Bioenergetic Module Energy Uptake

Energy uptake, assimilation and mobilization

For an individual in school $i$, the energy uptake from food at time step $t$ is described by a Holling’s type 1 functional response $f$ (Hol59) that depends on its somatic mass $w(i, t)$ (CW04, HJorgensen14, SC04) in two ways.

First, it determines on prey food biomass $P(i,t)$ available to an individual of school $i$ from the other schools present in the same grid cell $c(i,t)$ according to a minimum $R_{min}(s(i))$ and a maximum $R_{max}(s(i))$ predator to prey size ratio (SC04, TSJ+09):

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

where

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

and $\gamma(s(i), s(j))$ is the accessibility coefficient essentially determined by the position of potential prey schools $j$ in the water column relative to that of school $i$ and $B(j,t) = N(j,t) w(j, t)$ is the biomass of prey school $j$. Second, it sets the maximum possible ingestion rate according to an allometric function with a scaling exponent $\beta$. The energy uptake can then be written as:

$$ I(i,t) = f(P(i,t)) = min(P(i,t); I_{max}(i) \psi(i, t) w(i,t) ^{\beta} $$

where $I_{max}(i)$ is the maximum ingestion rate per mass unit at exponent $\beta$ (or mass-specific maximum ingestion rate) of individuals in school $i$ and $\psi(i,t)$ is a multiplicative factor that depends on their life stage according to:

$$ \psi(i, t) = \Biggl \{ { \theta \text{ if } a(i,t) < a_l \atop 1\text{ otherwise } } $$

where $a_l$ is age at the end of a fast growth period (e.g larval period or the larval and post-larval period) and $\theta$ is a factor accounting for higher mass-specific maximum ingestion rate at this stage. A portion $\xi$ of the energy uptake $I(i,t)$ is assimilated, a fraction $1 - \xi$ being lost due to excretion and faeces egestion.

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No reserves are modeled in EV-OSMOSE, hence all the assimilated energy can be directly mobilised. Mobilised energy $E_M$, referred to as active metabolic rate in the ecophysiology literature, fuels all metabolic processes such as maintenance, digestion, foraging, somatic growth, gonadic growth, etc… The mobilization of energy relies on the use of oxygen to transform the energy held in the chemical bonds of nutrients into a usable form, namely ATP Cla19. As a consequence, the maximum possible energy mobilised at a given temperature depends directly on dissolved oxygen saturation and, as temperature increases, on the capacity of individuals to sustain oxygen uptake and delivery for ATP production (see section 1.2.1 Basic principles for more details). The mobilised energy rate $E_M$ is thus described by

$$ E_M(i,t) = \xi I(i,t) \lambda \left([O_2](i, t)\right) \varphi_M(T(i, t)) $$

where $\lambda \left([O_2](i, t)\right)$ and $\phi_M(T(i, t))$ are the mobilisation responses to dissolved oxygen saturation and temperature encountered by school $i$ in the grid cell it occupies at time $t$. These are scaled between 0 and 1 such that, in optimal conditions, all assimilated energy $E_M(i,t) = \xi I(i, t)$ can be mobilised and, in suboptimal conditions, only a fraction of assimilated energy can be mobilised ($E_M(i,t) < \xi I(i, t)$). More precisely, the effect of dissolved oxygen is described by a dose-response function $\lambda$ TFSMC+19 which increases with the saturation of dissolved oxygen:

$$ \lambda([O_2]) = C_{O,1} \dfrac{[O_2]}{[O_2] + C_{O, 2}} $$

where $C_{O,1}$ and $C_{O, 2}$ are the asymptote and the slope of the dose-response function.

The effect of temperature $T$ is described according to the JL46 model (cf. PDS15):

$$ \varphi_{M}(T) = \Phi \dfrac {e^{\dfrac{-\varepsilon{}_M}{k_BT}}} {1 + \dfrac{\varepsilon_M}{\varepsilon{}_D - \varepsilon{}_M} e^{\dfrac{\varepsilon{}_D}{k_B}\left(\dfrac{1}{T_P} - \dfrac{1}{T}\right)} } $$

where $k_B$ is the Boltzmann constant, $\varepsilon_M$ the activation energy for the Arrhenius-like increase in mobilised energy with temperature $T$ before reaching its peak value at $T_P$, $\varepsilon_D$ is the activation energy for the energy mobilisation decline with $T$ after $T_P$, and

$$ \Phi = \left(1 + \dfrac{\varepsilon_M}{\varepsilon{}_D - \varepsilon{}_M}\right) {e^{\dfrac{\varepsilon{}_M}{k_BT}}} $$

where $\Phi$ is a standardising constant ensuring that $\psi_M(T_P) = 1$.

The thermal responses of the bioenergetic fluxes is summarised in figure below

Thermal responses of the bioenergetic fluxes from ingestion to tissue growth in Bioen-OSMOSE. The net energy rate dome-shaped curve (in red) conforms to the OCLTT theory and the principle of TPC. Food shortage impacts ingested energy and downstream fluxes. Hypoxia impacts mobilised energy and downstream fluxes. The maximum of the net energy rate (red curve) is called $T_{opt}$ hereafter.

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