Introduction
An Ecopath model requires the user to specify for each group \(i\) included in the model the following parameters:
- Biomass \(B_i\)
- Consumption \(Q_i\)
- Production \(P_i\)
- Fishery catch \(C_i\)
- The diet composition matrix \(DC_{ij}\)
One of the first three parameters above can be replaced by the “Ecotrophic Efficiency” \(EE\), because Ecopath can then deduce the missing parameter, see below.
We want to use this information to create a mizer model that in its steady state has the same values for the above parameters. The challenge is that mizer, which is a size-structured model, is parametrized differently. Instead of specifying properties at the group level, we specify parameters at the level of the individual and these parameters are size-dependent. In this document we will discuss how we deduce mizer parameters from the Ecopath parameters.
An Ecopath model usually includes a large number of ecosystem components (called groups), only some of which it would be appropriate to model explicitly in mizer. We will therefore need to select a subset of the Ecopath groups to include in the mizer model. Everything else we will treat as external forcing on the mizer model. So the groups that we do not include in the mizer model will still be sources of consumption and of mortality for the groups that we do include.
In cases where a species in the Ecopath model is split into several stanzas, the rates for the stanzas can be added together to get the total rate for the species. Mizer then decides itself how the rate is distributed over the mizer size classes.
The Ecopath parameters give us no direct information on the size structure of the populations. Therefore we supplement the Ecopath parameters with information about the size structure of the catches. This information is usually readily available from fishery-based observations.
We furthermore make some allometric assumptions, i.e., we assume that in the steady state the following parameters for an individual of weight \(w\) scale as a power of that weight:
- The consumption rate
- The metabolic respiration rate
- The mortality rate
- The rate at which a mature individual invests in reproduction
Of course, when we turn on the dynamics of the mizer model (which corresponds to going from Ecopath to Ecosim) and leave the steady state, the rates will change in response to changes in the abundances of prey and predators.
Definition of the Ecopath parameters
To start, we need to understand the definition of the Ecopath parameters and how to calculate their values in a mizer model. Each of these parameters has a value for each Ecopath component (or mizer species). We indicate that by an index \(i\).
Biomass
The biomass parameter \(B_i\) in Ecopath is the total biomass of all the individuals in the group \(i\) in the model. In mizer, the total biomass of a species is given by integrating over all sizes: \[B_i = \int_{w_0}^{w_{max}} N_i(w) w dw,\]where \(N_i(w)\) is the number density of the species \(i\) at size \(w\). This is calculated by mizer::getBiomass().
Consumption
The consumption rate parameter \(Q\) in Ecopath is the total rate at which all the individuals taken together consume food. In mizer, this is calculated as \[Q_i = \int_{w_0}^{w_{max}} q_i(w) N_i(w) dw,\]where \(q_i(w)\) is the consumption rate of an individual of species \(i\) and weight \(w\) (calculated with getFeedingLevel()) and \(N_i(w)\) is the number density of species \(i\) at weight \(w\). This is calculated with getConsumption().
Irrelevant detail: in mizer the individual consumption rate is split into two factors: \(q_i(w)=E_{e.i}(w)(1-f_i(w))\) where \(E_{e.i}(w)\) is the encounter rate of an individual of species \(i\) and weight \(w\), and \(f_i(w)\) is its feeding level.
Production
The production rate parameter \(P\) in Ecopath is the rate at which biomass would be accumulated if there were no mortality. Thus the production rate includes the somatic production rate that is due to growth of individuals and it includes the rate at which offspring biomass is produced during reproduction. It does not include the full rate of gonad production because most of that is lost during the inefficient reproduction process.
Somatic production
The rate at which biomass is created through somatic growth is
\[ P_{s.i} = \int_{w_0}^{w_{max}} g_i(w) N_i(w) dw, \]
where \(g_i(w)\) is the somatic growth rate of an individual of species \(i\) and weight \(w\) (calculated with getEGrowth) and \(N_i(w)\) is the number density of species \(i\) at weight \(w\). This is calculated with getSomaticProduction().
Offspring production
In mizer the rate at which offspring are produced is calculated from the rate of gonad production via a non-linear function. This rate is returned by getRDD(). To get the rate at which this offspring production adds biomass to the group this rate needs to be multiplied by the weight of an individual offspring. The resulting rate is returned by getOffspringProduction().
The total production obtained by summing the somatic production and the offspring biomass production is returned by getProduction().
Catch
The Catch rate parameter \(C_i\) in Ecopath is the rate at which biomass of group \(i\) is removed by fishing. This should include landings as well as the part of the discards that does not survive. In mizer, this rate is calculated with mizer::getYield().
The Ecopath model may disaggregate the catch into the catch by individual gears or fleets. This is calculated in mizer with mizer::getYieldGear().
Diet composition
The diet composition matrix \(DC_{ij}\) gives the proportion of the consumption of group \(i\) that is made up of group \(j\). So
\[ Q_i = \sum_j DC_{ij}\cdot B_j. \]
Therefore it also gives the proportion of the predation loss of species \(j\) that is due to predator species \(i\):
\[ M2_j\cdot B_j = \sum_i B_i\cdot DC_{ij}. \]
Master equations
Ecopath has two master equations that ensure that all biomass is accounted for. They are thus also referred to as mass balance equations. These are of course also true in mizer.
