control) or together (experimental batches) for 4
months. Page 124.
Table 11. Statistics on the small, large and mixed batches (fertilizations
Ff, 'small' and Fg, 'large'). Page 129.
Table 12. Results of quantile regressions for three values of τ, using
different growth models. Page 154.
17

List of tables
Table 13. Results of quantile regressions for three values of τ , using the
new growth model. Page 155.
Table 14. Results of quantile regressions for different values of τ, using
the new growth model constrained to the origin. Page 156.
18

List of equations
LIST OF EQUATIONS
Equation 1. Exponential growth model: population increase by a fixed
proportion. Page 43.
Equation 2. Exponential growth model: differential equation. Page 43.
Equation 3. Exponential growth model: equation. Page 43.
Equation 4. Logistic model: differential equation. Page 44.
Equation 5. Logistic model: equation. Page 44.
Equation 6. 4-parameter logistic model. Page 45.
Equation 7. Gompertz model: differential equation. Page 45.
Equation 8. Gompertz model: equation. Page 45.
Equation 9. von Bertalanffy 2 model: differential equation. Page 46.
Equation 10. von Bertalanffy 2 model: equation. Page 46.
Equation 11. von Bertalanffy 1 model. Page 46.
Equation 12. Richards model. Page 47.
Equation 13. Weibull model. Page 48.
Equation 14. Jolicoeur model. Page 49.
Equation 15. Johnson model. Page 50.
Equation 16. Preece-Baines model 1. Page 51.
Equation 17. Tanaka model. Page 52.
Equation 18. Definition of SIW. Page 100.
19

List of equations
Equation 19. Calculation of ds, the apparent mean density of the skeleton
of a sea urchin using DWs / IW relationship. Page 101.
Equation 20. Moving average smoothing applied to size-frequency
distributions. Page 117.
Equation 21. Objective function (deviance δ1) of the quantile regression.
Page 146.
Equation 22. Piece-wise linear function used to calculate the deviance δ1
in eq. 21. Page 147.
Equation 23. Definition of the relative time-scale t'. Page 149.
Equation 24. Function of size D with relative time t' for the set S in the
fuzzy model. Page 150.
Equation 25. Function of size D with relative time t' for the set L in the
fuzzy model. Page 151.
Equation 26. Membership function to the set L with relative time t'.
Page 151.
Equation 27. Membership function to the set S with relative time t'.
Page 151.
Equation 28. Defuzzification of the fuzzy model. Page 152.
Equation 29. Equation of the defuzzified model after simplification.
Page 152.
Equation 30. Definition of the relative size-scale D'. Page 155.
Equation 31. The new growth model with relative size D' and relative
time t'. Page 155.
Equation 32. Parameter l in function of τ in the envelope model.
Page 158.
20

List of equations
Equation 33. Parameters k1 and k2 in function of τ in the envelope
model. Page 158.
Equation 34. Parameter ∆D∞ in function τ in the envelope model.
Page 160.
Equation 35. Analytic function of the envelope model. Page 160.
Equation 36. Calculation of ˆ τ . Page 160.
Equation 37. Objective function (deviance δ2) for the quantile regression
modified for envelope models. Page 161.
Equation 38. Reparameterization of the 4-parameter logistic function.
Page 174.
Equation 39. Reparameterized 4-parameter logistic function after
simplification. Page 174.
Equation 40. A growth model that is both 'dimensional' and 'transitional'
at the same time. Page 175.
Equation 41. Equation defining the relation between metabolic time tM
and time t'. Page 175.
Equation 42. Generalized von Bertalanffy model. Page 175.
21

List of equations
22

List of symbols
LIST OF SYMBOLS
Variables and parameters are in italic; function names are in roman
type according to standard mathematical notation. For instance, e (in
italic) is the name of a variable and e (in roman) is the exponentiation
function as in y = e x and thus e equals Euler constant: 2.71282.
α, α, α, α, ββ
ββ
Parameters of the Huxley's allometric equation y = α·x β .
∆D∞ Maximum increase in diameter of the test of a sea urchin from a
defined initial value (usually, at metamorphosis) D0 to the asymptotic
maximum diameter D∞ (in mm).
∆Y∞ Maximum increase in size from a defined initial size (at birth, at
metamorphosis…) Y0 to the asymptotic size Y∞ (same unit as Y).
δδδδ1 Deviance of quantile regression, according to Koenker & Bassett
(1978) (same unit as the dependent variable y in the model).
δδδδ2 Deviance of quantile regression; modified version for envelope
models (same unit as the dependent variable y in the model).
δδδδs Apparent mean density of the skeleton of a sea urchin (in g/l).
ττττ Quantile (0 < τ < 1) of a distribution or of a quantile regression
(dimensionless).
ˆ ττττ Estimator of the quantile τ. Value calculated after a sample of the
distribution (dimensionless).
ωωωω Parameter corresponding to k·Y∞ in the von Bertalanffy equation
Y = Y∞·(1 – e -k·(t – t 0 ) ), as proposed by Gallucci et al (1979) to solve
the problem of intercorrelation between k and Y∞ (same unit as Y·t -1 ).
ξξξξ1 Value returned by a function of the form Y = f(t) at time t and for the
current solution for the parameters (same unit as Y).
23

List of symbols
ξξξξ2 Value returned by a function of the form Y = f(t, τ) (envelope model)
at time t and quantile τ and for the current solution for the parameters
(same unit as Y).
a, b, c, d, e Parameters in growth models with no particular functional
meaning or used in a context where the possible functional meaning
has no importance (fitting for descriptive purpose only) (units are
context-dependent).
D Diameter of the test of a sea urchin measured at the ambitus and
considered without spines (in mm).
D' Relative diameter of the test of a sea urchin: increase of the diameter
from a defined initial value (usually at metamorphosis) D0 to the
current value D (in mm).
D0 Diameter of the test of a sea urchin at time t = t0, usually at
metamorphosis (in mm).
D∞ Maximum asymptotic diameter of the test of a sea urchin
(determinate growth) (in mm).
DWs Dry weight of the skeleton of a sea urchin (in g).
Fx A reared batch of sea urchins issued from a single fertilization x used
in an experiment. If several batches issued from the same fertilization
are used, they are further labeled with numbers (Fx1, Fx2…).
fi
A frequency observed in the i th class of a size distribution
(dimensionless).
fsi Same as fi but after applying a moving average smoothing
(dimensionless).
GI Gonad index: the ratio between the weight of the gonads and the total
weight of a sea urchin, either in wet or in dry weight (dimensionless).
24

List of symbols
IW Immersed weight: the weight of a sea urchin measured when
immersed in seawater (in g). See also SIW.
k Kinetic parameter in a growth model. If there are different kinetic
parameters in the same model, they are further labeled with numbers:
k1, k2… (in day -1 ).
L Fuzzy set representing the largest possible size (in mm). See also ML
and S.
l Lag parameter in a growth model with an intraspecific competition
component. Indicate the length of the lag phase, i.e., the degree of
inhibition (dimensionless).
m Parameter in a dimensional model that indicates the power
transformation to apply to be in the best dimension for describing
growth (dimensionless?).
Md Mass density of seawater (in g/l).
MI Maturity index: the arithmetic mean of all maturity stages observed
in a batch. An eight-stage scale of maturity was defined by Spirlet et
al (1998a) for the sea urchin gonads (dimensionless).
ML Membership function for the set L in a fuzzy model with 0 ≤ ML ≤ 1
(dimensionless). See also L and MS.
MS Membership function for the set S in a fuzzy model with 0 ≤ MS ≤ 1
(dimensionless). See also S and ML.
n Number of observations in a dataset (dimensionless).
p Probability of a statistical test, or probability of a value according to
a statistical distribution with 0 ≤ p ≤ 1 (dimensionless).
S Fuzzy set representing the smallest possible size (in mm). See also
MS and L.
25

List of symbols
s Parameter of the envelope model. Slope of the linear relationship
l = s·(1 - τ). (dimensionless). See also l.
SCT Standard competence test. Determine if sea urchin larvae are ready to
metamorphose; adapted from Gosselin & Jangoux, 1996).
SIW Standard immersed weight. The apparent weight of a sea urchin in
seawater, standardized according to eq. 18 (in g).
t Chronological time with an arbitrary origin (in days).
t' Relative chronological time, that is, with a defined origin (usually
corresponding to the metamorphosis event) (in days).
t0
Time at a defined origin (usually corresponding to the
metamorphosis event) (in days).
tM Metabolic or physiologic time, that is, a time-scale that is modulated
by environmental parameters in the same proportions as they change
the metabolic rate of the organism (in days).
W A measure of the size of an animal using a weight measurement (in
g).
Y A measure of the size of an animal, using any kind of measurement.
Note that y corresponds to any kind of dependent variable, while Y is
a dependent variable that expresses the size of an individual, or of a
population (unit si context-dependent).
Y' Relative size: increase of the size from a defined initial value (at
birth, at metamorphosis…) Y0 to the current value Y (same unit as Y).
Y0 The size of an animal at time t = t0 (usually at birth or
metamorphosis) (same unit as Y).
Y∞ Maximum asymptotic size (determinate growth) (same unit as Y).
26

Introduction
27

Foreword
FOREWORD
This thesis is organized in four successive parts accordingly to a
logical progression in the scientific approach. Published or submitted
papers constitute the body of each section. Some supplemental material is
added when further exploration was made after the publication of the
papers.
In this document, results of some statistical tests are presented inline in
an abridged form, like: Student test, p < 0.001. Current tendency –at least
in international journals– is to favor a more detailed presentation, e.g.,
Student test, t = 3.858, df = 84, p < 0.001. We believe it does not bring
additional vital information, but it just surcharges text. A small table
summarizes much better statistical results where required.
Also I actively participated in scientific works that are marginal to the
main body of the present thesis. These works are shortly presented
(abstract of publications) as annexes (see Annex III).
29

Foreword
30

General introduction
GENERAL INTRODUCTION
This work was initiated in the context of the global overexploitation of
the natural resources of sea urchin fisheries (Allain, 1972a, 1972b; Le
Gall, 1987, 1990; Ledireac'h, 1987; Conand & Sloan, 1989; Hagen,
1996a). Recently this issue raised considerable interest in sea urchin
aquaculture (echiniculture) (Le Gall, 1990; Cellario & Fenaux, 1990; de
Jong-Westman, 1995a, 1995b; Fernandez, 1996; Hagen, 1996a; Blin,
1997; Kelly et al, 1998; Spirlet et al, 2000, 2001). As a consequence of
their differences [sea urchins are radically different than most marine
species usually farmed (finfishes, molluscs or crustaceans)], specific
rearing methods had to be developed. Our knowledge about the biology of
these animals was too fragmentary and several national or international
programs were established to lead ecophysiological studies of fished
echinoid populations and of the echinoids in culture. In this context, we
worked on two successive European contracts: FAR AQ2.530 BFE "Sea
urchins cultivation" and FAIR CT96-1623 BFN "Biology of sea urchins
under intensive cultivation (closed cycle echiniculture)".
This project presented an opportunity to build an experimental rearing
facility in the Marine Station of Luc-sur-mer, Normandy, France ("Centre
de Recherche et d'Etude Côtière"), where it was possible to grow
thousands of echinoids in strictly controlled food and environmental
conditions. If experiments conducted in this facility shed light on critical
aspects of echiniculture, they are also beneficial to fundamental research
because of the opportunity to experiment at a larger scale than in the
laboratory. This project allowed us to collect exhaustive data on sea urchin
grow when age and genetic origin (artificial fertilizations) are known.
These results revive the "organic growth" (sensu von Bertalanffy, 1938)
models in echinoids.
This introduction is organized in four parts. First, we provide a
summary of the sea urchin fishery, aquaculture potentials, and markets for
Paracentrotus lividus (Lamarck), in the economical context that motivated
31

General introduction
this study. Second, a brief review of selected aspects of its biology
provides essential background for this work. The third section reviews the
historical development of growth models. Finally, use of these growth
curves with sea urchins is summarized and discussed.
Economical interest of sea urchins
a. Sea urchin markets and fisheries
The most important market in the world is Japan. Sea urchin roe (both
male and female gonads, uni in Japanese) are marketed under different
forms: fresh (65%), but also dried, salted, frozen or cooked (35%) (Saito,
1992; Hagen, 1996a). According to various authors, the main species
exploited in Japan are Strongylocentrotus intermedius (A. Agassiz), S.
nudus (A. Agassiz), Heterocentrotus pulcherrimus (A. Agassiz),
Pseudocentrotus depressus (A. Agassiz), Anthocidaris crassispina (A.
Agassiz) and Tripneustes gratilla (L.) (Fuji, 1967; Fuji & Kamura, 1970;
Fernandez, 1996; Hagen, 1996a). Both Strongylocentrotus droebachiensis
(Müller) and S. franciscanus (A. Agassiz) are imported from North
America (Kato, 1972; Sloan, 1985), while Loxechinus albus Molina is
imported from Chile (Gonzalez et al, 1993; Lawrence et al, 1997). The
Japanese market is quite stable, around 60,000 tons of fresh echinoids per
annum (Hagen, 1996a), since several years and accounts for more than
95% of the whole world sea urchin market. Current landings in Japan
average 14,000 tons per year. Japanese imports total approximately 5,000
tons of roe under different forms, corresponding to 40 to 50,000 tons of
live sea urchins.
The average price of roe on the Japanese market ranges from 18.6 €/kg
(750 BEF/kg) for the local production (fresh animals considered as top
quality), to 7.9 €/kg (320 BEF/kg) of fresh imported echinoids (Hagen,
1996a). These figures equate to a total market of approximately 657
32

General introduction
millions €/y (26.5 milliard BEF/y), 397 millions €/y of which is
importated.
The second largest market is France. Its landings are much smaller:
about 1,000 tons of live echinoids per year in the 1960s and 1970s. Since
these peak levels harvests have dropped to 250 to 350 tons per annum
(Allain, 1972a; Ledireac'h, 1987; Le Gall, 1987, 1990). Spain, Ireland and
Greece export to France and compensate for the reduced local production,
so the market was kept at 500 to 600 tons from 1988 to 1990 (Fernandez,
1996). According to the same author, 185 tons transited by Rungis (Paris)
in 1991. The major species in the French market is Paracentrotus lividus
(Lamarck), but Psammechinus miliaris (Gmelin) and Sphaerechinus
granularis (Lamarck) are also sold. In France, most sea urchins are
consumed fresh during the period when gonads are in an adequate
reproductive stage, i.e., between December and March (Ledireac'h, 1987).
The season limits the importation market.
Wholesale prices in Rungis fluctuate according to the roe quality
(freshness, size, color, maturity stage and taste). It ranges from 4.5 €/y
(180 BEF/y) to 17.8 €/y (720 BEF/kg) (Fernandez, 1996; Grosjean et al,
1998, see Part I). Briton, and to a lesser extent, Irish sea urchins are most
valued. Mediterranean strains are of lower quality because they do not
withstand travel as well as Briton or Irish strains.
Fishing sea urchins is very profitable during the 5 to 10 years after
starting harvesting new stocks. But after that short period of time, wild
populations decline due to the high efficiency and selectivity of fishing
techniques: most exploitable natural stocks are easily picked by hand at
low tide, or at least using simple equipment at shallow depths (Allain,
1972b; Ledireac'h, 1987; Le Gall, 1987). Growth speed is also too low in
some harvested species to allow replacement of large adults (M. Russell,
pers. com.). Few rules exist to limit overexploitation. Indeed, the biggest
problem is that animals must be collected before they fully mature, and
they have no opportunity to spawn. A lack of recruitment results from
33

General introduction
intense fishing and, consequently, a rapid decline of the standing stock
(Allain 1971, 1972a; Le Gall 1990; Campbell & Harbo, 1991). In addition,
removing most adults from a site probably has a negative impact on the
survival of the remaining juveniles. The later could be more susceptible to
predation because they are no longer protected by the "spine canopy of
adults" (Tegner & Dayton, 1977).
An example of such a decline in landings is the Japanese fisheries
which produced 23 to 28,000 tons of whole live sea urchins per year from
1967 to 1982 (Hagen, 1996a). Landings dropped to 14,000 since 1991,
despite the establishment of hatcheries (to seed in the field) and of
artificial feeding of sea urchins in harvested areas (Saito, 1992). Chile and
the U.S.A. also produce less than before, and only Canada and Korea are
still increasing harvests (Fernandez, 1996) because the sea urchin fisheries
are more recent there. Average worldwide landings are still stable but are
obviously not sustainable in a near future. Aquaculture is a necessary
alternative in all countries with sea urchin fisheries.
b. Aquaculture potentials
Japan was the first country to address the issue of overexploitation, and
initiated stock enhancement programs very early (Saito, 1992; Hagen,
1996a). These techniques include habitat enhancement (artificial reefs),
artificial feeding, translocation and building of hatcheries that produce
several millions of seed a year that are transplanted to the field. For
instance, a single hatchery in Hokkaido produces 11 million juveniles per
year (Hagen, 1996a). Hatcheries may be a solution to ensure recruitment
where harvesting eliminates adults before they spawn, but good natural
habitats are required, like large tidal pools, to give enough protection to
juveniles released in the field (Saito, 1992, Hagen, 1996a).
Another way to enhance production is through gonad enhancement.
With an adequate artificial diet, it is possible to increase gonad size (de
Jong-Westman, 1995a; Spirlet, 1999; Spirlet et al, 2000), particularly with
34

General introduction
diets rich in proteins (Klinger et al, 1997; Spirlet, 2001). Gonad
enhancement in culture is a necessity in Canada because sea urchins are at
the right stage of maturity during the winter. At this time, the sea is frozen
and the collection of sea urchins under the ice by scuba divers is a painful
and dangerous activity. One solution is to collect animals during autumn,
store them in tanks, and feed them with an adequate diet before marketing
them (Motnikar et al, 1997).
The use of cages in sea ranching operations is also an alternative and
may be used in mono- or polycultures (Keats et al, 1983; Kelly et al,
1998). As for any mariculture activity, degradation of cages by waves and
storms is a major problem, and site location is critical. Suitable sites are
limited, and there is often strong competition for space with other
mariculture activities like salmoniculture or mytiliculture on long lines.
Because of their grazing activity, sea urchins erode the cage nets and are
also a direct cause of depredation which increases maintenance costs
(Kelly et al, 1998).
The ultimate step in the aquaculture production of sea urchin is
independence from natural resources, that is, to control the whole life cycle
in culture, from spawning to gonad enhancement (Le Gall, 1990; Hagen,
1996a). This is the goal we established for the experimental facility in
Normandy. It is called "closed-cycle echiniculture" (Grosjean et al, 1998,
see Part I). Somatic growth of juveniles untill they reach market size is a
process that requires major improvements in current technology and is key
to the successful development of closed-cycle echiniculture. The present
work is devoted to achieving this goal.
Overview of the biology of Paracentrotus lividus
The common European sea urchin, Paracentrotus lividus (Lamarck,
1816) (Echinodermata : Echinoidea : Echinidae) is a marine invertebrate
that lives along European coasts of the North Atlantic (Ireland, Brittany,
Spain) and troughout the Mediterranean Sea. It colonizes two types of
35

General introduction
habitats: intertidal (or sometimes subtidal) rocky shores (Atlantic,
Mediterranean sea) and Posidonia oceanica (L.) beds (Mediterranean sea)
(Fernandez, 1996).
Figure 1. Paracentrotus lividus in a tidal pool in Morgat, Brittany, France. Echinoids are
hardly visible (dark patches) being partly burrowed in cracks and holes in the rock and
hidden by stones, empty patellid shells and seaweed fragments (covering behavior).
In rocky shores, echinoids have a burrowing behavior (Fig. 1). The
holes they bore in the rock protect them from waves and predators.
Juveniles and adults are abundant in tidal pools close to algal fields
(Laminaria spp., but mainly Laminaria digitata Lamouroux in Briton and
Irish coasts). These sedentary populations feed on drift algae fragments
brought in the pools by waves and water currents. Some infratidal dense
populations also exist, but they exhibit the same sedentary and hidden
behavior (Grosjean, pers. obs.).
In contrast, in the Mediterranean Sea, most populations grow in
Posidonia oceanica beds where they exhibit a circadian cycle of activity.
36

General introduction
During the day, they stay near the roots of the plants, away from predators.
At night, they climb to the top of the fronds and feed on their softest parts
(Nedelec et al, 1981).
Like almost all echinoids, P. lividus is gonochoristic and fertilization is
external. When animals are ripe in early spring (Allain, 1975; Spirlet et al,
1998a) and in some localities in autumn (Crapp & Willis, 1975;
Fernandez, 1996), spawning is synchronized and triggered by an external
signal, e.g., temperature change or disturbance (Spirlet et al, 1998a;
Spirlet, 1999).
Figure 2. Paracentrotus lividus. A. Echinopluteus. B. Postlava a few days after
metamorphosis.
Homolecithal eggs are fertilized in the water column and develop into
free-swimming planktotrophic larva characteristic of echinoids: an
echinopluteus (Fig. 2A). This larva develops 4, 6 and then 8 arms
supported by calcareous skeletal rods. After a few weeks, the
echinopluteus will develop a rudiment inside the wall of an epidermic
invagination (the vestibule) located on the right-hand side of the body
(Strathmann, 1978). When the larva becomes competent, it seeks a solid
37

General introduction
substrate to settle and metamorphose. An adequate chemical stimulus is
required (Gosselin & Jangoux, 1996). Metamorphosis lasts less than one
hour: the echinoid rudiment is evaginated and most larval tissues are
resorbed. The postlarva resembles a miniaturized adult (Fig. 2B) but has
no mouth and no anus, and is thus endotrophic (Gosselin & Jangoux,
1998). After a week the postlava has undergone some major changes and
becomes an exotrophic juvenile with a fully developed and functional
digestive tract. It then begins foraging.
Figure 3. External anatomy of a regular sea urchin. A. Oral view. B. Aboral view. (after
Reid, W.M,. In: Ruppert & Barnes, 1994).
From an anatomical point of view, the body of the postmetamorphic
echinoid has a quasi-spherical shape (for regular sea urchins such as P.
lividus) with a pentaradial symmetry (Fig. 3). Its shape is constrained by
an endoskeleton, located just under the epidermis, composed of calcareous
ossicles sutured together in a solid test. This test supports movable spines
that cover the body of the animal and are the origin of the name
Echinoidea, "like a hedgehog (porcupine)" (Ruppert & Barnes, 1994). P.
38

General introduction
lividus reaches a maximal test diameter of 65 to 70 mm (Grosjean, pers.
obs.).
Figure 4. Internal anatomy of a regular sea urchin, side view. (modified after Reid, W.M., In:
Ruppert & Barnes, 1994).
The regular sea urchin body can be divided in two hemispheres: an oral
pole where the mouth opens, directed towards the substratum, and an
opposed aboral pole bearing the anus. The mouth opens in a short pharynx
surrounded by a complex scraping apparatus –recall that P. lividus is a
grazer– called Aristotle's lantern (Fig. 4). It is composed of 5 pyramids
radially arranged around the mouth and each holds one tooth (Fig. 3A).
The digestive tract forms two complete turns around the inner side of the
test wall, one in one way and the other one in the opposite direction,
leaving much space in the internal cavity for gonads (Fig. 4).
The anatomy of the reproductive organs reflects the radial symmetry of
the animal (Fig. 5). Five gonads open in genital pores close to the anus and
are disposed radially in the coelomic cavity along the ambulacral zones.
They start developing when the echinoid is still very small, around 4 to 6
39

General introduction
mm (Spirlet et al, 1994). P. lividus becomes mature when it reaches a test
diameter of 20 to 25 mm (Grosjean, pers. obs.).
Figure 5. Mature adult Paracentrotus lividus with oral region removed showing the five
gonads. The individual at the top is a male, the two others are females (gonads of brighter
color).
Like any echinoderm, sea urchins have a water-vascular system which
is used for locomotion. Tube feet (podia) elongate and retract and their
sucker-shaped tip can glue and unglue to the substratum (Flammang, 1996;
Flammang et al, 1998). Locomotion in any direction is sometimes aided by
the movements of spines. P. lividus exhibits a gregarious behavior and
lives in aggregates where small individuals tend to stay under larger ones
(including inside holes, for populations living in rocky shores). This
behavior was clearly identified as a protective mechanism against
predators (Tegner & Dayton, 1977). Many sea urchins species, including
P. lividus, also cover their exposed aboral surfaces with shells, stones, and
algae for camouflage (Crook et al, 1999; see also Fig. 1).
Echinoids are key-species in several ecosystems such as kelp forests,
barren grounds or Posidonia beds (Tegner & Dayton, 1981; Rowley, 1989;
Fernandez, 1996; Leinass & Christie, 1996). Emson (1984) suggested that
40

Brittany
General introduction
some features of echinoderms, namely the "position, size and the method
of formation of the skeleton" and "the substitution of collagenous tissues
for muscles", in addition to their low metabolic rate may have given them
competitive advantages over other animals. Among echinoderms, sea
urchins are the more mineralized ones, and their competitive advantage in
some marine ecosystems could probably be summarized by "bone idle – a
recipe for success" as proposed by Emson.
Morgat
Brest
Figure 6: Location of the sampled Paracentrotus lividus population.
Echinoids reared in the Luc-sur-mer facility originated from a single
population in Morgat, Brittany, France (Fig. 6, but see also Fig. 1). They
were collected at low tides from tidal pools. Individuals used in these
experiments were first or second generation sea urchins reared from the
egg.
41

Growth models
General introduction
The previous section on the biology of P. lividus was brief because we
do not need much information to model growth. Indeed, when modelling
growth, the animal is mainly regarded as a black box. We care about its
global change through time, but we do not have to detail all the complex
biochemical and physiological processes (feeding, digestion, assimilation,
respiration, excretion, etc…) or anatomical modifications that lead to such
changes. On the other hand, a good understanding of the various growth
curves and their respective shapes and properties is required. This section
is thus dedicated to a "portrait gallery" of all growth models that have been
used for sea urchins.
There are two types of growth models in biology: population and
individual. Population models describe the change in the number of
individuals through time. A typical population growth model is the logistic
curve (Verhulst, 1838). Another model often used to describe decreasing
number of individuals, i.e., mortality, is the Weibull function (Weibull,
1951).
Individual growth models represent changes in the size of a single
individual with time. A typical individual growth model is the von
Bertalanffy curve (von Bertalanffy, 1938, 1957). Although we will deal
exclusively with individual growth in this work, population growth models
are also often used to model individuals, particularly Gompertz or logistic
curves (for examples that used them for sea urchins, see Gage et al, 1986;
Gage, 1987; Ebert, 1999) and we will do so as well.
a. The exponential curve, a simple Malthusian growth model
In 1798, Thomas Malthus described a mathematical model for growth
of human populations. According to Murray (1993), this model was
previously suggested by Euler. Today this model is not used much,
however its historical significance should not be overlooked. It is the first
42

20
15
10
Y
5
Y 0
General introduction
mathematical formulation of one of the most fundamental aspects of
growth: its exponential nature (positive or negative). Malthus observed
that the U.S. population doubled every 25 years. He suggested that human
populations increase by a fixed proportion r on a given time interval, when
they are not affected by environmental or social constraints, and this
proportion is not dependent on the initial size of the population:
Yt+ 1 (1 r) Yt k Yt
= + ⋅ = ⋅ (1)
One obtains a continuous model by expression eq. 1 as a differential
equation:
dY () t
= Y'( t) = k⋅ Y( t)
(2)
dt
This differential equation results in the following solution:
() e kt ⋅
Yt = Y⋅
(3)
0
with Y0 being the initial size of the population at time t = 0 (Fig. 7). This 2parameter
model is also interesting because it demonstrates how to build a
growth model: write a differential equation of the variation of size with
time and solve it (dynamic modelling). Almost all existing growth models
have been constructed this way. They correspond, as a consequence, to a
simple differential equation.
0.5 1 1.5 2 2.5 3 t
Figure 7. Example of an exponential curve with Y0 = 1.5 and k = 0.9.
43

. The logistic function for asymptotic growth
Y
1
Y•
0.8
0.6
Y•ê2
0.4
0.2
General introduction
The exponential model describes infinite growth without constraints.
This is not a realistic hypothesis. In practice, growth is limited by available
resources. Verhulst (1838), working also on population growth, proposed a
model containing an auto-limitation term [Y∞ – Y(t)]/Y∞ that represents
some theoretical limiting resource:
dY () t Y∞ −Y
() t k
= k⋅ ⋅ Y t =− Y t + k⋅ Y t
(4)
dt Y Y
()
2
() ()
∞ ∞
Solving and simplifying this differential equation yields:
Y
∞ () −k⋅( t−t0) 1 e
Yt
= +
Eq. 5 is one form of the logistic function. The function has two
horizontal asymptotes at Y(t) = 0 and Y(t) = Y∞ (Fig. 8) and it is a
symmetrical sigmoid (the two limbs of the S are similar).
i
2 4 6 8 10 t
t
t
0
Figure 8. Example of a logistic curve with k = 1, Y∞ = 0.95, t0 = 5. This sigmoidal curve is
asymptotic in 0 and Y∞, and is also symmetrical around its inflexion point i at {t0, Y∞ /2}.
As a generalization of this model, it is easy to define a logistic function
whose lower asymptote is different from 0. If this lower asymptote is Y0,
we obtain equation:
(5)
44

General introduction
Y∞−Y Yt () = Y+ 1+ e
0
0 −k⋅( t−t0) We will refer to it as the 4-parameter logistic model.
c. The Gompertz model, an asymmetrical sigmoidal curve
Y
1
Y•
0.8
0.6
0.4
Y•êe
0.2
Gompertz (1825) empirically observed that survival rate often
decreases proportionally to the logarithm of survival. Although this model
is still used with survival data (Ebert, 1999), it has many applications for
growth data as well (Winsor, 1932, Ebert, 1999). The differential equation
of Gompertz model is:
which simplifies to:
dY () t
= k⋅[ lnY∞−ln Y( t) ] ⋅ Y( t)
(7)
dt
−k⋅( t−t0) e
−kt ⋅
e
t
b
∞ ∞ ∞
Yt () = Y.e = Y. a = Y. a
(8)
The last parameterization is simpler and used more often. The first one is
derived from the differential equation (eq. 7) and gives a better comparison
with the logistic model, since t0 also corresponds to the abscissa of the
inflexion point, which is not in a symmetrical position here (Fig. 9).
t 0
i
1 2 3 4 5 6
Figure 9. Example of a Gompertz curve with k = 1, Y∞ = 0.95, t0 = 1.5. Inflexion point i, at
{t0, Y∞ /e}, is lower than in the logistic curve.
t
(6)
45

d. The von Bertalanffy curves
General introduction
The von Bertalanffy model, sometimes called Brody-Bertalanffy
(according to works of von Bertalanffy, 1938, 1957, and Brody, 1945) or
Pütter (in Ricker, 1979), is the first growth model specifically designed to
describe individuals. It is based on a simple bioenergetic analysis. An
individual is regarded as a simple dynamic chemical reactor where inputs
(anabolism) compete with outputs (catabolism). The result of these two
fluxes is growth. Anabolism is more or less proportional to respiration, and
respiration is surface-proportional for many animals (von Bertalanffy,
1957). Catabolism is always proportional to the volume or weight. These
mechanistic relationships are collected together in the following
differential equation when Y(t) measures a volume or a weight with time:
dY () t
2/3 1/3 2/3
= aYt ⋅ () −bYt ⋅ () = 3 k⋅⎡Y∞⋅Yt () −Yt
() ⎤
dt
⎣ ⎦ (9)
Solving this equation, we obtain the von Bertalanffy model in weight, also
called "von Bertalanffy 2" in the next part of this work:
−k⋅( t−t0) ( )
3
Yt () = Y ⋅ 1− e (10)
∞
The simplest form of this model occurs when one measures a linear
dimension for the body size, since a linear dimension is the cubic root of a
volume or a weight (not considering a possible allometry). The von
Bertalanffy for linear measurements, called von Bertalanffy 1 in the
present work, is simply:
−k⋅( t−t0) ( )
Yt () = Y⋅
1− e (11)
∞
A graph of both models is shown in Fig. 10. Von Bertalanffy 1 model
has no inflexion point. Growth is fastest at the outset, gradually
diminishes, and finally reaches zero. Growth is determinate and size
cannot exceed the horizontal asymptote of the curve at Y(t) = Y∞. Due to
46

Y
Y
1
Y• 0.8
0.6
0.4
0.2
General introduction
the cubic power transformation of von Bertalanffy 1, von Bertalanffy 2 is
an asymmetrical sigmoid like the Gompertz model.
1 2 3 4 5 6
Figure 10. Both von Bertalanffy 1 (curve in bold) and von Bertalanffy 2 (plain curve) models
with k = 1, Y∞ = 0.95 and t0 = 0. Both models describe asymptotic growth, but von
Bertalanffy 1 has no inflexion point, while von Bertalanffy 2 is sigmoidal.
e. The Richards model, a flexible curve that contains many
others
The general scheme for von Bertalanffy models is:
t
−k⋅( t−t0) ( )
m
Yt () = Y⋅
1− e
(12)
∞
Von Bertalanffy (1938, 1957) set m at either 1 or 3. Richards (1959) lets m
vary freely and thus his model has an additional parameter. This curve is
very flexible (Fig. 11) and one can demonstrate several other growth
models are just special cases with different m values. We have already
observed it reduces to von Bertalanffy 1 when m = 1, and to von
Bertalanffy 2 when m = 3. It also reduces to the logistic curve when m = -1
and one can demonstrate it reduces to the Gompertz model when |m| → ∞
(Tomassone et al, 1993; Ebert, 1999).
47

Y
Y
1
Y•
0.8
0.6
0.4
0.2
General introduction
2 4 6 8 10 t
Figure 11. Shape of Richards curves depending on values of m. From left to right: m = 0.5,
1, 3, 6 and 10; with k = 0.5, Y∞ = 0.95 and t0 = 0 for all curves. The curve in bold, with
m = 1, is equivalent to the von Bertalanffy 1 model.
f. The Weibull model, a polyvalent and flexible function
Since it was introduced, the Weibull (1951) function was presented as
a polyvalent model. Originally, it was described as a statistical distribution.
It has many applications in population (negative) growth, and is used also
to describe survival in cases of injury or disease, or in population dynamic
studies (Fahrmeir & Tutz, 1994; Ebert, 1985, 1999). It is sometimes used
as a growth model (Tomassone et al, 1993). The most general form of this
model is:
m
∞ ∞
−kt ⋅
Yt ( ) = Y −d⋅ e with d= Y − Y
(13)
A 3-parameter model is used as well with Y0 = 0 (Tomassone et al, 1993).
The function is sigmoidal when m > 1, otherwise it has no inflexion point
(Fig. 12).
0
48

Y
1
Y• 0.8
0.6
Y•-d.e 0.4
-k
0.2
Y0
General introduction
1
i
2 4 6 8 10 tt
Figure 12. Examples of Weibull curves for m = 5, 2, 1 and 0.5 respectively, with k = 0.6,
Y∞ = 0.95 and Y0 = 0.05. In bold, the curve with m = 1, equivalent to a von Bertalanffy 1
model. All curves start from Y0 and pass by Y∞ - d·e -k which is also the inflexion point for the
sigmoidal curves when m > 1.
g. The Jolicoeur curve, another flexible model
Another curve that can possibly be sigmoid or not is Jolicoeur model
(Jolicoeur, 1985). It is derived from a logistic curve, but using the
logarithm of time instead of time itself:
Y∞
Yt () =
1+
bt ⋅
−m
(14)
Like the Weibull function, it is sigmoidal when m > 1, but with an
asymmetry between the limbs of the S that can vary independently,
according to the value of parameter b. When m ≤ 1, there is no inflexion
point (Fig. 13).
49

Y
1
Y•
0.8
0.6
Y•êH1+bL
0.4
0.2
General introduction
1
i
2 4 6 8 10 t
t
Figure 13. Examples of Jolicoeur curves with m = 3, 1 (bold curve) and 0.5 respectively from
highest to lowest curve; Y∞ = 0.95 and b = 0.9. When m > 1, the curve is sigmoidal.
h. The Johnson model, a heavily asymmetrical sigmoid
Y
Y
1
Y• 0.8
0.6
0.4
0.2
The Johnson growth curve (see Ricker, 1979) uses 1/t instead of t:
Yt () Y e
2 4 6 8 10 t
1 k⋅( t−t0) = ∞ ⋅ (15)
Figure 14. Example of a Johnson curve, with k = 0.7, Y∞ = 0.95 and t0 = 0.
It is sigmoidal with a very strong asymmetry, the inflexion point being
very low and close to 0 (and thus hardly visible in Fig. 14).
50

i. The Preece-Baines 1 model for human growth
Y
1
Y• 0.8
0.6
0.4
0.2
General introduction
Preece and Baines (1978) described various growth models specific to
human growth. These models combine two different exponential growth
phases to represent the gradual growth of infants followed by a faster
growth of adolescents, but becoming rapidly asymptotic (Fig. 15). These
are indeed very flexible models. Among these curves, model 1 was used
for a sea urchin by Gage and Tyler (1985) and is:
2( ⋅ Y∞−d) Yt () = Y −
e + e
∞ k1⋅( t−t0) k2⋅( t−t0) 2 4 6 8 10 t
Figure 15. Example of a Preece-Baines 1 curve with k1 = 0.19, k2 = 2.5, Y∞ = 0.95, d = 0.8
and t0 = 6.
j. The Tanaka model for indeterminate growth
(16)
All the previous models are asymptotic, except the exponential one
(but it is only usable for the initial stages of a growth process). They
describe determinate growth that can never exceed horizontal asymptote at
Y(t) = Y∞. Knight (1968) questioned whether it is a biological fact or just a
mathematical artifact. In the later case, growth seems to be determinate
only because mathematical models used to represent it are asymptotical.
To overcome this constraint, Tanaka (1982, 1988) elaborated a new model
that allows for indeterminate growth:
51

YY
3
2.5
2
1.5
1
0.5
General introduction
⎛ 1 ⎞
2 2
Yt () = ⎜ ⎟⋅ln2
b⋅( t− t0) + 2 b⋅( t− t0) + ab ⋅ + d
⎝ b ⎠
(17)
This complex 4-parameter model has an initial period of slow growth, a
period of exponential growth followed by an indefinite period of slow
growth (Fig. 16).
2 4 6 8 10 tt
Figure 16. Example of a Tanaka curve with a = 3, b = 2.5, d = -0.2 and t0 = 2.
Modelling sea urchins growth
Table 1 summarizes sea urchin studies using growth models.
Numerous works using growth rate only (final – initial size) are not
included. Most species of economic interest (Tripneustes gratilla,
Sphaerechinus granularis, Psammechinus miliaris, Paracentrotus lividus,
Strongylocentrotus droebachiensis, S. intermedius, S. nudus, S.
franciscanus, Hemicentrotus pulcherrimus) were considered by one or
more authors. They focused either on population dynamics aiming to
provide management criteria for fisheries, or on growth in cultivation.
Other species studied were either key-species in some biotopes (Diadema
antillarum Philippi, Echinus esculentus L.), or animals occupying some
particular biotopes [for instance, deep-sea urchins like Echinosigra phiale
(Thompson) or Hemiaster expergitus Loven].
52

a. Choice of the growth model for sea urchins
General introduction
Von Bertalanffy 1 is the model most often used. Among 69 studies of
sea urchins (regardless of species), this model was used 32 times, the
Richards model 17 times, the Gompertz model 9 times and other models
11 times (Table 1). However, in 13 studies where several models were
tested in addition to von Bertalanffy 1, the latter was considered as being
the best one only twice. The reason invoked to reject the von Bertalanffy
model was the initial lag phase in growth that is correctly represented
solely by a sigmoid like in the Richards, Gompertz or logistic models
(Yamagushi, 1975). In many studies where no other model was tested, it
seems that the von Bertalanffy 1 curve was just a default choice: it
represents the model "usually" fitted on such kind of data.
The Richards model was first proposed by Ebert (1973) as a better
alternative to the von Bertalanffy 1 curve to fit echinoid growth data. It
was intensively used by the same author (Ebert, 1973, 1980a, 1982, 1999;
Ebert & Russell, 1992, 1993) as well as by some others (Gage & Tyler,
1985; Russell, 1987; Kenner, 1992; Turon et al, 1995; Lamare &
Mladenov, 2000). The Gompertz model is also a favorite when there seems
to be a lag phase in growth and it has been used in various studies (Gage et
al, 1986; Gage, 1987; Cellario & Fenaux, 1990; Dafni, 1992; Turon et al,
1995; Ebert, 1999). Each of these models (i.e., Richards or Gompertz) was
preferred in 50% of the multi-model studies which considered them. The
logistic curve, although also sigmoidal, was systematically rejected in
multi-models studies of sea urchins, except for the irregular echinoid
Echinocardium pennatifidum Norman (Gage, 1987).
53

