1.7.6. Growth

Individuals of a given school are assumed to grow in size and weight at a given time only when the amount of food they ingested fulfill maintenance requirements, i.e., only when their predation efficiency at that time is greater than the predation efficiency ensuring body maintenance of school.

\[ \begin{align}\begin{aligned}G(s, a) = M_{\Delta}(s, a) \times \frac{S_R(s, a) - C_{S_R}(s)} {1 - C_{S_R}(s)}\ if\ S_R(s, a) \ge C_{S_R}\\G(s, a) = 0\ if\ S_R(s, a) < C_{S_R}\end{aligned}\end{align} \]

with \(C_{S_R}\) is the critical predation efficiency and \(S_R\) the predation success rate of the school.

\(M_{\Delta}(s, a) = \lambda \times \Delta L(a)\) is the maximum growth rate at age \(a\) and for species \(s\).

\(\Delta L(a) = L(a + 1) - L(a)\) is the mean length increase determined from a growth function (Von Bertalanffy or Gompertz growth function), while \(\lambda\) is a factor that allows to control the maximum length at a given age.

Table 1.31 Growth parameters

growth.java.classname.sp#	Class name of the age to length conversion
predation.efficiency.critical.sp#	Critical predation success (\(C_{S_R}\))
species.delta.lmax.factor.sp#	\(\lambda\) (default = 2)

1.7.6.1. Von Bertalanffy growth

When the growth.java.classname.sp# is equal to fr.ird.osmose.process.growth.VonBertalanffyGrowth.java, a von Bertalanffy growth function is used. It is the defaut one.

\[ \begin{align}\begin{aligned}L(a) = L_{egg}\ if\ a=0\\L(a) = L_{egg} + (L_{thres} - L_{egg}) \times \left(\frac{a}{a_{thres}}\right)\ if\ a>0\ \&\ a<a_{thres}\\L(a) = L_{\infty} \times \left(1 - \exp^{-K\left(age - t_0\right)} \right)\ else\end{aligned}\end{align} \]

with

\[L_{thres} = min\left[L_{egg}, L_{\infty} \times \left(1 - \exp^{-K\left(a_{thres} - t_0\right)}\right) \right]\]

A Von Bertalanffy model is used to calculate mean length increase above a threshold age \(a_{thres}\) determined for each HTL group from the literature. Below \(a_{thres}\), a simple linear model is used. The rationale behind this is that Von Bertalanffy parameters are usually estimated from data excluding youngs of the year or including only very few of them. Assuming a linear growth between age 0 and \(a_{thres}\) ensures a more realistic calculation of mean length increases for early ages of HTL groups ([Travers et al., 2009]).

Fig. 1.9 Von Bertalanffy growth curve

Table 1.32 Von Bertalanffy parameters

species.linf.sp#	\(L_{inf}\) (cm)
species.k.sp#	\(K\)
species.t0.sp#	\(t_0\)
species.vonbertalanffy.threshold.age.sp#	\(a_{thres}\) (years, default=1 year)

1.7.6.2. Gompertz growth

When the growth.java.classname.sp# is equal to fr.ird.osmose.process.growth.GompertzGrowth.java, a Gompertz growth function is used.

\[ \begin{align}\begin{aligned}L(a) = L_{egg}\ if\ a=0\\L(a) = L_{start} \times exp^{K_e \times a}\ if\ a>0\ \& a<a_{exp}\\L(a) = L_{exp} + (L_{gom} - L_{exp}) \frac{a - a_{exp}}{a_{gom} - a_{exp}}\ if\ a>a_{exp}\ \&\ a<a_{gom}\\L(a) = L_{inf} \times exp^{-exp^{-K_g (a - t_g)}}\ else\end{aligned}\end{align} \]

with

\[ \begin{align}\begin{aligned}L_{exp} = L_{start} \times exp^{K_e \times a_{exp}}\\L_{gom} = L_{inf} \times exp^{-exp^{-K_g (a_{gom} - t_g)}}\end{aligned}\end{align} \]

Fig. 1.10 Gompertz growth curve

Table 1.33 Gompertz parameters

growth.exponential.lstart.sp#	\(L_{start}\) (cm)
growth.exponential.ke.sp#	\(K_e\)
growth.gompertz.linf.sp#	\(L_{inf}\) (cm)
growth.gompertz.kg.sp#	\(K_g\)
growth.gompertz.tg.sp#	\(t_g\) (years)
growth.exponential.thr.age.sp#	\(a_{exp}\) (years)
growth.gompertz.thr.age.sp#	\(a_{gom}\) (years)