5 The energy balance of a box
Figure 1. Representation of Ecopath mass-balance (Ecopath “pies”^{[1]}) depicting how the consumption of an intermediary predator can be linked to the production of two prey groups.
Mass balance
Take a close look at Figure 1, it is key to understanding how Ecopath mass balance works. For three of the groups in the system, and intermediary predator, small pelagics and benthos their consumption is represented by a “pie”, which size is proportional to the consumption of the group. The predators consumption includes small pelagics and benthos in the proportion dictated by the diet composition of the predator – here that’s perhaps 55% for small pelagics and 45% for benthos. Within each of the three groups the consumption is broken into pieces of the pie, using this equation,
[latex]\text{Consumption = production + respiration + unassimilated part} \tag{1}[/latex]
where on Figure 1, Resp represents respiration and U the unassimilated food. This equation is in line with Winberg^{[2]} who defined consumption as the sum of somatic and gonadal growth, metabolic costs and waste products. The main differences are that Winberg (along with many other bioenergeticists) focused on measuring growth, where we focus on estimating losses, and that the Ecopath formulation does not explicitly include gonadal growth. How about predation then? On Figure 1, predation is split into its components, i.e.,
These two equations are so fundamental for understanding Ecopath that we call them “Master Equations”. Check out Figure 1, and consider what would happen if we don’t know the biomass of the intermediary predator? We would still know its diet composition, and the production of each of the two prey groups, we could then estimate a biomass for the predator, and see how much they would consume of the two prey groups, and if this was feasible. In that case, production of the prey set constraints for how much the predator potentially can eat. Alternatively, if we didn’t know the biomass of one (or both) of the prey groups, the consumption of the predator sets a demand for how much prey there has to be in order to meet the predators’ requirements. So, consumption by the predator sets constraints for how prey production. In summary, we use information about the predator consumption to provide constraints for prey production, and information about the prey production to set constraints for predator consumption. The process is called “mass balance”, and is conducted throughout the food web, see Figure 2.
Figure 2. Ecopath is a mass balance model where energy in has to equal energy out for each groups in the system. Energy out for a prey relates to energy in for its predators, which links groups in the system and provides constraints for the mass balance.
Essington^{[3]} evaluated Ecopath sensitivity to imprecise data inputs, and found that the mass balancing did not have any noticeable effect. The study used nine balanced Ecopath models, added parameter uncertainty and evaluated the degree to which the mass balance could retrieve the “true” parameters values. The study, however, did not recognize that the strength of mass balance is to weed out impossible parameter combinations, so when starting with balanced models those parameter combinations had already been excluded, and minor prediction errors (CV of 0.05 to 0.3) will not make the models sufficiently “unbalanced” compared to models developed from raw data, (which often have conversion errors that the mass balancing is good at pointing to). Our experience is clear, mass balance constrains the parameter space. The mass balance constraint implemented in the two master equations of Ecopath (Eq. 1 and Eq. 2) should not be seen as questionable assumptions, but rather as filters for mutually incompatible estimates of flow. One gathers all possible information about the components of an ecosystem, of their exploitation and interaction and passes them through the mass balance filter of Ecopath. The result is a possible (even plausible) representation of the energetic flows, the biomasses and their utilization. The more information used in the process and the more reliable the information, the more constrained and realistic the outcome will be. The possible representation of state variables and flows is all the Ecopath aims for. Once in the dynamic simulation modules, we can use routines to generate thousands of possible Ecopath models to evaluate impact of uncertainty on policy and research questions.
Parameters
The first Ecopath Master Equation (Eq. 1) can formally be expressed and values estimated from,
[latex]Q_i=P_i+R_i+U_i\tag{3}[/latex]
where the parameters are explained in Tables 2 and 3.
Notice that Eq. 3 uses absolute flow rates (t km^{-2} year^{-1}), but in the actual implementation, we estimate the production and consumption as Q_{i}=B_{i} (Q/B)_{i} and P_{i}=B_{i} (P/B)_{i}, respectively. The main reason for this is that the standing stocks (B_{i}) and instantaneous flow rates (Q/B) and (P/B) are those usually estimated, they are system size independent and therefore comparable between systems, and one can relate to them. That’s much more difficult for absolute values. Once inside Ecopath, it is, however, the absolute flow rates that are used in the calculations, but that’s a different story.