Table 1. Models used to fit sea urchins growth data (the one preferred by authors in multimodels
studies is in bold).
Species (1)
Growth model (2)
Family Cidaridae
Reference
Eucidaris tribuloides (Lamarck) von Bertalanffy1 McPherson, 1968
Family Diadematidae
Diadema setosum (Leske) Richards Ebert, 1980a
von Bertalanffy1, Richards, Gompertz, logistic Ebert, 1999
Diadema antillarum Philippi von Bertalanffy1 Ebert, 1975
Echinotrix diadema (L.) von Bertalanffy1 Ebert, 1975
Richards Ebert, 1982
Centrostephanus rodgersii (A. Agassiz) Richards Ebert, 1982
Family Stomopneustidae
Stomopneustes variolaris (Lamarck) Richards Ebert, 1982
Family Temnopleuridae
Salmacis belli Döderlein Richards Ebert, 1982
Family Toxopneustidae
Lytechinus variegatus (Lamarck) von Bertalanffy1 Ebert, 1975
Tripneustes gratilla (L.) von Bertalanffy1, Gompertz, logistic, Johnson Dafni, 1992
Tripneustes ventricosus (Lamarck) von Bertalanffy1 McPherson, 1965
Sphaerechinus granularis (Lamarck) von Bertalanffy1 Lumingas & Guillou, 1994
von Bertalanffy1 Jordana et al, 1997
Family Echinidae
Echinus esculentus L. Richards Ebert, 1973
logistic Nichols et al, 1985
logistic Sime & Cranmer, 1985
Gompertz Gage et al, 1986
von Bertalanffy1 Gage, 1992
Echinus acutus Lamarck logistic Sime & Cranmer, 1985
von Bertalanffy1, Gompertz, logistic Gage et al, 1986
Echinus elegans Düben & Koren von Bertalanffy1, Gompertz, logistic Gage et al, 1986
Echinus affinis Mortensen von Bertalanffy1, Richards, Gompertz,
logistic, Preece-Baines 1
Gage & Tyler, 1985
Gompertz Gage et al, 1986
linear Middleton et al, 1998
Psammechinus miliaris (Gmelin) von Bertalanffy1 Jensen, 1969a
von Bertalanffy1 Allain, 1978
von Bertalanffy1 Gage, 1991
Paracentrotus lividus (Lamarck) von Bertalanffy1 Allain, 1978
(von Bertalanffy1), Gompertz Cellario & Fenaux, 1990
Gompertz, logistic, Richards Turon et al, 1995
von Bertalanffy1 Sellem et al, 2000
von Bertalanffy1, von Bertalanffy2, Gompertz, Grosjean et al (submitted),
logistic, 4p-logistic, Weibull, original model see Part. IV
Loxechinus albus Molina von Bertalanffy1 Gebauer & Moreno, 1995
Family Strongylocentrotidae
Strongylocentrotus droebachiensis (Müller) von Bertalanffy1 Munk, 1992
von Bertalanffy1 Hagen, 1996b
logistic Meidel & Scheibling, 1998
Tanaka Russell et al, 1998
Strongylocentrotus intermedius (A. Agassiz) von Bertalanffy1, Gompertz Fuji, 1967
Strongylocentrotus nudus (A. Agassiz) von Bertalanffy1 Ebert, 1975
Strongylocentrotus purpuratus (Stimpson) von Bertalanffy1 Ebert, 1977
Richards Russell, 1987
Richards Kenner, 1992
Strongylocentrotus franciscanus (A. Agassiz) von Bertalanffy1 Ebert, 1977
Richards Ebert & Russell, 1992
Richards, Tanaka, Jolicoeur Ebert & Russell, 1993
Tanaka Ebert, 1998
Richards, Tanaka Ebert, 1999
Hemicentrotus pulcherrimus (A. Agassiz) von Bertalanffy1 Fuji, 1963
Allocentrotus fragilis (Jackson) von Bertalanffy1 Sumich & McCauley, 1973
General introduction
54

(Table 1, next part)
Species (1)
Growth model (2)
Family Echinometridae
Reference
Evechinus chloroticus (Valenciennes) von Bertalanffy1, Richards, Tanaka, Jolicoeur Lamare & Mladenov, 2000
Anthocidaris crassispina (A. Agassiz) von Bertalanffy1 Chiu, 1990
Heliocidaris erythrogamma (Valenciennes) Richards Ebert, 1982
Echinometra mathaei (de Blainville) von Bertalanffy1 Ebert, 1975
Richards Ebert, 1982
Echinometra oblonga (de Blainville) von Bertalanffy1 Ebert, 1975
Richards Ebert, 1982
Heterocentrotus mammillatus (Klein)
Heterocentrotus trigonarius (Lamarck)
Richards
Richards
Ebert, 1982
Ebert, 1982
Colobocentrotus atratus (L.) von Bertalanffy1 Ebert, 1975
Richards Ebert, 1982
Family Mellitidae
Mellita quinquiesperforata (Leske) von Bertalanffy1 Lane & Lawrence, 1980
Mellita grantii Mortensen von Bertalanffy1 Ebert & Dexter, 1975
Encope grantis L. Agassiz von Bertalanffy1 Ebert & Dexter, 1975
Family Pourtalesiidae
Echinosigra phiale (Thompson) von Bertalanffy1, Gompertz, logistic Gage, 1987
Family Hemiasteridae
Hemiaster expergitus Loven von Bertalanffy1, Gompertz, logistic Gage, 1987
Family Spatangidae
Spatangus purpureus Müller von Bertalanffy1, Gompertz, logistic Gage, 1987
Family Loveniidae
Echinocardium cordatum (Pennant) von Bertalanffy1 Duineveld & Jenness, 1984
Echinocardium pennatifidum Norman von Bertalanffy1, Gompertz, logistic Gage, 1987
(1) Classification according to Mortensen (1950) and Durham (1955).
(2) von Bertalanffy1: D = a·(1 - e -b·(t – c) ), von Bertalanffy2: D = a·(1 - e -b·(t – c) ) 3 , Richards: D = a·(1 - e -b·(t – c) ) d , Gompertz:
t
c
= ⋅ , logistic: D = a/(1 + e -b·(t - c) ), 4p-logistic: D = (a – d)/(1 + e -b·(t - c) ) + d, Johnson: D = a·e -1/b·(t – c) , Preece-Baines 1:
D a b
D = a – 2·(a – d)/(e b·(t – c) + e e·(t – c) c
−bt ⋅
), linear: D = a·t + b, Weibull: D= a−d⋅ e , original model: D = e + a·(1 - e -b·t )/(1 + d·e -c·t )
(see Part IV), Tanaka: D = (1/b 1/2 )·ln(|2b·(t – c) + 2·(b 2 ·(t – c) 2 + a·b) 1/2 | + d), Jolicoeur: D = a/(1 - c·t -b ).
General introduction
Questioning asymptotic growth in the largest regular sea urchin,
Strongylocentrotus franciscanus, Ebert & Russell (1993) introduced the
indeterminate growth model of Tanaka as a better representation of the
continuous growth of large individuals. However, this species seems to be
a special case, even inside the Strongilocentrotidae family (Lawrence et al,
1995). The Tanaka model was not used much for other species. Lamare &
Mladenov (2000) tested it on Evechinus chloroticus (Valenciennes), but
concluded it is not the more appropriate one in this particular case.
In an attempt to find a better model to fit echinoid growth data, various
"exotic" curves were also tested. They were sometimes successful, such as
in the works by Gage & Tyler (1985) that introduced the Preece & Baines
55

General introduction
model 1 for Echinus affinis Mortensen or by Dafni (1992) that proposed
the Johnson model to fit rapid growth with a very small initial lag phase of
the Toxopneustidae Tripneustes gratilla at Elat.
Based on this review (Table 1), it seems there is no ideal model to fit
growth data in sea urchins and, consequently, the use of one particular
model is more a question of personal preference. For instance, Ebert uses
the Richards model most of the time (Ebert, 1973, 1980a, 1982, 1999),
while Gage favors the Gompertz model (Gage & Tyler, 1985; Gage et al,
1986; Gage, 1987). This situation is problematic since parameters derived
from these models –and others– are not comparables. Using a single model
to fit all growth data would be preferable for comparison purposes (Turon
et al, 1995).
b. Fitting of growth models on real data for echinoids
In selecting a growth model, several conditions should be met when
fitting real data. First, animals sampled from a single homogeneous
population should be measured at various ages. Second, one particular
individual should be measured only once to ensure independence of the
errors (since authors consider individual variation as part of the error term:
they look for growth of a virtual "mean individual" among the population).
Third, as most regression methods assume no error on the dependent
variable –that is, time– (Sokal & Rohlf, 1981; Sen & Srivastava, 1990;
Draper & Smith, 1998; Zar, 1999), the age of each measured individual
should be known. Fourth, no interaction should exist between individuals
in the population. Such ideal conditions are so restrictive that they are
never met.
When the age of individuals can be determined precisely, such as for
sea urchins reared from the egg, it is common to measure the same
specimens several times (Bull, 1938; Michel, 1984; Cellario & Fenaux,
1990; Basuyaux & Blin, 1998; Lamare & Mladenov, 2000). Errors are
individual-dependent in such cases and this interaction is often ignored.
56

General introduction
Among various methods developed to fit growth curves (Walford, 1946;
Fabens, 1965; Allen, 1966; Causton, 1969; Ebert, 1980a; Kaufmann,
1981), the most powerful one is considered to be the nonlinear least-square
regression (Gallucci & Quinn, 1979; Vaughan & Kanciruk, 1982). All
studies listed in Table 1 use it, except Grosjean et al (submitted, see Part
IV). However, when using field-collected data, it is not possible to
determine precisely the age of individuals and thus it is estimated. Yet, the
same nonlinear least-square regression method is still used despite the
violation of the assumption that there is no error on the time variable. Two
methods coexist to estimate the age of echinoids in the field: cohort
separation and growth rings analysis.
The cohort separation method is commonly used with species
displaying annual recruitment (Hasselbald, 1966; McDonald & Pitcher,
1979; Ebert et al, 1993, 1999; Aksland, 1994; Smith et al, 1998). Cohorts
are separated by time increments of one year. Usually, the youngest
individuals recruited in the year form a well-defined cohort whose peak
displacement with time can be used to estimate mean growth rate
(Raymond & Scheibling, 1987; Dafni, 1992; Munk, 1992; Lumingas &
Guillou, 1994; Gebauer & Moreno, 1995). The cohorts are –implicitly–
assumed to be unimodal and normally, or at least, symmetrically
distributed. No authors working on sea urchins tested the validity of this
fundamental assumption, nor do they discuss implications of violations of
this assumption. When individuals interact, cohorts can be asymmetrical,
or even multimodal (Grosjean et al, 1996, see Part III). In this case, the
study is biased because several modes of a single cohort are interpreted
as multiple separate cohorts. Ebert (1968) observed wide variations in
growth rate of Strongylocentrotus purpuratus (Stimpson) in some habitats,
and attributed it to food limitation. However, there is also some evidence
of growth inhibition of juveniles in the field for Strongylocentrotus
droebachiensis (Himmelman, 1986). Kenner (1992), Munk (1992) and
Turon et al (1995) discuss Himmelman's conclusions for their biological
implications but do not question the validity of the cohorts separation
57

General introduction
method used. In addition, Levitan (1988) demonstrated that interactions
exist between adult Diadema antillarum as maximal size is densitydependent.
The second method to estimate age uses the natural growth bands. The
trabecules within the sea urchin skeleton are more or less densely packed
depending on growth rate (Pearse & Pearse, 1975). A succession of fast
and slow growth stages results in light and dark bands, respectively, in the
stereom of the ossicles (Jensen, 1969b; Pearse & Pearse, 1975; Sime,
1981; Gage, 1991, 1992; Lumingas & Guillou, 1994). It is postulated that
there is only one period of fast growth and another period of slow growth
per year. If this is true, counting these growth bands allows determining
the ages of the echinoids. If there is a single recruitment in a narrow time
window during the year (Ebert, 1983), precision is even better. Not all
authors agree with the validity of this method. Ebert (1986) questioned it
and Russell & Meredith (2000) experimentally demonstrated it is not valid
for Strongylocentrotus droebachiensis. However, Gage (1992) validated it
for Echinus esculentus with an experiment using echinoids kept in cages in
the sea during two years.
Many authors consider that if they use both methods simultaneously –
cohort separation and growth rings analysis– and get the same result, each
method is validated by the other one (Duineveld & Jenness, 1984;
Lumingas & Guillou, 1994; Gebauer & Moreno, 1995; Turon et al, 1995;
Jordana et al, 1997). Yet, if the number of growth rings is correlated with
the size, not the age, one would interpret a group of fast-growing
individuals as being older, and a group of slow-growing ones as being
younger and eventually mix animals of different age in a single cohort.
This would result in an agreement between both methods although
conclusions on size at age are incorrect.
Measuring relative growth (without knowing age) is an alternative to
calculating growth rate of individuals in the field. Animals are tagged,
field-released and captured again one year later (Ebert, 1977, 1988a;
58

General introduction
Russell, 1987; Rowley, 1990; Kenner, 1992; Ebert & Russell, 1992, 1993;
Russell et al, 1998; Lamare & Mladenov, 2000). This way, seasonal
variation in growth is also partly eliminated. For sea urchins, it is the
skeleton that is tagged using tetracycline (Kobayashi & Taki, 1969; Taki,
1972). Size increase of the ossicles can then be determined because a band
of tetracycline-labeled skeleton, visible under ultraviolet light, indicates its
size at tagging time. An allometric relationship between the size of the
given ossicles and the body size allow estimating the latter. To fit such
data, growth models need to be reworked, using a so-called Ford-Walford
representation, or Walford plot (Ford, 1933; Walford, 1946; Ebert, 1999)
where size at time t + 1 year (at recapture) is expressed as a function of
size at time t (at capture). Ebert (1999) reviewed such transformations for
von Bertalanffy, Gompertz, logistic, Richards and Tanaka models.
Sainsbury (1980) demonstrated the strong biases that could occur with
such a method when there is individual variation in growth in a population.
In fact, relative growth does not take the age into account, by definition.
Hence, due to individual variations in growth rate, some fast-growing but
young individuals have same size as slower-growing but older ones at a
given time. The method mixes all animals having the same size, no matter
their age, and calculates a mean growth rate at that size. This mixing of
age-cohorts is troublesome when growth variation is high in the
population. Rejection of a particular growth model could result from these
biases as Sainsbury evidenced, using a theoretical analysis with the von
Bertalanffy 1 model. As individual variation in growth pattern is probable,
one should be very careful using this method. Among all authors using
tagged sea urchins, Gage (1992) was the only one that cited Sainsbury's
work and care about a possible bias due to individual variation in growth
rate.
Clearly, there is no fool-proof method for modelling individual growth
of sea urchins in the field. In such a context, one can rely on experiments
conducted in aquaria, although it is evident that growth patterns observed
in artificial conditions could be very different to what happens in the field.
59

General introduction
Experiments in cages in the sea are closer to conditions of wild
populations, but in the case of P. lividus populations living in tidal pools,
they are impossible to perform in practice.
60

Aim of the thesis
AIM OF THE THESIS
The ultimate goal of this work is to formulate a mechanistic model –that is, whose
parameters are functionally interpretable– of somatic growth of the sea urchin
Paracentrotus lividus in cultivation. Artificial culture conditions offer the opportunity
to work with echinoids whose age, genetic origin, environmental and food conditions
are perfectly known. However, we need to set up a rearing protocol adapted to P.
lividus. We have also to define which is the best measurement of body size, and what
it exactly represents (an overall trend in growth, or just the growth of one
compartment or one organ of the echinoid). Once these issues are solved, we will
have to conduct experiments on reared sea urchins to reveal underlying processes that
influence growth. Using the results of these experiments we can then elaborate an
original mathematical model that functionally describes growth of P. lividus reared in
an aquaculture system, including those underlying processes.
61

Aim of the thesis
62

PART I
Set up of an experimental rearing procedure
for echinoids
63

PART I: SET UP OF AN EXPERIMENTAL
REARING PROCEDURE FOR ECHINOIDS
A 230 m 2 experimental facility (with ca. 20,700 l of circulating
seawater) was designed. A precise standard rearing method was set up for
P. lividus aiming to study different aspects of the biology of this species in
cultivation, ranging from larval biology to reproduction, including
metamorphosis, somatic growth, feeding and ecophysiological reactions. It
was adapted from Le Gall's method (Le Gall & Bucaille, 1989; Le Gall,
1990). It allows controlling the whole life cycle of the sea urchin (closedcycle)
in artificial conditions (that is, land-based systems with water
recirculation) on a pilot scale.
Performances of this rearing method were quantified. In particular,
surviving rates and timing of the different stages were recorded for a large
number of replicates (a grand total of ca. 65,000 echinoids were followed
in the rearing devices, some of them during 7 years; more than 225,000
measurements were performed). Somatic growth and production were also
recorded. Finally, gonadal production and maturation of the gonads were
examined. In the framework of the present work, all these measurements
were used to verify that the rearing method is in adequacy with the biology
of the sea urchin and ensures its correct development –not just its survival–
in the cultivation conditions.
One of the goals of the research was to provide a basic methodology to
be expanded on a commercial scale for industrial sea urchin farming
(echiniculture). Accordingly, implications, advantages and drawbacks of
the method on a large-scale are widely discussed in the paper. This
approach, however, is not of major concern in the present work. The latter
aims basically to obtain a well-calibrated rearing method for experimental
studies on somatic growth of postmetamorphic echinoids whose genetic
origin, age, environmental and food conditions are perfectly known.
Part I: Set up of an experimental rearing procedure for echinoids
65

Part I: Set up of an experimental rearing procedure for echinoids
66

Land-based closed-cycle echiniculture of Paracentrotus
lividus (Lamarck) (Echinoidea: Echinodermata): a long-term
experiment at a pilot scale
a. Abstract
b. Introduction
Ph. Grosjean, Ch. Spirlet, P. Gosselin, D. Vaïtilingon & M. Jangoux,
1998. Journal of Shellfish Research, 17(5):1523-1531
Today, most worldwide sea urchins fisheries must deal with
overexploitation. Better management of exploited field populations and/or
aquaculture is increasingly considered necessary to sustain sea urchin
production in the future. In this context, we evaluate the potential of landbased,
closed-cycle echiniculture. A long-term experiment with the edible
sea urchin Paracentrotus lividus has been done on a pilot scale. The
process allows total independence from natural resources because the
entire biological cycle of the echinoids is under control (closed-cycle
echiniculture) and all activities are performed on land. Furthermore, a
method has been set up to control the reproductive cycle with the aim to
produce marketable individuals all year long. Performances obtained on
each stage of the rearing process are quantified and analyzed. Overall, the
results of this experiment are promising; however, some problems remain
to be solved before we can claim profitability. The most important finding
is that land-based, closed-cycle echiniculture is a potential viable
supplement to fisheries to sustain worldwide sea urchin roe production.
Keywords: sea urchin, Paracentrotus lividus, aquaculture, larval culture,
metamorphosis, growth, roe enhancement.
Depending upon their respective gastronomic culture, people consider
sea urchin gonads (both male and female gonads are collectively referred
Part I: Set up of an experimental rearing procedure for echinoids
67

to as roe) as either a fine and delicate seafood or as absolutely inedible.
However, its economic value is well established given the price consumers
are willing to pay. The wholesale price of live sea urchins in France ranges
from 30 to 120 FF/kg (price range in the 1990s at Rungis, Paris), and fresh
roe in Japan from 6,000 to 14,000 ¥/kg (price in Japan in the 1990s, see
Hagen 1996a). Both market prices are roughly equivalent in terms of fresh
roe, making sea urchin roe one of the most valuable seafoods in the world.
In both markets, the lowest prices are those of imported sea urchins, which
are considered to be of poorer quality.
The most important market, Japan, imports approximately five
thousand tons of sea urchin gonads per year, the equivalent of 40 to 50
thousand tons of live sea urchins (Hagen, 1996a). According to the same
author, the Japanese consumes approximately 60,000 tons of whole sea
urchins per year. The second largest consumer is France, with an annual
consumption of approximately 1,000 tons of whole sea urchins (Le Gall,
1990).
Increasing demand for sea urchin roe and a steady rise in price have led
to worldwide intensification of sea urchin fisheries (Conand and Sloan,
1989; Le Gall, 1990; Saito, 1992) which has now (1998) probably reached
its maximum. This production cannot be sustained at current levels
because the declining productivity of overexploited existing stocks can no
longer be compensated by harvest of new stocks, as was possible over the
last three decades (most exploitable natural populations have already been
fished today). In Japan, this decline occurred despite the development and
implementation of extensive domestic fishery enhancement techniques
which include the annual release of 60 million juvenile sea urchins into the
wild (Saito, 1992; Hagen, 1996a). Consequently, the worldwide supply of
high quality sea urchin roe will be unable to meet market demand unless
commercial sea urchin aquaculture develops to partially replace the steady
decrease in natural captures.
Part I: Set up of an experimental rearing procedure for echinoids
68

Aquaculture of echinoderms, including sea urchins and sea cucumbers
is known as echinoculture (Le Gall & Bucaille, 1989; Le Gall, 1990;
Hagen, 1996a). We prefer to use the term echiniculture to describe sea
urchin aquaculture solely (Echinoidea); thus, it is more accurate in this
context. This activity is not yet fully developed. Maintenance or rearing of
sea urchins in the laboratory has been successfully performed for different
species (Hinegardner, 1969; Fridberger et al, 1979; Cellario & Fenaux,
1990). Several different processes are being experimented on a larger
scale, ranging from sea urchin ranching (cultivation in the field, see
Fernandez & Caltagirone, 1994; Fernandez, 1996), to land-based systems
(Le Gall & Bucaille, 1989; Le Gall, 1990; Fernandez, 1996) or polyculture
(sea urchins cultivated in cages with fish, see Kelly et al, 1998).
Nevertheless, considering the limited carrying capacity of natural sites that
are already largely exploited by fisheries, only land-based or cage
techniques will help to sustain worldwide sea urchin roe production.
Similarly, only cultivation processes totally independent of natural stocks,
that is, by controlling the complete life cycle of the echinoid, will lower
the pressure imposed by fisheries upon natural populations. In this context,
this paper presents a 7-year experimental rearing method to produce the
edible sea urchin P. lividus on a pilot scale, and discusses the biological
and technological issues that emerged from this cultivation method.
c. Material and methods
The aim of land-based, closed-cycle echiniculture is to get maximum
control over each phase of the sea urchin's life cycle by controlling the
major environmental parameters (temperature, photoperiod, water quality,
quality and quantity of food). A land-based system has advantages over
methods performed directly in the sea. The greatest of these is the ability
to control the whole life cycle of the animals (closed cycle), thus the sea
urchin never depends, at any of its stages, on a supply of animals
originating from the field.
Part I: Set up of an experimental rearing procedure for echinoids
69

The method used here is adapted from Le Gall (Le Gall & Bucaille,
1989; Le Gall, 1990) with some fine-tuning and modifications that allow a
routine output of sea urchins on a pilot scale. An experimental facility has
been set up at the "Centre de Recherche et d'Etude Côtière" (CREC,
Normandy, France) in which several generations of sea urchins have been
reared according to a thoroughly defined experimental procedure.
Pilot echiniculture facility
The experimental facility includes a hatchery (30 m 3 ) and a cultivation
room (160 m 3 ). The hatchery is equipped with 11 200-l larval rearing tanks
(see below) and a system for phytoplankton production (classical devices
for large-scale production).
The cultivation room is insulated, thermoregulated at 22°C ± 1°C,
correctly aerated, and exposed to a 12h/12h photoperiod. It is equipped
with 10 autonomous rearing structures with either three or six superposed
4-m long and 60-cm wide ponds called toboggans. Each set of toboggans
hangs over a reserve/settling tank of the same length, 80 cm wide and
80 cm deep. The water depth in the toboggans varies between 5 and 10 cm.
A centrifugal pump transfers water from the reserve tank to the top level
with a flow of 8 to 10 m 3 /h (4 to 5 m 3 /h for the pregrowth structure, see
below). The water then recirculates by gravity from one level to the other
(each toboggan has a gentle slope to help water run into it and is connected
to the previous and the next one at its opposite ends, see Fig. 17). This
device, specifically designed for sea urchin cultivation, optimizes both the
surface available for the postmetamorphic individuals and the water
current around them. It also facilitates access to the animals and their
visual control. The 10 rearing structures are organized as follows:
(1) One pregrowth structure of 3 toboggans with a capacity of 1500 l of
circulating water thermoregulated at 20°C ± 1°C. The water is renewed at
a rate of 150% per day. This structure can hold a biomass ranged between
0.2 and 1 kg/m 2 of toboggans.
Part I: Set up of an experimental rearing procedure for echinoids
70

6b. exploitation
KCl
water renewal
toboggans
KCl
6a. broodstock cond.
reserve tank
5. growth of subadults
1. fertilization
FERTILIZING
TUB
4. growth of juveniles
18 to 20°C
PREGROW TH & GROWTH STRUCTURES
Part I: Set up of an experimental rearing procedure for echinoids
2. larvae culture
200 l
20°C
LARVAL REARING TANK
3. metamorphosis
water
pump
Figure 17. Overview of the closed-cycle process and devices used to produce sea urchins on
land at a pilot scale.
(2) Two growth structures made of six toboggans each. The capacity of
each structure is 3000 l of circulating water thermoregulated at 18°C ± 1°C
and with a water renewal ranged between 100 and 600% per day,
depending upon the density of sea urchins present in the structures. These
structures can hold a maximum biomass of 7 kg of sea urchins per m 2
without supplemental filtration of the water.
71

(3) Seven experimental / conditioning structures of three toboggans each
with a capacity of 1500 l of circulating water. These are isolated from one
another so that they can be thermoregulated individually from 10°C to
25°C, and each has up to six different photoperiods (a dark separation
divides the toboggans in their center). An electronic system allows the
transition of light to darkness and vice versa, thus simulating dawn and
dusk. The rate of water renewal can be fixed between 50 and 600% per
day. Biomass varies following criteria imposed by the experiments.
Additional devices are grouped in a technical room containing a central
thermoregulation system (a thermorefrigerating pump providing either
cold or hot water to the heat exchangers that equip the rearing structures),
a water pumping and filtration system, an emergency electric generator
and a central alarm. The water is pumped directly from the sea at high tide
and is stored in a reservoir of 60 m 2 . It is filtered before being used (30 µm
mesh cartridge mechanical filtration, followed by a 14 m 3 biological filter
and two settling tanks of 8 m 3 each).
Origin of the animals
The species cultivated is Paracentrotus lividus (Lamarck, 1816). This
species is found all along the European coast from the northern Atlantic
Irish coast to the Mediterranean Sea. All individuals used come from a
single population located in Morgat, Brittany, France. Some were directly
collected in the small tidepools that spread all along the rocky shores of
Douarnenez Bay (emerged only during high coefficient tides). The
remaining animals come from artificial fertilizations in the laboratory and
were grown in the structures (cross fertilizations of first (F1) or second
(F2) generation of sea urchins collected in the field). By so doing, the age
and the parental origin of the F1 and F2 sea urchins are known precisely.
Rearing method
Aiming at closely matching the echinoid requirements along their life
history and minimizing technical constraints, the dissociation of the whole
Part I: Set up of an experimental rearing procedure for echinoids
72

earing cycle into seven stages is essential (Fig. 17). These stages are: (1)
fertilization, (2) larval culture, (3) metamorphosis, (4) growth of juveniles,
(5) growth of subadults and (6) growth of adults, which is further divided
into (6a) conditioning for the marketing of roe (exploitation) and (6b)
providing gametes (broodstock).
Stage 1: Fertilization is performed using gametes issued from healthy
animals that restored their gamete potential as described below in
broodstock conditioning (see stage 6b). The gametes are obtained by
stimulating the parents to spawn with 0.5 N KCl (injection of 50 µl per g
of body weight through the peristomial membrane). The gametes of each
individual are collected in a small jar of 50 ml in 20°C filtered natural
seawater (on a 1 µm cartridge filter, referred hereafter as "larval rearing
water").
When the spawning is over, the volume of the gametes is evaluated.
The ova of a single female are transferred in a fertilization tub, that is, a
shallow polyethylene container. The volume is brought to 800 ml with the
same water. One fifth of the spermatozoa of a single male is added to the
ova. The mixture is kept at 20 ± 1°C during 4 h and the tub is gently stirred
three or four times during that period. After that, the success of the
fertilization is checked and the fertilized eggs are counted (most often over
90% of the eggs are fertilized).
Stage 2: Rearing of the larvae is done in a 200-l polyethylene
cylindrical tank where larval rearing water is introduced 24 h beforehand
and stabilized at 20 ± 1°C. The embryos (in the gastrula stage) are
introduced at a concentration of 250 per liter. This density is low enough
to allow the entire rearing of the larvae to be conducted without renewing
the water. The food (Phaeodactylum tricornutum Bohlin issued from
cultivation in Erdschreiber medium) is introduced from the third day
postfertilization (acquisition of larval exotrophy). The larvae are fed once a
day with 600 ml algal cultivation (concentration around 10·10 6 cells/ml).
Part I: Set up of an experimental rearing procedure for echinoids
73

The whole is kept in dim light with a 12h/12h photoperiod and is gently
mixed and aerated by a central bubbling.
Stage 3: From the sixteenth day onward, competence to
metamorphosis is checked daily (Standard Competence Test or SCT,
adapted from Gosselin & Jangoux, 1996). One hundred larvae are
transferred in a clean SCT sieve (a 10 cm high, 20 cm 2 sieve with a bottom
mesh of 250 µm placed 1.5 cm above the water floor). This SCT sieve is
placed in the pregrowth structure in the presence of a metamorphosis
stimulating factor (living Corallina elongata Ellis & Sollander, freshly
collected from the field). The percentage of metamorphosed larvae is
determined 24 h later. If this value lies around 80%, the whole batch is
transferred in the pregrowth structure aiming at its fixation on one or two
metamorphosis sieves (similar to SCT sieves but each covering 1800 cm 2 ).
Batches containing large amounts of larvae exhibiting bad development,
abnormalities or too low metamorphosis ratios are discarded.
Stage 4: Growth of juveniles. The postmetamorphic period begins with
a short endotrophic stage. During this period, the postmetamorphic
individuals, also called postlarvae, reorganize their digestive tract
(Gosselin & Jangoux, 1998). The mouth and anus of the future juvenile are
not yet pierced. This postlarval stage lasts for up to 8 days, after which the
echinoids become exotrophic juveniles. One or two days before
development of exotrophy, 100 g fresh weight of Enteromorpha linza (L.)
Agardh collected in the field are distributed in each sieve. From this
moment onward, the same food quantity is given every time it is
completely consumed. Some Gammarus locusta L. are also introduced to
clean the sieves from decomposing parts of the algae.
The juveniles are left in these sieves until the mean individual size in
the batch reaches 2 mm. The entire batch is then transferred in 500 µm
mesh pregrowth sieves. A homogeneous bed of E. linza is maintained in
the sieves. The bottoms of the sieves are cleaned every week using filtered
seawater. Because the growth of the juveniles is not homogeneous
Part I: Set up of an experimental rearing procedure for echinoids
74

(Grosjean et al, 1996, see Part III), the animals are graded every month,
and those with a diameter larger than 5 mm are transferred into a 1-mm
mesh pregrowth sieve. The E. linza diet is maintained and the sieves
remain in the same pregrowth structure.
Stage 5: Growth of subadults. Every month, a sorting of size is done to
collect all individuals bigger than 10 mm. The individuals whose size
exceeds 10 mm but is below the minimum market size of around 40 mm
for P. lividus are defined as subadults. They are potentially mature but not
large enough for the market. Consequently, their somatic growth
performances must be promoted while their gonadal growth should be kept
as low as possible to optimize food allocation to the soma.
Subadults are placed in rectangular rearing baskets, with all sides made
out of 5-mm mesh. These rearing baskets are placed 1.5 cm above the
bottom of the toboggans and are just slightly narrower. This is important to
allow good water circulation around and inside them, and good elimination
of solid wastes produced by the sea urchins. Their surface ranges between
1200 and 2400 cm 2 . When the size of the animals increases above 15 mm
in test diameter, they are transferred in the same type of rearing baskets,
but with 10-mm mesh, which allows even better water circulation.
Subadults, inside their rearing baskets, are transferred to a growth
structure. From this time onward, and twice a week, they are fed ad libitum
with fresh kelp, Laminaria digitata. Cleaning of the baskets and toboggans
is also done twice weekly. Dead or dying animals are removed daily. Each
month, sorting by size is done to separate the batches into different size
categories from 5 to 5 mm. The entire cultivation is kept in 12h/12h
photoperiod.
Stage 6a: Conditioning of the adults for market. When the sea urchins
reach 40 mm, they are prepared to get marketable gonads in conditioning
structures. For the commercialization, it is of utmost importance that the
echinoids' gonadal cycle is synchronous, presents the right stage of
Part I: Set up of an experimental rearing procedure for echinoids
75

maturity (reproductive stages 4 and 5, growing and premature, according
to Spirlet et al, 1998a) and is of acceptable texture (firm and not leaking),
size (as large as possible), good color (yellow-orange to bright orange) and
taste. P. lividus has an annual reproductive cycle that tends to fade in
constant artificial conditions: lacking the "usual" stressors (low
temperature, lighting variation, lower quality or lack of food during
winter) the echinoids tend to bypass the growth phase of the gonads and
have permanent gametogenesis, giving rise to flabby gonads with few
nutritive phagocytes. Such gonads are unacceptable in the market. To
counteract this, the echinoids are starved at a temperature of 12-14°C and
at a 12h/12h photoperiod. This leads to consumption of the possible
content of the gonads, which also act as storage organs, in order for the
animals to get in phase regarding their reproductive cycle (reproductive
stages 1 to 3, spent and recovering, Spirlet et al, 1998a). When the content
of the gonads is fully consumed, that is, between 1 and 2 months later,
depending on their initial state, sea urchins are fed ad libitum with either
Laminaria digitata or an appropriate artificial food rich in proteins
(Klinger et al, 1994, 1997, 1998; Williams & Harris, 1998) at a higher
temperature (at least 16°C). The duration of this feeding stage is dictated
by the maturation of the gonads and lasts for 6 weeks to 3 months, mainly
depending on the food quality. Usually, both the size and the maturation
stage simultaneously reach acceptable values, and gonads are ready for the
market at the end of this starving-feeding treatment (see results).
Stage 6b: Conditioning broodstock. Maintaining mature broodstock of
P. lividus all year long is done by keeping individuals at high temperature
(between 18°C and 20°C) and under either a fixed photoperiod of 12h/12h
(directly in the growth structures) or, better, in total darkness (in a
conditioning structure), leading to the disruption of their reproductive
cycle. In these conditions, food is the most important factor to get large
quantities of good quality gametes. Feeding adults ad libitum with fresh
Laminaria digitata ensures both the quality and the quantity of sexual
Part I: Set up of an experimental rearing procedure for echinoids
76

output. The quality of gametes is often a little bit lower from December till
February, though still usable most of the time.
Measurements of reared sea urchins
Essentially two criteria are used to quantify the performances of the
rearing method: (1) the survival rate with time and (2) the growth rate, that
is, the change of test diameter of the urchins with time (gonadal size and
quality are taken into account only after the minimal market size has been
reached). The first is determined by counting survivals in a single batch
(issued from a single fertilization and a single larval rearing tank) at
various times. The counting of eggs, embryos and larvae is performed on
at least five samples of the homogenized batch (the volume chosen to
count each time is at least one hundred individuals) and the total amount is
estimated by extrapolating the mean concentration found to the whole
volume. The survival rate of competent larvae, postlarvae and juveniles is
determined by rearing subsamples of 50 to 100 individuals in SCT sieves.
Several replicates (at least five) are sacrificed and counted at each time.
All subadults and adults of a batch are counted and measured every 3
months (typically between a few hundred to a few thousand individuals in
each batch) during size sorting. Measurements of subadults and adults do
not induce additional stress or mortality other than those occurring during
the normal size grading operation (no additional manipulations). Mortality
caused by manipulations could thus be attributed to the rearing method
itself.
Size is evaluated by means of the diameter, which is measured to the
ambitus of the test (its largest part) considered without spines. To prevent
errors caused by a possible slightly oval shape, we measure two
perpendicular diameters, both to the ambitus, and only the average is
considered. The diameter of juveniles, after fixing them (glutaraldehyde
3%), is measured on digitized microphotographs using a specific image
analysis software (Grosjean et al, 1996, see Part III). The diameter of
subadults and adults is measured with a sliding caliper. Fresh weight, used
Part I: Set up of an experimental rearing procedure for echinoids
77

d. Results
to evaluate biomass, is measured after draining residual water on absorbent
paper during 5 minutes.
The relative size of the gonads is quantified by means of the fresh and
dry weight gonadal indices (GI, also called gonadosomatic indices). These
indices are defined as the ratio between the fresh (or dry) weight of the
gonads and the total fresh (or dry) weight of the urchins. First, fresh weight
of the urchins is determined after drying them for 5 minutes on absorbent
paper. The animals are then dissected, and the five gonads are extracted
and weighed. One gonad is fixed in Bouin's fluid for further determination
of its gametogenic stage (see below). The remaining four gonads are
weighed again, and the difference is computed to allow correction of the
dry weight for the missing gonad. The remaining gonads and the soma are
then dried at 70°C during 48 h (constant weight) before being separately
weighed. Dry weight gonad index is more accurate but has been found to
be less representative of the "filling" of the sea urchins (how much space
the gonads occupy inside the coelomic cavity), especially when comparing
various maturity stages and/or various diets (unpublished results). Both
indices are provided to allow comparisons.
The maturity stage is determined on histological sections of the fixed
gonad following an 8-stages scale defined by Spirlet et al (1998a). The
maturity index (MI) corresponds to the arithmetic mean of all the observed
maturity stages. Male and female data are pooled for both the GI and the
MI, since differences between sexes are not significant (Spirlet et al,
1998a, 1998b).
Table 2 shows the age, the density and the survival rate for each stage
described in rearing conditions. These data come from 29 fertilizations
studied during several years taking into account, among other things, the
seasonal variations. The survival rate for larvae is about 56%. Competence
is reached most often in 18 days (mode and median value), with an
Part I: Set up of an experimental rearing procedure for echinoids
78

average value of 19.5 days, a minimal time of 16 days and a maximal time
of 25 days. The mean metamorphosis rate is 80.4% when larvae are
competent. This rate was reached in almost two-third of the fertilizations
that attained the competent stage (non-symmetrical distribution). Thirty
percent of the larvae were discarded, either because of an incomplete
development or too low metamorphosis rate. The remaining larvae were
used for studies on postlarval or juvenile stages (and, thus, sacrificed
whenever measured) or were reared to the adult stage. Overall, the survival
rate is homogeneous from one fertilization to the other and for all stages,
except during and after the acquisition of exotrophy (transition from the
postlarval to the juvenile stage): the average rate is 54.5%, but extremes
are close to 0 and 100% (13% and 94.5% respectively). Whatever the
success of this transition, the most critical period for survival is the
juvenile stage, with a very low survival rate of 5%. Most of the mortality
occurs during the few first months of the juvenile's life (and even probably
during the few first weeks), with a progressive decrease around 8 to 9
months of age.
Table 2. Age, density, number, and survival rate of sea urchins at each rearing stage.
Rearing
stage
Developmental
stage
No.
replicated
fertilizations
1 embryos 29 (a)
2 competent larvae 29 (a)
3 postlarvae 18 (b)
4 juveniles 9 (b)
5 subadults 6 (c)
6a & b adults 5 (c)
Age Mean density Mean no. Survival from Mean
(min/ median / (no. ind./vol. individuals in previous stage global
max) or /surf. unit) 1 batch (%) survival
mean ± SD rate (%)
4 h 250 / l 50,000 - 100
16 d / 18 d / 25 d 141 / l 28,200 56.4 ± 11.6 56.4
idem + 1 d 6.5·10 4 / m 2
idem + 10 d 3.5·10 4 / m 2
2 (d)
ca. 9 months 4,000 / m
2 (d)
1.7 y / 2.6 y / 3.5 y 250 / m
Part I: Set up of an experimental rearing procedure for echinoids
22,700 80.4 ± 14.4 45.3
12,400 54.5 ± 26.8 24.7
600 4.9 ± 1.5 1.2
310 51.5 ± 3.0 0.6
(a)
Total number of larval rearing tanks: 103, from which 72 have produced enough usable competent larvae.
(b)
In the pregrowth structure, 5 to 15 replicates are measured at key times for each fertilization (see Material and Methods).
(c)
In the growth or conditioning structures. Each batch is issued from a single larval rearing tank and is followed over 2 to 7
years.
(d)
Densities in the rearing structures are adjusted during sorting operations according to both the individual size and the survival
rate.
79

survival rate (%)
test
diameter
(mm)
age (years)
Part I: Set up of an experimental rearing procedure for echinoids
nbr of individuals
Figure 18. Changes with time in the size distribution and survival rate of one fertilization
issued from a single larval rearing tank and followed over 7 years. Note the leading group
that singles out (represented by dark bars in the histograms).
Figure 18 shows both the survival rate and the size distribution over
time of a batch followed for 7 years, far beyond the minimal marketable
size and age. For the sake of clarity, only data taken every 6 months are
represented, although measurements were made every 3 months beginning
at 6 months of age. The trends observed on this single cohort are
representative of the way animals grow in cultivation, as confirmed by the
five other independent batches measured over 2 to 4 years (for an
illustrated example of another batch, see Grosjean et al, 1996, Part III).
Mortality (represented on the backwall of the 3-D box in Fig. 18)
remains very high until about 9 months of age in the pregrowth structure.
In the figured case, from around 12,400 juveniles issued from one rearing
tank, only 725 individuals where counted after 6 months and 507 remained
after another 3 months. Mortality dropped after this critical period, and 491
80

individuals were still alive 3 months later (1 year of age). This
corresponds, respectively, to a mortality of 94% (between the acquisition
of exotrophy by the juvenile to 6 months old), 30% (during the next 3
months) and 3% (after the following 3 months). The mortality rate of
subadults stabilizes around 5.4% per trimester until 6 years of age, but
ranges from 0.9% per trimester to 12.7% per trimester. Most of this
variation is correlated with season: mortality is higher during winter;
whereas, summer mortality nearly reaches 0%. Most of winter mortality
occurs by waves that start unpredictably and last for 2 to 3 days.
Juvenile's individual growth in test diameter is slow. It accelerates for
subadults but then scatters for intermediate sizes (15 to 35 mm), even
inside a presumably homogeneous batch. This scattering often results in
bimodal or trimodal size distributions (see Fig. 18 for an example and
Grosjean et al. 1996 –Part III– for an analysis). When echinoids approach
asymptotic size, their growth rate drops. Hence, the leading group is
eventually caught up by the trailers around or slightly above the minimal
market size. This minimal market size is attained between 1.7 and 3.5
years old (respectively 10% and 90% of the individuals are larger than 40
mm) with a median value of 2.6 years.
biomass (kg)
40
30
20
10
0
1 2 3 4 5 6 7
age (years)
Figure 19. Change with time in the biomass of a reared cohort of sea urchins (the same batch
as shown in Fig. 18).
Part I: Set up of an experimental rearing procedure for echinoids
81

