Bioenergetic Module New Tissue Production

New tissue production: somatic and gonadic growth

The net energy available for new tissue production $E_P$ is the difference between mobilised energy $E_M$ and maintenance $E_m$, defined as follows:

$$ E_P(i,t) = E_M(i,t) - E_m(i, t) $$

The net energy $E_P$ contributes to the production of new tissues with a proportion $\rho$ being allocated to the gonadic compartment $g(i,t)$ and a proportion $1 - \rho$ allocated to the somatic one $w(i,t)$. This proportion depends on the sexual maturity status $m(i,t)$ of the schools’ individuals and their somatic mass $w(i,t)$. Before sexual maturation, i.e., when $m(i,t) = 0$, $\rho$ is equal to 0 and, after maturation, i.e., when $m(i,t) = 1$, it is defined such that the annual mean gonado-somatic index of individuals $\dfrac{g(i,t)}{w(i,t)}$ is constant throughout their adult life-stage and equal to its genetically coded value $r(i)$ (BDE+14,LSA04,QASL08) :

$$ \rho(i,t) = m(i,t) \dfrac{r(i)}{\eta \overline{E_P}(i)} w(i, t)^{1 - \beta} $$

where

$$ \overline{E_P}(i) = \dfrac{\Delta t}{a(i, t)} \sum_{t=0}^{t= a(i,t)/\Delta t} E_P(i, t') $$

is the average net energy available per time step to individuals of school $i$ since their birth, with $\Delta t$ being the duration of a time step. The above equation differs from a deterministic continuous time version of the same model (BDE+14,LSA04,QASL08) where the current net energy $E_P(i,t)$ would be used instead of the average $\overline{E_P}(i)$. The averaging in a stochastic discrete time individual-based model such as EV-OSMOSE ensures a smooth increase of proportion $\rho$ as individuals grow by dampening strong variations in $E_P(i,t)$ and thus in $\rho(i,t)$ due to the stochasticity of prey encounter and hence of ingested energy $I(i, t)$.

According to the definition of $\rho$, the net energy $E_P$ is thus fully allocated to somatic growth before maturation and it is shared between somatic and gonadic growth after, and the proportion $\rho$ allocated to gonads increases with mass (BDE+14), which limits somatic growth as individuals become bigger. However, incase mobilised energy $E_M$ cannot fully cover maintenance $E_m$, i.e. when $E_P < 0$ , new tissue production is not possible and the gonadic compartment $g(i,t)$ can be resorbed to provide energy for sustaining maintenance.

Somatic growth is then defined as follows:

$$ \dfrac{dw}{dt}(i,t) = \begin{cases} (1 - \rho(i, t)) E_P(i,t) & \text{ if $E_P \geq 0$}\\ 0 & \text{ otherwise } \end{cases} $$

and gonadic growth as:

$$ \dfrac{dg}{dt}(i,t) = \begin{cases} \eta \rho(i, t) E_P(i,t) & \text{ if $E_P \geq 0$} \\ \eta E_P(i,t) & \text{ if $-g(i,t) \leq \eta E_P(i,t) < 0$}\\ -g(i,t) & \text{ if $\eta E_P(i,t) < -g(i, t)$}\\ \end{cases} $$

where $\eta$ is the ratio of energy density between somatic and gonadic tissues, and the second and third expressions account for maintenance coverage by energy reserves contained in gonads. In the former case, gonads’ energy can fully cover maintenance costs but in the latter it cannot, so that individuals undergo energetic starvation and incur additional starvation mortality (see Mortality).

The above equation mechanistically describes somatic mass increment at each time step. The length of an individual of school $i$ at time $t$ is then obtained from the length-mass allometric relationship:

$$ L(i,t) = k w(i,t)^{\alpha} $$

where $k$ and $\alpha$ are allometric parameters.

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BDE+14 David S Boukal, Ulf Dieckmann, Katja Enberg, Mikko Heino, and Christian Jørgensen. Life-history im- plications of the allometric scaling of growth. Journal of Theoretical Biology, 359:199–207, 2014

LSA04 NP Lester, BJ Shuter, and PA Abrams. Interpreting the von bertalanffy model of somatic growth in fishes: the cost of reproduction. Proceedings of the Royal Society of London. Series B: Biological Sciences, 271(1548):1625–1631, 2004.

QASL08 Christopher Quince, Peter A Abrams, Brian J Shuter, and Nigel P Lester. Biphasic growth in fish i: theo- retical foundations. Journal of Theoretical Biology, 254(2):197–206, 2008.