Reference Equation synthesis

Reference-Equation-synthesis

Samuel Dijoux June 24, 2026

Overview

This document simply aims to synthesize the equation system used in the Osmose model version 4.4 and its extended modules. The use of this document is to keep tract of all equations, mathematical terms, assumptions made in the model, and their definitions. All parameters with their respective configuration terms are listed at the end of the document (see Section 4). The current version is integrated as a page in the Osmose wiki in its .md format, but other .pdf and .html versions are available in the Osmose guideline repository to provide a better experience while navigating through the different equations and parameters.

[!NOTES]

Osmose users will notice changes compared to the main wiki pages. These changes differ in terms of how the processes are structured in the Wiki, in terms of how the equations are formulated and in terms of the terminology of the numerous parameters.

• The structure of this document slightly differs from the way the different processes and modules are presented in the wiki. Here, the different processes used across modules are gathered to improve the understanding of how the formulation of the processes (maturation, reproduction, growth, mortality) change depending on the module(s) implemented in the model.

• Related to the equations formulations and the parameter terminology. The reason for it is twice: 1) the two versions (wiki and this document) haven’t been merged yet. In the future, both version will be homogenized. 2) The changes made here answer a set of rules that matter when building a complex model (as Osmose) for targeting other ecologists, modelers and theoreticians:

• Avoidance of confusing and recurring terms across equations, especially when the terms refer to different processes and/or have different types (e.g. present as rates, proportions or multiplicative coefficients). See Table 1 for the denotations used.

• Simplification of the equations whenever possible. This helps readers to keep-track of the flow

• Any assumption made in the model formulation have to be addressed in the equation system

Therefore this document serves as a reviewing purpose for the equations and as an alternative workflow of the current structure in the wiki. Any feedback (comments, corrections to bring, etc.) are welcomed and appreciated.

Model state variables

OSMOSE is a discrete multispecies and individual-based dynamic model that focus on fish schools, cohorts of individuals belonging to the same species and that are characterized by the same age, size (length, weight), food requirements and the same geographical coordinates at any given time step.

The model states variables are described as follow:

<span>$$ \begin{array}{ll} t &amp; \mbox{time step in years}\\ x,y,z &amp; \mbox{combinations of latitude, longitude and optionnally depth defining the spatial cell grid _c_ of a studied system}\\ s &amp; \mbox{a given species composed of _N_ schools that differ in their traits}\\ i &amp; \mbox{a given school of multiple individuals from the same species sharing identical physiological traits (also called super-individual, or cohort)}\\ k &amp; \mbox{a specific trait class}\\ f &amp; \mbox{a given fishing gear (so-called fleet). This state variable is only accounted if the multispecies fisheries module is implemented} \end{array} \qquad(1)$$</span>

Key variables at (super-)individual level

At any given time and location, species schools are characterized by a set of key traits and characteristics variables $x(i,s,c,t)$ defined as follow:

$$ x(i,s,c,t) = \begin{cases} N(i,s,c,t) & \mbox{abundance (N individuals of the i-th school from k-th trait class)} \\ B(i,s,c,t) & \mbox{biomass} \\ L(i,s,c,t) & \mbox{length (or size)} \\ a(i,s,c,t) & \mbox{age}\\ w(i,s,c,t) & \mbox{somatic weight} \\ g(i,s,c,t) & \mbox{gonadic weight (only used in BioEn- and Ev-Osmose)} \end{cases}, \qquad(2)$$

with $x = N, B, a, L, w,\mbox{ or } g$.

Key variables at species level

The characteristics traits observed at super-individual level Eq. 2 are also present at species level but differ from school traits by the fact that species variables 1) are not forcefully dependent on a given location when targeting a whole studied area (including multiple cell locations); and they 2) sum all schools present over the trait gradient. Eq. 3 below illustrate this with an example of the abundance $N$.

$$ \begin{array}{lll} N(s,c,t) &= \sum_{a=0}^{a=a_\infty} N(i,s,c,t) & \mbox{in a given spatial cell }c\\ N(s,t) &= \sum_{c}N(s,c,t) = \sum_{c} \sum_{a=0}^{a=a_\infty} N(i,s,c,t) & \mbox{over the domain area} \end{array} \qquad(3)$$

Hence, the population size of a given species at a given time and spatial dimension is equal to the sum of all surviving schools in the chosen space dimension.

Fundamental relationships between variables

Within each organisational level, the model accounts for key relationships between variables to enable conversion from one trait into another:

• Relationship between biomass $B$, abundance $N$ and somatic weight $w$

$$ \begin{array}{ll} B = N \cdot w \\ N = \frac{B}{w} \end{array} \qquad(4)$$

• Allometric relationships between length and somatic weight

$$ \begin{array}{ll} w = k\cdot L^\alpha \\ L = \Big(\frac{w}{k}\Big)^{- \alpha} \end{array} \qquad(5)$$

Species population dynamics life-cycles processes

Osmose is a discrete multi-species individual-based population dynamic model. The model tracks the life cycle of numerous species schools from egg stage to death across the whole range of coordinates from a studied ecosystem. The model operates on a weekly to monthly time steps and runs up to 100 years.

At each incremental time step, the species population dynamics fluctuates regarding to changes affecting schools life cycles (foraging, growth, multiple mortality sources and reproduction) that occur between two time steps $\Delta t = (t+1) - t$. From hatching until death, species schools use their energy to forage by consuming resource species such as phytoplankton and zooplankton, or by predating on smaller species schools. The acquired energy is allocated to growing in size, to move toward new locations, and in species survival. Once schools reach maturity they participate in the population recruitment for the next generation.

Simply put, any species population dynamics can be described as:

$$ x(s, c, t+1) = x(t)*(1+ gains - losses) \qquad(6)$$

, with $x = N \mbox{ or } B$. Gains and losses terms refer to the different processes that positively or negatively influence the population size or biomass. For instance, reproduction and schools immigration from external studied cells positively affect population size at a given time, while predation, emigration and any mortality source will alter the population size. In the following sections, we review the different processes into more details.

At each time step, the different processes and life cycles occur in a predefined rank:

1. The incoming biomass flux $B_{incoming}(k,s,t)$ originating from outside the simulated domain. This phase is optional as it is used to simulate incoming biomass from species spending part of their lifespan within the studied domain area.

2. The initialization of species schools, a process during which all temporary variables are reset at the beginning of the new life cycle (e.g. $N_{M}(s,t)$ or $B_{eaten}(i,c,t)$, see Table 3 for more description).

3. Updates of the biomass within low trophic level resource species

4. Spatial distribution of the different schools across the studied domain

5. Application of species life-cycle: mortality, growth, and reproduction

6. Removal of dead schools

Incoming biomass flux

The incoming of biomass flux from species living outside the domain area refer to a set of school biomass from multiple traits classes $k$: $ B_{incoming}(s,t) = B_{incoming}(k,s,t) $. Example: an incoming of $1300t$ of fish school at $t = 2$, composed of 2 year old schools $B_{incoming}(a=2,s,t) = 500t$ and 3 year old schools $B_{incoming}(a=3,s,t) = 800t$.

Schools Initialization

Osmose initialization phase can defined in three different ways:

• from a pristine state, with no school species and all species populations start from egg development phase

• from a predefined structure of species populations, based on relative biomass

• from a previous simulation, by using NetCDF approach

Seeding approach

The pristine system state is initially defined by an absence of school in the domain area. During a seeding period $\Delta t_{seeding}$ taking place for multiple years, the model artificially releases species eggs in the domain, based on an initial seeding biomass or abundance input (see Section 3.8.3). The different eggs, spread into schools, integrate the different life cycles until being able to reproduce and participate in the new generations. This artificial seeding stops once species natural reproduction take over, and can start again if the spawning stock biomass becomes depleted.

Each school will be characterized as follow:

$$ s(i,0) : \begin{cases} \begin{array}{lll} N(s,0) & = N(s) &\mbox{Number of school for species } s\\ N(i,s,0) & = N_{eggs} \mbox{ or } N_{seeding} &\mbox{Number of individual per school } i\\ B(i,s,0) & = B_{seeding} \mbox{ or } B_{eggs} &\mbox{Biomass of school } i\\ a(i,0) & = 0 &\mbox{Age of school } i\\ L(i,0) & = L_{eggs} &\mbox{Length of school } i \end{array} \end{cases} \qquad(7)$$

Relative biomass

Each species in the system will be characterized by multiple

$$ s(i,k,0) : \begin{cases} \begin{array}{lll} N(i,k,s,0) & = \frac{B(i,k,s,0)}{w(i,k,s,0)} &\mbox{Number of school per class } k\\ B(i,k,s,0) &= \frac{B(k,s,0) \cdot 10^6}{N(k,s)} = \frac{ B(s) \cdot p_{B}(k,s) \cdot 10^6}{N(k,s)} &\mbox{School biomass for a given class } k\\ a(i,k,s,0) &= a(k,s) &\mbox{Age of schools from class } k \\ L(i,a,s,0) &= L_{eggs} \mbox{ if }a = 0; max(L(a,s)) \mbox{ otherwise} &\mbox{Length of schools from the size class } k\\ w(i,k,s,0) &= k \cdot L(i,a,s,0)^\alpha &\mbox{School weight for a given class } k \end{array} \end{cases} \qquad(8)$$

Consumption and predation process

Predation is a central part of Osmose as an opportunistic process. It plays a key role in species energy acquisition and allocation into survival, growth and reproduction for predator schools, and conversely as mortality source for preyed schools. When foraging, schools species feed on lower trophic level species (i.e. resources such as phytoplankton and zooplankton phyla) and smaller schools species based on three mains conditions:

• the vertical accessibility of prey school $j$ to predator school $i$ in the water column within a given spatial cell grid of the domain,

• range of predator to prey size ratio, and

• the trade off between preys availability to a school predator and predator efficiency to feed upon prey

Vertical overlap between predator and preys and resources

The vertical overlap in the water column between a given predator $i$ and a prey school $j$ determines the proportion of how much prey will be accessible to a predator. This is determined by taking into account the minimum and maximal depth boundaries of both species niches:

$$ \begin{array}{ll} p_z(i,j) &= \frac{\Delta z_{i,j} (min,max)}{\Delta z_{i} (min,max)} \\ &= \frac{ z_{j_{max}} - z_{i_{min}} }{z_{i_{max}} - z_{i_{min}}} \end{array}, \qquad(9)$$

with $z_{j_{max}} &lt; z_{i_{min}}$ to allow for both species overlap in the water column.

