Bioenergetic Module Mortality

Mortality

At each time step, a school experiences several mortality sources. The total mortality of a school $i$ is the sum of predation mortality caused by other schools, foraging mortality, starvation mortality, fishing mortality and a range of other natural mortalities (i.e. senescence, diseases and non-explicitly modelled predators). Within each time step , the different mortality sources impact a school $i$ in a random order so as to simulate the simultaneous nature of these processes.

The mortality induced by predation emerges from the energy uptake process previously described (see Energy Uptake) and thus is an explicit stochastic size-dependent process depending on the spatial co-occurrence between predators and prey. The predation mortality experienced by school $i$ is simply the sum of biomass losses due to the ingestion by all predator schools $j$ present in the same grid cell $c(i, t)$ at time step $t$ and with adequate minimum and maximum predator-to-prey size ratios ($R_{min}(s(j))$ and $R_{max}(s(j))$ ):

$$ \dfrac{dB}{dt} = \sum_j \dfrac{\gamma(s(j), s(i)) B(i,t)}{P(j, t)} I(j,t) $$

with

$$ j \in \left( j \lor (c(j,t)=c(i,t)) \left( \dfrac{L(i,t)}{R_{max}(i)} \leq L(j, t) \leq \dfrac{L(i,t)}{R_{min}(i)} \right) \right) $$

where the ratio $\dfrac{\gamma(s(j), s(i)) B(i,t)}{P(j, t)}$ represents the fact that predators prey on various prey schools according to their relative accessible abundance. Change in the number of individuals of school $i$ due to predation during the time step $t$ is then given by:

$$ N(i, t+\Delta t) = N(i, t) \left(1 - \sum_j \dfrac{\gamma(s(j), s(i))}{P(j,t)} I(j,t)\right) $$

Organisms face a trade-off between mortality and foraging activity (Man03) because more active foraging implies a higher exposure to predation, more unfavorable condition encounters (e.g. diseases) and/or increased oxidative stress. Assuming that variation in mass-specific maximum ingestion rate $I_{max}$ results from variation in foraging activity, this trade-off is modelled by including a foraging mortality that increases with $I_{max}$ and thus when foraging activity is more intense. The instantaneous foraging mortality rate experienced by school $i$ is defined as follows:

$$ M_f(i) = \mu_f I_{max}(i) $$

where $\mu_f$ is a conversion coefficient from foraging activity to mortality that measures the trade-off’s strength. Change in the number of individuals in school $i$ due to foraging mortality during time step $t$ is then obtained as:

$$ N(i, t+\Delta t) = N(i, t) e^{-M_f(i) \Delta t} $$

Starvation occurs when an individual cannot cover its maintenance needs, i.e. when net energy is negative, even by drawing energy from its gonadic reserves (see New tissue production for details). In this case, schools undergo a decrease in biomass equaling the energetic deficiency after accounting for the energy reserves contained in the gonadic compartment:

$$ B(i, t + \Delta t) = B(i, t) - N(i,t) \left( |E_P(i,t)| - \dfrac{g(i,t)}{\eta}\right) \text{ if $\eta E_P(i, t) < -g(i,t)$} $$

so that change in the number of individuals of school $i$ due to energetic starvation during time step $t$ is then given by:

$$ N(i, t + \Delta t) = N(i, t) \times \left(1 - \dfrac{|E_P(i,t)| - \dfrac{g(i,t)}{\eta}}{w(i,t)} \right) \text{ if $\eta E_P(i, t) < -g(i,t)$} $$

Starvation mortality at the time step $t+\Delta t$ relies on the net energy amount of time step $t$. During the first time step of a school, i.e. at the egg life stage, the school doesn’t feed to represent the energy provision by its yolk reserve. Therefore, starvation does not occur at the first and second time step, as starvation at the second step time relies on first time step.

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Man03 Marc Mangel. Environment and longevity: the demography of the growth rate. Population and Develop- ment Review, 29:57–70, 2003.