Biomass variations (Fig. 19, same batch as in Fig. 18) are correlated to
both the survival rate and the growth speed of reared echinoids. The higher
mortality observed in winter overrides growth speed, and biomass tends to
decrease slightly. Summer biomass is highest during the third and the
fourth years in this case. The first peak of biomass (around 3.5 years old in
the figured case, between 2.8 and 3.5 years old for the other batches
depending on the season) corresponds to reaching of the minimal market
size by more than 90% of the individuals and seems to be the best time to
commercialize them after conditioning their gonads (stage 6a) from a strict
biological point of view. At that time, between 35 and 40 kg of fresh
weight sea urchins are produced in a single batch. This represents an overall
yield per surface unit of the growth structures of 4 to 7 kg / m 2 of
toboggans / year. To obtain this result, roughly 400 kg of kelp was
provided to the sea urchins. Thus, over-all food conversion efficiency lies
around 10%.
Table 3. Gonadal and maturity indices of wild and reared sea urchins (pooled results for
males and females).
Origin Month Food Treatment Tempe-
rature
field March natural diet collected in Morgat (b)
Photoperiod
(L/D)
Wet w. GI (%)
mean ± SD
Part I: Set up of an experimental rearing procedure for echinoids
Dry w. GI (%)
mean ± SD
MI (a)
mean ±
SD
10°C 13h / 11h 11.6 ± 4.2 7.1 ± 2.6 4.3 ± 0.5
cultiv. June L. digitata 2 mo starving/3 mo feeding 16°C 12h / 12h 11.1 ± 2.6 7.3 ± 1.7 4.4 ± 0.5
cultiv. May pellets (c)
cultiv. June pellets (c)
cultiv. Oct. pellets (c)
2 mo starving/2 mo feeding 16°C 12h / 12h 11.2 ± 3.3 6.7 ± 2.1 4.2 ± 1.3
2 mo starving/3 mo feeding 16°C 12h / 12h 17.5 ± 2.4 11.3 ± 1.5 6.2 ± 1.0
2 mo starving/1.5 mo feeding 16°C 17h / 7h 13.9 ± 1.5 9.7 ± 1.0 4.8 ± 0.9
(a) Best MI values for the market range from 4 to 5 (growing and premature reproductive stages).
(b) Mean values obtained on samplings during 3 consecutive years.
(c) For the composition of this food, see Williams and Harris, 1998 (their Table 1, "new diet").
Table 3 presents some results obtained after conditioning the sea
urchins for the market with the starving-feeding method. To allow
comparisons, GI and MI of field echinoids issued from Brittany are also
provided. In the field, best GI was observed in March and reaches a mean
value of 11.6% in fresh weight. Sea urchins conditioned in cultivation
82

e. Discussion
show similar GI and MI. The feeding period must be extended to 3 months
when using L. digitata, whereas 2 months are sufficient with the artificial
diet to obtain the same results. Feeding 3 months with the pellets leads to a
remarkable mean GI of 17.5% in fresh weight. Such a GI has never been
observed in the field and corresponds to the complete filling of the
coelomic cavity with the gonads, the digestive tract, almost empty, being
compressed against the body wall. However, the MI is too high and the
gonads contain too many gametes to satisfy market criteria. Furthermore,
the color obtained with the pellets is too pale (white to beige) and the taste
does not match wild roe, whereas gonads produced with L. digitata are of
good quality. An out-of-season conditioning was initiated in July with a
long-day photoperiod (17h / 7h). Very large gonads (GI around 14%) with
an adequate MI were obtained in October, after only 6 weeks of feeding
with the artificial food. Hence, the starving-feeding method could be used
to produce marketable gonads all year long.
Both the increasing demand for roe and systematic overexploitation of
wild populations support the need for a sea urchin cultivation independent
of field resources. The method presented here is one design of a rearing
process that satisfies this criterion. It appears to be successful at any life
stages of P. lividus on a pilot scale.
Obtaining gametes of P. lividus in large amounts is an easy task, as is
the rearing of its larvae with the proposed method (rudimentary devices,
low maintenance and feeding with one of the easiest microalgae to grow:
Phaeodactylum tricornutum). Metamorphosis is a little bit more critical
but can be achieved with care and use of a good inductant (fresh coralline
algae). Rearing of juveniles, subadults and adults is feasible if five major
constraints are simultaneously respected. A specific design of the rearing
structures and baskets provides (1) correct water flow around the echinoids
(for gas exchanges and removal of solid wastes) and (2) sufficient bottom
Part I: Set up of an experimental rearing procedure for echinoids
83

surface on which those benthic animals can settle (stacked toboggans). The
maintenance of good water quality is ensured by (3) the adaptation of the
sea urchin density at each life stage and (4) water renewal fixed at a
sufficient rate to minimize pollution and avoid depletion in carbonates (see
rearing method for tolerable values for both parameters without
supplemental filtration). Finally, (5) providing adequate food ad libitum
promotes somatic and gonadal growth. Further regulation of resources
allocation is possible by diet (starving-feeding method), temperature and
photoperiod conditions, leading to good quality of the final product –the
roe–, which could be obtained all year long.
If all these five conditions are met, P. lividus behaves fairly well in
cultivation and seems highly resistant to diseases. The only cases of
disease observed (mainly necrosis on the test or spines) were attributed to
opportunistic bacterial or fungal infections attributable to poor rearing
conditions, that is, when one or several of these five parameters were
poorly controlled. Cannibalism was also observed when the quality of food
was low or when carbonates concentration or pH dropped ("foraging"
behavior to compensate the lack in calcium carbonates?) or on dying
animals, but never on healthy individuals maintained in good condition.
Growth is perfectly asymptotic, and the maximal size of 45 to 65 mm
(individual variation) is reached around 3.5 to 4 years old in the rearing
conditions mentioned above. This size is similar to that observed among
the field population of Morgat, from which reared sea urchins descend
directly or indirectly. In Brittany, the most precise estimation of size at age
for wild populations of P. lividus has been performed by Allain (1978) by
analysis of the growth bands in the skeleton. According to this author, wild
sea urchins reach the size of 40 to 50 mm in 4 years, which is a little bit
longer than in the present rearing conditions (between 2 and 3.5 years).
The gain could probably be attributed to the food distributed ad libitum all
year long and to the water temperature (heating of the water in the winter)
as already suggested by Le Gall (1990).
Part I: Set up of an experimental rearing procedure for echinoids
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The success of the present method leads to optimistic forecasting for
the future of echiniculture. However, we should probably expect slightly
different results with large-scale, intensive cultivation. With the experience
acquired during this long-term trial and some informal observations
performed at a larger scale, we can predict some problems that could
potentially arise when scaling up or when considering profit. These
problems can be ranged into four different categories: (1) loss of profit due
to high and/or uncontrolled mortality of juveniles and subadults; (2)
unevenly distributed growth rates due to intraspecific competition; (3) lack
of carbonates and accumulation of CO2 because of skeletogenesis in
intensive closed-circuit systems; and (4) problems linked to the quality of
food, water pollution or poor color and/or taste of gonads produced with
artificial diets.
Survival rates around the critical period when the postlarva acquires
exotrophy to become fully functional juvenile are highly unpredictable. To
get over the difficult phase of endotrophy, the larvae must store enough
reserves before undergoing metamorphosis. In addition, the early juveniles
must promptly find suitable food when their digestive tract becomes
functional. It seems that one or both parameters are not always optimal in
rearing conditions. In any way, with a mean 55% success rate, we obtain
over 12,000 viable juveniles per 200-l tank which is enough for our use but
can probably be improved. Indeed, gametes are not limited: a female of
40-mm diameter usually produces around 5 to 7 millions of eggs. Thus, the
50,000 embryos introduced in one larval rearing tank represent only about
1% of a whole spawn (about 0.2% of the sperm produced by a single
male). Hence, only a few dozen mature adults are necessary to produce
enough gametes for mass production of larvae.
However, after the critical phase of exotrophic acquisition, the
mortality of juveniles remains very high until they reach about 10 mm in
test diameter. To minimize this, juveniles are reared in specific structures
referred to as pregrowth structures where biomass is kept at a low level
and where water quality is of prime importance. Moreover, quality of the
Part I: Set up of an experimental rearing procedure for echinoids
85

immediate environment of juveniles is improved by use of a good "waterresistant"
diet (Enteromorpha linza) and by means of cleaners (Gammarus
locusta). In any case, the space occupied in the pregrowth structure by
juveniles and the total care they need remain much lower as compared to
subadults and adults (compare densities in Table 2). This minimizes the
cost of losing many juveniles from the point of view of the total
productivity of the cultivation.
More insidious is the effect of winter mortality of subadults and adults.
Its cumulative value is ten times lower than juvenile mortality, but its cost
is much higher, because it concerns individuals occupying a significant
space in the growth structures and having already consumed a significant
amount of food (drop of the overall yield per surface unit and food
conversion efficiency). However, the cause of this seasonal variability
cannot be explained. It could be because of lower quality of food (fresh
kelp with a seasonal variation in their composition, Gayral & Cosson,
1973; Abe et al, 1983), or to any pollution of the water probably induced
by the food itself (bad quality food is less ingested and decomposes more
easily), or to another undetermined cause. For the moment, waves of mass
mortality have not been correlated with either temperature variability of
the natural seawater, meteorological conditions (atmospheric pressure,
rain) or feeding. However, any correlation will be difficult to assess,
because of the scarcity of these mass mortality waves and the probable but
not quantified delay between the stress and the observed mortality. Total
productivity could undoubtedly be enhanced if this winter mortality was
lowered or eliminated. To suppress or minimize the winter decrease in the
biomass is also worth considering when one intends to produce marketable
gonads all year long.
Mortality is not the only problem inhibiting steady productivity:
widespread distribution of growth speed among individuals expands the
time interval when largest and smallest individuals are exploitable and
constrains to sort batches frequently. Growth of P. lividus is very slow at
the juvenile stage. This "lag-phase" has also been observed by Cellario &
Part I: Set up of an experimental rearing procedure for echinoids
86

Fenaux (1990) for the same species in cultivation and by Ebert & Russell
(1993) for wild populations of Strongylocentrotus franciscanus. When
growth initiates in term of test diameter, size distribution expands. This
individual variability is not genetic but is attributable to a reversible sizebased
intraspecific competition (Grosjean et al, 1996, see Part III) that
takes place rapidly, even in size-sorted batches, although sorting reduces
its effect. Presently, the exact impact of this competition on productivity
and the best way to avoid it (if it should be avoided at all) are still
unknown.
A third problem that will probably occur when considering further
intensification of echiniculture in closed or semiclosed systems is the
depletion of dissolved carbonates and the accumulation of CO2 in
seawater. In growing, the sea urchin builds a magnesium-calcite skeleton.
This skeleton represents an important fraction of the body weight: between
28% and 31% of the total fresh weight for P. lividus (measured on animals
issued from the reared strain, after digestion of organic tissues with a
12°Chl bleaching agent under gentle agitation, n = 356). Thus, for each kg
of fresh weight produced, about one-third has to be supplied in one or the
other form of calcium carbonate. However, P. lividus is unable to
assimilate efficiently carbonates provided as a solid substrate (calcareous
rocks, algae or cuttlefish bones for instance) because the pH of its
digestive tract is too high to dissolve large amounts of solid calcite
(between six and eight, for a review see Lawrence, 1982; for data
concerning P. lividus see Claerebout & Jangoux, 1985). The main usable
source of magnesium/calcium carbonates is thus present under a dissolved
form in seawater. If calcium and magnesium ions (respectively 400 mg
and 1,350 mg per kg seawater at a salinity of 35‰, Spotte, 1991) are not
limiting, the quantity of dissolved carbonates available could be consumed
very quickly in intensive closed or semiclosed systems (unpublished data).
Most of the carbonate alkalinity (about 2.3 - 2.4 meq / kg seawater,
corresponding to 140 mg of HCO3 - ) remains unavailable for
skeletogenesis, the pH dropping too much when sea urchins consume it
Part I: Set up of an experimental rearing procedure for echinoids
87

(carbonate and bicarbonate are the most important chemical components
that buffer pH in seawater, Stumm & Morgan, 1981). The actual fraction
the sea urchins can use is still unknown, but is probably under 10% of the
total carbonate alkalinity. To illustrate this, without supplemental chemical
filtration and with a usable fraction of 10% of the dissolved carbonates to
produce skeleton that final weight represents 30% of the total sea urchin
fresh weight, one must provide at least 24,500 m 3 of seawater per ton of
sea urchin fresh weight produced. However, this optimistic calculation
does not consider mortality that otherwise also exports carbonates.
Precipitation of bicarbonates (the main form of dissolved carbonates in
seawater at usual pH) into calcium carbonate is a dismutation reaction that
liberates a stoichiometric amount of carbonic acid in the water column.
This carbonic acid, together with the CO2 produced by the respiration of
sea urchins, algae and bacteria in the rearing structures tends to reach
rapidly undesired levels in a large-scale intensive cultivation. We have
observed sea urchins whose skeleton growth was totally inhibited in these
conditions. CO2 partial pressure was recorded to be 5 to 9 times higher
than usual in seawater (despite a strong aeration of the water) and was
presumed to be the direct cause of the inhibition of the skeletogenesis.
These limitations force us to choose either a flow-through system or to
provide a chemical filtration to level carbonates and carbonic acid
concentrations. The present method could be considered as a semiintensive,
semiclosed system where both sea urchin densities and water
renewals remain compatible with the equilibrium of the inorganic carbon
in seawater without supplemental filtration. However, such a trade-off
would not be compatible with a rearing strategy aiming to raise profit on a
large scale.
For the moment, fresh algae used as food form part of the natural diet
of P. lividus. The composition of this food is presumably correct, although
it might not be necessarily optimal (Frantzis & Grémare, 1992; Gonzalez
et al, 1993; Fernandez & Boudouresque, 1998). The major problem
Part I: Set up of an experimental rearing procedure for echinoids
88

encountered with food is its stability once put in the rearing structures
because this echinoid, being a grazer, ingests it slowly. Uneaten food could
easily give rise to undesired pollution. Hence, we recommend the use of a
stable diet (Enteromorpha linza) in the present rearing method instead of
higher quality algae (Laminaria digitata, L. saccharina Lamouroux or
Rhodymenia palmata (L.) Greville, unpublished results) for juveniles. We
also avoid using artificial diets at water temperature above 16°C without
the presence of an efficient biofilter in the rearing structures.
The use of fresh algae is not always possible or profitable on a large
scale (Fernandez, 1996). Hence, an artificial diet designed specifically for
sea urchins seems necessary for intensified echiniculture and is presently
under investigation by several authors (Fernandez & Caltagirone, 1994;
Klinger et al, 1994, 1997, 1998; de Jong-Westman et al, 1995a, 1995b;
Fernandez, 1996). Results obtained so far are encouraging, especially in
term of GI but the food we were able to test gave unsatisfactory results in
terms of color and palatability of the roe. Recent testing of semimoist diets
on Strongylocentrotus droebachiensis (Motnikar et al, 1997) seems to
confirm the positive effect of the artificial diet on the gonadosomatic index
and the failure to obtain high quality gonads in terms of color and taste.
Trials with carotenoids-enriched artificial food to enhance the color do not
yet produce high quality gonads (Goebel & Barker, 1998). Thus, a better
formulation of the food is basic to achieve a correct taste and color for
exploitation.
Finally we should mention that the rearing method described here is
labor-intensive. Hence, manpower cost could be too high when
considering profit. This would require some adaptation or mechanization
of the most time-consuming operations: feeding subadults and adults,
cleaning the growth structures, grading the batches or extracting the
gonads if sea urchins are not commercialized alive (exportation to Japan).
However, these are technical problems that could be solved by the
industry.
Part I: Set up of an experimental rearing procedure for echinoids
89

f. Conclusions
This rearing method constitutes a good working basis to design a
closed-cycle, land-based echiniculture. We suggest it could be used as a
standard method to evaluate improvement obtained by adaptations or
modifications aimed at intensification or profitability of echiniculture. This
method could possibly be adapted to other species, allowing better
comparisons of the biology of respective species as well as their
aquaculture potentials.
Latent remaining problems when scaling up and intensifying
cultivation, aiming at raising profit, should not be regarded as unavoidable
limitations, but should be considered as challenges to address in further
studies. Being "new" cultivated species, it is not surprising that these
obstacles mostly concern less known life stages or "biological features or
characteristics" of sea urchins: the transition between the endotrophic
postlarva and the exotrophic juvenile, the mechanism of the intraspecific
competition, the carbonate budget needed for skeletogenesis and the
biochemical pathways in gametogenesis and in stocking reserve material in
the gonads. Thus, it is probable that advances in fundamental biology of
echinoderms, and more particularly of echinoids, will suggest solutions to
these problems in the future.
It would seem that further development of closed-cycle, land-based sea
urchin cultivation is worthwhile and will undoubtedly promote
diversification of aquaculture and production of high quality seafood. This
will, secondarily, lead to the conservation of the natural environment by
limiting the fisheries impact on natural populations of sea urchins.
g. Acknowledgements
This study was conducted in the framework of EEC contracts in the
"AIR" and "FAR" aquaculture program (ref. AQ2.530 BFE & CT96.1623
BFN). This research was also supported by an EC research grant attributed
Part I: Set up of an experimental rearing procedure for echinoids
90

to Christine Spirlet (ref. ERB 4001 GT92 0223), in the framework of the
Sea Urchin Cultivation contract n° AQ 2.530 BFE. We thank Didier
Bucaille for his help in the laboratory work and the CREC (University of
Caen) for its financial contribution in the building of the specific sea
urchins facility. We are grateful to John Lawrence for providing the
artificial diet and to Addison Lawrence and the Wenger Company for
designing and producing it. Thanks to Raphaël Morgan for proofreading
the manuscript. This paper is a contribution to the "Centre
Interuniversitaire de Biologie Marine" (CIBIM).
Part I: Set up of an experimental rearing procedure for echinoids
91

Part I: Set up of an experimental rearing procedure for echinoids
92

PART II
Measurement for size in the sea urchin
93

PART II: MEASUREMENT FOR SIZE IN THE SEA URCHIN
Having a rearing method to grow P. lividus in aquaria, we still have to
decide how to measure growth, that is, size increase of the sea urchins
between various time intervals. Clearly the best measurement method
should be both rapid (to allow measuring hundreds of individuals in a
reasonable time) and as accurate and reproducible as possible. That
measurement should be also most representative of somatic growth.
Finally it should be harmless, since successive measures of the same
animals will be performed.
Various direct measurements of body size are available for sea urchins:
diameter or height of the test, volume, total fresh weight or 'immersed
weight'. Since sea urchin has a rigid endoskeleton, its shape is constrained
and it is easy to take a linear measurement: either the diameter or the
height of its test. However, mouth is not rigidly fixed to the test. A
measure of height, that is from mouth to anus, is thus less precise than a
measure of diameter, and we did not consider it. Volume measurement was
also eliminated for it is too inaccurate: the volume of 15 sea urchins
measured 8 times each using the 'displacement apparatus' described in
Comely & Ansell (1988, their Fig. 2) has an accuracy (expressed as
standard deviation in percent of the mean) of ca. 10.9% when accuracies
for diameter, fresh weight or immersed weight range from 0.6 to 2.5% (see
Table 4, p. 101).
Part II: Measurement for size in the sea urchin
95

Part II: Measurement for size in the sea urchin
96

Comparison of three body-size measurements for echinoids
a. Abstract
b. Introduction
Ph. Grosjean, Ch. Spirlet & M. Jangoux, 1999. Echinoderm Research
1998. M.D. Candia Carnevali & F. Bonasoro (eds). Balkema,
Rotterdam. Pp 31-35.
Several measurements can be employed to quantify the body size of
echinoids. We evaluate here the accuracy of three measurements on the sea
urchin Paracentrotus lividus (test diameter, fresh body weight and
immersed weight –the weight of the sea urchin when immersed in
seawater–) and discuss their respective potentials. The immersed weight
appears to be by far the most accurate, providing it is standardized, but
also the most time-costly measurement. Allometric relationships and
formula for calculating a standard immersed weight for P. lividus are also
provided.
In vivo determination of the body size is a basic approach used in many
fields in biology, including individual growth (Ebert, 1967; Grosjean et al,
1996; Régis, 1969), population dynamics (Allain, 1972a; Régis & Arfi,
1978), morphometry or biometry (Ebert, 1968, 1981, 1988b; Lawrence et
al, 1995; Moss & Meehan, 1968), physiology (through calculation of
gonadal or repletion indices, for instance Agatsuma & Sugawara, 1988;
Giese, 1966; Nedelec, 1983; Spirlet et al, 1998a). Various kinds of
measurements are available, from lengths to volumes or weights. For
echinoids, this task is facilitated by the presence of a rigid endoskeleton
that restrains both their external dimensions and their total
volume / weight. Hence, several measurements are accessible and used to
quantify the body size.
Part II: Measurement for size in the sea urchin
97

Few studies have compared and discussed the accuracy and suitability
of these various measurements, except for the classical relationship
between the test diameter and the body weight (Agatsuma & Sugawara,
1988; Allain, 1972a; Kaneko et al, 1981). Consequently, authors use
different body size measurements that are not always optimal for their
studies.
In the present study, the accuracy, reproducibility and suitability of
some measurements that can be performed in vivo on sea urchins were
assessed. The three selected measurements are test diameter, fresh body
weight and immersed weight.
c. Material and methods
A stratified sample of 224 Paracentrotus lividus sea urchins of various
sizes (20 to 25 individuals in each 5 mm size-class ranging from 10 to 60
mm in test diameter) was measured. Half of them where directly collected
in the field in Morgat, Brittany (France). The others where cultivated
specimens from the Marine Station of Luc-sur-Mer, Normandy (France)
(see Grosjean et al, 1998 –Part I– for the protocol), but which field parents
originated also from Morgat. In both field and cultivated individuals, half
population was measured in September, and half was measured in March
in order to assess a possible seasonal variation (Spirlet et al, 1998a).
- The diameter is measured to the nearest 0.1 mm with a sliding caliper to
the widest part of the sea urchin (the ambitus), and without considering the
spines. In case of a possible oval shape, it is the average of two
perpendicular diameters taken to the ambitus that is considered.
- The total fresh weight is measured to the nearest 0.001 g after leaving the
sea urchins for 5 minutes (stabilization of weight) on absorbent paper,
which prevents possible fluctuations due to residual water on the
integument surface.
Part II: Measurement for size in the sea urchin
98

- The immersed weight is a much less customary measurement and its use
is further discussed. In the present case, it corresponds to the apparent
weight of a sea urchin in seawater. It is assessed with a scale (precision of
0.1%) provided with a plate or basket immerged in a tank of seawater and
containing the individuals to be weighed (see Fig. 20).
1
Part II: Measurement for size in the sea urchin
2
3
4
EXCEL
Figure 20. Data acquisition system. A scale (1), an electronic sliding caliper (2) and a second
scale equipped with a plate immersed in a tank filled with sea water (3) are connected to a
computer (4) for fast data treatment (a software to acquire data directly from scales and
calipers into a PC and to calculate the SIW is freely available from the authors).
Each specimen was measured only once. However, to assess the
reproducibility of the 3 types of measurement and the possible variation
due to the experimenter, an additional 15 echinoids were measured 3 times
at a 4 h interval by 7 different people. The individuals were distributed
randomly to the experimenters. Data were collected by means of a data
acquisition system composed of an electronic sliding caliper and electronic
scales connected to a computer (Fig. 20).
99

d. Results and discussion
The ambital test diameter and the total fresh weight measurements are
exploitable directly. The values of the immersed weight can be compared
only if the density of the seawater (depending upon salinity and
temperature) is constant between measurements, otherwise a correction
factor must be introduced. The immersed weight is the resultant of 2
opposite forces, the weight and the buoyancy, which compensate each
other for organs of the same density as sea water: gonads, digestive tract,
and coelomic fluid (Stickle & Ahokas, 1974). Their apparent weight in
seawater is thus close to zero. Conversely, the calcareous skeleton has a
significant positive apparent weight which means that the immerged
weight is primarily a measure of the apparent weight of the skeleton in
seawater. This is also evidenced in Table 5, showing the immersed weight
is directly proportionate to the dry weight of the skeleton (allometric
coefficient = 1.00 = perfect isometry).
We define the standard immersed weight (SIW) as the immersed
weight that would have been measured in a liquid which density is strictly
equal to 1.00·10 3 g/l; we calculate it as follows:
2.80 −1.00
SIW = IW.
2.80 − Md /1000
where: - SIW is the standard immersed weight in g,
- IW is the measured immersed weight in g,
- Md is the mass density of sea water where echinoids are
measured, in g/l,
- 2.80 is the apparent mean density of the sea urchin skeleton in
10 3 g/l (δs in eq. 19).
Part II: Measurement for size in the sea urchin
(18)
The mass density of the seawater can be determined either directly
(with a densitometer), or by calculation (Cox et al, 1970; UNESCO, 1981).
In the second case, both the salinity and the temperature of the water are
needed.
100

The apparent mean density of the sea urchin skeleton δs is calculated
from the isometric relationship between the immersed weight measured at
a constant seawater density (1.023·10 3 g/l) and the dry weight of the
skeleton DWs (n = 63, R 2 = 0.999):
δ s
DWs = 1.576⋅ IW = ⋅IW⇔δs ≈2.80
δ −1.023
Part II: Measurement for size in the sea urchin
s
(19)
The SIW is usually 2 to 3% higher than the immersed weight actually
measured.
General comparison
The diameter is the fastest and easiest measurement in the field (see
Table 4). It is the most convenient parameter for separating the sea urchins
in size categories or for measuring individually large amounts of
echinoids. The others are suited more for batch evaluation (total biomass
for instance). Fresh weight and SIW take more time. Since several
individuals can be drought simultaneously for the fresh weight, time spent
for each measurement drops to around 40 s, instead of the overall 5 min.
Hence, the SIW is the longest measurement because the scale takes a while
to stabilize. However, it is both the most accurate and the less stressful
measurement, which can be of importance when working on sexually
mature individuals that can spawn when handled.
Table 4. Comparison of the three selected parameters.
Parameter Diameter Fresh weight SIW
Timing < 30s 40s (5min) ca. 2min
Accuracy (a) 1.33-2.52% (b)
> 1.31% < 0.62%
Possible bias (b)
yes no no
Stress medium medium low
Batch measure no yes yes
Field measure (c)
yes no no
(a)
Standard deviation expressed in percent of the mean.
(b)
Depending on the experimenter.
(c)
Easily usable in the field and underwater.
101

Reproducibility of measurements
The diameter of the test to the ambitus is reproducible when it is done
by the same person. A two-way ANOVA (measurement order versus
experimenter) indicates that there is no significant difference between
measures from a single experimenter and no interaction between
measurements and experimenters (p > 0.05 in both cases). However, the
experimenters have a great influence (p < 0.01) on the values recorded.
The same analysis done on fresh weight and SIW reveals there is only
one significant effect (p < 0.01): the order of measurements. The
difference between 2 successive measurements is steady for all animals, in
all cases. The SIW decreases by an average of –0.63% then –0.49%
between measurements which can be due to the accidental breaking and
loss of spines during handling. Conversely, there is an average increase in
the fresh weight of successively +1.70% and +0.64% between 2 series.
Such high variation in 4 h intervals can be explained only by slight
variations in the volume of the water confined in the echinoid (perivisceral
fluid and/or intradigestive fluid; protrusion more or less important of the
Aristotle's lantern). This is a drawback that would lower both the accuracy
and reproducibility of fresh weight measure in echinoids.
This analysis reveals that the SIW is by far the most reproducible and
thus reliable measurement. Its reliability is probably even higher than
shown in Table 4 where the loss due to broken spines between
measurements was not deduced from the overall recorded variation.
Allometry and measurements relationship
An ANCOVA on log-transformed data (p > 0.05) indicates there is no
effect of the origin (field or cultivated), or the seasons on the allometric
relationship between the three measurements. Hence, data are pooled.
Table 5 presents model I allometric relations between the 3 measurements
considered. All regressions are highly significant (R 2 ≥ 98%) for this
species in the size range explored. In all cases, the double log data
Part II: Measurement for size in the sea urchin
102

transformation leads to linear regressions with homoscedasticity of
variance and random distribution of the residuals. However, caution is the
rule as model I is not verified (the independent variable should be
measured without error which is not the case here). A non-biased model II
would be more adequate (Ebert, 1981, 1994; Laws & Archie, 1981;
Tessier, 1948) but only biased model II are available for such data sets
(Sokal & Rohlf, 1981, p. 549). Since the explained variance is higher than
98%, the bias remains negligible, whatever the model chosen. Thus, in this
case, a model I is to be preferred for prediction purposes with independent
regressions for reciprocal relationships (Sokal & Rohlf, 1981).
Table 5. Allometric relations (model I linear regressions on double log transformed data)
between parameters for Paracentrotus lividus from Morgat, n = 224. Verifications are needed
when applying on other strains, or out of the announced validity range.
Measured (x) Estimated (y) Allometry R 2
Std err
log(y)
Validity
Diameter (mm) Fresh weight (g) y = 5.50·10 -4 x 2.94
0.997 0.037 10 < x < 60
Diameter (mm) SIW (g) y = 2.40·10 -4 x 2.70
0.986 0.053 10 < x < 60
Fresh weight (g) Diameter (mm) y = 12.7 x 0.35
0.995 0.011 0.5 < x < 90
Fresh weight (g) SIW (g) y = 0.22 x 0.95
0.994 0.034 0.5 < x < 90
SIW (g) Diameter (mm) y = 22.1 x 0.37
0.984 0.019 0.1 < x < 15
SIW (g) Fresh weight (g) y = 4.95 x 1.05
0.994 0.036 0.1 < x < 15
SIW (g) Skeleton(dry w. g) (a)
y = 1.56 x 0.998 0.021 0.1 < x < 15
SIW (g) Soma (dry w. g) y = 1.74 x 0.98
0.999 0.010 0.1 < x < 15
(a)
Measured after digestion of the organic matter with sodium hypochloride 10% under gentle agitation
and drying for 48h at 70°C.
The SIW being a direct in vivo measurement of the skeleton weight of
the sea urchin, the latter can be calculated by the formula in Table 5. As
the soma is composed of ca. 90% of skeleton (in dry weight, Grosjean,
unpubl.), it is also a reasonably good in vivo estimation of the somatic dry
weight, after applying possibly a correction calculated after the SIW-soma
allometric relationship (Table 5). As such, it allows to follow most
accurately the somatic growth of the sea urchins. In some experiments
(Grosjean et al, in prep.), we were able to quantify somatic growth of sea
urchins within a 7-days period using the SIW, while it would require at
least a 1 or 2 months period to get the same accuracy with test diameter or
Part II: Measurement for size in the sea urchin
103

e. Conclusions
fresh weight! Caution must be taken, of course, when applying
conversions on echinoids in particular physiological state that lead to
variations in allometric relationships, such with starved individuals (Ebert,
1968; Kaneko et al, 1981).
The SIW should be used whenever possible both as a reference
measurement for indices (gonadal index, repletion index…) and for studies
involving somatic growth. Test diameter remains the fastest measure, and
thus preferred for measuring large number of individuals when accuracy is
not of prime importance, or for measures in the field. Fresh weight use
should be restricted to the determination of the biomass.
f. Acknowledgements
This work was supported by an EC research grand attributed to Ch.
Spirlet (ref. ERB 4001 GT92 0223), in the framework of the contract No.
AQ2.530 BFE ("Sea urchin cultivation"). We thank D. Bucaille, P.
Gosselin, F. Benard & F. Louise for help in measurements. This paper is a
contribution to the Centre Interuniversitaire de Biologie Marine (CIBIM).
Part II: Measurement for size in the sea urchin
104

Choice of measurement
Immersed weight is definitely the most accurate and less stressing
measurement of sea urchin body size. However, it is not possible, using
this method, to measure hundreds of echinoids in a reasonable period of
time. We thus used the test diameter, as the optimal compromise between
accuracy and speed.
The diameter has another advantage: it is the only parameter that can
be measured on very small animals of a few hundreds of microns (that is,
the size of P. lividus just after metamorphosis, see Fig. 2B, p. 37). To do
this, we have to kill and fix the individuals (3% glutaraldehyde) in order to
manipulate them and transfer them on a millimeter-graduated support to be
photographed. The picture is digitalized and analyzed with a custom image
analysis software (ShellAxis, available at http://www.sciviews.org). For a
description of the program, see Van Osselaer & Grosjean (2000) and for
extensive tests of accuracy and reproducibility of measurement with this
software, see Van Osselaer (2001).
Measuring body size with good accuracy is one aspect, knowing what
it really means in terms of relative sizes of the different organs during
growth is another one. Indeed, successive measurements would be really
representative of somatic growth if they were strongly correlated with
growth of all somatic organs.
To determine how different compartments of the sea urchin vary with
body size, a stratified sample (from 5 to 60 mm every 5 mm) of 440
animals was analyzed. 220 sea urchins were dissected in spring and 220 in
autumn, which correspond to two contrasting reproductive stages in the
field: empty or full gonads, in order to make sure maximum variance is
included in the dataset. Measurements done on each individual were:
immersed weight; two perpendicular diameters at the ambitus; height; total
fresh weight; fresh "drained" weight (that is, the test is opened by cutting
around the ambitus and the two resulting parts are left upside down on
Part II: Measurement for size in the sea urchin
105

principal axis 3
1
0.8
0.6
0.4
0.2
absorbent paper for 5 min to eliminate most of the coelomic fluid); fresh
and dry weight (drying at 70°C until constant weight) of the digestive tract
with its contents, of the gonads and of the integuments; dry weight of
Aristotle's lantern, spines and test after elimination of most organic part
with 12°Chl bleach under slow agitation.
0
0
0.2
0.4
principal
axis 2
0.6
0.8
1 1
Part II: Measurement for size in the sea urchin
0.8
0.6
0.2
0.4
principal
axis 1
Figure 21. Principal components analysis: orientation in the first 3D-space of the vectors
representing the 14 measurements. Different colors have been chosen to symbolize the 4
groups of measurements: for general body size measurements, for the integuments and
skeleton, for the digestive tract and its content and for the gonads.
Principal component analysis (PCA) indicated that all body size
measurements are well correlated with the first axis (trend corresponding
0
106

to general growth) that explains 55.0% of the whole variance (Fig. 21).
Integument and skeleton measurements are also well correlated with the
first axis. The second axis explains 30.2% of the total variance. Gonad
measurements are highly correlated with this axis that represents change
with the reproductive cycle. Note that body size measurements are not
much correlated with gonads measurements in any axis. Indeed, due to the
rigidity of the test, diameter, height and total weights are not affected by
the size of the gonads (when the sea urchin has small gonads, its general
cavity is filled with coelomic fluid with about the same density than the
gonads). This is convenient because body size measurements can be
considered as soma measurements, independently of the size of the
gonads. The third axis amounts for 12.9% of the total variance and is
slightly represented by digestive tract measurements. It should be due to its
contents, as a consequence of the feeding activity during the last day
before dissection. 98.2% of the variance is explained by the first three axes
that are kept (Table 6).
Table 6. Principal components analysis: contribution of the parameters to the three first axes.
Parameter Axis 1 Axis 2 Axis 3
Height 0.895 0.311 0.246
Diameter 0.869 0.369 0.269
Weight of Aristotle's lantern 0.852 0.365 0.323
Weight of spines 0.833 0.450 0.248
Weight of test 0.804 0.450 0.347
Immersed weight 0.799 0.510 0.298
Fresh weight of integuments 0.789 0.507 0.339
Dry weight of integuments 0.769 0.574 0.291
Total fresh weight 0.729 0.552 0.390
Drained weight 0.729 0.575 0.371
Dry weight of digestive tract and its content 0.699 0.407 0.567
Fresh weight of digestive tract and its content 0.629 0.454 0.626
Dry weight of gonads 0.360 0.900 0.231
Fresh weight of gonads 0.392 0.891 0.215
We are now confident that a body size measurement, e.g., the diameter
of the test, is an adequate representation of the growth achieved by all the
somatic organs. Sea urchin appears to be a good experimental subject for
studying growth because size is easy to measure with accuracy thanks to
the rigidity of its skeleton that constraints its shape; also because it exhibits
Part II: Measurement for size in the sea urchin
107

a relative homogeneous growth of all its somatic organs. For its model, the
sea urchin could be considered as a "system with three degrees of
freedom": (1) the body wall and most somatic organs, (2) the gonads and
(3) the digestive tract and its content. This means we can fully characterize
the echinoid by three, easily performed, measurements such as the body
size, the gonad index (Spirlet et al, 2000, 2001) and the repletion index
(Nedelec, 1983). With such three parameters, it should be possible to
calculate the size and the weight of all organs, once the corresponding
allometric relationships are established.
Part II: Measurement for size in the sea urchin
108

