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

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
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