All these accessibility proportions between prey are predator schools are joint into an accessibility matrix $A(j,i)$ that is structured with all prey and resource species in rows, and predator in columns

Size- dependent selection

Species school foraging

School predation by predatory schools depend on predator-to-prey body size ratio $R$. At any time t, a given school $j$ can become the prey of a predator school $i$ if its body length is within the foraging size range of predator, ca.. between $L_{j_{min}}$ and $L_{j_{max}}$. As it is relative to predator length $L_i$, the size ratio range within which predation occur results into the inverted relation:

$$ R_{max} < R \le R_{min}, \qquad(10)$$

with $R_{max} &lt; R_{min}$. Demonstration of Eq. 10 :

$$ \begin{array}{ccc} L_{j_{min}} \le & L_{j} &< L_{j_{max}} \\ \frac{L_{j_{min}}}{L_{i}} \le & \frac{L_j}{L_i} &< \frac{L_{j_{max}}}{L_i} \\ \frac{L_i}{L_{j_{max}}} < & \frac{L_i}{L_j} &\le \frac{L_i}{L_{j_{min}}}\\ R_{max} < & R & \le R_{min} \end{array} \qquad(11)$$

The minimal and maximal size thresholds of prey school are thus determined for any given predator size and the inverted ratios $\frac{1}{R}$ obtained from Eq. 10 :

$$ \begin{array}{lll} R &= \frac{L_i}{L_j} &\iff L_j = \frac{L_i}{R}\\ R_{max} &= \frac{L_i}{L_{j_{max}}} &\iff L_{j_{max}} = \frac{L_i}{R_{max}}\\ R_{min} &= \frac{L_i}{L_{j_{min}}} &\iff L_{j_{min}} = \frac{L_i}{R_{min}} \end{array} \qquad(12)$$

Resources foraging

Similarly to the foraging of prey school species (Eqs 9), the foraging on low-trophic level resource $Res$ by predatory school $i$ is also size-dependent. The size-dependent foraging process differs from the predator-prey size ratio (Eq 10) and relies on the assumption that resource groups are defined by a size range rather than a given size as it is the case for prey school. The accessibility of resource to consumer species is thus calculated as a proportion of the overlap between predator and resources relative to resource presence in the water column:

$$ \begin{array}{ll} R_{Res} &= \frac{\Delta L_{Res, Prey} }{\Delta L_{Res} }\\ &= \frac{ max(L_{Res_{min}}, L_{i_{min}}) - min(L_{Res_{max}}, L_{i_{max}}) }{ L_{Res_{max}} - L_{Res_{min}} } \mbox{ or }\\ R_{Res} &= log \Big( \frac{ max(L_{Res_{min}}, L_{i_{min}}) - min(L_{Res_{max}}, L_{i_{max}}) }{ L_{Res_{max}} - L_{Res_{min}} } \Big) \end{array} \qquad(13)$$

Classic predation: the trade off between prey availability and predator ability

The gain of biomass acquired during foraging for predator schools $i$, and conversely the mortality it indulges on prey schools $j$ is balanced between the total biomass of preys available for predator at a given time $B_{available}(i,c,t)$, and the amount of biomass a predator can actually eat at a given time based on its ability $B_{consumable}(i,c,t)$. Consequently, the biomass that will be eaten by predator school $B_{eaten}(i,c,t)$ therefore is the lowest amount of biomass between both.

At a given time, a predator can feed on all accessible prey schools in the water column of the same grid cell. The available biomass is the sum of accessible prey biomass. The total biomass a predator school can eat depend on it maximum ingestion rate $\mu_{I_{max}}$ and the biomass of predator $B(i,s,t)$. Such amount is standardized by the amount of mortality source implemented:

$$ \begin{array}{ll} B_{available}(i,c,t) &= \sum_{j} A(i,j) \cdot B(j,t) \\ B_{consumable}(i,c,t) &= \frac{I_{max} \cdot B(i,s,t)}{\Delta t_{mortality} }\\ B_{eaten}(i,c,t) &= min \big( B_{available}(i,c,t), B_{consumable}(i,c,t) \big) \end{array} \qquad(14)$$

Bioenergetic predation

The implementation of the bioenergetic module slightly change how predator schools $i$ prey on other schools $j$ (Section 3.3.3).

$$ \begin{array}{ll} B_{available, DEB}(i,s,t) &= \frac{\sum_{j} A(i,j) \cdot B(j,s,t)}{N(i,s,t)} \\ B_{consumable, DEB}(i,s,t) &= I_{max} \cdot w(i,t)^{\beta} \cdot \Psi(i,t)\\ B_{eaten, DEB}(i,s,t) &= I(i,s,t) = min \big( B_{available, DEB}(i,s,t), B_{consumable, DEB}(i,s,t) \big) \end{array} \qquad(15)$$

$\Psi(i,t)$ is a multiplicative factor determined by school development stage;

$$ \Psi(i,t) = \begin{cases} \begin{array}{ll} 1 & \mbox{if } a(i,t) \geq a_l\\ \theta & \mbox{otherwise; with } \end{array} \end{cases} \qquad(16)$$

with $\theta$ is a multiplicative factor accounting for higher mass-specific maximum ingestion rate during the fast and early larval growth. $\theta_\ell$ in the following equation is an intermediate term denoting the ratio between the averaged net energy acquired during the early-fast larval growth $c_{E_{N_{\ell}}}$ and the averaged net energy acquired during school lifespan $c_{E_N}$ (see Eq 30):

$$ \begin{array}{ll} \theta &= \frac{I_{max} + c_{E_N} \cdot (\theta_{\ell} - 1) }{I_{max}}\\ \theta_{\ell} &= \frac{c_{E_{N_{\ell}}}}{c_{E_N} } \end{array} \qquad(17)$$

Note: In the bioenergetic module, $B_{eaten, DEB}(i,s,t)$ is henceforth referring to the ingested biomass (or ingested energy) by predator school $I(i,s,t)$.

Predation success index

Predation efficiency during its foraging activity is measured by a metric called predation success index $p_{si}$, which defines the ratio of biomass consumed by predator relative to the total biomass available its consumption:

$$ p_{si}(i,t) = \frac{ B_{eaten}(i,c,t) }{ B_{available}(i,c,t) } \mbox{ or } \frac{ B_{eaten, DEB}(i,s,t) }{ B_{available, DEB}(i,s,t) } \qquad(18)$$

This index enables to tracks whether predator schools dispose of enough prey food or if they may suffer from starvation. This is performed by using a lower critical threshold value $p_{si_{Cr}}(i,t)$, fixed by default at $p_{si_{Cr}} = 0.57$ to ensure individuals body maintenance Shin and Cury (2001); Shin and Cury (2004). Predator school will be subject to a starvation-induced mortality rate if $p_{si}(i,t) \leq p_{si_{Cr}}(i,t)$ (see Eq. 36 in Section 3.5.5 for more detail).

Schools Dynamic energy budget, the Bioenergetic module (or BioEn-Osmose)

The Osmose model disposes of two implementation type for schools dynamic energy budget: a full implementation type that simulate all bioenergetic processes from ingestion to energetic allocation into growth and reproduction, and a lite implementation type which simplifies the bioenergetic expenditure into the different metabolic processes. We will first describe the full implementation before describing its simplified version.

Integral DEB model

Energy mobilization

The ingested energy during predation process (Section 3.3.4) is assimilated and mobilized to fuel all individual metabolic processes, i.e. maintenance, digestion, somatic growth, gonadic growth.

A fraction of the ingested energy $I(i,t)$ is not assimilated by the metabolism and is lost through feces egestion and excretion $I(i,t) \cdot (1- \xi_{M})$, while the resting fraction will be assimilated $I(i,t) \cdot \xi_{M}$. Temperature and dissolved oxygen in a given cell affect how schools mobilize the assimilated energy before using it for metabolic costs. The mobilized energy $E_M(i,t)$ will have the form:

$$ E_M(i,t) = I(i,t) \cdot \xi_{M} \cdot f( [O_{2}] (i,t) ) \cdot f_{M}( T (i,t) ) \qquad(19)$$

The effect of dissolved oxygen $f( [O_{2}] (i,t) )$ on $E_M(i,t)$ is described by the oxygen dose function (Thomas et al. (2019))

$$ f( [O_{2}] (i,t) ) = \frac{c_{O_1}\cdot [O_{2}]}{ [O_{2}] + c_{O_2}}, \qquad(20)$$

with $c_{O_1}$ and $c_{O_2}$ being respectively the asymptotic and the slope of the dose function.