First master equation
The first master equation expresses the fact that consumed biomass will either contribute to the biomass production or it will be used up in respiration or it will remain unassimilated:
\[ Q_i = P_i + R_i + U_i. \]
Here \(Q_i\) and \(P_i\) are the consumption and production rates already discussed above. \(U_i\) is the rate at which biomass remains unassimilated and \(R_i\) is the respiration rate, i.e., the rate at which biomass is lost. Both Ecopath and mizer assume that the unassimilated biomass is a fixed proportion of the consumed biomass. In mizer this parameter is called \(1-\alpha_i\) so that \(Q_i-U_i=\alpha Q_i\). Therefore the first master equation can be seen as an equation for the respiration rate:
\[ R_i=\alpha Q_i-P_i. \]
So respiration \(R\) contains all the losses of accumulated biomass.
In mizer there are two such sources: the metabolic respiration and the loss due to gonad production that does not result in offspring biomass.
The rate of metabolic respiration is calculated by getMetabolicRespiration() as
\[ K_i = \int_{w_0}^{w_{max}}k_i(w)N_i(w)dw, \]
where \(k_i(w)\) is the metabolic rate of an individual of species \(i\) of weight \(w\) and \(N_i(w)\) is the number density of such individuals.
The rate of loss due to reproduction is the difference between the rate of investment into reproduction (the gonadic production rate obtained with getGonadProduction()) and the rate of creation of offspring biomass (obtained with getOffspringProduction()). This difference is returned by getReproductionLoss().
The total respiration rate is returned by getRespiration().
Size-resolved first master equation
Ecopath does not explicitly include gonadal growth. Production is thought of essentially as somatic production and respiration is thought of essentially as metabolic respiration. The first master equation can then be written as \(P_{s.i}=\alpha Q_i - K_i\). If we similarly ignore investment into gonads in mizer, then mizer has an equivalent to this master equation acting at the level of the individual:
\[ g_i(w) = \alpha_i q_i(w) - k_i(w). \]
However because of investment into reproduction, which is proportional to growth in mizer, the first master equation at the level of the individual becomes
\[ g_i(w) = (1-\psi_i(w))(\alpha_i q_i(w) - k_i(w)), \]
i.e., the individual only invests a proportion \(1-\psi_i(w)\) of its surplus biomass into somatic growth and the the remaining proportion \(\psi_i(w)\) gets invested into gonadic growth. Unfortunately Ecopath tells us nothing about how to choose this proportion \(\psi_i(w)\).
Second master equation
The second master equation expresses that the produced biomass is either lost again to predation or to emigration or it is accumulated:
\[ P_i = M2_i\cdot B_i+C_i+BA_i+E_i+M0_i\cdot B_i. \]
Here \(M2_i\) is the predation mortality rate so that \(M2_i\cdot B_i\) is the rate at which biomass is lost to predation. This is calculated in mizer by getM2B(). Similarly \(M0_i\) is the rate of mortality due to causes other than the predation in the model and the resulting rate of biomass loss \(M0_i\cdot B_i\) is calculated in mizer by getM0B(). As discussed above, \(C_i\) is the rate at which biomass is lost to fishing. It is often expressed as \(F_i\cdot B_i\) where \(F_i\) is the fishing mortality rate. \(E_i\) is the rate at which biomass is lost due to emigration (if \(E_i\) is negative it would correspond to immigration) and \(BA_i\) is the rate at which biomass is accumulated.
We will simplify matter and restrict to models without emigration and biomass accumulation. Then this second master equation says that the production rate is equal to the biomass loss due to the various sources of mortality: \(P_i = Z_i\cdot B_i\) where \(Z_i = M2_i+F_i+M0_i\) is the total mortality rate. In fact, when estimating the production rate for an Ecopath model, one usually estimates the mortality rate instead.
Size-resolved second master equation
Mizer keeps track not only of how biomass is produced and then removed through mortality, it also keeps track of how biomass is moved from small sizes to larger sizes. Thus it has a size-resolved version of the second master equation:
\[ g_i(w)N_i(w) - \frac{d}{dw}\left(g_i(w)w N_i(w)\right) = \mu_i(w) w N_i(w), \]
where \(\mu_i(w)\) is the mortality rate of an individual. When integrated over all sizes the first term gives the somatic production \(P_{s.i}\) and the second term gives the offspring production.
Fixing mizer parameters
As we have seen, the Ecopath parameters Biomass, Consumption, Production and Catch are emergent properties in mizer and can be calculated only once the size-dependent densities \(N_i(w)\) are known. In the steady state these densities in turn can be calculated from the size-dependent growth rates \(g_i(w)\) and mortality rates \(\mu_i(w)\) as \[ N_i(w)=N(w_0)\frac{g(w_0)}{g(w)}\exp\left(-\int_{w_0}^w \frac{\mu_i(w')}{g_i(w)}dw'\right). \]
These rates \(g_i(w)\) and \(\mu_i(w)\) depend on the mizer parameters that we want to determine.
As we said in the introduction, we are making allometric assumptions that simplify the expressions for the rates in the steady state. For the mortality rates we have