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

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
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