PART III
Experimental studies of the intraspecific competition
109

110

PART III: EXPERIMENTAL STUDIES OF THE
INTRASPECIFIC COMPETITION
Working on P. lividus, we were puzzled by asymmetry and even
multimodality in size distributions of previously homogeneous batches of
reared sea urchins (see Part I). However, almost no author, except
Himmelman (1986) and Levitan (1988), considered that intraspecific
competition exists among populations of aggregative sea urchins (see the
general introduction).
In fact, field observations do not easily allow the study of intraspecific
competition because one can only describe size distributions of wild
populations. Shape of size distributions, being bimodal for instance, does
not tell which is the cause of this bimodality. One can speculate on
possible mechanisms involved without bringing evidences. Huston & De
Angelis identified, among other possible causes, at least 23 biological
mechanisms that can produce bimodality (1987, see their Table 1). In such
circumstances, only targeted experiments can bring clues on what really
happens. Our rearing system provided a good basis to start from. So, we
were able to explore a little deeper these puzzling asymmetric size
distributions that appear in batches of reared sea urchins.
Part III: Experimental studies of the intraspecific competition
111

Part III: Experimental studies of the intraspecific competition
112

Experimental study of growth in the echinoid Paracentrotus
lividus (Lamarck, 1816) (Echinodermata).
a. Abstract
b. Introduction
Ph. Grosjean, Ch. Spirlet & M. Jangoux, 1996. Journal of
Experimental Marine Biology and Ecology, 201:173-184.
Multimodal size frequency distribution (that is, a few individuals
growing very fast and a few individuals growing very slowly) among an
originally homogeneous cohort of juveniles Paracentrotus lividus is
observed in reared conditions when they are 6 to 24 months old. The
splitting of this cohort into homogeneous size-classed subgroups results in
an increased growth of the smaller animals that catch up with the bigger
ones in 4 months time. This indicates that the smaller animals are not
genetically less productive and suggests they were inhibited in their
growth due to the presence of larger ones. Supposing such growth
inhibition also occurs in the natural environment, the observed mechanism
could be very efficient in stabilizing field populations of aggregative
echinoid species by maintaining a protected pool of small individuals with
high growth potential but inhibited by the density of larger ones.
Keywords: Echinoids, growth, population dynamics, size frequency
distribution.
Echinoid surveys in the field are often based on size frequency
distribution studies which theoretically allow the separation of different
cohorts (Ebert, 1973; Kenner, 1992; Guillou & Michel, 1993). When
proceeding so, authors necessarily assume that the size frequency
distribution in a single cohort is normal or at least unimodal (Ebert, 1981;
Ebert et al, 1993; Botsford et al, 1994). When animals can be aged, the
Part III: Experimental studies of the intraspecific competition
113

assumption of normality can be tested because the different cohorts are
unambiguously separated. Since size is not a reliable indication of the age
of echinoid (Ebert, 1967; Levitan, 1988) and since the possibility to age
them with growth lines remains disputed (Ebert, 1986; Gage, 1992; Ebert
& Russell, 1993; Gebauer & Moreno, 1995), the interpretations of size
frequency distributions among natural populations of echinoids remain
rather speculative unless the assumption of normality can be verified. The
difficulty may be bypassed by rearing animals under controlled conditions
without the pressure of predators which should allow a good follow-up of
their growth and enable the observation of a cohort’s distribution. Indeed,
parameters like recruitment, mortality and age of the individuals can be
known precisely. Small-scale cultivation of echinoids under artificial
conditions has been successfully performed for several years (e.g.,
Hinegardner, 1969; Fridberger et al, 1979; Le Gall, 1990) and among the
various aspects tackled by different authors, growth seems to be a
privileged one (Ebert, 1975; Fridberger et al, 1979; Frantzis & Grémare,
1992). Yet, data remain rather limited and little interpretation of the size
distribution itself has been performed. Cellario and Fenaux (1990)
observed that there was a "wide size dispersion pattern with age progress"
in P. lividus rearing. They also observed that the relative spreading of size
(standard deviation of test diameters / average test diameter ratio)
increases rapidly in early postmetamorphic life, reaches a constant level at
2-3 mm of mean test diameter (the distribution is then at its widest) and
begins to decrease when the animals become larger than 10 mm in
diameter. As far as we know, no study treated more precisely the size
distribution shape of a cohort of reared echinoids.
The present paper focuses on how size distribution of reared
Paracentrotus lividus changes with time. It aims at testing if this
distribution is normal and attempts to determine the factors that lead to its
spreading.
Part III: Experimental studies of the intraspecific competition
114

c. Material and methods
All the echinoids used in this work were produced in laboratory. They
were cultivated with a method adapted from Le Gall (1990). The original
strain comes from the rocky shore of Morgat (Brittany, France).
Rearing procedure
Spawning was induced by injecting KCl 0.5 M in the body cavity of
adult individuals. Eggs of one female were transferred in a small plastic jar
containing 800 ml of seawater. A quantity of sperm equivalent to 0.5 ml of
milt was added to the eggs. The fertilization was controlled after 4 h and
the number of fertilized eggs was evaluated. The embryos (in the gastrula
stage) were then transferred in a 200-l larval rearing tank to a
concentration of 250 embryos per liter. Larvae were fed daily with
Pheodactylum tricornutum from the third day on. The water remained
unchanged for the whole larval period.
About twenty days later, competent larvae were transferred in clean
sieves with a 250-µm mesh. Sieves with larvae were placed in toboggans
(see Le Gall, 1990) with 10 cm depth of recirculating water providing a
gentle uniform current around them. Metamorphosis was induced by
introducing coralline algae in the sieves. Larval fixation and
metamorphosis took less than 24 hours.
The day before juveniles become exotrophic, 5 g (fresh weight) of
green alga Enteromorpha linza per 100 cm 2 sieve surface were distributed.
From this moment on, the same food quantity was given every 15 days.
The treatment remained identical during the first year, except that the sieve
diameter and mesh size were progressively increased according to the
growing diameter of the individuals. After one year, when the smallest
echinoids reached more than 5 mm in test diameter, all the individuals
were put in a basket with a 5 mm mesh and transferred in another
toboggan where stronger water current and higher echinoid biomass
Part III: Experimental studies of the intraspecific competition
115

occurred. From this moment on, and twice a week, individuals were fed ad
libitum with fresh kelp Laminaria digitata.
The entire rearing was carried out in natural dim light and to a constant
temperature of 18 ± 2°C all year long. The water was renewed
continuously at a rate of 200% to 300% of the total volume per day with
fresh natural seawater allowed to settle for at least 30 h beforehand.
Size frequency distribution just after the metamorphosis
All the larvae issued from a single fertilization (Fa) and reared as
described above in the same tank were induced to metamorphosis the same
day. Postmetamorphics were not fed. They were fixed (3% glutaraldehyde)
and photographed 7 days after metamorphosis. Pictures were transferred
on a Kodak photoCD and the test diameter of every individual computed
by image analysis software. Actual size was determined by using a
graduated background. The frequencies of observed sizes were tested
against a normal and a log-normal distribution with a Kolmogorov-
Smirnov test adapted by Lilliefors for intrinsic comparison (Sokal &
Rohlf, 1981).
Follow up of a single reared strain of juveniles during 30
months
A whole strain issued from a single fertilization (Fb) was cultivated
over 30 months. The test diameter of each individual was measured every
6 months with a sliding caliper. The first set of measurements was
performed when echinoids were 6 months old (no measurements were
done just after metamorphosis because of the extreme fragility of early
postmetamorphics). The total number of individuals was 536 at 6 months
old, and dropped progressively to 280 at 30 months old. The mean
mortality was thus 48% on a 2-year period.
Size frequency data obtained were then tested against a normal and a
log-normal distribution. Possible multimodality was checked by the
Part III: Experimental studies of the intraspecific competition
116

graphical method of Bhattacharya (1967). Since the number of individuals
is low, data need to be smoothed before applying the method:
fs = f + f + f
(20)
1 1 1
i 4 ( i− 1) 2 i 4 ( i+
1)
where: fi = frequency observed in the size class i,
fsii = smoothed frequency for the class i.
The minimal number of modes that match the observed size frequency
distribution (i.e: when a χ 2 test gives a probability higher than 0.05) was
determined by the technique of maximum-likelihood estimator using
NORMSEP (Hasselblad, 1966; modified by McDonald & Pitcher, 1979).
Effect of size sorting on the growth of juveniles and
interactions among them
Three additional fertilizations (Fc, Fd and Fe) were done at different
times with parents not genetically related. Produced individuals were
sorted twice before the beginning of the experiment so as to have several
homogeneous batches in terms of size. The experiment started with
populations of Fc, Fd and Fe being respectively 4, 6 and 8 months old.
Four batches (Fc1 to Fc4) and six batches (Fd1 to Fd6) of 50 size-sorted
echinoids were set up for Fc and Fd respectively. Mean-sized individuals
of Fe were separated into nine batches of 20 echinoids (Fe1 to Fe8 plus a
control batch randomly chosen).
Each Fc to Fe batch was cultivated in a sieve of 20 x 15 cm of surface
with a 2-mm mesh except the control batch of Fe whose 20 individuals
were reared separately in 20 different sieves. Individuals were fed ad
libitum with Enteromorpha linza exclusively. The seaweeds covered the
entire surface of the sieve so there was no competition for food.
Experiments lasted for 4 months and the final test diameter of all echinoids
from each batch was measured with a sliding caliper.
Part III: Experimental studies of the intraspecific competition
117

d. Results
Size distribution among early postmetamorphics and change
through time
Statistics on the distribution of the test diameters among early
postmetamorphics (Fa) is presented in Table 7. Temporal evolution in the
size frequency of all the individuals of the cohort followed during 30
months (Fb) is presented in Table 7 and Fig. 22. The test diameter of early
postmetamorphics distribute along a normal curve characterized by a mean
of 497 µm and a standard deviation of 56 µm (Kolmogorov-Smirnov /
Lilliefors test, p > 0.05).
At 6 months old, distribution is neither normal, nor log-normal
(p < 0.001, Kolmogorov-Smirnov / Lilliefors test); multimodality appears,
although not clearly yet. Two modes can be identified by both graphical
and numerical analysis. However, this kind of graph might also be
obtained with a unimodal distribution very skewed to the right (skewness =
0.748).
After 12 months, the distribution does not match a normal nor lognormal
one. This time, a "head" portion is clearly distinguishable. It
concerns 18 individuals, that is 5.1% of the total. At least two other classes
could be separated although not clearly isolated from each other. The
distribution is widely spread. The ratio between the 10% larger and the
10% smaller is 3.2 in test diameter and 30.9 in wet weight.
After 18 months, the general shape remains the same with a head
clearly detached, except that the two other classes seems to have merged.
The head is represented by 26 animals (8.7% of the total number) which
means there are newcomers. The distribution is always widely spread. The
mean size of the heading class is 39.6 mm, which is already more than the
mean size (36.9 mm) of all the individuals one year later, when they will
be 30 months old.
Part III: Experimental studies of the intraspecific competition
118

Table 7. Statistical analysis of the size frequency distribution of single cohorts of P. lividus at
different ages.
Age (months) 1 6 12 18 24 30
Fertilization Fa Fb Fb Fb Fb Fb
treatment Killed 7d. after
metamorphosis
Same cohort of reared echinoids followed in controlled conditions
General descriptive statistics on size frequencies (individual horizontal test diameter in mm)
Number of individuals 296 536 361 296 285 280
Median 0.501 4 15 23 31 36
Mean 0.497 4.81 17.1 24.6 32.3 36.9
Standard deviation 0.056 2.28 5.51 6.98 6.12 5.56
Skewness -0.275 0.748 0.851 0.599 0.136 0.042
Kurtosis 0.226 -0.044 0.373 0.094 -0.024 0.139
Intrinsic goodness of fit test to a normal curve (Kolmogorov-Smirnov / Lilliefors)
Maximum difference 0.046 0.172 0.125 0.097 0.077 0.064
Probability 0.127 **
0.000 0.000 0.000 0.000 0.011 *
Intrinsic goodness of fit to a log-normal curve (Kolmogorov-Smirnov / Lilliefors)
Maximum difference 0.068 0.130 0.075 0.064 0.059 0.095
Probability 0.002 0.000 0.000 0.005 0.018 *
0.000
Components analysis by the graphical Bhattacharya's method
Number of modes 1 2 3 or 4 2 3 or 4 1 or 2
Groups clearly
separated
- - 2 2 2 -
Characterization of the groups (maximum-likelihood estimator method, NORMSEP)
(minimal number of groups giving a χ 2 probability > 0.05)
Number of groups - 2 3 2 3 1
χ 2 value - 8.41 14.50 21.44 24.48 26.71
Probability - 0.077 **
0.488 **
0.554 **
Part III: Experimental studies of the intraspecific competition
0.140 **
0.181 **
Mean of group 1 3.43 13.3 23.1 18.4 36.9
SD group 1 1.15 2.49 5.39 1.82 5.56
Percentage in 1 58.2 53.2 91.3 2.3 100
No. of ind. in 1 312 192 269 7 280
Mean of group 2 6.73 20.2 39.6 27.7
SD group 2 2.07 3.61 1.97 2.54
Percentage in 2 41.8 41.7 8.7 34.4
No. of ind. in 2 224 151 26 98
Mean of group 3 31.2 35.3
SD group 3 1.34 5.00
Percentage in 3 5.1 63.3
No. of ind. in 3 18 179
* Fitting is significant (p > 0.01); ** Fitting is very significant (p > 0.05).
After 24 months, the individuals forming the head seem to be caught
up by the mean sized ones. However, some individuals do not follow this
general movement, and a tail remains amounting to 2.7% of the total
population. The shape of the distribution is completely changed: skewness
decreases (0.136) and the whole distribution matches possibly a lognormal
one (Kolmogorov-Smirnov / Lilliefors, p > 0.01).
119

Nber
of
ind.
120
100
80
60
40
20
0
0 5 10 15 20 25 30 35 40 45 50 55
Test diameter (mm)
6 months old
Part III: Experimental studies of the intraspecific competition
18 months old
12 months old
30 months old
24 months old
Figure 22. Evolution of a single cohort of P. lividus (Fb) reared in stable environmental
conditions according to time.
After 30 months, the latecomers forming the tail catch up the mean
sized ones, and the distribution loses its significant multimodal shape. The
shape becomes symmetrical: low skewness of 0.042 and approach a
normal curve (0.01 < p < 0.05, Kolmogorov-Smirnov / Lilliefors). The
spreading decreases also: the ratio 10% larger / 10% smaller drops to 1.8
in size and to 5.3 in wet weight.
Effect of size sorting on juveniles’ growth
Figure 23 and Table 8 show the initial and final size distributions of the
four Fc size-sorted batches. A Kolmogorov-Smirnov / Lilliefors test
applied on each distribution showed that 2 cases out of the 4 did not match
a Gaussian curve (p < 0.01). Since extremes were eliminated before the
experiment began (only mean-sized individuals were used), non normality
in the distribution of the whole cohort is not due to the spreading of these
extremes, producing "head" and "tail" classes, but could be the
120

consequence of differential growth rates among all individuals, including
mean-sized ones.
Table 8. Statistics on the four batches of Fc in the beginning of the experiment (initial) and
after 4 months (final).
Batch Fc1 Fc2 Fc3 Fc4
Treatment Size sorted batches of mean-sized individuals only (no "head" already
differentiated)
Initial mean size (mm) 6.0 6.0 6.0 6.0
Final mean size (mm) 14.3 15.1 14.0 14.9
Increase in size (mm) 8.3 9.1 8.0 8.9
Kolmogorov-Smirnov / Lilliefors on final sizes (mm)
Maximum difference 0.161 0.148 0.106 0.124
Probability 0.003 **
0.008 **
0.175 0.061
**
Does not fit a normal curve (p < 0.01)
A
Nber
of
ind.
B
Nber
of
ind.
50
40
30
20
10
0
50
40
30
20
10
0
5 7 9 11 13 15 17
Test diameter (mm)
5 7 9 11 13 15 17
Test diameter (mm)
Part III: Experimental studies of the intraspecific competition
Fc4
Fc3
Batch
Fc2
Fc1
Fc4
Fc3
Batch
Fc2
Figure 23. Size distribution of Fc juveniles in each batch in the beginning of the experiment
(A) (extreme individuals have been eliminated) and 4 months later (B).
Fc1
121

Table 9. Statistics on the six batches of Fd in the beginning of the experiment (initial) and
after 4 months (final).
Batch Fd1 Fd2 Fd3 Fd4 Fd5 Fd6
Treatment Size sorted batches ranging from "head" to mean-sized individuals
of a fertilization that already differentiated a heading group.
Initial mean size (mm) 10.5 9.5 8.5 7.0 6.0 6.0
Final mean size (mm) 14.0 14.0 13.2 12.7 13.5 13.6
Increase in size (mm) 3.5 4.5 4.7 5.7 7.5 7.6
Comparison of final mean sizes of the six batches (Kruskal-Wallis)
Statistic value 11.85 Associated probability 0.037 *
* Difference slightly significant (0.01 < p < 0.05).
A
Nber
of
ind.
B
Nber
of
ind.
50
40
30
20
10
0
50
40
30
20
10
0
5 7 9 11 13 15 17
Test diameter (mm)
5 7 9 11 13 15 17
Test diameter (mm)
Part III: Experimental studies of the intraspecific competition
Fd4
Fd3
Batch
Fd2
Fd1
Fd1
Fd6
Fd5
Fd6
Fd5
Fd4
Fd3
Batch
Figure 24. Size distribution of Fd individuals in each batch (ranging from "head" individuals,
Fd1, to mean-sized ones, Fd5 and Fd6) at the beginning of the experiment (A) and 4 months
later (B).
Figure 24 shows the size distributions of the Fd batches at the
beginning of the experiment (Fig. 24A) and after 4 months of controlled
Fd2
122

earing (Fig. 24B). Table 9 presents the statistics on these data. The
difference between the Fd batches after 4 months is only of slight
significance (Kruskal-Wallis, 0.01 < p < 0.05) although each batch of Fd
was made up initially of individuals from a different size class, going from
10.5 mm in Fd1 ("head" individuals) to 6 mm in Fd5 and Fd6 (mean-sized
individuals). Thus, it seems that size sorting eliminates the factor(s) that
maintained the difference in sizes between the batches so that smaller
individuals catch up rapidly to the larger ones.
Interactions between individuals
Table 10 presents the size frequency distributions of Fe batches
obtained after rearing them 4 months. The control shows the distribution of
the sizes that can be expected without interactions between the individuals
because each echinoid was cultivated independently. This distribution
matches very well a normal curve (intrinsic Kolmogorov-
Smirnov / Lilliefors test, p >> 0.05) characterized by a mean of 18.5 mm
(test diameter) and a standard deviation of 1.52 mm.
The distribution of the other eight experimental Fe batches is compared
to the theoretical normal curve fitted on the control with its estimated
mean and standard deviation. In the absence of interactions, no more than
5% of the individuals should be smaller than 15.5 mm or greater than 21.5
mm (considered as potential outliers, grayed zones in Table 10). The
number of small "outliers" is much more important than this prediction in
all eight batches, ranging from 11% to 35% of the total which indicates a
consistent tendency: some of the individuals reared together were
apparently inhibited in their growth. Moreover, whole batches Fe4, Fe6
and Fe7 do not match the control’s distribution (p < 0.05, extrinsic
Kolmogorov-Smirnov test). Consequent to this inhibition, the mean sizes
in the experimental batches are systematically lower than the mean size of
the control.
Part III: Experimental studies of the intraspecific competition
123

Table 10. Size distribution of Fe echinoids () after having been reared individually (control)
or together (experimental batches) for 4 months.
class size (mm) control experimental batches
min ≤ ≤ ≤ ∅ ∅ ∅ < max batch Fe1 Fe2 Fe3 Fe4 Fe5 Fe6 Fe7 Fe8
11 - 11.5
11.5 - 12
12 - 12.5
12.5 - 13
13 - 13.5
13.5 - 14
14 - 14.5
14.5 - 15 p = 5%
15 - 15.5
15.5 - 16
16 - 16.5
CD
16.5 - 17
17 - 17.5
17.5 - 18
18 - 18.5
18.5 - 19
19 - 19.5
19.5 - 20
20 - 20.5
20.5 - 21
21 - 21.5
21.5 - 22
22 - 22.5
22.5 - 23
23 - 23.5
p = 5%
23.5 - 24
mean size 18.5 17.5 18.3 17.8 18.1 18.2 17.2 17.5 18.0
standard dev. 1.52 2.28 2.52 2.37 2.49 3.07 2.05 2.38 2.23
Lilliefors test of normality Kolmogorov-Smirnov extrinsic test and probability to fit the control curve
max. dif. (K-S) 0.128 0.283 0.180 0.257 0.316 0.172 0.389 0.348 0.246
proba. to fit 0.543 **
0.066 0.614 0.177 0.042 *
0.637 0.003 **
0.015 *
0.170
Small "outliers" (5% expected) 30% 20% 24% 17% 18% 35% 11% 28%
Highlighted cells are the mean classes of each group. Grayed zones show 'outliers' (i.e., large and small sizes with p < 0.05
according to the control's fitted distribution CD). * Test significant; ** test highly significant.
e. Discussion
Growth of the regular echinoid Paracentrotus lividus was
experimentally studied using homogeneous cohorts kept in controlled
conditions. This homogeneity was obtained by using reared individuals
from the same parental origin, induced to metamorphosis the same day, fed
ad libitum with the same food and kept in the same environment. Size
frequencies observed among such a cohort distribute along a unimodal
normal curve just after metamorphosis but present a multimodal shape
with at least two, sometimes three, distinct subgroups (i.e.: "head", meansized
and "tail" individuals) when individuals were 6 to 24 months old.
Part III: Experimental studies of the intraspecific competition
124

Multimodality disappeared around 30 months of age and size frequency
distribution recovered a near-normal shape.
Such differences in growth between juveniles leading to non-normality
in the size frequency distribution could probably be attributed to every
individual, not only to extreme ones, as shown with mean-sized
individuals of Fc batches. Moreover, the difference in speed of growth
between heading and mean-sized echinoids cannot be attributed to their
respective genetic potentials, for size sorting eliminates rapidly the
differences in size between these subgroups (Fd). Hence, this difference is
the consequence of an intraspecific competition between echinoids having
different sizes. Furthermore, evidence of inhibition in the growth of
smaller individuals is provided by the comparison of the size distribution
of echinoids reared together (Fe1 to Fe8) or individually (Fe, control).
We observed that when echinoids of different sizes are reared together,
the smaller ones tend to insert themselves between the larger ones along
the walls and corners of the rearing baskets. In those aggregates, the water
is relatively stagnant and the pH there is lower than the one measured in
the running water of the tanks due to CO2 accumulation (pH NBS of 7.1-
7.7 and 7.8-8.0 respectively). Poor water quality could then contribute to
the slower growth rate of smaller juveniles. The difference in sizes
between large "inhibitor" and small "inhibited" individuals does not need
to be very important to engage the process of differential growth. Indeed,
the spreading in sizes that occurs from originally homogeneous batches
seems to be sufficient to trigger this intraspecific competition, as observed
in the eight experimental Fe batches reared together.
Whether a multimodal size distribution inside a single cohort of
juvenile echinoids could be possible in the field is worth questioning. If it
is the case, then all studies using analysis of size frequency distributions
and based on the separation of presumed unimodal cohorts from a whole
population could be biased. Aggregative behavior is currently observed
among various species of echinoids (Ebert, 1977: Strongylocentrotus
Part III: Experimental studies of the intraspecific competition
125

purpuratus; Dafni & Tobol, 1987: Tripneustes gratilla elatensis; Levitan
& Genovese, 1989: Diadema antillarum). In those cases, small juveniles
are often found under larger conspecifics or between their spines where
their rate of survival has proven to be higher thanks to the protection
provided against predators (Tegner & Levin, 1983; Levitan & Genovese,
1989). Yet, the chemical conditions in this environment could vary a lot, as
observed in our rearing devices, and growth could be greatly reduced for
some individuals, causing the spreading of the size distribution of the
cohort or transforming it into a multimodal one. The extension of the
critical period when the echinoid is small and thus vulnerable to predators
limits somewhat the benefits gained by the protection provided by the
adult’s spine canopy. However, if the adults’ density decreases, the
inhibition of a juvenile’s growth is then removed. Some of them can grow
very fast if food is available (as "head" ones in the currently studied
cohort) and rapidly replace missing adults. This mechanism could be very
efficient in stabilizing field populations of aggregative species of echinoids
by maintaining a protected pool of small individuals with high growth
potential but inhibited by the density of larger ones.
f. Acknowledgements
This study was conducted in the framework of an EEC contract in the
FAR aquaculture program (ref. AQ2.530). We thank Didier Bucaille for
his help in the laboratory work. We also thank the CREC and the
University of Caen for their contribution in building a specific echinoid
rearing facility. The "Centre Interuniversitaire de Biologie Marine" also
contributed to this study.
Part III: Experimental studies of the intraspecific competition
126

Intraspecific competition: an additional experiment
To investigate intraspecific competition in batches with a large initial
dispersion of sizes, two series of 6 replicates of 60 'small' sea urchins (7.5
to 9 mm test diameter; 720 individuals in the total) coming from the
heading group of a single fertilization (Ff), and two series of 6 replicates of
10 'large' sea urchins (20 to 24 mm test diameter; 120 individuals in the
total) extracted also from the heading group of another single fertilization
(Fg) were set up at the beginning of the experiment (Fig. 25). Small sea
urchins of Ff were 5 months old at the beginning of the experiment while
those of Fg were 13 months old. One series of small and one series of
large individuals were reared separately. The two remaining series of small
and large individuals were reared together ('mixed' series). The surface of
the rearing baskets used was the same for all batches (20 x 30 cm).
Figure 26 presents the change of size distributions in the three
experimental series. Table 11 presents corresponding statistics. The six
replicates from the same series are pooled after verifying that the
differences between them are not significant (Kruskal-Wallis test, p >>
0.05, except for the small individuals in the mixed batch at 2 months were
0.01 < p < 0.05, see Table 11). The presence or absence of smaller sea
urchins did not influence very significantly the growth of larger
individuals. The latter were thus not inhibited whatsoever. It is, however,
interesting to note that size distribution of larger echinoids did not spread
much with time, which means their density was low enough to avoid
competition among them and allowed a homogeneous growth of the whole
set. On the other hand, growth of smaller sea urchins was strongly affected
by the presence of larger individuals: their growth was limited and, as a
consequence, they remained more grouped than batches of small
individuals alone. In batches where small sea urchins occurred alone,
competition took place and size distribution spread. This is an effect
already identified in the previous experiments.
Part III: Experimental studies of the intraspecific competition
127

Nbr. of ind.
Nbr. of ind.
A. Size distribution of the "small" urchins just before the
experiment starts (fertilization Ff, 5 months old)
1000
900
800
700
600
500
400
300
200
100
0
0
0
3
1.5
6
3
Part III: Experimental studies of the intraspecific competition
4.5
6
7.5
Diameter (mm)
B. Size distribution of the "large" urchins just before the
experiment starts (fertilization Fg, 13 months old)
45
40
35
30
25
20
15
10
5
0
9
12
15
Diameter (mm)
Figure 25. Size distributions of the two different fertilizations used in the additional
experiment (Ff and Fg). Dark bars indicate the portion of animals that were actually used.
These were chosen in the heading part of the distribution for both fertilizations.
9
18
10.5
21
12
24
13.5
27
15
30
16.5
33
18
128

Table 11. Statistics on the small, large and mixed batches (fertilizations Ff, 'small' and Fg,
'large'): mean sizes (pooled replicates) and comparison of mean sizes of separate versus
mixed batches.
Time
(months)
Small
separate
mean size
(mm)
Small
mixed
mean size
(mm)
Kruskal-
Wallis
stat.
K.-W.
proba.
Large
separate
mean size
(mm)
Part III: Experimental studies of the intraspecific competition
Large
mixed
mean size
(mm)
Kruskal-
Wallis
stat.
Treatment Two fertilizations reared either separately or mixed during 6 months.
Then, 'large' individuals are discarded
and 'small' individuals are reared alone for another 6 months.
0 8.5 8.5 - - 21.7 21.7 - -
2 12.9 10.6 #
K.-W.
proba.
179.6 < 0.001 28.6 28 3.32 0.068
4 16.5 13.1 152.2 < 0.001 32.1 31.4 4.99 0.025 *
6 18.0 13.6 156.0 < 0.001 34.3 33.7 4.07 0.044 *
8 20.1 17.8 21.8 < 0.001
10 22.3 20.9 3.91 0.048 *
12 23.9 22.2 4.54 0.033 *
# Difference between the 6 replicates inside a group is slightly significant (Kruskal-Wallis,
0.01 < p < 0.05)
*
Difference slightly significant (0.01 < p < 0.05).
After 6 months (Fig. 27), large individuals were taken away, while the
smaller ones were left in place. Inhibited individuals of the mixed group
almost caught up with the others within 4 months when inhibition was
removed (Kruskal-Wallis, 0.01 < p < 0.05, test only slightly significant,
Table 11). Just after the large echinoids have been removed, the heading
group that rapidly formed in the mixed series had a remarkably high
growth speed (compare Fig. 26, 6 months to Fig. 27, 8 months: some
specimens grew from 18 to almost 30 mm within only two months, which
meant an individual weight increase from about four time, viz. from ca. 2.7
to ca. 11.0 g)! In the "mixed" series, when large sea urchins were removed,
sizes distribution spread much with a heading group that clearly detaches,
as it was the case for small animals alone, six months before. This
experiment demonstrates the high growth potential of inhibited
individuals, a potential that was expressed when inhibitors were removed.
129

Nbr.
of ind.
Nbr.
of ind.
Nbr.
d'ind.
70
60
50
40
30
20
10
70
0
60
50
40
30
20
10
70
60
50
5
0
5
40
30
20
10
0
5
Part III: Experimental studies of the intraspecific competition
10
15
20
Diameter (mm)
10
15
20
Diameter (mm)
10
15
20
Diameter (mm)
2 months
25
25
30
30
35
35
40
4 months
6 months
25
30
35
40
40
Large
Large
Large
Mixed
Mixed
Mixed
Figure 26. Change in size distributions of the sea urchin batches (large, mixed and small)
with time (replicates are pooled).
Small
Small
Small
Large
Small
Large
Small
Large
Small
130

Nbr.
of ind.
Nbr.
of ind.
Nbr.
of ind.
60
40
20
70
60
0
60
40
20
50
40
30
0
20
10
0
5
5
5
Part III: Experimental studies of the intraspecific competition
10
10
15
20
Diameter (mm)
15
8 months
20
Diameter (mm)
10
15
25
25
30
30
35
10 months
20
Diameter (mm)
25
30
35
12 months
35
40
40
40
Mixed
Small
Mixed
Mixed
Figure 27. Change in size distributions of the sea urchin batches with time. Large individuals
were removed from the mixed batches and the growth of small individuals (from either the
"mixed" or the small batches) was recorded during an additional 6 months (pooled
replicates).
Small
Small
131

Part III: Experimental studies of the intraspecific competition
132

PART IV
A growth model with intraspecific competition
133

134

PART IV: A GROWTH MODEL WITH
INTRASPECIFIC COMPETITION
We have now collected all elements required to describe the processes
implied in the somatic growth of P. lividus in cultivation. We can thus
elaborate a model. Since there is no growth model that includes a
component of intraspecific competition, we proposed an original approach
for modelling growth. Statistical methods for fitting growth curves were
also adapted.
Part IV: A growth model with intraspecific competition
135

Part IV: A growth model with intraspecific competition
136

A functional growth model with intraspecific competition
applied to a sea urchin, Paracentrotus lividus (Lamarck, 1816)
a. Abstract
Ph. Grosjean, Ch. Spirlet & M. Jangoux (submitted).
An original model obtained by defuzzifying a fuzzy model is fitted on
data from reared sea urchins, Paracentrotus lividus. Quantile regressions
are used instead of least-square, for they are insensitive to the dimension of
the measurement and accommodate more than just symmetrical
distributions. Quantile regressions allow comparison of fittings on various
parts of the size distributions, including large competitors versus small,
inhibited animals, in the presence of a size-based intraspecific competition.
The model has functionally interpretable parameters and allows
quantifying of the intensity of inhibition. An extension of this model,
called 'envelope model', fits the whole dataset at once, including size
distributions. Its parameters are constrained using information about
underlying biological processes involved, namely asymptotic growth with
inhibition in early ages due to intraspecific competition whose intensity
depends on the relative size of the individual in the cohort. The new model
appears most adequate to describe growth of Paracentrotus lividus and
probably of many other sea urchins species as well as other animals or
plants. It is an intermediary model in a hierarchy of asymptotic growth
models ranging from the simplest one (von Bertalanffy 1) to more complex
'dimensional' and 'transitional' groups. A general asymptotic growth
model, being both 'dimensional' and 'transitional', is proposed. Most other
models, including the new one, are just special cases of this general model.
However, it is only the visible tip of the iceberg. Many similar functions
can be designed by defuzzifying simple fuzzy models, including models
that do not derive from the von Bertalanffy curve. The family of so-called
Part IV: A growth model with intraspecific competition
137

. Introduction
fuzzy-remanent functions represents a powerful alternative to dynamic
modelling (using differential equations) for describing nonlinear
phenomena widely observed in sciences.
Keywords: growth model, intraspecific competition, fuzzy logic, quantile
regression, Paracentrotus lividus, sea urchin, population dynamic,
aquaculture.
For the last two centuries, after Malthus (1798) discovered the
exponential nature of growth, the diversity of growth models has steadily
increased (Gompertz, 1825; Verhulst, 1838; Winsor, 1932; von
Bertalanffy, 1938; Brody, 1945; Weibull, 1951; Richards, 1959; Preece &
Baines, 1978; Schnute, 1981; Tanaka, 1982; Jolicoeur, 1985). However,
these models compete rather than complement each other. Choosing a
growth model often remains arbitrary (Fletcher, 1974). Sea urchin
individual growth is an example of such a problem (Gage & Tyler, 1985;
Gage et al, 1986; Gage, 1987; Dafni, 1992; Ebert & Russell, 1993; Lamare
& Mladenov, 2000). Papers on growth models applied to sea urchins
compare the fit of different curves and are limited to their advantages and
drawbacks in representing the growth of a "mean individual" (Gage and
Tyler, 1985; Ebert & Russell, 1993; Lamare & Mladenov, 2000). They all
conclude that none of them is fully satisfactory. Ebert (1999, p. 200)
wrote: "searching for the [growth] model that will provide the "best fit"
can become a search for the Grail with all of the fun being in the search
because there may be no end."
The key problem with these models is that parameters are not all
comparable, and lack biological meaning (or lose it when applied to real
data). Moreover, elaborated growth models have three or more parameters
that are not independent from each other (intercorrelations). Thus it is not
possible to extract the estimator of one parameter from one fitting and
compare it with the value obtained for the same parameter with another
Part IV: A growth model with intraspecific competition
138

dataset. Indeed, its value depends on the estimation of all other parameters
in each respective fitting. This contrasts with linear models where slopes
can be compared and usually convey a biological or physical meaning
about the relationship between the considered variables (proportionality).
Some authors have tried to combine parameters into a single one.
Duineveld & Jenness (1984) used ω = k·Y∞ (Gallucci & Quinn, 1979) of
the von Bertalanffy equation Y(t) = Y∞·(1 – e -k·(t – t 0 ) ) to compare growth of
two populations of the irregular sea urchin Echinocardium cordatum
(Pennant). Growth being an emergent property of various physiological
processes like feeding, digestion, respiration, etc., it is dubious that it could
be summarized into a single coefficient, and such a practice is probably
error-prone.
A few authors (e.g., Richards, 1959) have built up flexible and general
growth models that include some other existing models as special cases.
Schnute (1981) developed a general model whose parameters have a better
biological meaning. None of these models have proved to be fully efficient
when fitting real data, partly due to the problem of intercorrelation
between parameters.
It is thus only possible either to carry on a global comparison of
various models for the same dataset, or to use a single model applied to
various datasets. Usually, either the R 2 value or the residual sum of squares
of the nonlinear least-square regression is used to evaluate how well a
model fits the data (Gage & Tyler, 1985; Cellario & Fenaux, 1990; Lamare
& Mladenov, 2000). A visual comparison of graphs is also commonly used
(Gage & Tyler, 1985; Dafni, 1992; Ebert & Russell, 1993). These two
techniques are not considered rigorous by statisticians. Among all papers
cited, only Lamare & Mladenov (2000) performed a complete residual
analysis. In this context, a growth model only summarizes the data cloud
into a "best-fit" curve supposedly representing the growth of a mean
individual. It is very difficult to use such a growth curve as a tool to
investigate underlying processes (e.g., to understand the impact of a
Part IV: A growth model with intraspecific competition
139

considered factor on the curve's shape). For instance, current models
hardly cope with a superimposed effect of intraspecific competition on
growth, though the latter was evidenced in sea urchins (Ebert, 1977;
Himmelman, 1986; Levitan, 1988; Grosjean et al, 1996) as in many other
species (Branch, 1974, for the limpet Patella cochlear Born; Timmons &
Shelton, 1980, for the largemouth bass Micropterus salmoides (Lacepede);
Kautsky, 1982, for the mussel Mytilus edulis L.).
One method to model growth in a functional way is by building
equations that directly represent underlying processes, i.e., by elaborating a
dynamic model that balances inputs (food, oxygen intake…) and outputs
(carbon dioxide, feces…) from which it is possible to calculate variation of
size with time, thus yielding an estimation of growth. This is the
bioenergetic and/or ecophysiologic approach, using the scope for growth
concept, which has proved very successful for filter feeders (Willows,
1992). Such an approach requires a lot of measurements and equations. It
is most often used in the simplest cases, where environmental conditions
are constant, or vary in a very predictable way, like in a protected reef
lagoon for cultivated pearl oysters (Pouvreau et al, 2000). Indeed, similar
studies on European oyster cultures in the intertidal zone –a very changing
and unpredictable environment– lead to a much more complex model
(Bacher et al, 1991). As far as we know, no such model has been
completely successful when applied to sea urchins because a large part of
the carbon or energy absorbed is lost as dissolved organic matter that is
hard to quantify and to enter in equations (Miller & Mann, 1973; Lawrence
& Lane, 1982).
Being a basic feature of life, growth has been widely explored but it
still remains unsatisfactorily modelled in a functional way, possibly
because of the approach used. Most growth models were elaborated from
their differential equations. Functions obtained by solving these equations
were then systematically used to determine growth of a "mean individual"
(by least-square regression) and no parameter of the model was
constrained using particular knowledge (such as size at birth or at
Part IV: A growth model with intraspecific competition
140

c. Material
metamorphosis) or hypotheses that can be formulated about changes in
individual growth (such as intra- or interspecific competition).
The aim of this paper is to propose a growth model taking into account
usually neglected aspects such as individual variations or intraspecific
competition. This means we will question the concept of "mean
individual" and tentatively build up a growth function with parameters
carrying high biological meaning.
The dataset we used results from a growth study of a single cohort of
sea urchins, Paracentrotus lividus, reared over a period of 7 years in a
controlled environment (see Grosjean et al, 1998, see Part I, for a detailed
description of the rearing protocol). Echinoids were never size-sorted, nor
individually tagged. All sea urchins in the cohort were measured every 3
months starting at 6 months old (younger echinoids are too fragile to be
measured alive) until 4.5 years old, and then every 6 months until 7 years
old (Fig. 28A, see also Annex II). Due to mortality, the total number of
individuals in the cohort dropped from 725 at 6 months old to 221 at 4.5
years old and to 67 at 7 years old (Fig. 29). Size is expressed by the
ambital test diameter D which corresponds to the external diameter of the
test at its largest region (the ambitus) excluding spines. D is measured with
an electronic sliding caliper at the nearest 0.1 mm (Grosjean et al, 1999,
see Part II) and recorded into 1 mm-wide size classes. Note that between
400 and 1200 days, size distributions were heavily skewed, or even
multimodal. This is the effect of an intraspecific competition (Grosjean et
al, 1996, see Part III).
Part IV: A growth model with intraspecific competition
141