The temperature-dependent function $f_{M}( T (i,t) )$

$$ f_{M}( T (i,t) ) = \phi \cdot \frac{ e^{\frac{-\varepsilon_M}{k_{B}\cdot T } } }{ 1+ \frac{\varepsilon_M}{\varepsilon_M - \varepsilon_D} \cdot e^{\frac{\varepsilon_D \cdot (T-T_{pk})}{k_{B}\cdot T \cdot T_{pk} } } }, \qquad(21)$$

with $\varepsilon_M$ and $\varepsilon_D$ the energy activation and energy deactivation for the energy mobilization function, $T_{pk}$ the peak temperature at which $f_{M}( T (i,t) )$ is maximal, $k_b$ the Boltzman constant, and $\phi$ a normalization constant that enable $f_{M}( T (i,t) ) = 1$ at peak temperature :

$$ \phi = \Big( 1 + \frac{\varepsilon_M}{\varepsilon_M - \varepsilon_D} \Big) \cdot e^{\frac{\varepsilon_M}{k_{B}\cdot T_{pk} } } \qquad(22)$$

Energy Maintenance

The energy maintenance $E_{m} (i,t)$ scales with schools somatic weight $w(i,s,t)$ with the same mass-specific exponent $\beta$ as the school maximum ingestion rate in Eq. 15.

$$ E_{m} (i,t) = c_m \cdot w(i,t)^{\beta} \cdot f_{m}( T (i,t) ) , \qquad(23)$$

with $c_m$ being the mass-specific maintenance rateand $f_{m}( T (i,t) )$ the temperature-dependent function of the energy maintenance, described by the van’t-Hoff Arrhenius relationship:

$$ f_{m}( T (i,t) ) = e^{\frac{-\varepsilon_m}{k_{B}\cdot T } }, \qquad(24)$$

$\varepsilon_m$ is the activation energy for the energy maintenance.

Net Energy production

The net energy available for new tissues production $E_{N}(i,t)$ is the remaining energy from the mobilized energy after covering all maintenance costs, described as follows:

$$ E_{N}(i,t) = E_{M} (i,t) - E_{m} (i,t) \qquad(25)$$

The resulting net energy is described by a right-skewed hump shaped curve across temperature gradient, defining species allocation performance in their thermal niche: null at their thermal boundaries ($T_{min}$ and $T_{min}$) and maximal at the optimum temperature $T_{opt}$.

• Average Net Energy production across individual lifetime $c_{E_N}(i,t)$: $$ c_{E_N}(i,t) = \frac{\Delta t}{a(i,t)} \cdot \sum_{t'= 0}^{t'=a(i,t)/\Delta t} E_N(i,t) \qquad(26)$$

Simplified DEB model

The full DEB model requires the estimation of multiple parameters that can be complex to obtain in absence of proper dataset. To overcome such potential obstacle, Osmose proposes a simplified alternative model by directly considering the net energy as a fraction of the assimilated energy $\xi_{N}$. By doing so, species mass-specific metabolic costs become implicit:

$$ E_{N}(i,t) = \xi_{N} \cdot E_{M} \cdot f_{M}( T (i,t) ) (i,t) = \xi_{N} \cdot \big( I(i,t) \cdot \xi_{M} \cdot f( [O_{2}] (i,t) \big) \cdot f_{M}( T (i,t) ) \qquad(27)$$

The temperature-dependent function differ from the full model, now defined by the Rosso function (as in Dijoux et al. (2024); Smalås et al. (2020); Rosso et al. (1995))

$$ f_{M}( T (i,t) ) = \frac{(T-T_{min})(T-T_{max})}{ (T-T_{min})(T-T_{max}) - (T-T_{opt})^2 }, \qquad(28)$$

with $f_{M}( T (i,t) ) \geq 0 \in [T_{min}, T_{max}]$ (i.e. $q=1$) and null for temperatures outside species thermal niche

Energy allocation

Any surplus energy after the mobilization and maintenance processes, when $E_N(i,t) &gt;0$, can be allocated to the production of new tissues for somatic or gonadic growth. A proportion $\rho$ of the net energy is allocated to gonadic growth. $\rho$ is determined by schools maturity status $m(i,t)$ (see Eq. 59), the gonado-somato index $r(i)$ kept constant throughout schools adult life-stage, $\eta(i)$ the energy density ratio between somatic and gonadic tissues, and the averaged net energy available to individual school $i$ since their birth $c_{E_{N}}$:

$$ \rho(i,t) = m(i,t) \cdot \frac{r(i)}{\eta(i) \cdot c_{E_N}(i,t)} \cdot w(i,t)^{(1-\beta)}, \qquad(29)$$

with

$$ \begin{cases} \begin{array}{ll} c_{E_{N}} &= \frac{\Delta t}{a(i,t)} \cdot \sum_{t=0}^{t=a(i,t)/\Delta t } E_N(i,t)\\ c_{E_{N_{\ell}}} &= \frac{ \big( L_\ell^{\alpha(1-\beta)} - L_{eggs}^{\alpha(1-\beta)} \big) \cdot k^{(1-\beta)}}{(1-\beta) \cdot \Delta t_{larval}} \end{array} \end{cases} \qquad(30)$$

$c_{E_{N_{\ell}}}$ is the averaged net energy available to individual school $i$ during the larval stage, $\Delta t$ refers to a time step.

Any investment in somatic growth $\Delta w(i,t)$ is directly used for tissue production and size increase, i.e. only when $E_{N}(i,t) &gt; 0$ (see Section 3.6.3 for more detail):

$$ \Delta w(i,t) = \begin{cases} (1-\rho(i,t)) \cdot E_N(i,t) & \mbox{if } E_N(i,t) > 0\\ 0 & \mbox{otherwise} \end{cases} \qquad(31)$$

Contrarily to $\Delta w(i,t)$, The energy allocated in reproductive output $\Delta g(i,t)$ is first stored in the reproductive compartment $g(i,t)$ before being converted into egg productions.

$$ \Delta g(i,t) = \begin{cases} \rho(i,t) \cdot E_N(i,t) & \mbox{if } E_N(i,t) > 0\\ 0 & \mbox{otherwise} \end{cases} \qquad(32)$$

This storage compartment is also used to cover the metabolic demands when these exceed the mobilized energy $E_M(i,t) &lt; E_m(i,t)$, resulting in $E_{N}(i,t) &lt; 0$. In such case (third and fourth cases in the following equation), the storage reserve change regarding of how much energy was previously stored in the compartment compared to the net energy of the present time. If the storage reserve at time $t$ is higher than the net energy at time $t+\Delta t$, the compartment $g(i,t+\Delta t)$ is depleted of an amount of energy equivalent to $E_{N}(i,t)$. If the energy stored is not enough to cover the net energy, then the storage compartment will be completely depleted, and individuals schools will face starvation-induced mortality (see Eq. for more):

$$ g(i,t+\Delta t) = \begin{cases} \begin{array}{ll} g(i,t) + \Delta g(i,t+\Delta t) = g(i,t) + \rho(i,t+\Delta t) \cdot E_N(i,t+\Delta t) & \mbox{when } E_N(i,t+\Delta t) > 0\\ g(i,t) & \mbox{constant as } E_N(i,t+\Delta t) = 0 \\ g(i,t) - E_N(i,t) & \mbox{when } E_N(i,t+\Delta t) < 0 \mbox{ but with enough energy to satisfy metabolic costs as } E_N(i,t+\Delta t) < g(i,t) \\ 0 & \mbox{when metabolic costs are not covered anymore, i.e. } g(i,t) < E_N(i,t+\Delta t) \end{array} \end{cases} \qquad(33)$$

Mortality

Within each time step, schools are subjected to multiple mortality sources, including predation by other schools, starvation, fishing-induced mortality, and diverse other natural mortality rate. These different mortality sources occur quasi simultaneous processes, but are computed consecutively in a random order trough sub-times $\Delta t_{M}$.

The total amount of dead individual for a given school is given by:

$$ N_{M} (i, s, t) = \sum_1^{\Delta t_{M}} N_{M_x} (i, s, \Delta t_{M}), \qquad(34)$$

with $N_{M} (i,s,t)$ being the number of individual dead from the mortality process $x$ ($x = P, Out, Starv, F, nat, add$). The symbol $M$ refers to the mortality process, while $\mu$ refers to rates. $\mu_M$ therefore refer to a mortality rate. In the following, we will go through the description of each of these mortality processes.