Diameter
D 20
in mm
Diameter D in mm
60
0 10 20 30 40 50 60
40
A
B
0
500
1000
1500
Time t in days
Part IV: A growth model with intraspecific competition
2000
2500
0
150
100
50
Nbr.
of
ind.
500 1000 1500 2000 2500
Time t in days
quantile 0.975
quantile 0.5 (median)
quantile 0.025
142

Figure 28. Left page. A. Histograms of size distributions of a cohort of reared P. lividus with
time. Only 6-month interval histograms are presented, although measurements were
performed every 3 months during the first 4.5 years (1600 days). Top of the box: a projection
of three quantiles (0.025, 0.5 and 0.075) issued from those size distributions. They are also
presented in (B) where points are unconditional quantiles extracted from each size
distribution considered separately, and lines are conditional quantile regressions, thus
considering the whole dataset (using all 6988 initial data points). Best fitting growth models
(according to δ1 values in Table 12) for each quantile are used: von Bertalanffy 1 for
τ = 0.975, 4-parameter logistic for τ = 0.5 and Weibull for τ = 0.025.
Nbr of individuals
100 200 300 400 500 600 700
500 1000 1500 2000 2500
Time t in days
Figure 29. Survival with time of the same reared cohort of P. lividus as in Fig. 28A. Line is a
spline interpolation of observed values.
Part IV: A growth model with intraspecific competition
143

d. Results
Theoretical considerations: use of quantile regression instead
of least-square regression
The most widespread method to fit a curve is the use of a least-square
method with one of the many minimization algorithms available [simplex,
(quasi-)Newton, etc.; Sen & Srivastava, 1990; Draper & Smith, 1998;
Nocedal & Wright, 1999]. The algorithm finds the combination of values
for the various parameters in the model (the solution) that leads to a
minimal value for the objective function, which is here the sum of the
square of the residuals (that is, the sum of squared distances between
observed values for the dependent variable and values predicted by the
model at the same levels for the independent variables).
Least-square regression has many advantages over other methods. In
particular, when partial first (gradient matrix) and second (Hessian matrix)
derivatives of the function are calculable for each parameter, convergence
through a solution is accelerated and can be verified (at least for a local
solution, Nocedal & Wright, 1999). In the counterpart, that regression
supposes that the fluctuation around the model (called the error term) is
additive, independent, normally distributed and with a constant standard
deviation (heteroscedasticity). It is also very sensitive to outliers because it
uses the squared residuals. Those constraints, even if not strictly met every
time, particularly in many nonlinear phenomena like growth, appear to be
of minor importance for many authors. Indeed, outliers are eliminated, or
weighing methods are applied to limit their impact.
Yet, there are two arguments against the least-square regression used in
the framework of growth models, particularly when individuals' growth is
influenced by the presence of conspecifics or of other species (indeed a
general case to be verified in each study, except when a single individual is
grown alone in a cage or an aquarium!): first it is sensitive to the
Part IV: A growth model with intraspecific competition
144

dimension of the size measurement, and second it can only model mean
individuals.
If least-square regression is influenced by the dimension of the
dependent variable used to quantify size in time, what is a good
measurement of size? Is it a linear measurement (height, width, diameter,
etc…) or is it a volume, a weight, or any other tri-dimensional measure? It
could also be a two-dimensional measure such as, e.g., the surface covered
by a colony of sponges or corals. Is there a privileged dimension (1, 2 or
3D) to measure growth? Except for changes in the curve shape due to
distortion introduced by a power transformation –compare von
Bertalanffy's model in size and in weight (von Bertalanffy, 1957, his
Fig. 3, and Fig. 10 p 47)–, no criterion exists for choosing the best
dimension to describe growth. Accordingly, a length, a surface or a
volume/weight can each be acceptable, and the final choice will be
dictated by practical considerations: which measurement is the easiest to
obtain with the highest possible accuracy (see Grosjean et al, 1999, see
Part II, for a discussion of this problem in P. lividus). Thus if a regression
method is highly sensitive to power transformation, it will be less desirable
because results of the regression will vary according to the measure used.
By contrast, the median, as well as the quantiles, are insensitive to power
transformation. Hence, quantile regression is generally insensitive to any
transformation by a monotonous function (Koenker, 2001) and will not be
influenced by the dimension (1, 2 or 3D) of the dependent measurement.
Accordingly, a median individual in a regression of length against time
will remain the same median individual in a regression of a volume (as
length 3 , or any allometry coefficient) against time with a quantile
regression, while this is not true for a mean individual using a least-square
regression. Using median/quantiles instead of mean allows remaining the
independence of the dimension of size measurement in a context where the
dimension of the studied phenomenon (growth) is undetermined. As
suggested by von Bertalanffy (1938, 1957), growth could be a
multidimensional process, since it is a balance between anabolism (with
Part IV: A growth model with intraspecific competition
145

some two-dimensional processes like respiration or digestion, i.e.,
exchanges across surfaces) and catabolism (proportional to the volume of
the animal, and thus a process quantified in 3D).
The second argument against the least-square regression method is its
limitation to model a "mean individual" in the presence of individual
variations in the dataset. Yet, the concept of mean individual lacks
meaning when the distribution of the error is not symmetrical and even,
sometimes, multimodal. Considering a bimodal distribution with two
similar, but well-separated modes, the mean value is located just in the
middle of the two groups, where there are no individuals! In this example,
the median is located at the same place and will also be a poor
representation of the dataset. Quantiles, however, can be used more
efficiently (1 st and 3 rd quartiles will be located around each of the two
modes). Clearly, a regression that can fit a curve on another part of the
distribution than a symmetrical position can be useful in situations where
the distribution is asymmetrical or multimodal. For instance when sizebased
intraspecific competition occurs, the largest animals are inhibitors,
while the smallest ones represent the most inhibited fraction. Growth
curves fitted on large and small individuals thus contain information about
the impact of the interspecific competition (by comparison). This
information is not available using a least-square regression, but it is when
two quantile regressions are used, based on large and small quantiles
respectively.
Quantile regression, as defined by Koenker & Bassett (1978) is an
extension of the least-absolute deviation regression that fits a function for a
median individual. In quantile regression, the objective function (called
here deviance δ1) to be minimized is:
with:
Part IV: A growth model with intraspecific competition
n
δ 1= ∑ ρτ( Di−ξ1) (21)
i=
1
146

- ξ1 being the solution returned by the model,
- Di being each actual observation (diameter of a sea urchin) with i = 1…n
observations.
- ρτ (u) being a piece-wise linear function defined as:
Part IV: A growth model with intraspecific competition
ρ ( u) = u( τ − I( u<
0))
(22)
τ
and where τ is the quantile and I(u < 0) equals 1 if (u < 0) is true and 0 if it
is false. Thus, quantile τ defines the fraction of all observations that lie
beneath the curve. If τ = 0.5, half of the observations are beneath, and the
other half are above the curve, and the objective function (eq. 21)
simplifies to half the least-absolute deviation
∑
1 δ 1= 2 | Di−ξ1| . With τ
values different than 0.5 (0 < τ < 1), it is possible to fit a curve that will
represent another fraction of the population. For instance, τ = 0.975 fits a
curve that represents the 2.5% largest individuals in the cohort; here they
are the fastest growing competitors. Similarly, using τ = 0.025 fits a curve
that represents the 2.5% smallest individuals, here, the slowest growing
fraction that undergoes the strongest competition. The surface between
these two curves represents 95% of the whole batch. A third curve fitted
on median individuals using τ = 0.5 indicates asymmetry in the
distribution. If this last curve is located in the middle of the two previous
ones, the distribution is symmetrical. If it is closer to the lowest curve, the
distribution is skewed toward small individuals. If it is closer to the highest
curve, it is skewed toward large individuals. This is the representation we
will keep for the following sections in this paper; it is more descriptive
than just modelling a median (or a mean) individual.
As far as we know, only one method is currently available to fit a
nonlinear model using quantile regression (Koenker & Park, 1996), using a
particular interior point algorithm (Nocedal & Wright, 1999). Presently,
only R (Ihaka & Gentleman, 1996), a fast S-Plus clone freely available
under the GNU license (http://cran.r-project.org) proposes a package that
147

implements this method ('nlrq' package, by R. Koenker & Ph. Grosjean,
see Annex I) from the same site. It runs on several platforms (several Unix,
Linux, Windows, MacOS). Analyses in this paper are performed using this
package, and also some experimental R code developed by the authors and
available at http://www.sciviews.org/_phgrosjean/growth/index.htm. A
script for complete treatment of the example presented in this paper is also
provided.
Elaboration of a new growth model which includes
intraspecific competition
Fuzzy logic can deal efficiently with complex nonlinear problems
(Zimmermann, 1991; Passino & Yurkovich, 1998; Cox, 1999) and is thus
another possible approach for creating growth models than dynamic
modelling (differential equations) commonly used in this field, though, it
has not been employed yet so far (Salski et al, 1995). Fuzzy systems can
often be formulated rather intuitively in a linguistic way (Zimmermann,
1991) and we will start this way, introducing mathematics subsequently.
For individual growth without competition, a trivial semantic
description could be: "a young, small individual gradually becomes larger
with age". In terms of fuzzy sets, this translates into a temporal transition
between two sets: 'small' and 'large' (or S and L). Young animals belong to
the S set, old ones belong to the L set, and "middle-aged" individuals
belong partly to each of these sets. The "degree of membership" to each set
depends upon the amount of growth achieved and thus gradually shifts
with time form S to L sets. This change is characterized by a membership
function to each set, thus MS and ML respectively. The sum of all
membership values at any given time is one, since we deal with a single
individual in its integrity.
The next step is to incorporate the concept of growth inhibition to
represent the effect of intraspecific competition. Grosjean et al (1996, see
Part III) showed that this competition is size-based in the case of reared P.
Part IV: A growth model with intraspecific competition
148

lividus. 10 to 15% of the largest individuals (the inhibitors) in the
populations grow at their maximal speed (i.e. the growth speed they would
have if they were alone in the same food and environmental conditions).
The growth of others depends upon their relative size in the population
(the smaller they are, the slower they grow). The cause of this inhibition is
not known yet, but it does not seem to be food-related: in the experiments,
all individuals had access to food ad libitum. Inhibition progressively fades
out when larger individuals reach their asymptotic size and are caught up
with smaller ones that are still growing.
According to these observations, a semantic formulation of the
problem becomes: "a young, small individual is potentially inhibited in its
growth, but gradually reaches its maximum size with age". It can also be
represented by two sets and one transition, but now the S set is the minimal
size with time (with maximum inhibition) and L set is the size at the age
where the growth speed is maximal (with no inhibition at all). The
transition is now the expression of a progressive release of the inhibition,
instead of a representation of the entire growth process. The difficulty
resides in the proper characterization of sets and membership functions
with age.
First of all, we use a time-scale t' with a well-defined origin that really
coincides with the initiation of the growth process. Until now, we used
(time-scale t) the age of sea urchins (since fertilization, thus including
larval life). However, growth of postmetamorphic sea urchins really starts
after metamorphosis. Knowing the age at metamorphosis (t0), t' is simply:
Part IV: A growth model with intraspecific competition
t' = t − t0
(23)
For the studied dataset, metamorphosis was artificially induced for all
echinoids in the batch at the same time at 30 days old. t' scale is thus
shifted to the left by t0 = 30 days.
Set S corresponds to the minimum possible growth, that is simply no
growth at all. Thus, in set S, size remains constant at its minimum initial
149

value just after metamorphosis (D0, see Fig. 30). In this model, we do not
consider negative growth (observed by Régis, 1979, on P. lividus; Ebert,
1967, on Strongylocentrotus purpuratus; Levitan, 1988, on Diadema
antillarum) because echinoids are fed ad libitum and therefore size
shrinking should not occur. Moreover, negative growth was observed only
for large, full-grown adults but not for juveniles that die instead of
reducing size in the absence of enough food to maintain their basic
metabolism (unpubl. res.). Equation for S set is:
10
8
6
4
2
0.5
D
0
0
0
2
Membership
1
0
2
Part IV: A growth model with intraspecific competition
D( t') = D
(24)
4
4
D∞
D0
0
Dmax
D0
6
6
large (L)
8
8
∆Dmax
∆D∞
small (S)
fuzzy
large (ML)
small (MS)
Figure 30. Construction of the fuzzy growth model. Two sets, called 'small' and 'large' (or S
and L) are used. Actual size will always be located between these two curves ('fuzzy'). The
membership functions (ML and MS) model the belonging to each set with logistic functions. If
a membership is close to 1, like MS for young juveniles or ML for large adults, the resulting
fuzzy curve is close to the corresponding set. When memberships are 0.5 for each set, here at
t' ≈ 3.9, the value returned by the fuzzy model is in the middle of values returned by both sets.
10
10
t'
t'
150

Set L describes the largest size reached by sea urchins at maximum
growth speed with time. P. lividus having a determinate or asymptotic
growth (see Fig. 28A), final increase of size ∆D∞ = D∞ – D0 is finite for
t' → ∞. However, maximum size cannot be reached instantaneously. If the
largest individuals in the actual dataset are not inhibited at all, they can be
used as a reference for the whole cohort to define this maximum growth
curve. Supposing that a von Bertalanffy 1 curve best fit the largest fraction
of the size distributions (see next section), it is an appropriate model for set
L. Because we use the t' time-scale here, we choose a parameterization of
the model such as D'(t' = 0) = D0:
Part IV: A growth model with intraspecific competition
−k1⋅t' Dt' ( ) = D +∆D⋅(1− e )
(25)
0
∞
Membership to the L set with time, ML(t'), is modelled with a logistic
function (a classical model for a transition in fuzzy sets, Cox, 1999)
(Fig. 30):
1
M ( t')
=
1+ l ⋅e
L −k2⋅t' (26)
Membership to the S set with time, MS(t'), is complementary so that MS
and ML add up to one:
1
M ( t') = 1 − M ( t')
= 1− 1+ l ⋅e
S L −k2⋅t' (27)
Thus, full-growing individuals belong to the L set from the beginning. The
stronger the growth inhibition, the longer other individuals remain in the S
set before gradually shifting to the L set. The fuzzy model integrates the
effect of intraspecific competition (or any other inhibition mechanism
having a similar effect on growth) as a delayed transition from S to L sets,
as explicitly quantified by parameter l (the lag or position of the inflexion
point in the membership curves, see Fig. 30). If l = 0, there is no inflexion
point and ML(t') = 1; growth occurs at maximum speed from start and
follows set L, that is, a von Bertalanffy curve. While parameter k1
151

quantifies maximum speed growth, k2 represents the speed at which
inhibition is released with time. All parameters in this model carry a clear
biological meaning, considering hypotheses that were formulated to build
it.
Usually, a fuzzy model is treated with fuzzy arithmetic. The output is
then "defuzzified" by one of several methods (Cox, 1999) to provide a
crisp number (the most probable size of an individual at a determined age).
Being simple enough, the current model can also be transformed into a
classical analytic equation:
D( t') = M ( t') ⋅ S( t') + M ( t') ⋅ L( t')
(28)
S L
which gives, after combination of eqs 24-28 and simplification:
Dt' ( ) = D +∆D
Part IV: A growth model with intraspecific competition
0
1−e 1+ l ⋅e
−k1⋅t' ∞ −k2⋅t' (29)
This way the model can be treated with classical (crisp) arithmetic that
offers a larger panel of statistical tools than fuzzy arithmetic.
Fitting the dataset
Since echinoids are not tagged individually, it is not possible to track
animals across measurement sets. Consequently, one will consider virtual
individuals according to their relative position in the entire size
distribution at each sampled time, that is, virtual individuals corresponding
to fixed quantiles (or percentiles) in each size distribution. It should be
noted also that, if mortality is not randomly distributed among individuals,
actual growth speed could be different from the one calculated on virtual
individuals. This means that if mortality preferably affects small
individuals, growth speed is overestimated; conversely, if mortality affects
rather larger animals, growth speed is underevaluated. In absence of
individual tagging, we will thus consider the apparent growth speed of the
virtual individuals as defined here above.
152

Another feature of this dataset is that the error terms are time- and
individual-dependents. Since the same (surviving) individuals are
measured at each sampling time, this constitutes a time-series thus having
autocorrelated errors. Moreover, even if artificial rearing conditions are
kept as constant as possible (Grosjean et al, 1998, see Part I), some
seasonal variations are possible, partly because animals are fed with
freshly field-collected kelp whose chemical composition is seasondependent
(Abe et al, 1983). To be rigorous, autocorrelation terms and
some seasonal variation should be introduced into the model. However, to
simplify the model as much as possible, and also because these effects are
very limited (see further), we decide to ignore them here and we will thus
fit growth curves without autocorrelation terms.
Table 12 and Fig. 28B illustrate quantile regressions on P. lividus
dataset with some usual growth models. 4-parameters models fit all 3
quantiles while 3-parameters models seem adequate for some quantiles
only. Logistic function yields unreliable results in all cases. Criteria to
decide which model best fits the data (deviance δ1 and visual impression
on a graph) are neither rigorous nor discriminant. Two to four models
among the six tested seem adequate in each situation with these criteria.
Indeed, the increasing lag-phase for quantiles 0.5 and 0.025 compared to
quantile 0.975 (more pronounced S-shape, see Fig. 28B) was
experimentally demonstrated to be an inhibition in growth (Grosjean et al,
1996, see Part III). None of these models, no more than many others like
Richards (1959), Preece & Baines (1978), Johnson (Ricker 1979), Schnute
(1981), Tanaka (1982), Jolicoeur (1985)… contain explicit parameters that
quantify such an inhibition and thus none of them are really adequate in
this case. Good fitting of data does not imply that the model is correct.
Part IV: A growth model with intraspecific competition
153

Table 12. Results of quantile regressions with different growth models for quantiles
τ = 0.975, 0.5 and 0.025.
model (a)
a b c d deviance δδδδ1 fitting (b)
τ = 0.975
Gompertz 63.1 7.06·10 -2 0.998 . 2247 -
von Bertalanffy 1 68.1 1.44·10 -3 81.8 . 2186 + +
von Bertalanffy 2 63.9 2.24·10 -3 -178 . 2232 -
logistic 61.8 3.45·10 -3 525 . 2334 - -
4-param. logistic 67.3 1.55·10 -3 -1488 -761 2188 + +
Weibull 67.1 1.04·10 -3 1.05 74.0 2186 + +
τ = 0.5
Gompertz 57.5 1.39·10 -2 0.977 . 14964 +
von Bertalanffy 1 68.4 9.67·10 -4 185 . 15750 - -
von Bertalanffy 2 59.4 1.87·10 -3 -54.3 . 14980 +
logistic 54.6 3.77·10 -3 776 . 15328 - -
4-param. logistic 56.9 2.70·10 -3 626 -13.9 14854 + +
Weibull 56.5 5.79·10 -6 1.77 57.0 14870 + +
τ = 0.025
Gompertz 47.3 4.74·10 -4 0.998 . 1869 + +
von Bertalanffy 1 60.9 8.14·10 -4 296 . 2020 - -
von Bertalanffy 2 49.7 1.95·10 -3 114 . 1848 + +
logistic 47.0 4.21·10 -3 929 . 2152 - -
4-param. logistic 47.9 3.06·10 -3 809 -8.18 1853 + +
Weibull 47.0 2.16·10 -7 2.21 48.2 1843 + +
(a) Gompertz model is
t
c
D = ab ⋅ , von Bertalanffy 1 is D = a·(1 - e -b·(t – c) ), von Bertalanffy 2 is
D = a·(1 - e -b·(t – c) ) 3 , logistic is D = a/(1 + e -b·(t - c) ), 4-parameter logistic is D = (a – d)/(1 + e -b·(t - c) ) + d,
c
−bt ⋅
and Weibull is D = a−d⋅ e (parameters are not all comparable between models).
(b) 'Fitting' is a visual impression of adequacy of the model on a graph (as presented in Fig. 28B for
models with lowest deviance δ1, in bold in the table).
Fitting of the original model which includes intraspecific competition
(eq. 29) on the same data and for the same quantiles τ = 0.975, 0.5 and
0.025 is presented in Table 13. Deviance δ1 and visual inspection on a
graph (not shown but very close to Fig. 28B) indicate that this model is
one of the most adequate, with all cautions previously formulated about
these criteria. The major difference between this model and previous ones
(in Table 12) is the better biological meaning of its parameters. However,
unconstrained regression leads to meaningless estimations of some
parameters: all D0 values are negative, meaning a negative size at
Part IV: A growth model with intraspecific competition
154

metamorphosis! Biological meaning of parameters in theory does not
imply meaningful estimates when using unconstrained regression on real
datasets. Constraints should be formulated according to background
knowledge or hypotheses about the phenomenon studied. For instance, it is
logical to constrain D0 to be a positive value, since negative size is
meaningless.
Table 13. Results of quantile regressions for three values of τ using the new growth model
(eq. 29). Curves are not shown in a graph, but they are quasi identical to those in Fig. 28B.
ττττ D0 ∆D∞ k1 k2 l deviance δδδδ1
0.975 -8.34 74.6 5.42·10 -3 2.03·10 -3 326 2183
0.5 -5.09 61.5 8.26·10 -3 2.95·10 -3 693 14827
0.025 -3.79 52.3 2.59·10 -3 2.86·10 -3 749 1855
Constraining parameters of the model
It is possible to do a little better for D0 than just forcing it to be
positive. If it is known, it can be replaced in eq. 29 by its real value.
Ambital test diameter was measured on a large number of sea urchins
(n = 296) just after metamorphosis in similar rearing conditions by
Grosjean et al (1996, see Part III). Its values are normally distributed, with
a mean of 0.497 mm and a standard deviation of 0.056 mm. Since this
spreading in initial sizes is negligible compared to the size-scale during the
whole growth process (compare 0.056 mm with 0.50 to 50-65 mm), one
can simplify the model and consider that the initial size of echinoids just
after metamorphosis, D0, is about 0.5 mm for any individual. Accepting
this simplification, it is possible to eliminate D0 from the equations by
working with size increase D' instead of absolute size D:
and:
Part IV: A growth model with intraspecific competition
D'( t') = Dt' ( ) − D
(30)
D'( t') =∆D
1−e 1+ l ⋅e
0
−k1⋅t' ∞ −k2⋅t' (31)
155

This way, one obtains a 4-parameter model with an origin constrained to
be D'(t' = 0) = 0, size increase being null just after metamorphosis for all
quantiles.
Fitting the modified growth model of eq. 31 with quantile regression
for various quantiles is shown in Table 14 and Fig. 31. The model is
flexible enough to accommodate any quantile. Adding a constraint on one
parameter leads to a slightly poorer fitting according to δ1 and visual
impression, which is not surprising.
Table 14. Results of quantile regressions for different values of τ, using the new growth model
constrained to the origin (eq. 31). Curves for quantiles τ = 0.025, 0.5 and 0.975 (in bold) are
also presented graphically in Fig. 31. Relations between some parameters of the model and τ
are shown in Fig. 32.
ττττ ∆D∞ k1 k2 l deviance δδδδ1
0.975 64.5 1.97·10 -3 1.89·10 -3 0.806 2251
0.95 68.2 1.25·10 -3 7.51·10 -3 1.60 4048
0.9 60.9 2.16·10 -3 2.80·10 -3 1.96 7105
0.85 59.8 2.39·10 -3 2.76·10 -3 2.65 9542
0.8 58.8 2.31·10 -3 2.82·10 -3 3.09 11448
0.75 58.3 2.39·10 -3 2.82·10 -3 3.77 12871
0.7 57.5 2.35·10 -3 2.84·10 -3 4.07 13898
0.65 57.9 1.88·10 -3 3.02·10 -3 3.99 14577
0.6 57.0 2.21·10 -3 2.89·10 -3 4.93 14968
0.55 57.2 1.70·10 -3 3.33·10 -3 5.04 15094
0.5 56.6 1.89·10 -3 3.11·10 -3 5.38 14943
0.45 55.9 1.93·10 -3 3.15·10 -3 6.16 14616
0.4 55.8 1.96·10 -3 3.17·10 -3 7.00 14049
0.35 55.6 1.73·10 -3 3.38·10 -3 7.28 13296
0.3 55.1 1.74·10 -3 3.43·10 -3 8.11 12338
0.25 55.6 1.44·10 -3 3.65·10 -3 8.25 11152
0.2 55.4 1.35·10 -3 3.84·10 -3 9.57 9702
0.15 55.4 1.25·10 -3 3.90·10 -3 9.80 7993
0.1 57.7 9.92·10 -4 4.58·10 -3 13.8 5948
0.05 54.0 1.03·10 -3 4.83·10 -3 19.4 3429
0.025 53.3 9.24·10 -4 5.84·10 -3 39.9 1943
Part IV: A growth model with intraspecific competition
156

Diameter increase D' in mm
0 10 20 30 40 50 60
500 1000 1500 2000 2500
Time t' in days
Part IV: A growth model with intraspecific competition
quantile 0.975
quantile 0.5 (median)
quantile 0.025
Figure 31. First step of constraining parameters of the new growth model (eq. 31, origin
forced to {t0, D0}). At this stage, fitting seems slightly poorer than with some unconstrained
models (compare with Fig. 28B). As a consequence of independence of the three quantile
regressions, curvatures do not appear "homogeneous" between curves.
Some inconsistencies are observed for extreme quantiles. This is partly
due to a less satisfactory convergence of the quantile regression because a
lower number of data points have a greater influence on the fitting (due to
ρτ(u), see eq. 22) for large and small quantiles.
As a consequence of independence of the fitting for the different
quantiles, curves do not appear very harmonious in Fig. 31. This can be
solved by linking regressions, that is, by adding relationships between the
4 parameters and τ. Intuitively, there should be a relationship between each
of these curves because they originate from a single dataset with a single
conditional distribution and also because they represent growth of virtual
157

individuals related to the presence of other virtual individuals (inhibitorsinhibited
interactions). With current model (eq. 31) and quantile regression
method (eqs. 21 and 22), it is only possible to fit one curve at a time. An
extension or adaptation of both the model and the regression method are
required to link curves for different quantiles.
In the case of parameter l, we have already mentioned that we expect
l = 0 for τ = 1, according to the hypothesis that larger animals in the batch
are not inhibited at all (see above, construction of the model). We would
also expect a monotonous increase of l with a decrease of τ because
fractions of smaller individuals should be more inhibited than fractions of
larger ones (recall this is a size-based competition mechanism). Fig. 32A
highlights a linear relationship between l and τ, except for the 10% smaller
fraction. Consequently, we constrain l(τ) as:
where s is the slope of the linear relation.
Part IV: A growth model with intraspecific competition
l( τ ) = s⋅(1
− τ )
(32)
k1 and k2 appear negatively correlated in Fig. 32B but are quite
constant along τ values, except for extreme quantiles. Moreover, large
values of k2 for the three rightmost points in the graph at Fig. 32B seem
associated with a potential overestimation of corresponding l values in
Fig. 28A (outliers). In regard with these considerations, reasonable
relationships between k1/k2 and τ could be:
k1( τ ) = cste = k1; k2( τ ) = cste = k2
(33)
with k1 probably different (and lower) than k2.
158

k
l
45
40
35
30
25
20
15
10
5
0
0.008
0.007
0.006
0.005
0.004
0.003
0.002
0.001
0
l
l (outliers)
model
l = 11.644 (1 - τ )
R 2 = 0.9595
0 0.2 0.4 0.6 0.8 1
1 - τ
k2
k1
0 0.2 0.4 0.6 0.8 1
1 - τ
Figure 32. A. Variation of l as a function of 1-τ for several quantile regressions performed
separately with the new growth model constrained to the origin (eq. 31, Table 14 and
Fig. 31). A simple linear relationship appears suitable to define l in function of τ , except for
the 10% smallest individuals (black dots, "outliers"). 'Model' is a least-square linear
regression, after eliminating these outliers, with s = 11.6 (R 2 = 0.960). It is just indicative. B.
Variation of k1 (black squares) and k2 (white triangles) as functions of 1-τ.
Part IV: A growth model with intraspecific competition
A
B
159

Finally, ∆D∞ should follow a normal distribution, as size distributions
when approaching asymptotic maximum size are normal or close to
normal (see Fig. 28A, t > 1500 days, and also Grosjean et al, 1996, see
Part III):
with µ D∞
Part IV: A growth model with intraspecific competition
∆D ( τ) ∼ N ( µ , σ )
(34)
∞ ∆D ∆D
∆ being the mean and σ ∆D∞
normal distribution of ∆D∞(τ).
∞ ∞
Replacing eqs. 32-34 into eq. 31, we get:
−k1⋅t' 1+ e
D'( t', τ) =∆D
( τ)
1 −s⋅(1 −τ) ⋅e
being the standard deviation of the
∞ −k2⋅t' (35)
which links curves for all quantiles 0 < τ < 1 and has 5 parameters to be
estimated: k1, k2, s, µ D∞
∆ and σ ∆ D∞
. It includes individual variations into
the model and is a kind of 3D-surface that envelops data (see Fig. 33). For
this reason, it will be called an 'envelope model'.
The quantile regression method is modified as follows. Considering
that every individual present in the batch is measured at each sampling
time, unconditional quantiles at each size distribution can be regarded as
estimators of conditional quantiles τ at corresponding time t' in eq. 35
(note that this is fundamentally different than the previous quantile
regression method in eqs. 21-22 where unconditional quantiles were not
used at all in the regression). Estimators of τ, noted ˆ τ , are then calculated
as:
ˆ τ =
i
nt' ( i )
∑
j=
1
( D'j t'i < D'i)
I ( )
nt' ( )
i
(36)
where t'i is t' corresponding to the i th observation, n(t'i) is the total number
of individuals measured at time t'i, D'j(t'i) is the j th observation among all
160

measures made at time t'i and I(u < v) returns 1 if true and 0 if false, as in
eq. 22. Parameters of the envelope model to fit, ξ2, are estimated by
minimizing the following objective function δ2:
δ 2 =
Part IV: A growth model with intraspecific competition
n
∑
i=
1
| D' −ξ
2( t' , ˆ τ )|
i i i
n
(37)
δ2 is indeed the mean absolute deviation between observed and predicted
sizes for all observations. A robust simplex minimization algorithm is used
to converge to the solution (Nelder & Mead, 1965; Nocedal & Wright,
1999).
Fitting of the envelope model (eq. 35) by minimizing δ2 is presented in
Fig. 33. This graph emphasizes how individual variation is now included
in the model itself. Gain is obvious by comparing it to Fig. 28A, where the
same dataset is summarized into less informative 2D-curves. Parameters of
the model are: k1 = 1.53 10 -3 , k2 = 3.65 10 -3 , s = 12.7, µ ∆ D = 57.0 and
∞
σ ∆ D = 4.28, deviance δ2 = 1.18.
∞
Fig. 34 is a diagnostic of this regression. Fig. 34A shows residuals (as
differences between observed and predicted values) using a contour plot.
There are only small patches of residuals above 2 mm or below –2 mm,
attesting a good fitting of the model. Residuals are not randomly
distributed. This is probably due to some autocorrelation in the dataset, to
some subtle environmental fluctuations in the rearing system (seasonal
variations…), or possibly to some lack of fit of the model.
161

60
Diameter
increase
D'
in mm
40
20
0
500
1000
1500
Time t' in days
Part IV: A growth model with intraspecific competition
2000
2500
Figure 33. Envelope model (eq. 35) fitted (upper surface) to the whole dataset (lower
surface). The dataset is the same as the histograms of Fig. 28A, but represented differently
here to facilitate comparison with the model. Elevations (z-values, number of individuals) in
the model's surface are weighted according to the number of individuals surviving with time
in the dataset (spline interpolation of actual values, see Fig. 29).
0
50
100
150
Nbr.
of
ind.
162

Nbr of individuals
0 20 40 60 80 100
A
Diameter increase D' in mm
60
50
40
30
20
10
1
0 10 20 30 40 50 60
Diameter increase D' in mm
Figure 34. Diagnostic of the envelope model (eq. 35) fitted in Fig. 33. A. Contour plot of the
residuals as [observed D' - predicted D'] (shades according to the scale at right, in mm).
Quantiles 0.025, 0.5 and 0.975, obtained from the initial dataset (points) and corresponding
curves extracted from the envelope model by fixing τ (lines) are superimposed. Three "slices"
are cut in the 3D-surfaces of Fig. 33 at t' = 300 (1), 600 (2) and 1800 (3) days (vertical dotted
lines in Fig. 34A) and represented as size distributions in B. For each pair of distributions,
the smoothest one is the model. Differences are clearly visible and correspond to positive or
negatives patches in the residuals in A.
Part IV: A growth model with intraspecific competition
500 1000 1500 2000 2500
Time t' in days
2 3
quantile 0.975
quantile 0.5
quantile 0.025
B
163
4
2
0
-2
-4

e. Discussion
Fig. 34B shows three sections across the surfaces of Fig. 33 at three
given times t' = 300, 600 and 1800 days. Overall size spreading and
asymmetries in the original dataset are quite well respected by the model.
The higher peak in the first section of the model at 300 days is partly a
consequence of considering D0 as strictly equivalent for all individuals
while, in reality, it is normally distributed. With the simple relation
between l and τ, as established in eq. 32, multiples modes are just
approximated by a unimodal, skewed distribution. This is most visible in
the second section, at 600 days. A more complex model would be required
to fit multimodal distributions.
Fitting methods
We have argued in favor of quantile regression instead of least-square
regression in modelling growth. Distribution of the "error", that is, mainly
individual variation in the present case, can be either asymmetrical or
multimodal and this is a violation of basic assumptions of the least-square
model. In the same circumstance, a mean effect obtained by least-square is
less representative of the tendency. Quantile regression fits any part of the
distribution, including extremes, and accommodates any kind of size
distribution. It is also robust against any power transformation and it is a
guarantee that the regression remains independent from the dimension of
measurements for size (linear versus surface versus volume/weight). Leastsquare
regression is very sensitive to any power transformation, and thus
to the dimension of the variables.
For purely descriptive fittings, we introduced the triple
τ = 0.975 / 0.5 / 0.025 quantile regression representation as an informative
summary of the data. 5% is a commonly used critical level in statistics.
The curves for τ = 0.975 and t = 0.025 materialize a kind of two-tailed 5%
nonparametric conditional confidence interval of the dataset: 95% of data
Part IV: A growth model with intraspecific competition
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are inside this interval. Additionally, a τ = 0.5 median curve visualizes
possible asymmetry and its change with time. When a representative
sample of the whole size distribution at each time is available, typically
with n > 50, points representing unconditional quantiles can be superposed
on the graph to show how well quantile regressions match them, as we did
in Figs. 28B, 31 and 34A. There is no residuals analysis for this kind of
quantile regression, at least in the way it is conceived for least-square
regression.
Nonlinear quantile regression using interior point algorithm proposed
by Koenker & Park (1996) is compatible with many growth models. For
the cases where representative samples are measured at each time interval,
we have proposed a modified method to use with so-called 'envelope
models', that is, models of the form Y = f(t, τ) as eq. 35. These models
calculate a 3D-surface enveloping data (Fig. 33). Curves for all quantiles
0 < τ < 1 are calculated at once. They contain most of the information in
the initial dataset, including individual variability. They are particularly
useful when individual variability in itself expresses some aspects of the
phenomenon studied, like growth in the presence of an intraspecific
competition. We have developed basic tools to fit and diagnose them
(namely, basic graphical residuals analysis, Fig. 34). Tests of significance
of the fitting can be elaborated as extensions of some tests for
unconditional size-distributions, like χ 2 or Kolmogorov-Smirnov tests or
by other means (Koenker & Machado, 1999). Confidence intervals for the
parameters do not yet exist. Tests for comparing various models fitted on
the same dataset, or to compare a single model fitted on various datasets
remain also to be developed. However, one difficulty in conceptualizing
such tests for envelope models is that one of the "independent variables",
τ, is estimated according to the dependent variable y (eq. 36) and is thus
not really independent.
Flexible, unconstrained envelope models can be incredibly complex,
with dozens of parameters. They are impossible to fit in practice.
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Constraining parameters as we did in eq. 30-34 leads to a double benefit.
First, it simplifies the model. In the present case, we started with a 5parameter
unconstrained classical model (eq. 29) and we ended with a 5parameter
constrained envelope model (eq. 35). Yet, the latter contains
much more information than the former. Second, if constraints are
formulated according to some knowledge about the underlying
phenomenon (value of the intercept corresponding to actual initial size) or
to some reasonable hypotheses (relation between l and τ, in regard with
information in the literature), parameters remain meaningful in the fitted
model. A correct formulation of both the initial unconstrained model and
of superimposed constraints is a bit of an art. It requires many trials and
errors, much patience and perseverance. But at the end, it pays off with a
model whose parameters are fully functionally interpretable, even on real
data.
Fitting and individual variations in growth
A good fitting is not a criterion for deciding if a model is adequate
(Fletcher, 1974). Choosing an inadequate model that fit the data very well
is not problematic if that model is just used for descriptive purposes. It
turns out to be a problem when parameters are functionally interpreted or
if it is used in population dynamic simulations. In this case, individual
variation should not be simply considered as an independent, normally
distributed "error" whenever it is not. Individual variation has often been
overlooked in the literature. The most aberrant calculations could result
from such mistakes. For instance, Basuyaux & Blin (1998) extrapolated
over 4 years a growth model for P. lividus where size distributions were
supposed to be normal and using measurements from 7 to 23 months only.
They calculated the fraction of the size distribution that would reach 40
mm (the minimal market size) with time on basis of this extrapolation.
Many techniques for population dynamic analyses are based on the
assertion that cohorts should distribute normally, including most recent
ones (Smith & Botsford, 1998; Morgan et al, 2000) and could be also
Part IV: A growth model with intraspecific competition
166

iased for the same reason. On the contrary, the envelope model is an
elegant alternative that considers non-symmetrical individual variations in
the model itself.
Not considering individual variation and asymmetries in the size
distributions could lead to the rejection of the von Bertalanffy model
(Sainsbury, 1980). In the case of P. lividus, Cellario & Fenaux (1990) for
reared and Turon et al (1995) for wild populations both rejected the von
Bertalanffy model in favor of the Gompertz curve. We would conclude to
the same rejection of the von Bertalanffy 1 model if we considered only
median quantile regression with τ = 0.5 in the present study (see Table 12).
However, taking intraspecific competition into account using the new
growth model leads to a different conclusion when inhibition is eliminated.
Functional interpretation of the von Bertalanffy model in sea
urchins
From a functional point of view, von Bertalanffy growth implies that
metabolism should be surface-proportional (see von Bertalanffy, 1957, his
Table 6). This is also known as the Rubner's surface rule of metabolism
(Fletcher, 1974). The relation between body-weight (W) and respiratory
rate (R), which is proportional to metabolic rate in aerobic organisms,
should thus vary as R = α·W β , with β ≈ 0.67 (surface:volume). There is no
indication of β for P. lividus in the literature but for other sea urchins:
β = 0.620-0.685 (Percy, 1972), β = 0.708-0.866 (Miller & Mann, 1973) for
Strongylocentrotus droebachiensis; β = 0.65 (Webster & Giese, 1975) for
Strongylocentrotus purpuratus. Lawrence & Lane (1982), after
summarizing similar studies on echinoderms in general, concluded: "most
values of β […] are between 0.6 and 0.8 regardless of body form or
taxonomic group". In a review, Shick (1983) gives a value of 0.64 for
echinoids and considers that, among the echinoderms, they are closest to
the expected value of 0.67. It should be noted that the prediction of
β = 0.67 in von Bertalanffy's theory does only account for somatic growth.
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It does not consider respiration associated with gonadal growth or
gametogenesis. There are two possibilities for mature individuals: either
gonadal growth competes with somatic growth (and the later should be
lower than predicted while β remains 0.67) or it just adds to it (and β
should be somewhat larger). Giese et al (1966) measured no difference in
the respiration of S. purpuratus in function of gonad index. This could
indicate a competition between somatic and gonadal growth in the
presence of a limiting factor. However, it should imply a different somatic
growth model for juveniles and adults. In the present case, a single model
fits both juvenile and adult stages for reared P. lividus. Measurements of
respiration and a more accurate comparison between both phases are
required to evidence a possible effect of maturity on the somatic growth
curve. In any case, β-values between 0.6 and 0.8 do not contradict von
Bertalanffy's law. Thus, the present study rehabilitates its theory that was
too often rejected after observation of a sigmoidal growth (and now we
know it could result from an inhibition).
Functional analysis of the constrained parameters
Constraining the model to the origin is very easy, in theory. Most
models in Table 12 and also eq. 29 have a free intercept. It could be
formally expressed: D0 in eq. 29, or it could be hidden in the
parameterization: a⋅(1 – e bc ) for von Bertalanffy 1, a - d for Weibull, for
models of Table 12. Unconstrained intercept means the parameter
representing size when the growth process initiates is estimated at the same
time as all others, and is thus influenced by their values (intercorrelation).
In real life, the initial size can influence following growth (for some
experimental studies on P. lividus, see Vaïtilingon et al, 2001). We believe
that a meaningful model should follow the same logic: initial size is fixed
first and parameters that characterize growth are estimated afterwards.
This is done in eqs. 30-31. Of course, "initial" size just after
metamorphosis is the result of another growth process during larval life but
the model describes postmetamorphic growth, not larval growth.
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If a model does not fit correctly after fixing the origin, it means that it
is not adapted to describe growth in this case. The only good reason to
avoid constraint is when time (t0), size (D0) or both are unknown at the
origin of the growth process. It is unfortunately common with data
collected in the field (Ebert, 1973, 1980a), when it is not possible to
estimate age accurately (Ebert, 1998; Russell & Meredith, 2000). In this
case, only relative growth can be studied and the problem of origin is thus
eliminated de facto. However, the model must be reworked to fit relative
growth data.
In the present case, we have the information necessary to characterize
the whole size distribution at the origin because we worked in aquaria:
metamorphosis was artificially induced (same t0 for all individuals, see
Grosjean et al, 1998, see Part I), and we have measurements of initial sizes
just after it (Grosjean et al, 1996, see Part III). However, by using actual
distribution instead of approximating D0 by the mean value for all
quantiles, this parameter cannot be eliminated from eq. 31. The model is
still viable, and perhaps a little bit more accurate for small sizes (see
profile 1 in Fig. 34B). Yet, we preferred to keep the simplest model in the
present case.
At the other extreme of the growth process, a single parameter
characterizes its completion when growth is asymptotic in all models. It is
parameter a in models of Table 12 and ∆D∞ in eqs. 29, 31 and 35. Several
authors questioned whether asymptotic growth is a biological reality, or
just a mathematical artifact. Ricker (1979) wrote a section untitled
"asymptotic growth: is it real?" in a chapter of a book; Knight (1968)
devoted a whole article to demonstrate it is a biological non-sense. Some
models with infinite growth appeared (for instance, Tanaka 1982, 1988).
They were also tested on sea urchins (Ebert, 1998, 1999; Ebert & Russell,
1993). Some P. lividus were reared in our installations for 15 years. They
reached their maximum size at 4 to 5 years old. They thus kept exactly the
same size for more than 10 years, proving asymptotic growth is a fact for
Part IV: A growth model with intraspecific competition
169

this species. For other species, where no plateau is observed, lifetime could
be simply too short to reach it. Yet, it is then impossible to tell if growth is
determinate or indeterminate. Anyway, if maximum size is not actually
reached, it is very difficult to estimate the corresponding parameter in the
model.
We constrained ∆D∞ to be normally distributed in the envelope model
(eq. 35). It is in agreement with the analysis of size distributions for fullgrown
animals (Grosjean et al, 1996, see Part III; current dataset). It is also
a consequence of the genetic homogeneity of the batch as all individuals
are issued from a single artificial fertilization, i.e., from one male and one
female. In case where D0 is also considered as normally distributed, the
model relates individuals with largest D0 with individuals with largest
∆D∞. But remember these are virtual individuals. This could be the case
for real echinoids or not. We cannot verify it without tagging individuals to
track them through time in the cohort.
As a consequence of fixing k1 (eq. 33), ∆D∞ is the only parameter to
contain information on relative growth potential of the individuals among
the cohort in eq. 35. The kinetic parameter k1 could be viewed as
environment-dependent (temperature, food, water quality, etc…). Since
these are the same for all animals because they are in the same aquarium,
they are fed ad libitum and have access to the food the same way, it
appears logical to fix k1. Fixing k2 is motivated by a similar reason: we
want it to express one global aspect of the inhibition. When homogeneous
batches of animals of same age and same genetic origin are reared
together, speed at which inhibition is released is supposed to be about the
same for all individuals. This way, only l quantifies changes between
virtual individuals (inhibitors versus inhibited). Of course, many other
variants are possible, but at the cost of an increasing complexity of the
model.
Indeed, as discussed by Grosjean et al (1996, see Part III), water
quality is not exactly the same for all echinoids in culture because they
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170

tend to form aggregates where a pH gradient is measurable from outside to
inside. Smaller animals are more likely found inside and larger sea urchins
outside. This was described as a protective behavior against predators in
the field (Tegner & Dayton, 1977; Tegner & Levin, 1983; Levitan &
Genovese, 1989; Ebert, 1998). But then, parallel gradients in both pH and
size distribution could be the explanation of the size-based inhibition of
growth: a lower pH possibly lowers the speed of skeletogenesis and,
consequently, the growth rate (the soma of sea urchins is made of ca. 90%
of mineralized tissues). No matter what the cause of the inhibition may be,
in such simple experimental conditions, the relationship between l and τ
appears amazingly simple. A precise measure of the pH gradient among
aggregates and the quantification of skeletogenesis speed with pH drop
should bring some more insight into these relations and causalities.
Lacking such information (the only experiment on sea urchins growth
related to pH was performed on larvae, Bouxin, 1926), eq. 32 is currently
the only way to quantify the degree of inhibition.
Even in the present example, eq. 32 seems to be a very simplified
relationship between l and τ. The 10% smallest animals do not follow it.
Profile 2 of Fig. 34B at 600 days shows that the model overestimates the
smallest fraction (and thus underestimates the peak of mid-sized animals)
at this age. Adaptation of l, or even k2, is perhaps necessary to better fit the
smallest fraction. Again, the simplest possible model was presented but
many variations can be conceived.
Relation between l and τ could be even more complex in other
circumstances: different rearing methods, large and small animals of
different ages and/or genetic origins maintained together, periodic size
sorting in culture, etc. Profiles of l in function of τ must be studied in each
particular case. It is also probably very different in the field, or for other
species. The model still needs to be reformulated and tested before being
used with field-collected data (mainly cohort separation, or mark-
Part IV: A growth model with intraspecific competition
171

ecapture, McDonald & Pitcher, 1979; Baker et al, 1991; Francis, 1995;
Ebert 1999) to confirm it.
An interesting potential of this model, thanks to variations in the
relations between l and τ, is the possibility of predicting growth of the
remaining fraction after elimination of largest animals (fisheries or
harvesting of largest fraction in aquaculture). Virtual individuals just
below minimum harvesting size suddenly become the largest fraction and
will exhibit a very rapid catch up growth to reach the maximum growth
speed curve (as evidenced by Grosjean et al, 1996, see Part III). This goes
far beyond the scope of this paper.
Relations with other growth models
Several authors have formulated general growth models, of which
many other models are just particular instances. Richards' model
D = a·(1 – e -b·(t – c) ) d is an extension of a von Bertalanffy 1 to a von
Bertalanffy 2 model with a variable exponent as an additional parameter d
(Richards, 1959; Ebert, 1980a). Depending on the value of d, it reduces to
one of the two von Bertalanffy's models (d = 1 or d = 3), to a logistic (d = -
1), or to a Gompertz curve (|d| → ∞) (Ebert 1999). Schnute (1981), from a
formulation of the derivative of growth rate with time, developed a
sophisticate model that contains most other ones, and also some
unexplored functions. These studies are most useful to show relations
between models that are sometimes hidden by different parameterizations.
For instance, it is hard to tell which is the relation between the Gompertz
(1825) curve and other models in Table 12 just by looking at their
equations.
Starting from von Bertalanffy 1 as the simplest asymptotic growth
model with no inflexion point, one possible contrasting classification of
derived models is 'dimensional' versus 'transitional'. A typical
'dimensional' model is Richards'. Parameter d is an exponent that changes
the dimension of the value returned by the function. Hence, with d = 1, we
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172

have the basic von Bertalanffy 1 model that was designed for linear
measurements of size (1938, his eq. 26). With d = 3, we have the cube of a
linear measurement, that is, a volume or a weight (not considering possible
allometries) and Richards model reduces to von Bertalanffy model 2,
designed for weight measurements. Weibull's model (Weibull, 1951)
belongs probably to this category too, though the exponent c applies only
to time t. When c = 1, it also reduces to the basic von Bertalanffy 1 model,
with a slightly different parameterization. The effect of the exponent,
being d in Richards or c in Weibull, is to transform the von Bertalanffy 1
curve into a S-shaped one, or sigmoid, by means of a power
transformation.
'Transitional' models fit the S-shape as a transition between two states.
The logistic function describes a transition between two constant states
corresponding to its two horizontal asymptotes: D = 0 and D = a. In regard
to the results obtained in Table 12, this model is not adapted here and it
has no affinity with the von Bertalanffy 1 model. This model was initially
designed to model population growth, not individual growth (Verhulst,
1838). The 4-parameter logistic is another 'transitional' model and we will
demonstrate later its relation with von Bertalanffy 1. With the current
parameterization, it also represents a transition between two constant states
materialized by two horizontal asymptotes at D = a and D = d. It fits P.
lividus data very well.
The original growth model of eq. 29 is a third 'transitional' model and
is our missing link as a general equivalent for 'transitional' models to the
Richards' curve for 'dimensional' model (if we except d = -1 and d → ∞
that are physically and biologically meaningless, and thus probably
mathematic artifacts in this context). When l = 0, it reduces to the von
Bertalanffy 1 model. We now have to demonstrate it is a generalization of
the 4-parameter logistic function. If, in D = d + (a - d)/(1+e -b(t-c) ) we
perform the following replacements: t ⇒ t', a ⇒ D0 + ∆D∞, b ⇒ k,
c ⇒ ln(l)/k and d ⇒ D0 – ∆D∞/l, we obtain:
Part IV: A growth model with intraspecific competition
173

∆D∞
D0 +∆D∞ − D0 +∆D∞
/ l
Dt (') = D0−
+ ⇔
−kt ⋅ '+ ln( l)
l 1+ e
Dt (') = D +
−kt ⋅ '
∆D∞⋅( −1−l⋅ e + 1 + l)
0 −kt ⋅ '
l⋅ (1+ l⋅e
)
which gives, after further simplification:
Dt (') = D +∆D
1−e 1+ l ⋅e
−kt ⋅ '
0 ∞ −kt ⋅ '
Part IV: A growth model with intraspecific competition
(38)
(39)
Eq. 39 is equivalent to eq. 29 where k1 = k2 = k. Thus the 4-parameter
logistic is another parameterization of the new growth model where both
growth speed constants k1 and k2 are equal. With this new
parameterization, reduction to a von Bertalanffy 1 model when l = 0 is
now obvious. The 4-parameter logistic model also represents a transition
between same sets S and L as our fuzzy model, but with k1 = k2. It is a
mathematical coincidence that the same model represents also, with
another parameterization, a transition between two constant states,… a
misleading coincidence as it hides its affinity with the von Bertalanffy 1
model!
Whether eq. 29 or eq. 39 is more appropriate to describe P. lividus
growth is hard to tell. From a biological point of view, we do not see any
reason why k1 should equal k2, but we perhaps miss it. Without
confidence intervals on parameters, it is not possible to show if k1 is
significantly different of k2 for the dataset studied (see Fig. 32B).
The distinction between 'dimensional' and 'transitional' models does not
help to explain why one model fits the data better than another. On the
contrary, both types fit data very well. Indeed, dimension change
('dimensional' models) and inhibition of growth ('transitional' models) both
have the same effect on the shape of the von Bertalanffy 1 function: they
transform a curve without inflexion point into a sigmoid. P. lividus growth
data are S-shaped for low τ values because of an inhibition caused by an
intraspecific competition (according to background knowledge on the
174

involved processes). It is thus appropriately described by a 'transitional'
model. However, by fitting data, only the shape that is matched or not by
the model is considered. Consequently, 'dimensional' models fit equally
well, although their mathematical formulation does not match biological
observation.
Considering this distinction, we can now formulate a model that is both
'dimensional' and 'transitional':
⎛ −k1⋅t' 1−e m
⎞
0 ∞ −k2⋅t' Yt (') = Y+∆Y⎜ ⎟
⎝1+ l ⋅e
⎠
Part IV: A growth model with intraspecific competition
(40)
Y being any kind of measurement of size and m (corresponding to d in the
Richards model) indicating the power transformation required to be in the
best 'dimension of growth'. To further generalize eq. 40, we could also
replace k1·t' (the chronological time modulated by a constant kinetic
parameter) with tM, the metabolic –or physiologic– time (Brody, 1937),
using:
t = f( t, x , x ,..., x )
(41)
M 1 2 n
where x1-n are environmental variables that modulate growth, e.g., season
(Cloern & Nichols, 1978) or temperature (Muller-Feuga, 1990). This
gives:
−t
m
M ⎛ 1−e ⎞
= 0 +∆ ∞ ⎜ −kt ⋅ ⎟ M
Yt (') Y Y
⎝1+ l ⋅e
⎠
(42)
where k = k2/k1. This is a general functional model for an asymptotic
growth that derives from the von Bertalanffy 1 curve. Therefore, we call it
a generalized von Bertalanffy model. This model is impossible to fit
without some precautions because the two effects, 'dimension' (m) and
'transition' (l and k), are impossible to separate with solely a shape
criterion. From a fitting point of view, this model is overparameterized
(Draper & Smith, 1998). The dimension parameter m must first be
175

evaluated on basis of biological knowledge: determine the relation
between the dimension of growth and that of the measurement Y that
evaluates it.
But is there a privileged dimension when measuring growth? We have
already asked this question and must come back to it now, because two
concurrent observations suggest that the best dimension to describe growth
in the case of P. lividus is linear. First, using a linear measurement (the
diameter D), basic shape of growth curve, in absence of inhibition, is
exactly the von Bertalanffy 1 model, without inflexion point. With a
weight or a volume to evaluate size, growth of the largest (not inhibited)
fraction in the batch would have followed a more complex von Bertalanffy
2 sigmoidal model and it would have been difficult to distinguish the Sshape
due to dimension from the S-shape due to inhibition. With a linear
measurement, everything is clear: no S-shape means no inhibition and Sshape
means inhibition. Second, the diameters are normally distributed
before (at t' = 0) and after the growth process (when the asymptotic size is
reached that is, above 1600 days, see Fig. 28A). Normal distribution for
linear measurement means that size distribution of corresponding weight
or volume measures must be asymmetrical. In this circumstance, the
envelope model (eq. 35) would have been more difficult to fit because
eq. 34, normal distribution of ∆Y∞(t), is not correct any more. Hence, the
preferred dimension to describe growth of P. lividus seems linear. Using a
linear measurement of size, like the diameter D, we are in the right
dimension and parameter m in eq. 40 equals one, and thus, the model
reduces to eq. 29. If we had chosen to measure weight, we would have
been in a "wrong dimension" to describe growth and a transformation to
the right dimension would imply m ≈ 3 (or more precisely, the allometric
coefficient between weight and diameter).
Part IV: A growth model with intraspecific competition
176

f. Conclusions
The new growth model with intraspecific competition is a very flexible
one. It can accommodate different situations and has meaningful
parameters that allow exploring and quantifying various aspects of growth.
Using a quantile regression method, modified for envelope modelling, and
constraining parameters ensures the meaning of the latter is saved into the
fitted model. It takes also individual variability into account. This is
particularly useful to model growth of P. lividus and probably of many
other sea urchins species and other animals or plants. It is original in many
aspects, including the way it was designed, by defuzzifying a fuzzy model
where most of the other growth models were built from their differential
equations. It is a general 'transitional' growth model. By distinguishing
'dimensional' and 'transitional' growth models, a duality in sigmoidal
growth curves comes to light. A preferred dimension for modelling growth
seems to exist. It is linear in the case of P. lividus but it should be most
interesting to check it for other species.
A generalized von Bertalanffy growth model, which is both
'dimensional' and 'transitional' and includes varying environmental effects
on growth, thanks to the use of metabolic time, was proposed (eq. 42). It
is, however, the visible tip of the iceberg. Fuzzy sets and transitions
(membership functions) can be combined in countless ways to create many
other similar models. In the present work, we studied models deriving
from von Bertalanffy 1 curve, because the latter seemed to be a good basis
for describing growth of P. lividus. Other models incorporating an
inhibition component or any other 'transitional' feature can be derived from
other growth models, including non-asymptotic ones, and would perhaps
be more adapted for other species. We propose to call this family of
functions 'fuzzy-remanent' models. From their fuzzy origin, they keep
nothing in appearance, but the biological meaning of their parameters is
still there. Recalling the fuzzy model they come from, one has a much
clearer idea of how various components –sets and membership functions–
interact to produce the final result. Fuzzy logic is closer to the way human
Part IV: A growth model with intraspecific competition
177

ain conceptualizes complex objects. Statistical tools handle defuzzified
analytic functions more conveniently. By their bivalence, fuzzy-remanent
functions promise to be powerful tools for modelling complex nonlinear
phenomena, like growth, in a functional way.
g. Acknowledgments
We thank the CREC and the University of Caen for their contribution
in building a specific sea urchin rearing facility. We are grateful to Didier
Bucaille who performed the laboratory measurements. We thank also the
Prof. Michael Russell for constructive criticisms and Prof. Jean-Pierre Van
Noppen for proofreading the manuscript. This study was conducted in the
framework of the European Contracts AQ2.530, "Sea urchins cultivation"
and FAIR-CT96-1623, "Biology of sea urchins under intensive cultivation
(closed cycle echiniculture)". This is a contribution of the "Centre
Interuniversitaire de Biologie Marine".
Part IV: A growth model with intraspecific competition
178

General conclusions
179

180

General conclusions
GENERAL CONCLUSIONS
By focusing on variability and interactions in individual growth of the
reared sea urchin Paracentrotus lividus, we raised questions on the
adequacy of existing models and methods. To resolve these problems we
developed a new growth model with incorporating interspecific
competition by defuzzifying a fuzzy model and a quantile regression
method was adapted to account for individual variability (envelope
modelling). The model appears to be an appropriate functional description
of the process, as it is in agreement with all experimental results. It allows
quantifying the degree of growth inhibition in the P. lividus echinoids in
cultivation.
Similarly, one should question the validity of growth models, of fitting
methods and of calculation of size at age (growth ring analysis, sizefrequency
data analysis, mark and recapture) in all studies on either reared
or wild echinoids. Yet, if there is some interaction between individuals or
if individual variability is large (as both can be suspected in most if not all
cases), all these methods could lead to biased estimations of growth and,
consequently, to erroneous inferences about population dynamics.
Adequate tools remain to be developed for field data where age is not
measurable without error. A modification of the new model for relative
growth rate data would be a logical starting point.
The new model clearly has application both in sea urchin aquaculture
and in fisheries management. Indeed, the experiment with mixed Ff and Fg
batches (see Part III, p. 130) indicated a great growth potential of smaller,
inhibited individuals in a heterogeneous population. Yield per surface unit
should improve in cultivation when using mixed batches because they are
almost as productive as small plus large batches reared separately and thus,
on the double of the surface. If the mechanism of inhibition/catch up
growth also occurs in the field, a bimodal size distribution could be the
most efficient configuration to maintain a wild population of sea urchins.
When the largest fraction of the population is harvested, some mid-sized
181

General conclusions
individuals, whose inhibition is suddenly eliminated, could quickly replace
missing adults. Two conditions should be met, however, to obtain this
result on the long term. First, mortality of small and mid-sized individuals
should not increase when large adults are removed (indeed, when
removing large adults, the passive protection of small individuals against
predators disappears). If necessary, part of the adults should be left in
place during harvesting. Second, recruitment should not be a limiting
factor. By harvesting large adults before they spawn, recruitment is de
facto lowered. An artificial production of a large amount of seed in
hatcheries is one way to maintain recruitment levels. Beyond these general
considerations, it is difficult to define rules for sustainable fishery
practices. If the growth model with intraspecific competition were
calibrated against field population data, it would be possible to quantify
the impact of various fishery methods, and to provide objective criteria for
sustainable sea urchin fisheries (Grosjean & Jangoux, 2000).
From a theoretical point of view, the new growth model rehabilitates a
60-year old theory of growth elaborated by von Bertalanffy (1938). Since
then, several authors have questioned its validity (Knight, 1968; Roff,
1980; Frontier & Pichot-Viale, 1993). Many other works have indicated
that the von Bertalanffy 1 model is probably not acceptable to describe the
growth of sea urchins (Gage & Tyler, 1985; Gage et al, 1986; Gage, 1987;
Dafni, 1992; Ebert, 1980a; Ebert & Russell, 1993; Lamare & Mladenov,
2000), including P. lividus (Cellario & Fenaux, 1990; Turon et al, 1995).
Here we have shown that sigmoidal growth does not necessarily means
that von Bertalanffy's theory is invalid. An S-shape could result from
inhibition of growth at small sizes/ages. P. lividus follows the von
Bertalanffy's law for its somatic growth when it is not inhibited. In a
cohort, the non-inhibited fraction amounts for less than 10% of all the
individuals. Using least-square regression leads to the rejection of the von
Bertalanffy 1 model. Using quantile regression validates it for the largest
fraction. Using an envelope model with intraspecific competition
182

General conclusions
component validates it as the basic process for the whole cohort and shows
how intraspecific competition delays actual growth.
Growth is the result of many complex mechanisms that interact:
feeding, digestion, respiration, building up of new somatic tissues,
maintenance, reproduction, etc. A general consensus is that growth is too
complicated and could only be reliably described by complex models.
Indeed, it is surprising that a simple 2-parameter model like von
Bertalanffy 1 (D(t) = D∞·[1 – e -k·t ]) could represent individual growth of
many organisms. It is also surprising that, considering interactions and
individual variability in addition to the basic processes, a quite simple 5parameter
model (our envelope curve, eq. 35 p 160) represents growth of a
whole cohort of reared sea urchins.
Most of the simple models of the first half of the twentieth century,
attempt to interpret, in a functional way, either individual growth (von
Bertalanffy, 1938, 1957; Brody, 1945) or size/shape at age (Huxley, 1932;
Teissier, 1934, 1948; d'Arcy Thompson, 1961). In the last half of the
century, these models have been replaced by more complex, but purely
descriptive and/or speculative ones (Richards, 1959; Schnute, 1981;
Tanaka, 1982, 1988; Jolicoeur, 1985). Clearly, it is worth revisiting old
concepts using new tools, e.g., fuzzy logic or quantile regression. Such
revisitation was accomplished for instance by van Osselaer & Grosjean
(2000) in their study of the suture of coiled shells reflecting the ontogeny
of molluscs. Despite the conclusion of Tursh (1998) that the shape of the
suture is too complex to be modelled with a simple equation, they
demonstrated that a simple 4-parameter helicospiral model was the best
descriptor for most coiled shells and that some methodological errors
prevailed when using much more complex 8- to 16-parameter models (for
a review, see Stone, 1996). This example demonstrates another case where
growth is less complex in reality than in theory. The discipline (call it
"ontogenology") of describing individual growth or ontogeny with models
that are both reasonably simple and functional may reveal one day how
183

Root models:
y’ = ay m -by n
(exponential,
von Bertal. 1,
logistic)
General conclusions
various mechanisms constrain "organic growth" of metazoans. These
constraints, models, and "rules of growth and development" may turn out
to be simpler than expected.
As a final conclusion, we propose an original classification of growth
models based on their functional features rather than their pure mathematic
affinities (though similar functions are often expressed by similar
equations). In this typology, general growth models derive from simpler
predecessors thanks to one or more additional features having a biological
(or physical) meaning (Fig. 35).
transitional
dimensional
metabolic time
diauxic
Preece-Baines
polyphasic
4-p. logistic Fuzzy-reman.
von Bertal. 2 Richa rds
Weibull?
Seasonal VB
T(°C) VB
General. logis.
monophasic
von Bertal.
with
t M = f(t, x i ..)
Go mpe rtz
?
Johnson
Generalized
von Bertal.
with t M
Jolicoeur
?
Tanaka
…
Figure 35. A classification of growth models based on their functional features (see text for
further explanations).
184

General conclusions
In this classification, all models derive from simplest forms described
by the general differential equation y' = ay m - by n , that is, a damped
exponential growth with the first term being the limiting factor and the
second one representing the exponential growth (Fletcher, 1974; Mueller-
Feuga, 1990). The various forms of basic growth models are obtained from
different particular values for m and n. With m = 1 and n = 1, we obtain the
exponential growth, which is a simple indeterminate growth model. If
m = 0 and n = 1, we get the von Bertalanffy 1 model (von Bertal. 1),
which seems to be the simplest determinate growth model for individuals.
If m = 1 and n = 2, we obtain the logistic curve, which is probably the
simplest determinate growth model for populations (Verhulst, 1838). It is
not clear if other solutions of the differential equation could be considered
as useful roots in the classification. Some solutions correspond to more
complex models that are best placed in a stem instead of as a root in this
classification. For instance, using m = 2/3 and n = 1, we obtain the von
Bertalanffy 2 model that we prefer to position inside the "von Bertalanffy
1 family" (see Fig. 35). Some authors have generalized this differential
equation (Fletcher, 1974; Schnute, 1981) to a point that it describes almost
all existing models. Our concern here is just to derive the simplest models
as roots of our classification tree and consider a di- or a polychotomous
system to position more complex models in the tree as generalizations of
the simplest root models. One should consider each of the simplest models
as the root of a distinct tree. However, in our presentation (Fig. 35), we
mixed all existing models in a single tree for conciseness (because, except
for the "von Bertalanffy 1 family", the trees would not been much
populated).
Starting from those simplest models, one could consider a first
dichotomous separation: monophasic versus polyphasic models. While
the group of monophasic models is more populated (because they probably
represent more common growth processes), the second group contains two
items:
185

General conclusions
- The diauxic growth model of Liquori et al (1981). This is a biphasic
model used to describe the increase in cell numbers in an embryo that
results in a slow and a fast division processes.
- The Preece-Baines (1978) model that describes human growth (and
perhaps also the growth of some other mammals) that we presented in
the introduction, p. 51.
In the monophasic growth group, we have various sub-groups that
correspond each to a distinct feature added to the basic model. In Part IV,
we discussed the two antagonist features that transform a von Bertalanffy 1
model without inflexion point into a sigmoid: the dimensional and
transitional sub-groups. Models in the dimensional sub-group account for
a change in the dimension of size measurement (length, surface or
volume/weight) thanks to an additional parameter m. Von Bertalanffy 2
model (von Bertal. 2) is a particular case with m = 3 and Richards model
is the general equation of this sub-group. In the transitional sub-group,
there is an inhibition of growth that is progressively released with age.
This is probably the kingdom of 'fuzzy-remanent functions' as they are
convenient descriptors for transitions. Both a reparameterized form of the
4-parameter logistic function (4-p. logistic, eq. 39, p. 174) and the model
with intraspecific competition component that we propose here for
modelling the growth of reared P. lividus (Fuzzy-reman., eq. 29, p. 152)
belong to this category. The former being a special case of the latter, with
both kinetic parameters k1 and k2 being equal.
Another sub-group can be created for models that incorporate the effect
of environmental variation on growth, through the concept of metabolic
time. One such model was proposed by Cloern & Nichols (1978) for
considering seasonal variations (as a global summary of various
environmental variables with a seasonal cycle like temperature,
photoperiod, food availability…) in the von Bertalanffy 1 growth model
(Seasonal VB). Mueller-Feuga (1990) proposed another model of this
group which account for temperature only (T(°C) VB). Both of these
186

General conclusions
models obviously belong to a more general family where metabolic time tM
is a function of time t as well as several other environmental
(meta)variables [von Bertal. with tM = f(t, xi…)].
There are perhaps other unidentified sub-groups. The Weibull model is
a generalization of von Bertalanffy 1 with an exponent applied to time t.
As such, it could belong to both the dimensional and the metabolic time
sub-groups. We do not see any functional meaning of them, but it could
exist. A generalized logisitic model was proposed by Nelder (1961) and
Turner et al (1969). It is a logistic function where an additional parameter
m is applied as a global exponent. From its analytic form, it is a
dimensional model, which makes sense only when it is applied to
individual growth. Turner (1969) indicates that, when applied to
populations, this model accounts for growth with a maximum population
size that is allowed to vary. It this context, it may belong to another
unidentified sub-group.
The generalized von Bertalanffy with tM (eq. 42, p. 175) is at the
same time a 'dimensional', a 'transitional' and a 'metabolic time' model. It
represents the highest level of generalization in the "von Bertalanffy 1"
family but it spans a large space for deriving other kinds of models.
Finally, there are some unclassifiable models, such as Gompertz,
Johnson, Jolicoeur and Tanaka. Either they are purely descriptive
models that just mimic the shape of some functional models, or their
affinity is not established yet. In the first case, they have clearly no place
in the proposed functional classification, as it is the case for other purely
descriptive models (e.g., Rao's polynomial growth model; Rao, 1965;
Basu, 1999). In the other case, a reparameterization or a good example of a
functional use is probably required to reveal their real nature.
Much space is left empty in this typology for adding new functional
models when growth of other animals and plants will be described in a
functional way. Such a classification should help choosing the right growth
187

General conclusions
model not on a shape criterion (does it fit data according to the R 2 , or sum
of square of residuals, or to an analysis of residuals, or to a visual
inspection on a graph), but on its ability to represent at best underlying
processes of growth, as they are experimentally evidenced.
188

References
REFERENCES
Abe E., M. Kakiuchi, K. Matsuyama & T. Kaneko, 1983. Seasonal variations in
growth and chemical components in the blade of Laminaria religiosa
Miyabe in Oshoro Bay, Hokkaido. Sci. Rep. Hokkaido Fish. Exp. St.,
25:47-60.
Agatsuma, Y. & Y. Sugawara, 1988. Reproductive cycle and food ingestion of
the sea urchin, Strongylocentrotus nudus (A. Agassiz), in Southern
Hokkaido. II. Seasonal changes of the gut content and test weight. Sci.
Rep. Hokkaido Fish. Exp. St., 30:43-49.
Aksland, M., 1994. A general cohort analysis method. Biometrics, 50:917-932.
Allain, J.-Y, 1971. Note sur la pêche et la commercialisation des oursins en
Bretagne nord. Trav. Labo. Biol. Halieut. Univ. Rennes, 5:59-69.
Allain, J.-Y., 1972a. Structure des populations de Paracentrotus lividus
(Lamarck) (Echinodermata, Echinoidea) soumises à la pêche sur les côtes
nord de Bretagne. Rev. Trav. Inst. Pêches Marit., 39(2):171-212.
Allain, J.-Y., 1972b. La pêche aux oursins dans le monde. La Pêche Maritime,
1133:625-630.
Allain, J.-Y., 1975. Structure des populations de Paracentrotus lividus soumises à
la pêche sur les côtes nord de Bretagne. Rev. Trav. Inst. Pêches Marit.,
39(2):171-212.
Allain J.-Y., 1978. Age et croissance de Paracentrotus lividus (Lmk) et de
Psammechinus miliaris (Gmelin) des côtes nord de Bretagne
(Echinoidea). Cah. Biol. Mar., 19(1):11-21.
Allen, R.K., 1966. A method of fitting growth curves of the von Bertalanffy type
to observed data. J. Fish. Res. Board Can., 23(2):163-179.
Bacher, C., M. Héral, J.M. Deslous-Paoli & D. Razet, 1991. Modèle énergétique
uniboite de la croissance des huîtres (Crassostrea gigas) dans le bassin de
Marennes-Oléron. Can. J. Fish. Aquat. Sci., 48:391-404.
189

References
Baker, T.T., R. Lafferty & T.J. Quinn II, 1991. A general growth model for markrecapture
data. Fish. Res., 11:257-281.
Basu, T.K., 1999. Application of Rao's polynomial growth curve model to
analyse the length growth data of a freshwater major carp of India, Catla
catla (Ham) during the early development period. Fish. Res., 42:303-307.
Basuyaux, O. & J.L. Blin, 1998. Use of maize as a food source for sea urchins in
a recirculating rearing system. Aqua. International, 6(3):233-247.
Bhattacharya, C.G., 1967. A simple method of resolution of a distribution into
Gaussian components. Biometrics, 23:115-135.
Blin, J.-L., 1997. Culturing the purple sea urchin, Paracentrotus lividus, in a
recirculation system. Bul. Aqua. Assoc. Canada, 97(1):8-13.
Botsford, L.W., B.D. Smith & J.F. Quinn, 1994. Bimodality in size distributions:
the red sea urchin Strongylocentrotus franciscanus as an example. Ecol.
Applic., 4:42-50.
Bouxin, H., 1926. Action des acides sur le squelette des larves de l'oursin
Paracentrotus lividus. Influence du pH. C. R. Soc. Biol., 94:453-455.
Branch, G.M., 1974. Intraspecific competition in Patella cochlear Born. J. Anim.
Biol., 44:263-281.
Brody, S., 1937. Relativity of physiologic time and physiologic weight. Growth,
1:60-67.
Brody, S., 1945. Bioenergetics and growth. Reinhold, New York.
Bull, H.O., 1938. The growth of Psammechinus miliaris (Gmelin) under
aquarium conditions. Rep. Dove Mar. Lab., 3(6):39-42.
Campbell, A. & R.M. Harbo, 1991. The sea urchin fisheries in British Columbia,
Canada. In: Y. Yanagisawa et al (eds). Biology of Echinodermata.
Balkema, Rotterdam. Pp. 191-199.
Causton, D.R., 1969. A computer program for fitting the Richards function.
Biometrics, 25:401-409.
190

References
Cellario, Ch. & L. Fenaux, 1990. Paracentrotus lividus (Lamarck) in culture
(larval and benthic phases): Parameters of growth observed during two
years following metamorphosis. Aquaculture, 84:173-188.
Chiu, S.T., 1990. Age and growth of Anthocidaris crassispina (Echinodermata:
Echinoidea) in Hong Kong. Bull. Mar. Sci., 47(1):94-103.
Claerebout, M. & M. Jangoux, 1985. Conditions de digestion et activité de
quelques polysaccharidases dans le tube digestif de l'oursin Paracentrotus
lividus (Echinodermata). Biochem. Syst. Ecol., 13(1):51-54.
Cloern, J.E. & F.. Nichols, 1978. A von Bertalanffy growth model with a
seasonally varying coefficient. J. Fish. Res. Board Can., 35:1479-1482.
Comely, C.A. & A.D. Ansell, 1988. Population density and growth of Echinus
esculentus L. on the Scottish West Coast. Estuar. Coast. Shelf Sci.,
27:311-334.
Conand, C. & N.A. Sloan, 1989. World fisheries for echinoderms. In: J.F. Caddy
(ed.). Marine invertebrate fisheries: their assessment and management.
Wiley & Sons, New York. Pp. 647-663.
Cox, E., 1999. The fuzzy systems handbook, 2 nd ed. Academic Press, San Diego.
Cox, R.A., M.J. McCartney & F. Culkin, 1970. The specific gravity/salinity/
temperature relationship in natural sea water. Deep-Sea Res., 17:679-689.
Crapp, G.B. & M.E. Willis, 1975. Age determination in the sea urchin
Paracentrotus lividus (Lamarck), with notes on the reproductive cycle. J.
Exp. Mar. Biol. Ecol., 20:157-178.
Crook, A.C., E. Verling & D.K.A. Barnes, 1999. Comparative study of the
covering reaction of the purple sea urchin, Paracentrotus lividus, under
laboratory and fied conditions. J. Mar. Biol. Assoc. U.K., 79(6):1117-
1121.
Dafni, J., 1992. Growth rate of the sea urchin Tripneustes gratilla elatensis. Israel
J. Zool., 38:25-33.
191

References
Dafni, J. & R. Tobol, 1987. Population structure patterns of a common Red Sea
echinoid (Tripneustes gratilla elatensis). Israel J. Zool., 34:191-204.
d'Arcy Thompson, W., 1961. On growth and form, 4 th ed. Cambridge University
Press, Cambridge.
de Jong-Westman, M., B.E. March & T.H. Carefoot, 1995a. The effect of
different nutrient formulations in artificial diets on gonad growth in the
sea-urchin Strongylocentrotus droebachiensis. Can. J. Zool., 73:1495-
1502.
de Jong-Westman, M., P.-Y. Quian, B.E March & T.H. Carefoot, 1995b.
Artificial diets in sea urchin culture: effects of dietary protein level and
other additives on egg quality, larval morphometrics, and larval survival
in the green sea urchin, Strongylocentrotus droebachiensis. Can. J. Zool.,
73:2080-2090.
Draper, N.R. & H. Smith, 1998. Applied regression analysis, 3 rd ed. Whiley &
Sons, New York.
Duineveld, G.C.A. & M.I. Jenness, 1984. Differences in growth rates of the sea
urchin Echinocardium cordatum as estimated by the parameter ω of the
von Bertalanffy equation applied to skeletal rings. Mar. Ecol. Prog. Ser.,
19:65-72.
Durham, J.W., 1955. Classification of Clypeasteroid echinoids. University of
California, Museum of Paleontology. Pp. 73-192.
Ebert, T.A., 1967. Negative growth and longevity in the purple sea urchin
Strongylocentrotus purpuratus (Stimpson). Science, 157(3788):557-558.
Ebert, T.A., 1968. Growth rates of the sea urchin Strongylocentrotus purpuratus
related to food availability and spine abrasion. Ecology, 49:1075-1091.
Ebert, T.A., 1973. Estimating growth and mortality rates from size data.
Oecologia, 11:281-298.
Ebert, T.A., 1975. Growth and mortality of post-larval echinoids. Amer. Zool.,
15:755-775.
192