Migration-induced mortality

Species schools spending part of their cycles outside the simulated domain cannot be properly tracked at time and, therefore, cannot be subjected to the accounted mortality sources implemented in the system. They are however subjected to an annual outside mortality rate $\mu_{Out}$ which simulate different sources of mortality schools may encounter outside the domain area. The number of dying individuals outside the simulated domain $N_{Out}$ is calculated as follow:

$$ N_{Out} = N(i,t) \cdot (1-e^{-\mu_{Out}} ) \qquad(35)$$

Foraging-induced mortality

When species dynamic energetic budget is taken into account (i.e. when the bioenergetic module is implemented) schools face a trade-off between foraging activity and mortality. A higher foraging activity implies higher predation exposures, increased risks of unfavorable condition encounters (e.g. diseases) and increased oxidative stress. Assuming that variation in mass-specific maximum ingestion rate $\mu_{I_{max}}$ results from variations in schools foraging activity, this trade-off is modeled by including a foraging-induced mortality rate that increases with $\mu_{I_{max}}$, when schools foraging activity is more intense. The instantaneous foraging mortality rate experienced by school is defined as follows:

$$ \mu_{M_{Foraging}} = c_{Foraging} \cdot \mu_{I_{max}} \qquad(36)$$

Predation-induced mortality

The Number of individuals from school $i$ dying from predation $N_{M_{P}}(i,t)$ is equals to

$$ N_{M_{P}}(i,t) = \frac{ B_{M_{P}}(i,t) }{ w(i,t) }, \qquad(37)$$

with $B_{M_{P}}(i,t)$ the biomass loss in school $i$ during the predation by multiple predator schools $j$ at time $t$. It correspond to the prey biomass that is accessible to predators, relative to the amount of biomass that has been invested by predator given all other prey schools:

$$ B_{M_{P}}(i,t) = \sum_{j, (j\ne i)} B(i,t) \cdot A(i,j) \cdot \frac{ B_{eaten}(j,c,t) }{B_{available}(j,c,t)} \qquad(38)$$

Bioenergetic predation-induced mortality

When the bioenergetic module is accounted, $N_{M_{P}}(i, s, \Delta t_{M})$ is replaced by $N_{M_{P, DEB}}(i, s, \Delta t_{M})$ (Section 3.5.3). This process is pretty similar to $N_{M_{P, DEB}}(i, s, \Delta t_{M})$.

The Number of individuals from school $i$ dying from predation $N_{M_{P}}(i,t)$ is equals to

$$ N_{M_{P,DEB}}(i,t) = \frac{ B_{M_{P,DEB}}(i,t) }{ w(i,t) }, \qquad(39)$$

with $B_{M_{P}}(i,t)$ the biomass loss in school $i$ during the predation by multiple predator schools $j$ at time $t$. It correspond to the prey biomass that is accessible to predators, relative to the amount of biomass that has been invested by predator given all other prey schools:

$$ B_{M_{P,DEB}}(i,t) = \sum_{j, (j\ne i)} B(i,t) \cdot A(i,j) \cdot \frac{ B_{eaten, DEB}(j,c,t) }{B_{available, DEB}(j,c,t)} \qquad(40)$$

Starvation-induced mortality

When schools predation success index is lower than the critical threshold $p_{si_{Cr}}$ face starvation risk when their efficiency reach a critical threshold to individuals their efficiency

$$ \mu_{M_{Starv}} = \mu_{Starv_{max}} \cdot (1- \frac{p_{si} }{p_{si_{Cr}}}) \mbox{ if } p_{si} \leq p_{si_{Cr}} \qquad(41)$$

The number of individuals dying from starvation $N_{M_{Starv}}(i, s, \Delta t_{M})$:

$$ N_{M_{Starv}}(i, s, \Delta t_{M}) = N(i,s,t) \cdot e^{-\mu_{M_{Starv}}} \qquad(42)$$

$p_{si}$ and $p_{si_{Cr}}$ refer to the predation success index and critical threshold value (see Eq. Equation 18 in section Section 3.3).

Bioenergetic starvation-induced mortality

When the bioenergetic module is accounted, $N_{M_{Starv}}(i, s, \Delta t_{M})$ is replaced by $N_{M_{Starv, DEB}}(i, s, \Delta t_{M})$. Similarly to the starvation-induced mortality $N_{M_{Starv}}(i, s, \Delta t_{M})$ (Section 3.5.5), individuals are subject to mortality if they cannot fulfill their metabolic demands. This happens when the mobilized energy is lower than the maintenance energy ($E_M(i,t) < E_m(i,t)$). The Potential energy $E_P(i,t)$ to allocate to somatic and gonadic growth becomes negative, and schools gonadic reserves is used to cover the imbalance. Somatic growth stop ($\Delta w(i,t) = 0$) are insufficient to cover the metabolic costs (as seen in Eq. 32 and Eq. 33).

$$ N_{M_{Starv, DEB}}(i, s, \Delta t_{M}) = \frac{B_{M_{Starv, DEB}}(i,t) }{w(i,t)} = \frac{ E_N(i,t) - g(i,t) \cdot }{w(i,t)} \qquad(43)$$

Fisheries-induced mortality

Species-specific fisheries

Species-specific fisheries are implemented in two ways. The first approach is to consider an annual mortality rate applied homogeneously across all fish school of class $k$ (age or length) or heterogeneously across size or age classes.

$$ \begin{array}{ll} N_{M_{F}}(i,k,t) = N(i,k,t) \cdot (1-e^{-\mu_{F} } )\\ N_{M_{F}}(i,k,t) = N(i,k,t) \cdot (1-e^{-\mu_{F_k} } ) \end{array} \qquad(44)$$

The second approach is similar to the predation-induced mortality on species school $i$ (Eqs. 37, Section 3.5.3), by considering the number of individual impacted by fisheries as the minimum amount of individual between the school size at a given time and a proportion of the school size that can be retrieved from fisheries based on the observed catch:

$$ \begin{array}{ll} N_{M_{F}}(s,t) &= min \big( N(s,t); N_{fishable}(s,t) \big) \\ N_{M_{F}}(i,k,s,t) &= min \big( N(i,k,s,t); N_{fishable}(i,k,s,t) \big) \\ \end{array} \qquad(45)$$

with $N_{fishable}(i,s,t)$ defined as follow:

$$ \begin{array}{ll} N_{fishable}(i,s,t) &= N(i,t) \cdot \frac{ B_{fished}(s,t) }{ B_{fishable}(i,t) } \big)\\ N_{fishable}(i,k,s,t) &= N(i,t) \cdot \frac{ B_{fished}(s,k,t) }{ B_{fishable}(i,k,t) } \big) \end{array} \qquad(46)$$

[!NOTE] $B_{fishable}(i,t)$ is not currently well detailed in the wiki. It may refer to either 1) the amount of fish that can be retrieved depending on the fisheries characteristics (similar to $B_{consumable}$ for predation), or 2) the amount of fish permissible (such as biomass controlled by laws and regulations). Since the next small section described gear-specific fisheries, $B_{fishable}(i,t)$ should refer to the second option.

Gear-specific fisheries, the fisheries module

When the multi-species fisheries module is implemented, fishing-induced mortality on species schools $\mu_F$ are defined by a set of characteristic specific to fishing gears $f$ and is distributed across multiple target species, species size spectrum, and over time and the whole domain.

Any fishing gear $f$ is defined by:

• the spatial distribution of the fishing effort $c_{effort}(x,y,t)$. $c_{effort}(x,y,t)$ represents binary data over the whole domain that can be constant/varying over time,

• the fishing catchability $p_F(s,f)$, defining the specificity of a given fishing gear on a given species. All these catchability proportions between the studied species and fishing gears are joint into a catchability matrix $C(s,f)$ that is structured with all target species in rows and the fishing gears in columns,

• The gear selectivity $S(L,s,f)$, defining the cumulative fishing pressure across species size spectrum,

• The time-dependent fishing effort $E(f,t)$, modulated by seasonal and periodic fishing activities and shifts in intensity pressure

$$ \mu_F(s, f, c, t) = F(s,L,x,y,f,t) = c_{effort}(x,y,t) \cdot p_F(s,f) \cdot S(L,s,f) \cdot E(f,t) \qquad(47)$$

In the following, we decompose the equation to explain each term

Fishing size selectivity

Fishing selectivity is gear-specific and required a proper application of the selectivity regarding to the type of fishing. Osmose can implements eight main selectivity type:

• Knife-edge type
• Sigmoid type
• Gaussian type
• log-normal
• Double gaussian
• 3 additional non-parametric selectivity

Formulate as follow:

$$ S(L,s,f) = \begin{cases} \begin{array}{ll} 1 & \mbox{if } L(s,t) \geqslant L_{50} & \mbox{(Knife-edge selectivity)} \\ \frac{1}{1+e^{S_1 - S_2 L} } & \mbox{with } S_1 = \frac{L_{50} \cdot ln3}{L_{75} - L_{50}}; S_2 = \frac{S_1}{L_{50}} & \mbox{(Sigmoid selectivity)} \\ \frac{F(L)}{F(L_{50})} & \mbox{with } F(L) = e^{- \frac{L - L_{50}}{2 \sigma^2} }; \sigma = \frac{L_{75} - L_{50}}{q_{75}} & \mbox{(Gaussian selectivity)}, \end{array} \end{cases} \qquad(48)$$

with $L_{50}$ being the length at which the cumulative probability reach $50 %$ of the distribution, $L_{25}$ and $L_{75}$ the lengths at which the cumulative probability reach $25 %$ and $75 %$ of the distribution, and $q_{75}$is the inverse cumulative standard normal distribution for the $75^{th}$ percentile.

and illustrated as follow:

Time-dependency of the fishing effort

The fishing pressure of a given fleet $f$ can shift over time $t$ depending on event occurring within and/or across years. Osmose implements this differential changes through the product of three components that occur at the seasonal, periodic and yearly level:

$$ E(t,f) = F_{tot} = c_{F_{base}}(t) \cdot \mu_F(\Delta t_{period},t) \cdot \mu_F(\Delta t_{season},t) \qquad(49)$$

[NOTES!] This equation corresponds to $F(tot)$ in the wiki $F(t) = F_{base}(t) \cdot F_{period}(t) \cdot F_{season}(t)$

at interannual changes

Changes of fishing pressure are modulated between years. This is accounted with $c_{F_{base}}(\Delta t_{Regime},t)$, a multiplicative coefficient that is either constant through years or changing between years, thus marking a regime shift in fisheries pressure at the given time step $\Delta t_{Regime}$. In the later case, the time step

intrannual changes

• at seasonal level

The fisheries-induced mortality rate is firstly defined regarding the number of seasonal activity $N_{season}$ within a given year. The mortalities are defined for each seasonal time step $\mu_F(\Delta t_{season})$, with $\Delta t_{season} = \frac{N_{\Delta t,y}}{N_{season}}$ and $N_{\Delta t,y}$ are the number of time step within a year.