References
Ebert, T.A., 1977. An experimental analysis of sea urchin dynamics and
community interactions on a rock jetty. J. Exp. Mar. Biol. Ecol., 27:1-22.
Ebert, T.A., 1980a. Estimating parameters in a flexible growth equation, the
Richards function. Can. J. Fish. Aquat. Sci., 37(4):687-692.
Ebert, T.A., 1980b. Relative growth of sea urchin jaws: an example of plastic
response allocation. Bull. Mar. Sci., 30(2):467-474.
Ebert, T.A., 1981. Estimating mortality from growth parameters and a size
distribution when recruitment is periodic. Limnol. Oceanogr., 26:764-769.
Ebert, T.A., 1982. Longevity, life history, and relative body wall size in sea
urchins. Ecol. Monographs, 52(4):353-394.
Ebert, T., 1983. Recruitment in echinoderms. In: M. Jangoux & J.M. Lawrence
(eds). Echinoderm studies , vol. 1. Balkema, Rotterdam. Pp. 169-203.
Ebert, T.A., 1985. Sensitivity of fitness to macroparameter changes: an analysis
of survivorship and individual growth in sea urchin life histories.
Oecologia, 65:461-467.
Ebert, T.A., 1986. A new theory to explain the origin of growth lines in sea
urchin spines. Mar. Ecol. Prog. Ser., 34:197-199.
Ebert, T.A., 1988a. Calibration of natural growth lines in ossicles of two sea
urchins, Strongylocentrotus purpuratus and Echinometra mathaei, using
tetracycline. In: R.D. Burke, P.V. Mladenov, P. Lambert, R.L. Parsley
(eds). Echinoderm biology. Balkema, Rotterdam. Pp. 435-443.
Ebert, T.A., 1988b. Allometry, design and constraint of body components and of
shape in sea urchins. J. Nat. Hist., 22:1407-1425.
Ebert, T.A., 1994. Allometry and model II non-linear regression. J. Theo. Biol.,
168:367-372.
Ebert, T.A., 1998. An analysis of the importance of Allee effects in management
of the red sea urchin Strongylocentrotus franciscanus. In: R. Mooi & M.
Telford (eds). Echinoderms: San Francisco. Balkema, Rotterdam.
Pp. 619-627.
193

References
Ebert, T.A., 1999. Plant and animal populations. Methods in demography.
Academic Press, San Diego.
Ebert, T.A. & D.M. Dexter, 1975. A natural history study of Encope grandis and
Mellita grantii, two sand dollars in the northern Gulf of California,
Mexico. Mar. Biol., 32:397-407.
Ebert, T.A., S.C. Schroeter & J.D. Dixon, 1993. Inferring demographic processes
from size-frequency distributions: effect of pulsed recruitment on simple
models. Fish. Bull. U.S., 91:237-243.
Ebert, T.A. & M.P. Russell, 1992. Growth and mortality estimates for red sea
urchin Strongylocentrotus franciscanus from San Nicolas Island,
California. Mar. Ecol. Prog. Ser., 81:31-41.
Ebert, T.A. & M.P. Russell, 1993. Growth and mortality of subtidal red sea
urchins (Strongylocentrotus franciscanus) at San Nicolas Island,
California, USA: problems with models. Mar. Biol., 117:79-89.
Emson, R.H., 1984. Bone idle – a recipe for success? In: B.F. Keegan & B.D.S.
O'Connor (eds). Echinodermata. Balkema, Rotterdam. Pp. 25-30.
Fabens, A.J., 1965. Properties and fitting of the von Bertalanffy growth curve.
Growth, 29:265-289.
Fahrmeir, L. & G. Tutz, 1994. Multivariate statistical modelling based on general
linear models. Springer-Verlag, New York.
Fernandez, C., 1996. Croissance et nutrition de Paracentrotus lividus dans le
cadre d’un projet aquacole avec alimentation artificielle. PhD Thesis,
Université de Corse, France.
Fernandez, C. & C.-F. Boudouresque, 1998. Evaluating artificial diet for small
Paracentrotus lividus (Echinodermata: Echinoidea). In: R. Mooi & M.
Telford (eds). Echinoderms: San Francisco. Balkema, Rotterdam. Pp.
651-656.
Fernandez, C. & A. Caltagirone, 1994. Growth rate of adult sea urchins,
Paracentrotus lividus in a lagoon environment: the effect of different diet
194

References
types. In: B. David, A. Guille, J.-P. Féral & M. Roux (eds.). Echinoderms
through Time. Balkema, Rotterdam. Pp. 655-660.
Flammang, P., 1996. Adhesion in echinoderms. In: M. Jangoux & J.M. Lawrence
(eds). Echinoderm studies, vol. 5. Balkema, Rotterdam. Pp. 1-60.
Flammang, P., P. Gosselin & M. Jangoux, 1998. The podia, organs of adhesion
and sensory perception in larvae and post-metamorphic stages of the
echinoid Paracentrotus lividus (Echinodermata). Biofouling, 12(1-3):161-
171.
Fletcher, R.I., 1974. The quadric law of damped exponential growth. Biometrics,
30:111-124.
Ford, E., 1933. An accounting of the herring investigations conducted at
Plymouth during the years from 1924-1933. J. Mar. Biol. Assoc. U.K.,
19:305-384.
Francis, R.I.C.C., 1995. An alternative mark-recapture analogue of Schnute's
growth model. Fish. Res., 23:95-111.
Frantzis, A. & A. Gremare, 1992. Ingestion, absorption and growth rates of
Paracentrotus lividus (Echinodermata: Echinoidea) fed different
macrophytes. Mar. Ecol. Prog. Ser., 95:169-183.
Fridberger, A., T. Fridberger & L.-G. Lundin, 1979. Cultivation of sea urchins of
five different species under strict artificial conditions. Zoon, 7:149-151.
Frontier, S. & D. Pichot-Viale, 1993. Ecosystèmes: structure, fonctionnement,
évolution, 2 nd ed. Masson, Paris.
Fuji, A., 1963. On the growth of the sea urchin, Hemicentrotus pulcherrimus (A.
Agassiz). Bull. Jap. Soc. Sci. Fish., 29(2):118-126.
Fuji, A., 1967. Ecological studies on the growth and food consumption of
Japanese common littoral sea urchin, Strongylocentrotus intermedius (A.
Agassiz). Mem. Fac. Fish. Hokkaido Univ., 15(2):83-160.
Fuji, A. & K. Kawamura, 1970. Studies on the biology of the sea urchin. IV.
Habitat structure and regional distribution of Strongylocentrotus
195

References
intermedius on a rocky shore of southern Hokkaido. Bull. Jap. Soc. Sci.
Fish., 36(8):755-762.
Gage, J.D., 1987. Growth of the deep-sea irregular sea urchins Echinosigra phiale
and Hemiaster expergitus in the Rockall Trough (N.E. Atlantic Ocean).
Mar. Biol., 96:19-30.
Gage, J.D., 1991. Skeletal growth zones as age-markers in the sea urchin
Psammechinus miliaris. Mar. Biol., 110:217-228.
Gage, J.D., 1992. Natural growth bands and growth variability in the sea urchin
Echinus esculentus: Results from tetracycline tagging. Mar. Biol.,
114:607-616.
Gage, J.D. & P.A. Tyler, 1985. Growth and recruitment of the deep-sea urchin
Echinus affinis. Mar. Biol., 90:41-53.
Gage, J.D., P.A. Tyler & D. Nichols, 1986. Reproduction and growth of Echinus
acutus var. norvegicus Düben & Koren and E. elegans Düben & Koren
on the continental slope off Scotland. J. Exp. Mar. Biol. Ecol., 101:61-83.
Gallucci, V.F. & T.J. Quinn, 1979. Reparameterizing, fitting, and testing a simple
growth model. Trans. Am. Fish. Soc., 108:14-25.
Gayral, P., J. Cosson, 1973. Exposé synoptique des données biologiques sur la
laminaire digitée Laminaria digitata. In: Synopsis FAO sur les pêches,
n°89.
Gebauer, P. & C.A. Moreno, 1995. Experimental validation of the growth rings of
Loxechinus albus (Molina, 1782) in Southern Chile (Echinodermata:
Echinoidea). Fish. Res., 21:423-435.
Giese, A.C., 1966. Changes in body-component indexes and respiration with size
in the purple sea urchin Strongylocentrotus purpuratus. Physiol. Zool.,
40:194-200.
Giese, A.C., A. Farmanfarmaian, S. Hilden & P. Doezema, 1966. Respiration
during the reproductive cycle in the sea urchin, Strongylocentrotus
purpuratus. Biol. Bull., 130:192-201.
196

References
Goebel, N. & M.F. Barker, 1998. Artificial diets supplemented with carotenoid
pigments as feeds for sea urchins. In: R. Mooi & M. Telford (eds).
Echinoderms: San Francisco. Balkema, Rotterdam. Pp. 667-672.
Gompertz, B., 1825. On the nature of the function expressive of the law of human
mortality and a new mode of determining the value of life contingencies.
Phil. Trans. Roy. Soc., 115:513-585.
Gonzalez, M.L., M.C. Pérez, D.A. López & C.A Pino, 1993. Effects of algal diet
on the energy available for growth of juvenile sea urchins Loxechinus
albus (Molina, 1782). Aquaculture, 115: 87-95.
Gosselin, P. & M. Jangoux, 1996. Induction of metamorphosis in Paracentrotus
lividus larvae (Echinodermata, Echinoidea). Oceano. Acta, 19(3-4):293-
296.
Gosselin, P. & M. Jangoux, 1998. From competent larva to exotrophic juvenile: a
morphofunctional study of the perimetamorphic period of Paracentrotus
lividus (Echinodermata, Echinoidea). Zoomorphology, 118:31-43.
Grosjean, Ph. & M. Jangoux, 2000. Growth of the sea urchin Paracentrotus
lividus: model and optimization. Green sea urchin workshop, Moncton,
Canada. Inline: http://crdpm.cus.ca/oursin/pdf/gros.pdf (last consulted
Sept. 8 th 2001).
Grosjean Ph., Ch. Spirlet & M. Jangoux, 1996. Experimental study of growth in
the echinoid Paracentrotus lividus (Lamarck, 1816) (Echinodermata). J.
Exp. Mar. Biol. Ecol., 201:173-184.
Grosjean, Ph., Ch. Spirlet & M. Jangoux, 1999. Comparison of three body-size
measurements for echinoids. In: M.D. Candia Carnevali & F. Bonasoro
(eds). Echinoderm Research 1998, Balkema, Rotterdam. Pp. 31-35.
Grosjean, Ph., Ch. Spirlet, P. Gosselin, D. Vaïtilingon & M. Jangoux, 1998.
Land-based closed cycle echiniculture of Paracentrotus lividus Lamarck
(Echinoidea: Echinodermata): a long-term experiment at a pilot scale. J.
Shellfish Res, 17(5):1523-1531.
197

References
Guillou, M. & Ch. Michel, 1993. Reproduction and growth of Sphaerechinus
granularis (Echinodermata: Echinoidea) in the Glenan archipelago
(Brittany). J. Mar. Biol. Ass. U.K., 73:172-193.
Hagen, N.T., 1996a. Echinoculture: from fishery enhancement to closed cycle
cultivation. World Aquaculture, Dec. 1996, pp. 7-19.
Hagen, N.T., 1996b. Tagging sea urchins: a new technique for individual
identification. Aquaculture, 139:271-284.
Hasselblad, V., 1966. Estimation of parameters for a mixture of normal
distributions. Technometrics, 8:431-444.
Himmelman, J.H., 1986. Population biology of green sea urchins on rocky
barrens. Mar. Ecol. Prog. Ser., 33:295-306.
Hinegardner, R.T., 1969. Growth and development of the laboratory cultured sea
urchin. Biol. Bull., 137:465-475.
Huston, M.A. & D.L. De Angelis, 1987. Size bimodality in monospecific
populations: a critical review of potential mechanisms. Amer. Nat.,
129(5):678-707.
Huxley, J.S., 1932. Problems of relative growth. Metuen and Co, London.
Ihaka, R. & R. Gentleman, 1996. A language for data analysis and graphics. J.
Comput. Graphic. Stat., 5(3):299-314.
Jensen, M., 1969a. Breeding and growth of Psammechinus miliaris (Gmelin).
Ophelia, 7:65-78.
Jensen, M., 1969b. Age determination of echinoids. Sarsia, 37:41-44.
Jolicoeur, P., 1985. A flexible 3-parameter curve for limited or unlimited somatic
growth. Growth, 49:271-281.
Jordana, E., M. Guillou & L.J.L. Lumingas, 1997. Age and growth of the sea
urchin Sphaerechinus granularis in Southern Brittany. J. Mar. Biol. Ass.
U.K., 77:1199-1212.
198

References
Kaneko, I., Y. Ikeda & H. Ozaki, 1981. Biometrical relations between body
weight and organ weights in freshly sampled and starved sea urchin. Bull.
Jap. Soc. Sci. Fish., 47(5):593-597.
Kato, S., 1972. Sea urchins: a new fishery develops in California. Mar. Fish.
Rev., 34(9-10):23-30.
Kaufmann, K.W., 1981. Fitting and using growth curves. Oecologia, 49:293-299.
Kautsky, N., 1982. Growth and size structure in a Baltic Mytilus edulis
population. Mar. Biol., 68:117-133.
Keats, D.W., D.H. Steele & G.R. South, 1983. Food relations and short term
aquaculture potential of the green sea urchin (Strongylocentrotus
droebachiensis) in Newfoundland. MSRL Tech. Rep., 24:1-24.
Kelly, M.S., J.D. McKenzie & C.C. Brodie, 1998. Sea urchins in polyculture: The
way to enhanced gonad growth? In: R. Mooi & M. Telford (eds).
Echinoderms: San Francisco. Balkema, Rotterdam. Pp. 707-711.
Kenner, M.C., 1992. Population dynamics of the sea urchin Strongylocentrotus
purpuratus in a Central California kelp forest: recruitment, mortality,
growth and diet. Mar. Biol., 112:107-118.
Klinger, T.S., J.M. Lawrence & A.L. Lawrence, 1994. Digestive characteristics of
the sea urchin Lytechinus variegatus (Lamarck) (Echinodermata:
Echinoidea) fed prepared feeds. J. World Aqua. Soc., 25(4):489-496.
Klinger, T.S., J.M. Lawrence & A.L. Lawrence, 1997. Gonad and somatic
production of Strongylocentrotus droebachiensis fed manufactured feeds.
Bull. Aqua.. Assoc. Canada, 1:35-37.
Klinger, T.S., J.M. Lawrence & A.L. Lawrence, 1998. Digestion, absorption, and
assimilation of prepared feeds by echinoids. In: R. Mooi & M. Telford
(eds). Echinoderms: San Francisco. Balkema, Rotterdam. Pp. 713-721.
Knight, W., 1968. Asymptotic growth: an example of non-sense disguised as
mathematics. J. Fish. Res. Board Canada, 25(6):1303-1307.
199

References
Kobayashi, S. & J. Taki, 1969. Calcification in sea urchins. I. A tetracycline
investigation of growth of the mature test in Strongylocentrotus
intermedius. Calcif. Tiss. Res., 4:607-621.
Koenker, R., 2001. Quantile regression. In: S. Fienberg & J. Kadane (eds).
International encyclopedia of the social sciences. (in press)
Koenker, R. & G. Bassett, 1978. Regression quantiles. Econometrica, 46:33-50.
Koenker, R. & J. Machado, 1999. Goodness of fit and related inference processes
for quantile regression. J. Am. Stat. Assoc., 94:1296-1310.
Koenker, R. & B.J. Park, 1996. An interior point algorithm for nonlinear quantile
regression. J. Econometrics, 71(1-2):265-283.
Lamare, M.D. & P.V. Mladenov, 2000. Modelling somatic growth in the sea
urchin Evechinus chloroticus (Echinoidea: Echinometridae). J. Exp. Mar.
Biol. Ecol., 243:17-43.
Lane, J.E.M. & J.M. Lawrence, 1980. Seasonal variation in body growth, density
and distribution of a population of sand dollars, Mellita
quinquiesperforata (Leske). Bull. Mar. Sci., 30(4):871-882.
Lawrence, J.M., 1982. Digestion. In: M. Jangoux & J.M. Lawrence (eds.).
Echinoderm nutrition. Balkema, Rotterdam. Pp. 283-316.
Lawrence, J.M. & J.M. Lane, 1982. The utilization of nutrients by
postmetamorphic echinoderms. In: M. Jangoux & J.M. Lawrence (eds).
Echinoderm nutrition. Balkema, Rotterdam. Pp. 331-371.
Lawrence, J.M., S. Olave, R. Otaiza, A.L. Lawrence & E. Bustos, 1997.
Enhancement of gonadal production in the sea urchin Loxechinus albus in
Chile fed extruded feeds. J. World Aqua. Soc., 28:91-96.
Lawrence, J.M., B.D. Robbins & S.S. Bell, 1995. Scaling of the pieces of the
Aristotle's lantern in five species of Strongylocentrotus (Echinodermata:
Echinoidea). J. Nat. Hist., 29:243-247.
Laws, E.A & J.W. Archie, 1981. Appropriate use of regression analysis in marine
biology. Mar. Biol., 65:13-16.
200

References
Ledireac'h, J.-P., 1987. La pêche des oursins en Méditerranée: historique,
techniques, législation, production. In: C.F. Boudoresque (ed). Colloque
international sur Paracentrotus lividus et les oursins comestibles. GIS
Posidonie Publ., Marseille. Pp. 335-362.
Le Gall, P., 1987. La pêche des oursins en Bretagne. In: C.F. Boudoresque (ed).
Colloque international sur Paracentrotus lividus et les oursins
comestibles. GIS Posidonie Publ., Marseille. Pp. 311-324.
Le Gall, P., 1990. Culture of echinoderms, In: G. Barnabé (ed.). Aquaculture,
Vol. 1. Ellis Horwood, New York. Pp. 443-462.
Le Gall, P. & D. Bucaille, 1989. Sea urchins production by inland farming. In: N.
De Pauw, E. Jaspers, H. Ackefors & N. Wilkins (eds.). Aquaculture - a
biotechnology in progress. European Aquaculture Society, Bredene,
Belgium. Pp. 53-59.
Leinass, H.P. & H. Christie, 1996. Effects of removing sea urchins
(Strongylocentrotus droebachiensis): stability of the barren state and
succession of kelp forest recovery in the East Atlantic. Oecologia,
105:524-536.
Levitan, D.R., 1988. Density-dependent size regulation and negative growth in
the sea urchin Diadema antillarum Philippi. Oecologia, 76:627-629.
Levitan, D.R. & S.J. Genovese, 1989. Substratum-dependent predator-prey
dynamics: Patch reefs as refuges from gastropod predation. J. Exp. Mar.
Biol. Ecol., 130:111-118.
Liquori, A.M., A. Monroy, E. Parisi & A. Tripiciano, 1981. A theoretical
equation for diauxic growth and its application to the kinetics of the early
development of the sea urchin embryo. Differentiation, 20:174-175.
Lumingas, L.J.L. & M. Guillou, 1994. Growth zones and back-calculation for the
sea urchin Sphaerechinus granularis from the bay of Brest, France. J.
Mar. Biol. Assoc. U.K., 74:671-686.
Malthus, T.R., 1798. An essay on the principal of population. Reedition (1970),
Penguin Books, New York.
201

References
McDonald, P.D. & T.J. Pitcher, 1979. Age-groups from size-frequency data: A
versatile and efficient method of analyzing distribution mixtures. J. Fish.
Res. Board Can., 36:987-1001.
McPherson, B.F., 1965. Contributions to the biology of the sea urchin Tripneustes
ventricosus. Bull. Mar. Sci., 15(1):228-244.
McPherson, B.F., 1968. Contributions to the biology of the sea urchin Eucidaris
tribuloides (Lamarck). Bull. Mar. Sci., 18(2):400-443.
Meidel, S.K. & R.E. Scheibling, 1998. Size and age structure of the sea urchin
Strongylocentrotus droebachiensis in different habitats. In: R. Mooi & M.
Telford (eds). Echinoderms: San Francisco. Balkema, Rotterdam. Pp.
737-742.
Michel, H.B., 1984. Culture of Lytechinus variegatus (Lamarck) (Echinodermata:
Echinoidea) from egg to young adult. Bull. Mar. Sci., 34(2):312-314.
Middleton, D.A.J., S.C. Gurney & J.D. Gage, 1998. Growth and energy allocation
in the deep-sea urchin Echinus affinis. Biol. J. Lin. Soc., 64:315-336.
Miller, R.J. & K.H. Mann, 1973. Ecological energetics of the seaweed zone in a
marine bay on the Atlantic coast of Canada. III. Energy transformations
by sea urchins. Mar. Biol., 18:99-114.
Morgan, L.E., L.W. Botsford, S.R. Wing & B.D. Smith, 2000. Spatial variability
in growth and mortality of the red sea urchin, Strongylocentrotus
franciscanus, in northern California. Can. J. Fish. Aquat. Sci., 57:980-
992.
Mortensen, T., 1950. A monograph of the Echinoidea. 5 vol., Reitzel edit.,
Copenhagen.
Moss, M.L. & M. Meehan, 1968. Growth of the echinoid test. Acta Anat., 69:409-
444.
Motnikar S., P. Bryl & J. Boyer, 1997. Conditioning green sea urchins in tanks :
the effect of semi-moist diets on gonad quality. Bull. Aqua. Assoc.
Canada, 97-1:21-25.
202

References
Muller-Feuga, A., 1990. Modélisation de la croissance des poissons en élevage.
Rapp. Sci. Tech. IFREMER, n° 21.
Munk, J.E., 1992. Reproduction and growth of green sea urchin
Strongylocentrotus droebachiensis (Müller) near Kodiak, Alaska. J.
Shellfish Res., 11(2):245-254.
Murray, J.D., 1993. Mathematical biology. 2 nd ed. Springer-Verlag, Berlin.
Nedelec, H., 1983. Sur un nouvel indice de réplétion pour les oursins réguliers.
Rapp. Comm. Int. Mer Médit., 28(3):149-151.
Nedelec, H., M. Verlaque & A. Dialopoulis, 1981. Preliminary data on Posidonia
consumption by Paracentrotus lividus in Corsica (France). Rap. Comm.
Int. Mer Médit., 27(2):203-204.
Nelder, J.A., 1961. The fitting of a generalization of the logistic curve.
Biometrics, 17:89-110.
Nelder, J.A. & R. Mead, 1965. A simplex algorithm for function minimization.
Comput. J., 7: 308-313.
Nichols, D., A.A.T. Sime & G.M. Bishop, 1985. Growth in populations of the sea
urchin Echinus esculentus L. (Echinodermata: Echinoidea) from the
English Channel and Firth of Clyde. J. Exp. Mar. Biol. Ecol., 86:219-228.
Nocedal, J. & S.J. Wright, 1999. Numerical Optimization. Springer-Verlag, New
York.
Passino, K.M. & S. Yurkovich, 1988. Fuzzy control. Addison-Wesley, Menlo
Park.
Pearse, J.S. & V.B Pearse, 1975. Growth zones in the echinoid skeleton. Amer.
Nat., 15:731-753.
Percy, J.A., 1972. Thermal adaptation in the boreo-arctic echinoid,
Strongylocentrotus droebachiensis (O.F. Muller 1776). I. Seasonal
acclimatization of respiration. Physiol. Zool., 45(4):277-289.
203

References
Pouvreau, S., C. Bacher & M. Héral, 2000. Ecophysiological model of growth
and reproduction of the black pearl oyster, Pinctada margaritifera:
potential applications for pearl farming in French Polynesia. Aquaculture,
186(1-2):117-144.
Preece, M.A. & M.J. Baines, 1978. A new family of mathematical models
describing the human growth curve. Ann. Hum. Biol., 5:1-24.
Rao, C.R., 1965. The theory of least squares when parameters are stochastic and
its application to the analysis of growth curves Biometrika, 52:477-458.
Raymond, B.G. & R.E. Scheibling, 1987. Recruitment and growth of the sea
urchin Strongylocentrotus droebachiensis (Müller) following mass
mortalities off Nova Scotia, Canada. J. Exp. Mar. Biol. Ecol., 108:31-54.
Régis, M.-B., 1969. Premières données sur la croissance de Paracentrotus lividus
Lmk. Téthys, 1(4):1049-1056.
Régis, M.-B., 1979. Croissance négative de l'oursin Paracentrotus lividus
(Lamarck) (Echinoidea-Echinidea). C. R. Acad. Sci. Paris D, 188:355-
358.
Régis, M.-B. & R. Arfi, 1978. Etude comparée de la croissance de trois
populations de Paracentrotus lividus (Lamarck), occupant des biotopes
différents, dans le golfe de Marseille. C. R. Acad. Sc. Paris D, 286:1211-
1214.
Richards, F.J., 1959. A flexible growth function for empirical use. J. Exp. Bot.,
10(29):290-300.
Ricker, W.E., 1979. Growth rates and models. In: Fish physiology, vol. 8.
Academic Press, New York.
Roff, D.A., 1980. A motion for the retirement of the von Bertalanffy function.
Can. J. Fish. Aquat. Sci., 37:127-129.
Rowley, R.J., 1989. Settlement and recruitment of sea urchins
(Strongylocentrotus spp.) in a sea urchin barren ground and a kelp bed:
204

References
are populations regulated by settlement or post-settlement? Mar. Biol.,
100:485-494.
Rowley, R.J., 1990. Newly settled sea urchins in a kelp bed and urchin barren
ground: a comparison of growth and mortality. Mar. Ecol. Prog. Ser.,
62:229-240.
Ruppert, E.E. & R.D. Barnes, 1994. Invertebrate zoology. 6 th ed. Saunders
College Publishers, Philadelphia.
Russell, M.P., 1987. Life history traits and resource allocation in the purple sea
urchin Strongylocentrotus purpuratus (Stimpson). J. Exp. Mar. Biol.
Ecol., 108:199-216.
Russell, M.P., T.A. Ebert & P.S. Petraitis, 1998. Field estimates of growth and
mortality of the green sea urchin, Strongylocentrotus droebachiensis.
Ophelia, 48(2):137-153.
Russell, M.P. & R.W. Meredith, 2000. Natural growth lines in echinoid ossicles
are not reliable indicators of age: a test using Strongylocentrotus
droebachiensis. Invert. Biol., 119(4):410-420.
Sainsbury, K.J., 1980. Effect of individual variability on the von Bertalanffy
growth equation. Can. J. Fish. Aquat. Sci., 37:241-247.
Saito, K., 1992. Japan's sea urchin enhancement experience. In: C.M. Dewees
(ed.). The Management and enhancement of sea urchins and other kelp
bed resources: a Pacific rim perspective. Publication no. T-CSGCP-028,
California Sea Grant College, University of California, La Jolla, CA
92093-0232. Pp. 1-38.
Salski, A., O. Franzle & P. Kandzia (eds), 1995. Fuzzy logic in ecological
modelling. Papers presented at a workshop held in Kiel, Germany, 12-13
October 1993. Ecol. Model., 85:1-97.
Schnute, J., 1981. A versatile growth model with statistically stable parameters.
Can. J. Fish. Aquat. Sci., 38:1128-1140.
205

References
Sellem, F., H. Langar & D. Pesando, 2000. Age et croissance de l'oursin
Paracentrotus lividus Lamarck, 1816 (Echinodermata-Echinoidea) dans le
golfe de Tunis (Méditerranée). Oceanol. Acta, 23(5):607-613.
Sen, A. & M. Srivastava, 1990. Regression analysis. Theory, methods, and
applications. Springer-Verlag, New York.
Shick, J.M., 1983. Respiratory gas exchange in echinoderms. In: M. Jangoux &
J.M. Lawrence (eds). Echinoderm studies, vol. 1. Balkema, Rotterdam.
Pp. 67-110.
Sime, A.A.T., 1981. Growth ring analyses in regular echinoids. Prog. Underwat.
Sci., 7:7-14.
Sime, A.A.T. & G.J. Cranmer, 1985. Age and growth of North Sea echinoids. J.
Mar. Biol. Ass. U.K., 65:583-588.
Sloan, N.A., 1985. Echinoderm fisheries of the world. In: B.F. Keegan & B.D.F.
O'Connor (eds). Proceedings of the fifth international echinoderm
conference, Galway. Balkema, Rotterdam. Pp. 109-124.
Smith, B.D. & L.W. Botsford, 1998. Interpretation of growth, mortality, and
recruitment patterns in size-at-age, growth increment, and size frequency
data. In: G.S. Jamieson & A. Campbell (eds). Proceedings of the North
Pacific workshop on invertebrate stock assessment and management. Can.
Publ. Fish. Aquat. Sci., 125:125-139.
Smith, B.D., L.W. Botsford & S.R. Wing, 1998. Estimation of growth and
mortality parameters from size frequency distributions lacking age
patterns: the red sea urchin (Strongylocentrotus franciscanus) as an
example. Can. J. Fish. Aquat. Sci., 55:1236-1247.
Sokal, R.R. & F.J. Rohlf, 1981. Biometry, 2 nd ed. Freeman and co, New York.
Spirlet, Ch., 1999. Biologie de l'oursin comestible (Paracentrotus lividus):
contrôle du cycle reproducteur et optimalisation de la phase de
remplissage gonadique. PhD Thesis, Université Libre de Bruxelles,
Belgium.
206

References
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1994. Differentiation of the genital
apparatus in a juvenile echinoid (Paracentrotus lividus). In: B. David,
A. Guille, J.-P. Féral & M. Roux (eds). Echinoderms through Time,
Balkema, Rotterdam. Pp. 881-886.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1998a. Reproductive cycle of the
echinoid Paracentrotus lividus: analysis by means of the maturity index.
Invert. Reprod. Develop., 34(1):69-81.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1998b. Optimizing food distribution in
closed-circuit cultivation of edible sea urchins (Paracentrotus lividus:
Echinoidea). Aquat. Living Resour., 11(4):273-277.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 2000. Optimization of gonad growth by
manipulation of temperature and photoperiod in cultivated sea urchin,
Paracentrotus lividus (Lamarck) (Echinodermata). Aquaculture, 185:85-
99.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 2001. Cultivation of Paracentrotus
lividus (Echinodermata: Echinoidea) fed extruded feeds: digestion
efficiency, somatic production and gonadal growth. Aqua. Nutr., 7(2):91-
99.
Spotte, S., 1991. Captive seawater fishes. Science and technology. Wiley & Sons,
New York.
Stickle, W.B. & R. Ahokas, 1974. The effects of tidal fluctuation of salinity on
the perivisceral fluid composition of several echinoderms. Comp.
Biochem. Physiol. A, 47:469-476.
Stone, J.R., 1996. The evolution of ideas: a phylogeny of shell models. Amer.
Nat., 148:904-929.
Strathmann, R.R., 1978. Larval settlement in echinoderms. In F.S. Chia & M.E.
Rice (eds). Settlement and metamorphosis of marine invertebrate larvae.
Elsevier, New York. Pp. 235-246.
207

References
Stumm, W. & J.J. Morgan, 1981. Aquatic chemistry. An introduction
emphasizing chemical equilibria in natural waters, 2 nd ed. Wiley & Sons,
New York.
Sumich, J.L. & J.E. McCauley, 1973. Growth of sea urchin, Allocentrotus
fragilis, off Oregon coast. Pacif. Sci., 27(2):156-167.
Taki, J., 1972. A tetracycline labelling observation on growth zones in the test
plates of Strongylocentrotus intermedius. Bull. Jap. Soc. Sci. Fish.,
38:117-121.
Tanaka, M., 1982. A new growth curve which expresses infinite increase. Publ.
Amakusa Mar. Biol. Lab., 6:167-177.
Tanaka, M., 1988. Eco-physiological meaning of parameters of ALOG growth
curve. Publ. Amakusa Mar. Biol. Lab., 9:103-106.
Tegner, M.J. & P.K. Dayton, 1977. Sea urchin recruitment patterns and
implication of commercial fishing. Science, 196:324-326.
Tegner, M.J. & P.K. Dayton, 1981. Population structure, recruitment and
mortality of two sea urchins (Strongylocentrotus franciscanus and S.
purpuratus) in a kelp forest. Mar. Ecol. Prog. Ser., 5:255-268.
Tegner, M.J. & L.A. Levin, 1983. Spiny lobsters and sea urchins: analysis of a
predator-prey interaction. J. Exp. Mar. Biol. Ecol., 73:125-150.
Tessier, G., 1934. Dysharmonies et discontinuités dans la croissance. Exposés de
biométrie et de statistique biologique, vol. 95. Hermann and Cie, Paris.
Tessier, G., 1948. La relation d'allométrie. Sa signification statistique et
biologique. Biometrics, 4:14-48.
Timmons, T.J. & W.L. Shelton, 1980. Differential growth of largemouth bass in
West Point reservoir, Alabama-Georgia. Trans. Amer. Fish. Soc.,
109:176-186.
Tomassone, R., C. Dervin & J.P. Masson, 1993. Biométrie: modélisation de
phénomènes biologiques. Masson, Paris.
208

References
Turch, B., 1998. A simple shell model: applications and implications. Apex,
13:161-176.
Turner, M.E., Jr., B.A. Blumenstein & J.L. Sebaugh, 1969. A generalization of
the logistic growth. Biometrics, 25:577-580.
Turon, X., G. Giribet, S. Lopez & C. Palacin, 1995. Growth and population
structure of Paracentrotus lividus (Echinodermata: Echinoidea) in two
contrasting habitats. Mar. Ecol. Prog. Ser., 122:193-204.
UNESCO, 1981. Tables océanographiques internationales, Vol. 3. UNESCO
Tech. Pap. Mar. Sci., 39.
Vaïtilingon, D., R. Morgan, Ph. Grosjean, P. Gosselin & M. Jangoux, 2001.
Influence of delayed metamorphosis and food intake on the
perimetamorphic period of the echinoid Paracentrotus lividus. J. Exp.
Mar. Biol. Ecol., 262(1):41-60.
Van Osselaer, Ch., 2001. Differentiation morphométrique et génétique de Helix
pomatia L., 1758 et Helix lucorum L., 1758. PhD Thesis, Université Libre
de Bruxelles, Belgium.
Van Osselaer, Ch. & Ph. Grosjean, 2000. Suture and location of the coiling axis
in gastropod shells. Paleobiology, 26(2):238-257.
Vaughan, D.S. & P. Kanciruk, 1982. An empirical comparison of estimation
procedures for the von Bertalanffy growth equation. J. Cons. Int. Explor.
Mer, 40:211-219.
Verhulst, P.F., 1838. Notice sur la loi que suit la population dans son
accroissement. Cah. Math. Phys., 10:113-121.
von Bertalanffy, L., 1938. A quantitative theory of organic growth (inquiries of
growth laws II). Hum. Biol., 10(2):181-213.
von Bertalanffy, L., 1957. Quantitative laws in metabolism and growth. Quart.
Rev. Biol., 32(3):217-231.
Walford, L.A., 1946. A new graphic method of describing the growth of animals.
Biol. Bull., 90:141-147.
209

References
Webster, S.K. & A.C. Giese, 1975. Oxygen consumption of the purple sea urchin
with special reference to the reproductive cycle. Biol. Bull., 148:165-180.
Weibull, W., 1951. A statistical distribution function of wide applicability. J.
Appl. Mech., 18:293-296.
Williams, C.T. & L.G. Harris, 1998. Growth of juvenile green sea urchins on
natural and artificial diets. In: R. Mooi & M. Telford (eds). Echinoderms:
San Francisco. Balkema, Rotterdam. Pp. 887-892.
Willows, R.I., 1992. Optimal digestive investment: a model for filter feeders
experiencing variable diets. Limnol. Oceanogr., 37(4):829-847.
Winsor, C.P., 1932. The Gompertz curve as a growth curve. Proc. Nat. Acad.
Sci., 18:1-8.
Yamaguchi, M., 1975. Estimating growth parameters from growth rate data.
Oecologia, 20:321-332.
Zar, J.H., 1999. Biostatistical analysis, 4th ed. Prentice Hall, London.
Zimmermann, H.J., 1991. Fuzzy set theory and its applications. 2 nd ed. Kluwer
Academic, Boston.
210

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211

212

Annexes
ANNEXES
Annex I: R code for fitting growth models.
Annex II: dataset of the cohort measured during seven years.
Annex III: abstracts of publications and symposia.
213

Annexes
214

Annex I: R code for fitting growth models
Annexes
Code (as well as dataset presented in annex II) is available at:
http://www.sciviews.org/_phgrosjean/growth/index.htm. This code
runs under the free (GNU Public License) statistical software R, which
is downloadable at: http://cran.r-project.org. It is available for almost
all plateforms (Unixes, Linux, Windows, MacOS). The 'nlrq' package
for nonlinear quantile regression is also downloadable from there.
Rem: LaboKit and ShellAxis used to assist in measurements of sea
urchins are available for free (GPL) at: http://www.sciviews.org.
a. Code for analyzing data and fitting envelope models
Main script file
This script runs a complete analysis of the dataset presented in annex
II, and discussed in Part IV. The dataset is first explored (distribution of
sizes, growth pattern…). Then, quantile regressions are fitted with
traditional models and with the original growth model. Finally, the
envelope model is designed, tested, and fitted on the same dataset.
## Demonstration of using R for analyzing growth data as in the paper:
# A functional growth model with intraspecific competition applied to
# sea urchins. Grosjean, Ph., Ch. Spirlet & M. Jangoux (in preparation)
#
# version 1.0 (30/08/2001)
#
# by Ph. Grosjean (phgrosjean@sciviews.org)
# GNU Public License v. 2 or above at your convenience
# Use at your own risks!
# You need:
# R v. 1.3.0 or above (tested only under Windows, please, report other)
# libraries nls, nlrq, akima (see http://cran.r-project.org)
# files Plividus.txt, GrowthFun.R, nlModels.R
# (see http://www.sciviews.org/_phgrosjean/growth/index.htm)
# Put all files in a common directory
# In R, change current directory to that one
# Enter: source("Growth.R", print.eval=TRUE)
cat("\n\n ===== DEMONSTRATION OF ANALYSIS OF GROWTH DATA =====\n")
# To do: put here a more detailed introduction!!!
library(nls)
library(nlrq)
source("GrowthFun.R")
source("nlModels.R")
215

# If no graphic device currently open, create one for the demo
if (is.null(dev.list())) windows()
# Define a pause function
Pause