• at period level

The fishing period defines when a fleet is active. It is defined by the number of periods within one year. In concept, it is pretty identical to the seasonal activity, but the difference between both are that $\mu_F(\Delta t_{season})$ enables to better modulate the fishing pressure per time steps, with heterogeneity between and within seasons, while $\mu_F(\Delta t_{period})$ will be homogeneous over a given active period.

Parameters	Units	Description	Link to Equations	Term in Osm.config
$c_{F_{base}}$	$-$	Multiplicative coefficient of the fishing regime	Eq. 49	fisheries.rate.base.fsh
$\mu_F(\Delta t_{period})$	$year^{-1}$	Vector of all mortality rate for each fishing period	Eq. 49	$-$
$\mu_F(\Delta t_{season})$	$year^{-1}$	Vector of all mortality rate for each fishing seasonal time step	Eq. 49	$-$
$\Delta t_{Regime}$	$year$	Time step in year denoting a shift in fishing regime (change in $c_{F_{base}}$)	$-$	fisheries.rate.base.shift.fsh
$N_{F_{period}}$	$-$	Number of fishing periods per year	$-$	fisheries.period.number.fsh
$N_{F_{season}}$	$-$	Number of fishing season per year	$-$	fisheries.season.number.fsh
$a_{50}$	$year$	Age at which 50% of the population is harvested	Eq. 48	fisheries.selectivity.a50.file.fsh
$L_{25}$	$cm$	Length at which 25% of the population is harvested	Eq. 48	fisheries.selectivity.l50.file.fsh
$L_{50}$	$cm$	Length at which 50% of the population is harvested	Eq. 48	fisheries.selectivity.l50.file.fsh
$L_{75}$	$cm$	Length at which 75% of the population is harvested	Eq. 48	fisheries.selectivity.l75.file.fsh

Growth

Individuals of a given school are assumed to grow in size and weight at a given time when the amount of food they ingested fulfill individual maintenance requirements, i.e. when school predation success index $p_{si}(i,t)$ is greater than the critical value threshold of predation success index $p_{si_{Cr}}$. Individuals won’t grow in size, and will even be subject to starvation, if $p_{si}(i,t) &lt; p_{si_{Cr}}(i,t)$ (Section Section 3.5.5, Eqs. 36 - Equation 42).

$$ \Delta L(i) = L(i, t+1) - L(i, t) = \begin{cases} \begin{array}{ll} 0 & \mbox{if } p_{si}(i,t) < p_{si_{Cr}} \\ \frac{ c_{\Delta L_{max}} \cdot (p_{si}(i,t) - p_{si_{Cr}} ) }{ 1- p_{si_{Cr}} } L(i,a) & \mbox{otherwise } \end{array}, \end{cases} \qquad(50)$$

with $c_{\Delta L_{max}}$ being the multiplicative growth coefficient.

Osmose disposes of two types of growth model for its classic feature (von Bertalanffy and Gompertz growth models), and a third type when accounting for the bioernergetic module (the biphasic growth model also called Quince-Boukal-Diekmann growth model).

von Bertalanffy growth model

The model accounts three lifestage growing phases: the egg phase when there is no growing, the early but fast growing (or larval) phase, and individual growing phase with no distinction between adults and juveniles.

$$ L(i, t) = L(i, a, t) = \begin{cases} \begin{array}{ll} L_{egg} & \mbox{when } a = 0 \\ L_{egg} + (L_{\ell} - L_{egg}) \frac{a}{a_{\ell}} & \mbox{when } 0 < a < a_{\ell}\\ L_{\infty} (1- e^{-K(a - t_0)}) & \mbox{when } a_{\ell} < a \end{array}, \end{cases} \qquad(51)$$

with

$$ \begin{array}{cl} L_{a} &= L_{\infty} (1- e^{-K(a - t_0)}) \mbox{ when } a_{\ell} < a \\ a &= \frac{-1}{K} \cdot log(1- \frac{L_{a}}{L_{\infty}})+t_0 \end{array} \qquad(52)$$

The von Bertalanffy growth model von Bertalanffy (1957) is used to estimate the mean length increase above the threshold age $a_{\ell}$, denoting the age threshold ending the early-life fast-growth period for each higher-trophic level groups (by default fixed at 1 year old):

$$ L_{\ell} = max \Big( L_{egg}, 1-e^{-K(a_{\ell} - t_0)} \Big) \qquad(53)$$

For younger ages lower than $a_{\ell}$, the estimation of length-at-age $L(i,a,t)$ is done with a simple linear model. The rationale behind this is that Von Bertalanffy parameters are usually estimated from data that exclude youngs of the year or include only very few of them. Assuming a linear growth between age 0 and $a_{\ell}$ ensures a more realistic calculation of mean length increases for early ages of HTL groups Travers et al. (2009).

Gompertz growth model

The Gompertz model Gompertz (1825) implements 4 growing phases: an eggs phase with no growing, an exponential phase once individuals hatched until they reached the size $L_{exp}$, followed by slowing declining growth up to $L_{gom}$, and a final exponential growing phase:

$$ L(i, t) = L(i, a) = \begin{cases} \begin{array}{ll} L_{egg} & \mbox{when } a = 0 \\ L_{start} e^{ K_{e} \cdot a} & \mbox{when } 0 < a < a_{exp}\\ L_{exp} + (L_{gom} - L_{exp}) \frac{a - a_{exp} }{a_{gom} - a_{exp}} & \mbox{when } a_{exp} < a < a_{gom}\\ L_{\infty} e^{ -e^{-K_{g} \cdot (a - t_g) } } & \mbox{when } a_{gom} < a \end{array}, \end{cases} \qquad(54)$$

with

$$ \begin{array}{ll} L_{exp} &= L_{start} e^{ K_{e} \cdot a_{exp}} \\ L_{gom} &= L_{\infty} e^{ -e^{-K_{g} \cdot (a_{gom} - t_g) } } \end{array} \qquad(55)$$

Biphasic growth model

The growth process implemented with the bioenergetic module encompass the growth models initially implemented in the classic Osmose submodels (see Section Section 3.6). The growth trajectories of individuals in length ($L(i,t)$) and weight ($w(i,t)$) are defined by the biphasic growth model, also called Qunice-Boukal-Dieckamnn model as follow:

$$ w(i, t+\Delta t) = w (i,t) + \Delta w(i,t) \qquad(56)$$

with the length at age equivalent to:

$$ w (t) = w (a) = \begin{cases} \bigg( w_{a_\ell}^{1-\beta} + c_{E_{N_\ell}}(a-a_\ell)(1-\beta) \bigg)^{\frac{1}{1-\beta} }, & \text{if } m(a) = 0\text{ (immature juveniles) } \\ \bigg( w_{\ell}^{ (1-\beta) } \cdot (1+r(1-\beta))^{(a_m-a_\ell)} + \frac{c_{E_N}(i,t) }{k^{(1-\beta)} \cdot \bigr( 1- a_{mat}(1-r(1-\beta)) \bigr)^{a_{mat} - a} }\bigg)^{ \frac{1}{(1-\beta)} } & \text{if } m(i,t) = 1 \text{ (mature adults) } \end{cases} \qquad(57)$$

and the length at age:

$$ L(i, t) = L(i, a, t) = \begin{cases} \bigg( L_{\ell}^{\alpha(1-\beta)} + \frac{c_{E_{N}} \cdot a(i,t)(1-\beta)}{k^{(1-\beta)} } \bigg)^{\frac{1}{\alpha(1-\beta)} } & \text{if } m(i,t) = 0\text{ (immature juveniles) } \\ \bigg( l_{\ell}^{ \alpha(1-\beta) } \cdot (1+r(1-\beta))^{(a_m-a_\ell)} + \frac{c_{E_N}(i,t) }{k^{(1-\beta)} \cdot \bigr( 1- a_{mat}(1-r(1-\beta)) \bigr)^{a_{mat} - a} } \bigg)^{ \frac{1}{\alpha(1-\beta)} } & \text{if } m(i,t) = 1 \text{ (mature adults) } \end{cases} \qquad(58)$$

**Note: ** I am not entirely sure that the two set of equations above are correct. I tried to crosscheck with Boukal et al. 2014 and the function R script

Maturation process

Schools maturity denote a shifting phase in individuals behavior, growth and energy allocation. Mature individuals start allocating more of their energy into reproduction

Maturation mode

Deterministic maturation mode

When the deterministic approach is chosen, the maturation is determined by the size and age at maturation $L_{mat}$ and $a_{mat}$:

$$ \begin{array}{cl} L_{mat} &= L_{\infty} (1- e^{-K(a_{mat} - t_0)}) \\ a_{mat} &= \frac{-1}{K} \cdot log(1- \frac{L_{a_{mat}}}{L_{\infty}})+t_0 \end{array} \qquad(59)$$

Stochastic maturation mode

When the deterministic approach is chosen, the size at maturation follows a normal distribution $\mathcal{N}(\mu, \sigma^2)$, with mean $\mu$ and variance $\sigma$:

$$ \begin{array}{cl} L_{mat} &\sim \mathcal{N}(\mu, \sigma^2) \\ \mu &= L_{50} \\ \sigma &= \frac{L_{75}-L_{50}}{q_{75}} \end{array}, \qquad(60)$$

with $L_{50}$ and $L_{75}$ referring to the size at which $50%$ and $75%$ of individuals have matured. They are extracted from the cumulative distribution function of school length ($F(L)$). $q_{75}$ is the inverse distribution function (or quantile function) for the 75th percentile.