Pause()
# Rem: other graphs not used here...
#hist(Pl$sizes[, s

lines(edatq[, 1], predict(edat.025, newdata=list(age=edatq[, 1])), col=4)
legend(1500, 20, c("quantile 0.975", "quantile 0.5 (median)", "quantile 0.025"),
col=c(1,2,4), lty=1, pch=1)
#Rem: for the graph of Fig. 1B in the paper, it is:
#windows(8, 6)
#plot(edatq[, "age"], edatq[, "0.975"], ylim=c(0,65), xlab=expression(paste("Time ",
italic("t"), " in days")), ylab=expression(paste("Diameter ", italic("D"), " in mm")),
pch=2)
#points(edatq[, "age"], edatq[, "0.5"], pch=1)
#points(edatq[, "age"], edatq[, "0.025"], pch=6)
#lines(edatq[, 1], predict(edat.975, newdata=list(age=edatq[, 1])), lty=2)
#lines(edatq[, 1], predict(edat.5, newdata=list(age=edatq[, 1])), lty=1)
#lines(edatq[, 1], predict(edat.025, newdata=list(age=edatq[, 1])), lty=4)
#legend(1700, 18, c("quantile 0.975", "quantile 0.5 (median)", "quantile 0.025"),
lty=c(2,1,4), pch=c(2,1,6))
Pause()
cat("\n ---- Quantile regression with the fuzzy-remanent growth function ----\n\n")
cat("... it takes some time, please, be patient...\n")
# Rem: not able to calculate self-starting conditions with this data set... give
initial plausible values!
# starting values are very important here! We often get stuck in a local minimum or
the algorithm fails!
# Rem: time-scale starts at metamorphosis, and is thus shifted by 30 days (see paper)
edatf.975

plot(edat0q[, "age"], edat0q[, "0.975"], ylim=c(0,65), xlab="Time elapsed from
metamorphosis in days", ylab="Diameter increase in mm", main="Quantile regressions
with SSfuzremOrig1")
points(edat0q[, "age"], edat0q[, "0.5"], col=2)
points(edat0q[, "age"], edat0q[, "0.025"], col=4)
lines(edat0q[, 1], predict(edat0f.975, newdata=list(age=edat0q[, 1])), col=1)
lines(edat0q[, 1], predict(edat0f.5, newdata=list(age=edat0q[, 1])), col=2)
lines(edat0q[, 1], predict(edat0f.025, newdata=list(age=edat0q[, 1])), col=4)
legend(1500, 20, c("quantile 0.975", "quantile 0.5 (median)", "quantile 0.025"),
col=c(1,2,4), lty=1, pch=1)
#Rem: for the graph of Fig.4 in the paper, it is:
#windows(8, 6)
#plot(edat0q[, "age"], edat0q[, "0.975"], ylim=c(0,65), xlab=expression(paste("Time ",
italic("t'"), " in days")), ylab=expression(paste("Diameter increase ", italic("D'"),
" in mm")), pch=2)
#points(edat0q[, "age"], edat0q[, "0.5"], pch=1)
#points(edat0q[, "age"], edat0q[, "0.025"], pch=6)
#lines(edat0q[, 1], predict(edat0f.975, newdata=list(age=edat0q[, 1])), lty=2)
#lines(edat0q[, 1], predict(edat0f.5, newdata=list(age=edat0q[, 1])), lty=1)
#lines(edat0q[, 1], predict(edat0f.025, newdata=list(age=edat0q[, 1])), lty=4)
#legend(1700, 18, c("quantile 0.975", "quantile 0.5 (median)", "quantile 0.025"),
lty=c(2,1,4), pch=c(2,1,6))
Pause()
#Rem: the following code takes very long to run, so it is deactivated
#just remove comment marks to run it...
#cat("\n ---- Calculating parameters estimation for quantiles every 5% step ----\n\n")
#cat("... it takes some time, please, be patient...\n")
#edat0f.pars

SimRes

Thetas2[Thetas2 < 0.002] 0.998]

Annexes
Code of functions required to run the script
These functions perform various data manipulations and implement the
object-oriented R code for fitting envelope models. Utilities to convert
data, to generate artificial datasets and to run simulations are also
provided.
#===============================================================================#
# #
# Envelope Growth Models v. 1.0. #
# #
#===============================================================================#
#
# by Ph. Grosjean, 2001 (phgrosjean@sciviews.org)
#
# Parameters estimation, analyses and simulations of envelope growth models
# including the fuzzy-remanent growth curve presented in:
# Grosjean, Ph., Ch. Spirlet & M. Jangoux, A functional growth model with
# intraspecific competition applied to sea urchins (in preparation).
# Regression methods include nonlinear quantile regression with an interior
# point algorithm of Koenker, and a custom nonlinear quantile regression
# over the whole quantile range 0 -> 1 specifically developped for envelope
# growth models (models that envelop size distributions with time).
#
# This is a free software distributed under the terms of the GNU Public
# License version 2 or above at your convenience (see licence.txt).
#
#This version is still in development. Use at your own risks!
# To run these samples:
# Source this file in R, version 1.3.0 or higher
# Change current directory to the one containing Plividus.txt and Growth.R
# and enter source("Growth.R", print.eval=TRUE) in R
#(see Growth.R for more details)
###==== Basic data manipulation ===========================================
# Data are issued from Plividus.txt and stored in variable 'dat':
# dat is: - col #1: class (1 by 1 mm in test diameter)
# - col #2: mean diameter for each class (ex: class 1 = 0 to 1 mm, mean diam.
= 0.5 mm)
# - cols #3 to 27: counts of individuals in each class at various increasing
ages (Xxxx where xxx is age from fertilisation in days)
## Transformations of raw data
# Sum per column
colsum

# otherwise, the latter function must be overloaded (a better alternative => should be
done later!!)
expfreq

plot(cumsum(dat[,columns[1]])/sum(dat[,columns[1]])*100, type="s",
col=cols[1],...)
for (i in 1:length(columns-1)) {
lines(cumsum(dat[,columns[i]]/sum(dat[,columns[i]])*100),
type="s", col=cols[i])
}
} else { # Relative==FALSE
plot(cumsum(dat[,columns[1]]), type="s", col=cols[1],...)
for (i in 1:length(columns-1)) {
lines(cumsum(dat[,columns[i]]), type="s", col=cols[i])
}
}
}
###=== Basic functions for envelope model =================================
# Data are stored in an EGMData presentation which is a list containing:
# - x: the time values in day, from birthday (or metamorphosis day) being 0
# time is assumed to be known without error (follow up of animal born
# in captivity and tagged for instance).
# - y: the vector of all classes upper bounds (lower bound of first class is
# always 0, and of course, lower bound of following classes are upper
# bounds of the preceeding classes). Classes limits are: ] x ].
# y is the size increase, and is thus Y - Yini where Y is the measured
# size and Yini is the initial size at birth
# - freq: the table of all frequencies of individuals measured at time t and
# whose y is included in the corresponding class y
# - n: the vector of the number of individuals counted in the batch at each
# sampled time. It only correspond to the number of individuals in freq
# if ALL the batch is measured at each sampled time (if possible!).
# - sizes: (facultative) the table of all measured sizes (or its approximation
# as back-calculated from freq table
# - units: a list with time and size units, ex: c("days", "mm")
# - desc: (facultative) a description of the data set
# - info: (facultative) information about sizes. How it was obtained (measured
# or back-calculated (and with which algorithm), and do the data in a
# row correspond to the same animal (individual tagging) or not.
# An example dataset is a whole batch of reared sea urchins Paracentrotus
# lividus grown in aquaculture, and is presented in Grosjean et al (in prep.)
# The next function create the EGMData corresponding to this dataset
# Usage: Pl

EGMData.Sizescalc

for the CI!!!
Annexes
SEMean

### Some additional graphes to visualize raw data
# Plot the mortality curve for the data
plotmort

}
Annexes
# - y, a vector of classes upper bound of length j
# - freq, a matrix of i columns and j rows with
# frequencies in each class at each time
#
# Return a list with:
# - xtable being x repeated along j rows
# - ytable being y repeated along i columns
# - freq as in freq input
# - theta being quantiles for each distribution
# at each class upper bound and at each time
# Rem: does not check sizes of x, y and freq!!!
i

###===Plots for data visualisation=============================================
# Plot an "image" of the frequency table (color of each cell according to
log(frequency))
# Note: one can superpose data points and model lines on this graph using adddual=T.
In this case only, fuzres must be provided
plotimage

plot(x, y, type="l", col=col, lty=lty, xlab="Time (in days)",
ylab="Size (in mm)", main=paste("Growth model for quantile", theta))
}
}
# Plot original points corresponding to a quantile.
# This function requires fuzdata$sizes, obtained using fuzgenerate(ReturnSizes=T)
plotpoints

sizes

}
Annexes
}
results[i, 1:5]

# library. See the R documentation and also:
# Pinherio, J.C. & D.M. Bates, 2000. Mixed-effects models in S and Splus.
# Springer, New York. Appendix C, p. 511-521.
# Since R v. 1.2.3 there is also SSweibull and SSgompertz in nls library
library(nls)
# This is for the artificial data generators
AddError

}
.value
Annexes
dimnames(.grad)

lrc

#{
# # This is adapted from SSasympOrig
# xy

.actualArgs

Exp.ival

SSexpAB

SSallo

}
.value
Annexes
.grad[, "y0"]

.expr4

.expr4

.expr9

.expr9

. The 'nlrq' package for nonlinear quantile regression
Annexes
This package was developed by R. Koenker and Ph. Grosjean and it is
now available as part of the official distribution of the R software.
Description file
Package: nlrq
Version: 0.1-1
Date: 2000/05/14
Title: Nonlinear quantile regression
Author: Roger Koenker ,
Philippe Grosjean
Maintainer: Roger Koenker
Depends: R (>= 1.2.3)
Description: Nonlinear quantile regression routines
License: GPL version 2 or later
URL: http://www.econ.uiuc.edu/~roger/research/nlrq/nlrq.html
This package contains functions and methods for nonlinear quantile regression
coef.nlrq extract coefficients
deviance.nlrq deviance at solution
fitted.nlrq response of the fitted model
formula.nlrq formula used in the nlrq object
nlrq nonlinear quantile regression
nlrq.control construct a control list for using with nlrq
predict.nlrq predict data according to the model
residuals.nlrq extract residuals
summary.nlrq display summary of an nlrq object
tau.nlrq quantile used in the nlrq object
Online manual pages
nlrq package:nlrq R Documentation
Function to compute nonlinear quantile regression estimates
Description:
Usage:
This function implements an R version of an interior point method
for computing the solution to quantile regression problems which
are nonlinear in the parameters. The algorithm is based on
interior point ideas described in Koenker and Park (1994).
nlrq(formula, data=parent.frame(), start, tau=0.5, control, trace=FALSE)
Arguments:
formula: formula for model in nls format; accept self-starting models
data: an optional data frame in which to evaluate the variables in
`formula'
start: a named list or named numeric vector of starting estimates
tau: a vector of quantiles to be estimated
control: an optional list of control settings. See `nlrq.control' for
246

Annexes
the names of the settable control values and their effect.
trace: logical value indicating if a trace of the iteration progress
should be printed. Default is `FALSE'. If `TRUE'
intermediary results are printed at the end of each
iteration.
Details:
Value:
An `nlrq' object is a type of fitted model object. It has methods
for the generic functions `coef' (parameters estimation at best
solution), `formula' (model used), `deviance' (value of the
objective function at best solution), `print', `summary',
`fitted' (vector of fitted variable according to the model),
`predict' (vector of data points predicted by the model, using a
different matrix for the independent variables) and also for the
function `tau' (quantile used for fitting the model, as the tau
argument of the function). Further help is also available for the
method `residuals'.
A list consisting of:
m: an `nlrqModel' object similar to an `nlsModel' in package nls
data: the expression that was passed to `nlrq' as the data
argument. The actual data values are present in the
environment of the `m' component.
Author(s):
Based on S code by Roger Koenker modified for R and to accept same
models as nls by Philippe Grosjean.
References:
See Also:
Examples:
Koenker, R. and Park, B.J. (1994). An Interior Point Algorithm for
Nonlinear Quantile Regression, Journal of Econometrics, 71(1-2):
265-283.
`nlrq.control' , `residuals.nlrq'
# Example using model defined in the nls library
library(nls)
# build artificial data with multiplicative error
Dat

nlrq.control package:nlrq R Documentation
Set control parameters for nlrq
Description:
Usage:
Annexes
Set algorithmic parameters for nlrq (nonlinear quantile regression
function)
nlrq.control(maxiter=100, k=2, big=1e+20, eps=1e-07, beta=0.97)
Arguments:
maxiter: maximum number of allowed iterations
k: the number of iterations of the Meketon algorithm to be
calculated in each step, usually 2 is reasonable,
occasionally it may be helpful to set k=1
big: a large scalar
eps: tolerance for convergence of the algorithm
beta: a shrinkage parameter which controls the recentering process
in the interior point algorithm.
See Also:
`nlrq'
residuals.nlrq package:nlrq R Documentation
Return residuals of an nlrq object
Description:
Usage:
Get residuals from an nlrq (nonlinear quantile regression) object
residuals.nlrq(nlrqObject, type = c("response", "rho"), ...)
Arguments:
nlrqObject: an `nlrq' object as returned by function `nlrq'
type: the type of residuals to return: "response" is the distance
between observed and predicted values; "rho" is the weighted
distance used to calculate the objective function in the
minimisation algorithm as tau * pmax(resid, 0) + (1 - tau) *
pmin(resid, 0), where resid are the simple residuals as above
(with type="response").
See Also:
`nlrq'
R code
###===Nonlinear quantile regression with an interior point algorithm===
# see: Koenker, R. & B.J. Park, 1996. An interior point algorithm
# for nonlinear quantile regression. J. Econom., 71(1-2): 265-283.
# adapted from nlrq routine of Koenker, R.
# to be compatible with R nls models
# by Ph. Grosjean, 2001 (phgrosjean@sciviews.org)
# large parts of code are reused from the nls library of R v. 1.2.3
248

# It is made available under the terms of the GNU General Public
# License, version 2, or at your option, any later version
#
# This program is distributed in the hope that it will be
# useful, but WITHOUT ANY WARRANTY; without even the implied
# warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR
# PURPOSE. See the GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public
# License along with this program; if not, write to the Free
# Software Foundation, Inc., 59 Temple Place - Suite 330, Boston,
# MA 02111-1307, USA
# TO DO:
# - nlrq should return a code 0 = convergence, 1 = lambda -> 0, etc..
# - Extensive diagnostic for summary() (Roger, what would you propose?)
# - Calculate with a list of tau values at once (currently accept 1 value)
# - When providing several tau values, allow calculating a single value
# for one or more parameters across all models fitted to all tau values
# ...but I have another idea for doing that more efficiently.
"nlrq.control"

gradSetArgs[[2]]), call("[", gradSetArgs[[1]], gradSetArgs[[2]],
gradSetArgs[[2]]), call("[", gradSetArgs[[1]], gradSetArgs[[2]],
gradSetArgs[[2]], gradSetArgs[[3]]), call("[", gradSetArgs[[1]],
gradSetArgs[[2]], gradSetArgs[[2]], gradSetArgs[[3]],
gradSetArgs[[4]]))
getRHS.varying

}
": ", format(getPars()), "\n"), Rmat = function() qr.R(QR),
predict = function(newdata = list(), qr = FALSE) {
Env

model.step

object$m$deviance()
"tau.nlrq"

Annexes
254

Annex II: dataset of the cohort measured during seven years
Age
(days)
Annexes
210
306
364
456
546
636
726
818
911
1006
Size class
(mm)
Count of the no. of individuals in each size class
0 – 1 65 5
1 – 2 142 30 11
2 – 3 114 29 22
3 – 4 115 46 19
4 – 5 65 45 27 3
5 - 6 62 50 32 3
6 - 7 31 47 42 10
7 - 8 34 56 46 13
8 - 9 31 31 44 18 2
9 - 10 20 27 38 17 4
10 - 11 17 22 28 28 6
11 - 12 18 19 22 26 10 1
12 - 13 7 21 25 33 22 2
13 - 14 4 20 16 33 31 8 2
14 - 15 12 16 21 30 13
15 - 16 10 15 29 35 23 3
16 - 17 12 23 37 29 26 10 1
17 - 18 10 15 26 35 45 6 1
18 - 19 11 13 26 25 29 12
19 - 20 4 11 19 31 28 20 4 1
20 - 21 5 16 25 27 25 6
21 - 22 6 22 14 27 26 8 2
22 - 23 10 19 21 29 36 19 1 1 1
23 - 24 5 13 27 20 36 15 5
24 - 25 15 19 22 35 31 6 1
25 - 26 4 17 19 10 23 15 2 1
26 - 27 6 7 21 23 28 18 7
27 - 28 9 8 21 27 28 15 5 2 1 1
28 - 29 5 12 6 18 29 12 6 1
29 - 30 8 3 5 18 27 23 13 4 1
30 - 31 3 8 8 16 19 29 14 9
31 - 32 3 7 8 7 17 22 20 16 1 1
32 - 33 2 6 9 4 17 33 18 9 2
33 - 34 8 4 7 15 23 16 16 3 1
34 - 35 5 5 7 12 17 16 9 3
35 - 36 7 4 10 9 16 16 13 10 1
36 - 37 4 10 9 8 19 26 17 10 1 1
37 - 38 3 9 8 8 11 28 15 8 1 1 1
38 - 39 5 7 10 15 20 13 9 5 1 1 1 1 1 1 1
39 - 40 6 10 12 29 28 10 5 1
40 - 41 2 11 6 13 9 21 16 9 10 1
41 - 42 1 1 12 3 14 22 12 8 5 8
42 - 43 2 6 16 10 19 37 19 8 3 1
43 - 44 8 11 18 14 25 16 7 1 5 1 1
44 - 45 1 7 12 17 17 30 17 15 15 4 6 2
45 - 46 1 13 10 12 21 31 8 13 6 3 3 2 1
46 - 47 2 10 9 13 19 28 16 15 11 7 2 1 2 1 1
47 - 48 7 15 10 12 24 28 16 16 8 2 4 3 3 1
48 - 49 3 17 8 20 30 23 9 13 9 6 3 2 1
49 - 50 2 7 11 14 22 33 34 17 12 10 1 2 2
50 - 51 1 2 14 5 16 24 22 19 21 8 5 3 1 2
51 - 52 2 2 8 12 15 16 22 26 25 10 6 4 1 2
52 - 53 15 8 17 16 22 28 14 8 7 5 2
53 - 54 1 10 13 10 6 14 13 16 12 1 4 3
54 - 55 1 7 11 12 14 15 21 19 14 13 8 12
55 - 56 1 1 12 13 13 10 17 12 10 12 13 5
56 - 57 1 8 11 8 8 9 9 4 7 7 11
57 - 58 4 7 7 9 9 6 6 8 13 8
58 - 59 1 4 2 6 5 5 6 6 7 5
59 - 60 2 2 4 10 11 4 3 5 5 5
60 - 61 1 2 3 7 1 3 4 2
61 - 62 1 5 4 5 3 1 2 3
62 - 63 1 1 2 1 1 1 1 1
63 - 64 1
64 - 65 1 2 1 2 1
65 - 66 1 2 1
66 - 67 1
Total no. 725 507 491 467 461 437 403 386 387 371 324 315 309 275 233 223 221 137 92 85 82 67
1097
1188
1281
1369
1461
1551
1642
1825
2008
2190
2377
255
2554

Annexes
256

Annex III: abstracts of publications and symposia
Annexes
Since my scientific activity was very diversified, a part of my work
was included in this thesis. Here are the abstracts of all publications and
symposia where I have participated.
a. International journals
Vaïtilingon, D., R. Morgan, Ph. Grosjean, P. Gosselin & M. Jangoux.
Influence of delayed metamorphosis and food intake on the
perimetamorphic period of the echinoid Paracentrotus lividus. J. Exp.
Mar. Biol. Ecol., 262(1):41-60.
ABSTRACT: Effect of delayed metamorphosis and food ration on late
(competent) larvae and postlarvae of Paracentrotus lividus were
investigated. Metamorphosis of competent larvae was either not delayed or
delayed from 1 up to 4 days. Larvae were starved or submitted to two
different food rations of the algal species Phaeodactylum tricornutum.
Larvae during the prolonged competence period and the resulting
postlarvae were characterized by: (1) the size of the larval body, (2) the
size of the rudiment, (3) the rate of metamorphosis, (4) the size of
postlarvae 24 h after metamorphosis, (5) the rate of opening of mouth and
anus, (6) the rate of survival, and (7) the growth rate of early
postmetamorphic individuals. Both the width of the larval body and the
diameter of the echinus rudiment grew in competent larvae that were fed.
Unfed larvae did not grow. There was no significant difference in growth
between the two food rations. The rate of metamorphosis was higher with
larvae that metamorphosed soon after they became competent. Lower
capacity of larvae to metamorphose during the delay period was associated
with treatments yielding a greater larval width and rudiment diameter
during the same period. Postlarval development was affected by a delayed
metamorphosis treatment inflicted on competent larvae before
257

Annexes
metamorphosis. Acquisition of exotrophy happened earlier in postlarvae
issued from prolonged competent larvae whatever the larval food rations.
The delay treatment negatively affected the development of the digestive
tract through it positively affected the growth of early postmetamorphic
individuals during the first 6 days following metamorphosis. However,
selective mortality occurred afterwards as bigger individuals died
preferentially.
KEYWORDS: Larvae, metamorphosis, Paracentrotus lividus, plutei,
sea urchin.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 2000. Cultivation of
Paracentrotus lividus (Echinodermata: Echinoidea) fed extruded feeds:
digestion efficiency, somatic production and gonadal growth.
Aquaculture Nutrition, 7(2):91-99.
ABSTRACT: This study assessed the use of extruded feeds, in the
form of pellets, for growing of the echinoid Paracentrotus lividus within a
closed culture system. Two feeds types, one with soybean protein, the
other with both soybean and fish protein were compared to dried Lessonia
sp. and fresh Laminaria sp. as food sources. Pellets present a very high
conversion efficiency (about 80%) against about 50% for Laminaria and
35% for Lessonia. However, since pellets are less absorbed, somatic
growth is statistically equivalent for the sea urchins fed with pellets and
Laminaria: between 2 and 2.2%.g of soma.day -1 . Sea urchins fed pellets
produced significantly more gonadal tissue in a shorter time. Resulting in a
gonadal index twice higher (6.5%) than Laminaria (3%) in the second
month of the experiment. Dry Lessonia does not promote gonadal growth.
This study shows that extruded feeds are well assimilated by P. lividus and
promote both somatic growth and production of gonadal tissue.
258

Annexes
KEYWORDS: Sea urchin, aquaculture, artificial food, somatic
growth, roe, digestion.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 2000. Optimization of gonad
growth by manipulation of temperature and photoperiod in cultivated
sea urchin, Paracentrotus lividus (Lamarck) (Echinodermata).
Aquaculture, 185:85-99.
ABSTRACT: A starvation and then feeding method was developed to
produce about 100% marketable sea urchins, Paracentrotus lividus, in 3 ½
months. This method is needed because the reproduction cycle is
desynchronized in the conditions imposed during the somatic growth stage
in land-based closed systems. The major advantages of starving the
animals are resetting the reproductive cycle to the spend stage (gonads
almost devoid of sexual cells) and stressing the individuals so that they
mobilize and restore the nutritive phagocytes, filling them with nutrients.
Batches of sea urchins starved for 2 months beforehand were fed ad
libitum for 45 days with enriched food under eight combinations of four
temperatures (12°C, 16°C, 20°C and 24°C) and two photoperiods (9 and
17 h daylight). In our system, the best combination was 24°C and 9 h
daylight for growth as well as for gonad quality. The gonadal indices
obtained (in dry weight) were over 9% at 16°C and over 12% at 24°C,
which are better than what is found in the field for this population.
KEYWORDS: Gonad, growth, temperature, photoperiod, sea urchin,
Paracentrotus lividus.
Van Osselaer, Ch. & Ph. Grosjean, 2000. Suture and location of the
coiling axis in gastropod shells. Paleobiology, 26(2):238-257.
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ABSTRACT: The general allometric equations for the logarithmic
helicospiral can fit many extraconical shapes, but the isometric conditions
traditionally used limits study only to conical growth. We present evidence
to show that in real gastropod shells, the logarithmic helicospiral equations
fit the suture. Poor location of the coiling axis and/or an inappropriate pole
for the logarithmic helicospiral has often led to the rejection of this model.
The differences between the errors associated with measurements or
previously available models, are discussed. Two methods, based on suture
trace measurements, are proposed to locate the coiling axis both in apical
and lateral views. The first is a graphical method based on an elementary
property of the logarithmic spiral. The second, computational method is
based on iterative reprojections of the suture. It is shown that the
protoconch and the teleloconch must be treated separately. The precision
of the new methods (especially the computing method) enables deviations
from logarithmic helicospiral trajectory to be identified and differentiated
from irregularities of the shell and sequential growth phases. Application
of these methods may be useful not only for other gastropod
morphological features, but also for other taxa such as brachiopods and
other molluscs.
KEYWORDS: Coiled shell, coiling axis, gastropod, mollusc,
morphometry, suture.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1998a. Reproductive cycle
of the echinoid Paracentrotus lividus: analysis by means of the maturity
index. Invert. Reprod. Develop., 34(1):69-81.
ABSTRACT: The gonad maturity index cycles of the echinoid
Paracentrotus lividus and their relations with environmental abiotic
parameters are assessed after 2 years of observation in southern Brittany,
France. The gonadal cycle is briefly described and eight gonadal stages are
characterized. The annual cycle, the time of spawning and the period of
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gonadal growth are well established, suggesting they are controlled
externally. The reproductive cycle has three main phases: the growing
phase (late autumn and winter) when gonads accumulate reserve material;
the maturation phase (spring and early summer) in which gametogenesis
then spawning take place; and the spent/regenerating phase when relict
gametes are resorbed by the nutritive phagocytes, the gonads being
virtually devoid of sexual cells. The maturity index based on the
histological diagnosis of gonads and the use of circular data and polar
graphical representation make it possible to reliably determine the
spawning period, the rate of gametogenesis and the synchronization of
males and females among the echinoid population. From this analysis, we
can reasonably say that the gonadal cycle (represented by the gonad
index), the rate of gametogenesis, and the end of the spawning period are
influenced by temperature whereas the first spawning event appears to be
triggered by day length.
KEYWORDS: Echinoid, reproduction, maturity index, gonadal cycle,
abiotic parameters.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1998b. Optimizing Food
distribution in closed-circuit cultivation of edible sea urchins
(Paracentrotus lividus: Echinoidea). Aquat. Living Resour., 11(4):273-
277.
ABSTRACT: In the framework of echinoid cultivation, whose
objective is to succeed in continuously producing large amounts of edible
sea urchins (Paracentroutus lividus) under controlled conditions
(aquaculture), gonadal growth is to be optimized. Among the various
parameters influencing the production of roe, the quantity of food
distributed was tested for optimization. After a 1-month fast, echinoids
were fed artificial food pellets (enriched in soybean and fish proteins) for
different periods of time over 48 h, the food thus being available ad
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libitum for 8, 16, 24, 32, 40 and 48 h; the cycles were repeated for a
month. The results show that the quantity of food intake and the gonad
index peak after about 35 h of food availability. This suggests food should
be distributed discontinuously for optimal gonad production and minimal
waste.
KEYWORDS: Aquaculture, food ration, gonad growth, artificial diet,
sea urchin.
Grosjean, Ph., Ch. Spirlet, P. Gosselin, D. Vaïtilingon & M. Jangoux,
1998. Land-based closed cycle echiniculture of Paracentrotus lividus
(Lamarck) (Echinoidea: Echinodermata): a long-term experiment at a
pilot scale. J. Shellfish Res., 17(5):1523-1531.
See Part I.
Grosjean, Ph., Ch. Spirlet & M. Jangoux, 1996. Experimental study of
growth in the echinoid Paracentrotus lividus (Lamarck, 1816)
(Echinodermata). J. Exp. Mar. Biol. Ecol., 201:173-184.
See Part III.
b. Reports and other publications
Jangoux, M., Ph. Grosjean, D. Vaïtilingon, C. Cam, J. Cosson, Ch.
Billard, D. Bucaille, J.M. Ouin, C. Rebours, N.T. Hagen, C. Solberg &
H.H. Ludvigsen, 2000. Biology of sea urchins under intensive
cultivation (closed-cycle echiniculture). European Contract FAIR
CT96-1623 BFN, final report. 163 pp.
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EXECUTIVE SUMMARY: The ultimate objective of the project is to
control every life stage of the most valuable species of European edible sea
urchins (Paracentrotus lividus and Strongylocentrotus droebachiensis)
under intensive cultivation (closed-cycle echiniculture) to produce high
quality gonads (roe, i.e., the edible part of the animal) at a pilot scale. The
obstacles that prevent the intensification of echiniculture have been clearly
identified: (1) post-settlement survival and growth rate need to be
improved, and (2) the carrying capacity of the rearing structures needs to
be increased by bypassing main limiting factors, i.e., depletion in dissolved
carbonates and accumulation of carbonic acid. Moreover, the quality
control of gonads and optimization of gonad growth are key factors that
have to be addressed. The proposed work aims to investigate aspects of the
biology of cultivated sea urchins related to these obstacles, to finalize
technical enhancements of the cultivation procedure in either eliminating
or bypassing these obstacles, and to adapt the rearing method presently
used for Paracentrotus lividus to Strongylocentrotus droebachiensis.
Grosjean, Ph., Ch. Spirlet & M. Jangoux, 1999. Comparison of three
body-size measurements for echinoids. In: M.D. Candia Carnevali &
F. Bonasoro (eds). Echinoderm Research 1998, Balkema, Rotterdam.
Pp. 31-35.
See Part II.
Jangoux, M., P. Gosselin, Ph. Grosjean, M. Larsonneur, Ch. Spirlet,
D. Bucaille, M. Catoira Gomez, 1996. Sea-urchin cultivation.
European Contract FAR AQ 2.530 BFE, final report. 103 pp +
annexes.
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EXECUTIVE SUMMARY: The aim of the research is to succeed in
continuously producing large amount of edible sea urchins with high
gonadal productivity under controlled conditions (aquaculture). The
selected species is Paracentrotus lividus (viz. the edible echinoid species
from Europe). The research focuses on the following aspects:
a. Investigation on metamorphic events, with a special interest in
the morphological changes affecting metamorphic larvae and in
the metamorphosis inducing factors (this approach needs to
perform routinely mass-cultivation of larvae);
b. Optimization of juveniles' somatic growth and adults' gonadal
productivity under rearing conditions;
c. Study of reproductive periodicities in field and cultivated
individuals (investigations on parameters responsible for the
cycling of reproduction and the duration of the spawning period)
and attempt for a continuous reproduction under aquaculture
conditions.
Spirlet, Ch., Ph. Grosjean & M. Jangoux, 1994. Differentiation of the
genital apparatus in a juvenile echinoid (Paracentrotus lividus). In: B.
David, A. Guille, J.-P. Féral & M. Roux (eds). Echinoderms through
Time, Balkema, Rotterdam. Pp. 881-886.
ABSTRACT: Gonad and genital pore development was observed on
field and cultivated juveniles of the echinoid Paracentrotus lividus. The
gonad condition was evaluated by counting the number of acini per gonad
following dissection. Presence of the genital pores was determined for
each individual, plate by plate, after partial digestion of the tissues.
Progressive and relative development was determined. The gonads first
appear as a filament which starts to bud and develop acini that eventually
fill up with genital material. The pores are pierced from the inside out
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when the gonads have reached a certain growth stage. Both gonads and
pores do not develop simultaneously but in a certain order. In addition,
statistical analysis shows that size has more influence on the condition of
the genital apparatus than age.
KEYWORDS: Echinoid, gonads, growth, reproduction, development.
c. International symposia
Green Sea Urchin Workshop, Moncton, Canada, 2000. Talk. Ph.
Grosjean & M. Jangoux. Growth of the sea urchin Paracentrotus
lividus: model and optimization.
Inline: http://crdpm.cus.ca/oursin/pdf/gros.pdf
(last consulted Sept. 8 th 2001).
ABSTRACT: Echiniculture, or sea urchin aquaculture, is more and
more considered as an alternative or a complement to fisheries but it is not
implemented yet on a commercial scale. This is because rearing methods
still have to be optimized. We developed a mathematical model to simulate
and predict sea urchins' production according to various rearing methods
and exploitation strategies. Simulations using this model demonstrate how
the variation of production has a complex and sometimes counterintuitive
relationship with both the rearing method and the exploitation strategy.
Intraspecific competition and spread in sizes in reared batches are taken
into account in the model. Hence, an accurate estimation of the time
required to grow a whole cohort of reared echinoids to the market size is
obtained. This tool allows a rapid determination of best rearing methods
and should speed up the optimization process. At a latter date, this model
could help in rationalizing stock management in future sea urchins farming
activities. By applying this model to field sea urchins populations, it could
also help in the establishment of sustainable fishery policies.
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KEYWORDS: Somatic growth, dynamic model, aquaculture,
intraspecific competition.
5 th European Echinoderm Colloquium, Milano, 1998. Poster. Ph.
Grosjean, Ch. Spirlet & M. Jangoux. Comparison of three body-size
measurements for echinoids and their use in growth and
gonadosomatic calculations.
See Part II.
Aquaculture '98, Las Vegas, 1998. Talk in a special session. Ph.
Grosjean, Ch. Spirlet & M. Jangoux. Is land-based closed cycle
echiniculture (sea urchins aquaculture) a viable alternative to fisheries
today?
ABSTRACT: Today, most world sea urchins fisheries have to deal
with overexploitation or yields drop problems. Better management of
exploited field populations and/or aquaculture is more and more
considered as necessities to sustain sea urchins’ production in the near
future. In this context, we evaluate here the potentials of land-based closed
cycle echiniculture.
A long-term experiment with the edible violet sea urchin
(Paracentrotus lividus) has been done at a pilot scale in France. The
process used allows total independence against natural resources, since the
whole biological cycle of the echinoids is under control (closed cycle
echiniculture) and all activities are performed on land. Also, a method has
been set up to gain control over the reproductive cycle of these animals
and to produce marketable individuals all year long.
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Overall conclusions of this experiment reveal great potentials, but also
point out some pitfalls that remain to be eliminated before pretending for
profitability. The most critical pitfalls identified are (1) poor control of
extremely variable growth rates due to intraspecific competition, (2) poor
control on inorganic carbon in closed or semi-closed systems due to a high
demand in carbonates for skeletogenesis and (3) needs for increased
quality of gonads (the edible part of the urchins) thanks to a specific
artificial diet that remains to be formulated.
One important aspect comes to light: land-based closed cycle
echiniculture should have a very low impact on other mariculture or
touristic activities that usually compete strongly for space on the coastline
in many places. This should be a major advantage considering tomorrow’s
aquaculture diversification.
KEYWORDS: Sea urchin, Paracentrotus lividus, aquaculture, larval
culture, metamorphosis, growth, roe enhancement.
3 th International Symposium on Nutritional Strategies and
Management of Aquaculture Waste, Porto, 1997. Talk. Ph. Grosjean,
Ch. Spirlet, J. M. Lawrence & M. Jangoux. Optimizing somatic growth
of the edible sea urchin (Paracentrotus lividus Lmk) (Echinodermata:
Echinoidea) in closed-circuit cultivation with artificial diet.
ABSTRACT: The closed-circuit cultivation of sea urchins offers the
opportunity to optimize their growth by controlling the rearing parameters,
but the question whether water pollution would result from the waste
produced is critical. Little is known about the requirements of cultivated
sea urchins in terms of food composition. The effect of two prepared feeds
(soybeans and soybeans-fish pellets) versus fresh and dried kelp (the
natural food of Paracentrotus lividus) on feeding, digestion and somatic
growth has been investigated under semi-intensive cultivation. The total
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amount of waste produced at different levels of feeding, digestion, and
assimilation has been measured. Both prepared feeds are used as
efficiently as fresh kelp for somatic growth, but wastes are reduced by
20% due to a much higher conversion efficiency. These preliminary results
suggest that a drastic improvement of feeding strategies in the culture of
sea urchins would be obtained soon by appropriate formulations of
prepared feeds. On-land aquaculture of sea urchins with prepared feeds
provides a means of controlling the wastes produced. It would prevent
environmental pollution and would allow recovery of the large amounts of
material still rich in organic material that may be used for other purposes.
KEYWORDS: Artificial food, digestion, somatic growth, sea urchin,
aquaculture.
9 th International Echinoderm Conference, San Francisco, 1996. Talk
in a special session. Ph. Grosjean, Ch. Spirlet & M. Jangoux. Closedcircuit
cultivation of the edible sea urchin Paracentrotus lividus:
optimization of somatic growth through the control of abiotic
environment.
ABSTRACT: Among the technological choices to develop
aquaculture of new species, open-sea versus "on-land" cultivation is a
major one. On-land based systems are more expensive but offer the
possibility to control the environmental conditions, possibly leading to
better performance. In the case of echinoderms, ecophysiological
responses are insufficiently understood to decide at the present time which
is the best strategy. To investigate this crucial question, one should (1) set
up a good cultivation system, (2) develop an experimental methodology
adapted to the specificity of echinoderm biology, and (3) quantify the
responses of the animals against gradients of environmental parameters.
As an illustration of the promising perspectives this approach offers, the
case of Paracentrotus lividus is discussed.
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A pilot system has been set up to control all life stages of reared sea
urchins, from fertilization to gonad filling; the used protocol is being
carefully standardized.
A rapid (3 weeks) but accurate method has been developed to measure
feeding, digestion, and somatic growth of reared sea urchins under various
environmental conditions. Both technical and statistical improvements
have been made to increase the significance of the results.
Two of the most important abiotic factors (photoperiod and
temperature) have been investigated and let us to model their effects on sea
urchins. Photoperiod has an impact on the feeding rate, but not on
absorption or somatic growth. Optimal temperature for juveniles appears
to be higher than for adults (respectively 23-24°C and 19°C). Moreover,
juveniles are more sensitive to departure from this optimum. Hence, a
strict control of temperature is a more critical issue for juveniles than for
adults.
Integration of our results in a wider model concerning the entire
rearing structure shows that the apparent food conversion efficiency results
in several complex phenomena: feeding and digestion of course, but also
degradation of food that in addition depends on temperature. In cultivation,
the highest productivity is obtained by making a compromise between a
high somatic growth at optimal temperature and a high apparent food
conversion efficiency at a lower temperature.
KEYWORDS: Echinoid, aquaculture, food conversion, temperature,
photoperiod.
4 th European Echinoderm Colloquium, London, 1995. Poster. Ph.
Grosjean, Ch. Spirlet & M. Jangoux. Establishment and presumed
causes of the multimodal distribution commonly observed in cultivated
populations of juvenile Paracentrotus lividus.
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ABSTRACT: Multimodal distribution (i.e., few individuals growing
very fast and few individuals growing very slowly among an originally
homogeneous group of sea urchins P. lividus of the same strain) is often
observed in controlled cultivation. The splitting of this group into
homogeneous size-classed subgroups induces an increased growth of the
smaller individuals that achieve the same size than the others in 3 months.
This indicates that the smaller animals are not genetically less productive
and suggests they are inhibited in their growth due to some environmental
constraints.
KEYWORDS:
competition, growth.
Echinoid, population dynamics, intraspecific
3 rd European Aquaculture Symposium, Bordeaux, 1994. Poster. Ph.
Grosjean, Ch. Spirlet & M. Jangoux. First approach of the
performances of a closed-circuit sea urchin rearing structure.
ABSTRACT: Within the context of an ECC financed research
program on echinoid cultivation which objective is to succeed in
continuously producing large amounts of edible sea urchins
(Paracentrotus lividus) under controlled conditions (aquaculture), the
performances of a closed-circuit rearing structure was tested. The rearing
structure consists of toboggans measuring 4 m long and 60 cm wide on 3
levels, the whole overhanging a reserve/settling tank of the same length, 80
cm wide and 80 cm high. The sea urchins are placed on the toboggans in 5
to 10 cm running seawater. A 2.5 months follow-up of 2 structures do not
show any significant increase of the biomass, mortality being barely
compensated. An assessment of the quality of the water, done
simultaneously reveals that CO2 is continuously oversaturated from 300 to
400 %. This factor could be likely the cause of the poor growth of the
echinoids.
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KEYWORDS: Sea urchin, aquaculture, closed-circuit system, growth,
CO2.
8 th International Echinoderm Conference, Dijon, 1993. Poster. Ph.
Grosjean & M. Jangoux. Effect of light on feeding in cultivated
echinoids (Paracentrotus lividus).
ABSTRACT: Within the context of a research on sea urchin
cultivation (Paracentrotus lividus), the effect of light on the amount of
food ingested and of faeces produced per echinoid per day has been
investigated. Three sets of 20 adults were subjected to particular light
conditions (constant light, constant darkness or 12 hours of light per day)
for 6 days. Measurements were done for each of the 60 investigated
echinoids. Calculation of the mean daily feeding and absorption rates for
each set of individuals indicates that the highest values were obtained for
echinoids at constant darkness and the lowest for echinoids subjected to
light/darkness alternation (values for the three sets of individuals are
significantly different).
KEYWORDS: Echinoid, feeding rate, absorption rate, light,
photoperiod.
271