The probability for a school to mature $p_{m}$ is determined by the cumulative distribution function for the normal distribution of length:

$$ p_{m} = \frac{F(L_{t+\Delta t})-F(L_{t})}{1 - F(L_{t})} \qquad(61)$$

Linear maturation reaction norm LMRN

Age and size at maturation vary strongly between individuals due to phenotypic plasticity. This plasticity in maturation is modeled by a deterministic linear maturation reaction norm (LMRN) that represents all the age-length combinations at which an individual can become mature . In this framework, individuals become sexually mature when their growth trajectory in term of body length intersects the LMRN, so that the maturity status of individuals of school at time step is then described as:

$$ L_{mat} = m_{0} + m_{1} \cdot a(i,t) \qquad(62)$$

with $m_{0}$ and $m_{1}$ the intercept and slope of the LMRN.

Maturation status

The maturation status is determined as follow:

• Deterministic and Bioenergetic modes

$$ m(i,t) = \begin{cases} 1 & \mbox{(adults) if } L(i,t) \geq L_{mat}\
0 & \mbox{(juveniles) if otherwise} \end{cases} $$ #eq-maturation-status-deterministic

• Stochastic mode

$$ m(i,t) = \begin{cases} 1 & \mbox{(adults) if } p_{m} < \delta \mbox{, } \delta \sim \mathcal{U}(0,1)\\ 0 & \mbox{(juveniles) if otherwise} \end{cases} \qquad(63)$$

Reproduction process

Reproductive mode

The model simulate the dynamics of viviparous and oviparous species.

• viviparous reproductive mode (e.g. blue sharks, Bull sharks, Swordtails)

  • embryos development within the mother body, with their metabolic needs fulfilled by the mother maintenance

  • birth of partially or fully developed juveniles, marking the begining of the metabolic separation from the mother

• oviparous reproductive mode (default mode, adapted for most fish)

  • embryos development outside the mother, layed in eggs in the natural system

  • After hatching, early juveniles (hatchlings) need to forage to sustain their metabolic needs

Egg production

Viviparous and Oviparous egg production

The amount of egg produced at each time step by each species $N_{eggs}(s,t)$ are similar between oviparous and viviparous species, with small differences between both reproductive modes. In both reproductive modes, the number of eggs produced is obtained by the number of females $\phi_{f:m}$ that are mature $m(i,t)$ based on the spawning activity $\phi_{spawning}$ during the spawning season $\Delta t_{spawning}$ multiplied by either the spawning stock biomass $SSB$ (i.e. the biomass of all mature schools, Eq 66) and the population relative fecundity $\lambda_{rel}$ for oviparous species, or by the spawning stock abundance $SSA$ (i.e. abundance of mature individuals, Eq. 65) and the population absolute fecundity $\lambda_{abs}$ for viviparous species. In case of depletion in the spawning stock biomass or abundance, a seeding biomass or abundance ($B_{seeding}$, $N_{seeding}$ correspondingly) is used in order to avoid the species population to collapse on its own. This seeding approach is also used at the beginning of the simulations as an artificial egg release process for a few years $\Delta t_{seeding}$ until the reproduction process takes over.

$$ N_{eggs}(s,t) = \begin{cases} \begin{array}{ll} SSB(s,t) \cdot \lambda_{rel} \cdot m(i,t) \cdot p_{f:m} \cdot p_{spawning} &\mbox{ for oviparous species}\\ SSA(s,t) \cdot \lambda_{abs} \cdot m(i,t) \cdot p_{f:m} \cdot p_{spawning} &\mbox{ for viviparous species} \end{array} \end{cases} \qquad(64)$$

The spawning stock biomass and abundance:

• for viviparous species

$$ SSA(s,t) = \begin{cases} \begin{array}{ll} N_{mat}(s,t) = \sum_{a \geq a_{mat}}N(a,s,t) \\ N_{seeding} & \mbox{if } N_{mat}(s,t) = 0 \mbox{ and/or } t \in \Delta t_{seeding} \end{array} \end{cases} \qquad(65)$$

• for oviparous species

$$ SSB(s,t) = \begin{cases} \begin{array}{ll} B_{mat}(s,t) = \sum_{a \geq a_{mat}}B(a,s,t) \\ B_{seeding} \cdot 10^{6} & \mbox{if } B_{mat}(s,t) = 0 \mbox{ and/or } t \in \Delta t_{seeding} \end{array} \end{cases} \qquad(66)$$

Bioenergetic eggs production

The gonadic weight storage compartment $g(i,t)$ remains null until schools reach their maturation, and then change across time depending on the net energy and the metabolic needs (see Section 3.4.3). During spawning season, the gonadic reserve will be used for egg production relative to the number of mature schools and the proportion of female in the system.

$$ \begin{array}{ll} N_{eggs,DEB}(s,t) &= SSA(s,t) \cdot m(i,t) \cdot p_{spawning} \\ SSA(s,t) &= \sum_{a \geq a_{mat}} N_{eggs,DEB}(i, a, s,t) \\ &= \sum_{a \geq a_{mat}} N(i,t) \cdot p_{f:m} \cdot \frac{g(i,t)}{w_{egg}} \end{array} \qquad(67)$$

Schools formation

At any given time, the Osmose model determine the amount of schools to form $N(s,t)$ from the number of eggs produced $N{eggs}(s,t)$ by fixing the upper amount of schools to produce per generation; this parameter can be species-specific $N_{school}(s)$ or common to all species $N_{school}$.

If the amount of eggs produced at a given time is lower than the fixed critical threshold, only a single school is produced and all eggs will form the equivalent amount of individuals in this school. In the opposite case, the number of school to produce is equivalent to the fixed threshold and the number of eggs produced are dispersed across the newly formed schools:

$$ \mbox{if } \begin{cases} \begin{array}{ll} N_{eggs}(s, t) < N_{school}(s) \mbox{; } N(s,t) = 1 \mbox{; } N(i,t) = N_{eggs}(s,t) \\ N_{eggs}(s, t) > N_{school}(s) \mbox{; } N(s,t) = 1 \mbox{; } N(i,t) = N_{eggs}(s,t) \end{array} \end{cases} \qquad(68)$$

Evolutive module Ev-Osmose

Genetic inheritance

Left empty for now

Genetic structure

Left empty for now

Traits genetic determinism and expression

This section is still a working progress.

It concerns the following evolving traits: the mass-specific ingestion rate $I_{max}$, the intercept $m_{0}$ and slope $m_{1}$ of the maturation norm, and gonado-somatic index $r$.

Each evolving trait of individuals from school $i$ $Z_{i}$ requires the definition of four parameters: the initial mean of genotypic value $\overline{G_{Z_{i}} }(0)$, the initial additive genetic variance $\sigma^2_{A,Z_{i}}(0)$, the expression noise variance $\sigma^2_{e,Z_{i}}$, the number of functional loci $l_{Z_{i}}$, and the number of allelic values $n_{Z_{i}, l}$ for each of them.

In addition, each evolving trait may influence the instantaneous foraging mortality rate of schools $i$ $M_f(i,t)$, defined as $M_f(i,t) = k_1 \cdot e^{k_2 \cdot (I_{max}(i) - \overline{I_{max}}(0) ) }$ with $k_1$ the foraging mortality of an individual with Imax(i) = $\overline{G_{I_{max}}}(0)$ and $k_2$ the exponential slope of the change in foraging activity.

Economic module Eco-Osmose

Left empty for now

Synthesis of parameters

Symbols denotation

Table 1: Summary of Temporary variables

Parameters	Units	Description
$I,P,M$	$-$	Capital letters, used for important processes
$\mu_{x}$	$time^{-1}$	Rate of a given process $x$
$p_{x}$	$-$	Proportion or ratio for process $x$
$c_{x}$	$-$	Multiplicative coefficient for a process $x$

State variables

Table 2: State variables and species key traits

Parameters	Units	Description	Link to Equations
$t$	$-$	Time step in years	Eq. 1
$c$	$-$	Spatial cell grid location (combination of latitude and longitude)	Eq. 1
$i$	$-$	school, super-individual, or cohort variable	Eq. 1
$s$	$-$	Species variable	Eq. 1
$N(x,t)$	$-$	Abundance of $x$ organisation and space dimensions and time $t$	Eq. 2
$B(x,t)$	$g.area^{-2}$	Biomass	Eq. 2
$L(x,t)$	$cm$	length (or size)	Eq. 2
$a(x,t)$	$year$	Schools age	Eq. 2
$w(x,t)$	$g$	Somatic weight	Eq. 2
$g(x,t)$	$g$	Gonadic weight (only used in BioEn- and Ev-Osmose)	Eq. 2

Temporary variables

Some variables are used temporarily by the model to track schools abundance and biomass through the different processes. They are reset at each time step

Table 3: Summary of Temporary variables

Parameters	Units	Description	Link to Equations
$a_{Death}(i,s,t)$	$year$	Age of death of school $i$	$-$
$B_{Preyed}(i,s,t)$	$tons$	Biomass of prey fish ingested by school $i$ at time $t$	$-$
$B_{available}(i,s,t)$	$tons$	Total available prey biomass for a predator school $i$ at time $t$	Eq. 14
$B_{consumable}(i,s,t)$	$tons$	Biomass that can be ingested by predator school $i$ at time $t$	Eq. 14
$B_{eaten}(i,s,t)$	$tons$	Ingested biomass by predator school $i$ at time $t$	Eq. 14
$N_{M}(i,s,t)$	$-$	Number of dead fish in the current time step for each mortality cause	$-$, Eq. 34
$B_{M}(s,t)$	$tons$	Biomass of dead fish in the current time step from all mortality causes	$-$
$B_{M_x}(s,t)$	$tons$	Biomass of dead fish in the current time step for each mortality cause	$-$
$B_{Fished}(s,t)$	$tons$	Fished biomass at time $t$	$-$
$B_{Discarded}(s,t)$	$tons$	Discarded biomass from the simulation	$-$

Model parameters

Table 4: Summary of Model parameters

Parameters	Units	Description	Link to Equations	Term in Osm.config
$k$	$g.cm^{-\alpha}$	Allometric length-somatic mass relationship coefficient	Eq. 5	species.length2weight.condition.factor.sp
$\alpha$	$-$	Allometric length-somatic mass relationship exponent	Eq. 5	species.length2weight.allometric.power.sp
$B_{seeding}$	$ton$	Seeding biomass	Eq. 7, Eq. 66	population.seeding.biomass.sp
$N_{seeding}$	$-$	Seeding abundance	Eq. 7, Eq. 65	population.seeding.abundance.sp
$N_{sch}(s)$	$-$	Upper threshold amount of new school to produce per generation for a given species $s$	Eq. 7, <a href="#eq-reproduction-school-creation"
class="quarto-xref">Eq. 68	simulation.nschool.sp
$N_{sch}$	$-$	Upper threshold amount of new school to produce per generation	Eq. 7, [Eq. <a href="#eq-reproduction-school-creation"
class="quarto-xref">Equation 68	simulation.nschool
$B(s)$	$tons$	Species initial biomass	Eq. 8	population.initialization.biomass.sp
$a(k,s)$	$year$	Age to attribute to each size-class (vector of size equal to $k$ class +1	Eq. 8	population.initialization.age.sp
$TL(k,s)$	$-$	Trophic level of each size class (vector of size equal to $k$ class +1)	Eq. 8	population.initialization.tl.sp
$N(k)$	$-$	Number of class $k$ per species (vector of size equal to $k$ class +1)	Eq. 8	population.initialization.size.sp
$N(k,s)$	$-$	Number of new schools produced for each size class $k$	Eq. 8	population.initialization.nschool.sp
$p_{B}(k,s)$	$-$	Proportion of biomass to attribute to each size-class	Eq. 8	population.initialization.relativebiomass.sp
$w_{egg}$	$g$	Egg mass	Eq. 8, Eq. 67	species.egg.weight.sp
$L_{egg}$	$cm$	Egg size	Eq. 7, Eq. 8, Eq. 30, Eq. 51	species.egg.size.sp
$Restart.file$	$-$	Netcdf restart file	$-$	simulation.restart.file
$Population.Init.file$	$-$	Population initialization file	$-$	population.initialization.file
$\Delta t_{M}$	$-$	Number of sub-time steps equivalent to the number of mortality processes implemented in the model	Eq. 34	mortality.subdt
$R_{max}$	$-$	Maximal predator-prey size ratio	Eq. 10, <a href="#eq-predation-PPMR-demonstration"
class="quarto-xref">Eq. 11, Eq. 12	predation.predPrey.sizeRatio.max.sp
$R_{min}$	$-$	Minimum predator-prey size ratio	Eq. 10, <a href="#eq-predation-PPMR-demonstration"
class="quarto-xref">Eq. 11, Eq. 12	predation.predPrey.sizeRatio.min.sp
$L_{Res_{min}}$	$cm$	Minimum size of resource (or low-trophic level) species	Eq. 13	species.size.min.sp
$L_{Res_{max}}$	$cm$	Maximum size of resource (or low-trophic level) species	Eq. 13	species.size.max.sp
$p_{si_{Cr}}$	$-$	Critical predation success threshold value (by default, eq. to 0.57)	Eq. 41, Eq. 36, Eq. 50	predation.efficiency.critical.sp
$A(i,j)$	$-$	Accessibility matrix	Eq. 14, Eq. 15, Eq. 38	predation.accessibility.file
$\mu_{Out}$	$year^{-1}$	Annual mortality rate for species moving outside the simulated area	Eq. 35	mortality.out.rate.sp
$c_{Foraging}$	$g^{\beta-1}$	Conversion coefficient of trade-off costs during foraging	Eq. 36	species.bioen.forage.k_for.sp
$\mu_{I_{max}}$	$g.g^{-1}.year^{-1}$	Maximum mass-specific ingestion rate	Eq. 14	predation.ingestion.rate.max.sp
$\mu_{I_{DEB,max}}$	$g.g^{-\beta}.\Delta t^{-1}$	Maximum mass-specific ingestion rate		predation.ingestion.rate.max.bioen.sp
$\mu_{Starv_{max}}$	$year^{-1}$	Maximum rate of starvation mortality	Eq. 42, Eq. 36	mortality.starvation.rate.max.sp
$\mu_{M_{F}}$	$year^{-1}$	Annual fishing mortality rate	Eq. 44	mortality.fishing.rate.sp
$\mu_{M_{F}}$	$\Delta t^{-1}$	Fishing mortality rate by time step	Eq. 44	mortality.fishing.rate.byYear.file.sp
$\mu_{F_a}$	$\Delta t^{-1}$	Age-specific fishing rate	Eq. 44	mortality.fishing.rate.byDt.byAge.file.sp
$\mu_{F_{\ell}}$	$\Delta t^{-1}$	Length-specific fishing rate	Eq. 44	mortality.fishing.rate.byDt.bySize.file.sp
$B_{fished}(s,t)$	$ton$	Species annual catches	Eq. 46	mortality.fishing.catches.sp
$B_{fished}(s,\Delta t)$	$ton$	Species catches by time step	Eq. 46	mortality.fishing.catches.byYear.file.sp
$B_{fished}(a,s,\Delta t)$	$ton$	Species catches by time steps and age class	Eq. 46	mortality.fishing.catches.byDt.byAge.file.sp
$B_{fished}(\ell,s,\Delta t)$	$ton$	Species catches by time steps and length class	Eq. 46	mortality.fishing.catches.byDt.bySize.file.sp
$G.class.model$	$-$	Class of Growth model chosen: fr.ird.osmose.process.growth.VonBertalanffyGrowth.java for vB growth model, and fr.ird.osmose.process.growth.GompertzGrowth.java for Gompertz growth model	Eq. 51	growth.java.classname.sp
$c_{\Delta L_{max}}$	$-$	Growth coefficient (by default, eq. to 2)	Eq. 50	species.delta.lmax.factor.sp
$K$	$-$	vB Growth rate coefficient	Eq. 51	species.k.sp
$K_e$	$-$	Gompertz Growth rate coefficient for $L_{exp}$	Eq. 54	growth.exponential.ke.sp
$K_g$	$-$	Gompertz Growth rate coefficient for $L_{gom}$	Eq. 54	growth.gompertz.kg.sp
$L_{start}$	$cm$	Starting size in the Gompertz growth model	Eq. 54	growth.exponential.lstart.sp
$L_{a_{mat}}$	$cm$	Size at maturation	Eq. 52	species.maturity.size.sp
$L_{\infty}$	$cm$	Asymptotic size	Eq. 51, Eq. 54	species.linf.sp, growth.gompertz.linf.sp
$L_{exp}$	$cm$	Length reach at $a_{exp}$	Eq. 55	$-$
$L_{gom}$	$cm$	Length reach at $a_{gom}$	Eq. 55	$-$
$t_0$	$year$	Age at which null length is observed	Eq. 51	species.t0.sp
$t_g$	$year$	Age at which null length is observed in Gompertz model	Eq. 54	growth.gompertz.tg.sp
$a_{\ell}$	$year$	Age threshold ending the early-life fast-growth period (by default = 1 year)	Eq. 51, Eq. 53, Eq. 16	species.vonbertalanffy.threshold.age.sp
$a_{mat}$	$year$	Age at maturation	Eq. 52	species.maturity.age.sp
$a_{exp}$	$year$	Age at the ending exponential growing phase	Eq. 54	growth.exponential.thr.age.sp
$a_{gom}$	$year$	Age at the gom growing phase	Eq. 54	growth.gompertz.thr.age.sp
$-$	$-$	Maturity mode (legacy or stochastic)	$-$	species.maturity.mode
$L_{50}$	$cm$	Length at which $50%$ of the population is mature	Eq. 60	species.maturity.l50.sp
$L_{75}$	$cm$	Length at which $75%$ of the population is mature	Eq. 60	species.maturity.l75.sp
$m_0$	$cm$	Intercept of the PMRN midpoint curve	Eq. 62	species.bioen.maturity.m0.sp
$m_1$	$cm.year^{-1}$	Slope of the PMRN midpoint curve	Eq. 62	species.bioen.maturity.m1.sp
$-$	$-$	Reproductive mode (oviparity or viviparity)	$-$	species.reproduction.mode.sp
$\Delta t_{seeding}$	$-$	Seeding period in years	Eq. 65- Eq. 66	population.seeding.year.max
$\Delta t_{spawning}$	$-$	Spawning season	Eq. 65, Eq. 66	derived from reproduction.season.file.sp
$p_{spawning}$	$-$	Spawning probability	Eq. 64	derived from reproduction.season.file.sp
$p_{f:m}$	$-$	Females/males sex ratio	Eq. 64, Eq. 67	species.sexratio.sp
$\lambda_{rel}$	$g^{-1}$	Population relative fecundity: Number of eggs per gram of mature female	Eq. 64	species.relativefecundity.sp
$\lambda_{abs}$	$-$	Population absolute fecundity: Number of eggs release per mature female	Eq. 64	species.absolutefecundity.sp
$DEB.model$	$-$	DEB model implementation type (full or lite)	$-$	species.bioenergetics.model.sp
$\beta$	$-$	Scaling exponent of maximum ingestion & maintenance rates with body mass	Eq. 15, Eq. 23	species.beta.sp
$\Psi$	$-$	Multiplicative factor of maximum mass-specific ingestion rate	Eq. 15, Eq. 16	$-$
$\theta$	$-$	Multiplicative factor of maximum mass-specific ingestion rate at larval rate	Eq. 16, Eq. 17	$-$
$\theta_{\ell}$	$-$	Ratio between the larval net energy acquisition and the net energy acquisition	Eq. 17	$-$
$c_{E_N}$	$-$	Annual mean mass specific net energy acquisition	Eq. 17, Eq. 29, Eq. 30	predation.c.bioen.sp
$c_{E_{N_{\ell}}}$	$-$	Net energy acquisition rate during the larval stage	Eq. 17, Eq. 30	predation.coef.ingestion.rate.max.larvae.bioen.sp
$\xi_{M}$	$-$	Proportion of ingested energy that is assimilated	Eq. 19, Eq. 25, [Eq. eq-DEB-ENet-light]	species.bioen.assimilation.sp
$c_{O_1}$	$-$	Asymptotic of the dose-response function to dissolved oxygen saturation	Eq. 20	species.oxygen.c1.sp
$c_{O_2}$	$-$	Slope of the dose-response function to dissolved oxygen saturation	Eq. 20	species.oxygen.c2.sp
$\varepsilon_M$	$eV$	Activation energy of the energy mobilization-temperature function	Eq. 21, Eq. 22	species.bioen.mobilized.e.mobi.sp
$\varepsilon_D$	$eV$	Deactivation energy of the energy mobilization-temperature function	Eq. 21, Eq. 22	species.bioen.mobilized.e.d.sp
$T_{pk}$	$K$	Temperature of peak value in energy mobilization	Eq. 21, [Eq. <a href="#eq-DEB-mobilization-fT-phi"
class="quarto-xref">Equation 22	species.bioen.mobilized.tp.sp
$k_{b}$	$eV.K^{-1}$	Boltzmann constant $8.62*10^{-5} eV.K^{-1}$	Eq. 21, Eq. 22, Eq. 24	$-$
$\Phi$	$-$	Normalization constant of the energy mobilization temperature function	Eq. 21, Eq. 22
$\varepsilon_m$	$eV$	Activation energy of the maintenance-temperature function	Eq. 24	species.bioen.maint.e.maint.sp
$c_{m}$	$g.year^{-1}$	Mass-specific maintenance rate	Eq. 23	species.bioen.maint.energy.c_m.sp
$\xi_{N}$	$-$	Proportion of mobilized energy transferred to net energy	[Eq. eq-DEB-ENet-light]	species.netenergy.scaling.sp
$T_{min}$	$K$	Minimal viable temperature, where $ E_N(i,s,t) = 0$	Eq. 28	species.temperature.tmin.sp
$T_{opt}$	$K$	Optimal temperature of peak value in energy mobilization	Eq. 28	species.temperature.topt.sp
$T_{max}$	$K$	Maximal viable temperature, where $ E_N(i,s,t) = 0$	Eq. 28	species.temperature.tmax.sp
$r(i)$	$-$	Gonado-somatic index (by default $r=1$)	Eq. 29	species.bioen.maturity.r.sp
$\eta(i)$	$-$	Energy density-ratio between somatic and gonadic tissues (by default $\eta=1$)	Eq. 29, <a href="#eq-DEB-Allocation-GonadicGrowth"
class="quarto-xref">Eq. 32	species.bioen.maturity.eta.sp\
$\rho(i,t)$	$-$	Energy Fraction allocated to gonadic growth	Eq. 29, <a href="#eq-DEB-Allocation-GonadicGrowth"
class="quarto-xref">Eq. 32	$-$
$L_{\ell}$	$cm$	Length at first age	Eq. 30, Eq. 58, Eq. 57
$w_\ell$	$g$	Weight at first age	Eq. 58, Eq. 57
$a_{\ell}$		$year$	Age threshold ending the early-life fast-growth period, also called Nominal age (and first age)	Eq. 58, Eq. 57
$c_{F_{base}}$	$-$	Multiplicative coefficient of the fishing regime	Eq. 49	fisheries.rate.base.fsh
$\mu_F(\Delta t_{period})$	$year^{-1}$	Vector of all mortality rate for each fishing period	Eq. 49	$-$
$\mu_F(\Delta t_{season})$	$year^{-1}$	Vector of all mortality rate for each fishing seasonal time step	Eq. 49	$-$
$\Delta t_{Regime}$	$year$	Time step in year denoting a shift in fishing regime (change in $c_{F_{base}}$)	$-$	fisheries.rate.base.shift.fsh
$N_{F_{period}}$	$-$	Number of fishing periods per year	$-$	fisheries.period.number.fsh
$N_{F_{season}}$	$-$	Number of fishing season per year	$-$	fisheries.season.number.fsh
$a_{50}$	$year$	Age at which 50% of the population is harvested	Eq. 48	fisheries.selectivity.a50.file.fsh
$L_{25}$	$cm$	Length at which 25% of the population is harvested	Eq. 48	fisheries.selectivity.l50.file.fsh
$L_{50}$	$cm$	Length at which 50% of the population is harvested	Eq. 48	fisheries.selectivity.l50.file.fsh
$L_{75}$	$cm$	Length at which 75% of the population is harvested	Eq. 48	fisheries.selectivity.l75.file.fsh

References

Dijoux, Samuel, Aslak Smalås, Raul Primicerio, and David S. Boukal. 2024. “Differences in Tri-Trophic Community Responses to Temperature-Dependent Vital Rates, Thermal Niche Mismatches and Temperature-Size Rule.” Ecology Letters 27 (11): e70022. https://doi.org/10.1111/ele.70022.

Gompertz, Benjamin. 1825. “On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies.” Philosophical Transactions of the Royal Society of London 115: 513–83.

Rosso, L, J R Lobry, S Bajard, and J P Flandrois. 1995. “Convenient Model To Describe the Combined Effects of Temperature and pH on Microbial Growth.” Applied and Environmental Microbiology 61 (2): 610–16. https://doi.org/10.1128/aem.61.2.610-616.1995.

Shin, Yunne-Jai, and Philippe Cury. 2001. “Exploring Fish Community Dynamics Through Size-Dependent Trophic Interactions Using a Spatialized Individual-Based Model.” Aquatic Living Resources 14: 65–80. https://doi.org/10.1016/S0990-7440(01)01106-8.

———. 2004. “Using an Individual-Based Model of Fish Assemblages to Study the Response of Size Spectra to Changes in Fishing.” Can. J. Fish. Aquat. Sci. 61 (3): 414–31. https://doi.org/10.1139/f03-154.

Smalås, Aslak, John F. Strøm, Per-Arne Amundsen, Ulf Dieckmann, and Raul Primicerio. 2020. “Climate Warming Is Predicted to Enhance the Negative Effects of Harvesting on High-Latitude Lake Fish.” Edited by Jani Heino. Journal of Applied Ecology 57 (2): 270–82. https://doi.org/10.1111/1365-2664.13535.

Travers, M., Y.-J. Shin, S. Jennings, E. Machu, J. A. Huggett, J. G. Field, and P. M. Cury. 2009. “Two-Way Coupling Versus One-Way Forcing of Plankton and Fish Models to Predict Ecosystem Changes in the Benguela.” Ecological Modelling 220 (21): 3089–99. https://doi.org/10.1016/j.ecolmodel.2009.08.016.

von Bertalanffy, Ludwig. 1957. “Quantitative Laws in Metabolism and Growth.” The Quarterly Review of Biology 32 (3): 217–31. https://doi.org/10.1086/401873.