The production equation, aka Master Equation 2 (Eq. 2) can similarly be expressed as,
[latex]P_i=M2_i \cdot B_i+C_i+BA_i+E_i+M0_i \cdot B_i\tag{4}[/latex]
where M2_{i} is the predation mortality (year^{-1}), and M0_{i} is an “other mortality” instantaneous rate (year^{-1}), both of which becomes flow rates (t km^{-2} year^{-1}) when multiplied with biomass (t km^{-2}). The parameters are again explained in Tables 1 and 2.
“Other mortality” is often called M0 in some models – dating back to the MSVPA (and probably Andersen and Ursin’s North Sea model), and we have adopted this convention.
Other mortality includes mortality due to diseases, starvation, etc.. The animals or plants concerned will become flow to detritus. In addition, mortality caused by predator groups not explicitly included in the model are included in the M0 term. This mortality term is in the MSVPA called M1, while in EwE it is included in M0 as Ecopath models traditionally would be descriptive and inclusive (and hence M1 is likely to be small).
For MICE type models, one should be aware that the M1-part of the M0 flow doesn’t actually go to detritus, but is being consumed by predators not included in the model. Given that MICE models are focused on a specific research question this is not likely to be of concern.
The “other mortality” is the difference between total production and the sum of export, biomass accumulation, net migration, and predation mortality.
The “other mortality” thus expresses the mortality terms that the Ecopath model does not include, it could for instance be fish dying of diseases or old age, or mortality due to predators not considered in the model. It follows that 1-M0_{i} expresses the proportion of the production for which the fate is described in the model. We call that entity the “ecotrophic efficiency” (EE_{i}) in tradition with Polovina’s first Ecopath model^{[4]}, and it can be expressed,
[latex]EE_i=\frac {M2_i \cdot B_i+C_i+BA_i+E_i}{P_i}\tag{5}[/latex]
In Eq. 4 and Eq. 5 all terms are expressed as flow rates (t km^{-2} year^{-1}). If these flow terms are made relative to biomass (t km^{-2}), and considering that F_{i}=C_{i}/B_{i}, they become rates (year^{-1}), and as F_{i}=C_{i}/B_{i}, Eq. 4 can be re-expressed as,
[latex](\frac PB)_i =M2_i +F_i+\frac{BA_i}{B_i}+ \frac {E_i}{B_i}+ M0_i\tag{6}[/latex]
An interesting twist to Eq. 6 is that the Ecopath mortality form (at Ecopath > Output > Mortality) actually shows this equation.
This equation is important, study it carefully. We describe production as the sum of predation mortality plus fishing mortality plus net migration plus biomass accumulation plus “other mortality”.
Oh, that’s actually the second Master Equation (Eq. 2), we’re back where we started, neat.
The following table provides an overview of the input parameters for Ecopath models.
Table 1. Basic input parameters for Ecopath models
Input parameter | Name | Default value | Unit |
---|---|---|---|
B_{i} | Biomass | t km^{-2} | |
(P/B)_{i} | Production/biomass ratio | year^{-1} | |
(Q/B)_{i} | Consumption/biomass ratio | year^{-1} | |
EE_{i} | Ecotrophic efficiency (EE_{i} = 1 - M0_{i}) | (proportion) | |
BA_{i} | Biomass accumulation | 0 | t km^{-2} year^{-1} |
E_{i} | Net migration (immigration - emigration) | 0 | t km^{-2} year^{-1} |
Table 2. Other input parameters for Ecopath models
Input parameter | Name | Default value | Unit |
---|---|---|---|
DC_{ji} | Proportion of prey i in diet of predator j | 0 | (proportion) |
U_{i} | Unassimilated part (excretion + egestion) | 0.2 | (proportion) |
C_{i} | Catches by fleet | 0 | t km^{-2} year^{-1} |
Table 3. Estimated parameters for Ecopath models
Parameter | Name | Unit |
---|---|---|
P_{i} | Production (P_{i}=B_{i} (P/B)_{i}) | t km^{-2}year^{-1} |
Q_{i} | Consumption (Q_{i}=B_{i} (Q/B)_{i}) | t km^{-2}year^{-1} |
g_{i} | Gross food conversion efficiency g_{i} = P_{i} / Q_{i}, can be an input in which case either (P/B)_{i} or (Q/B)_{i} is estimated | (proportion) |
R_{i} | Respiration (= Q_{i} - P_{i} - unassimilated food) | (proportion) |
F_{i} | Fishing mortality (F_{i} = C_{i} / B_{i}) | year^{-1} |
M0_{i} | Other mortality (M0_{i} = 1 - EE_{i}) | year^{-1} |
M2_{i} | Predation mortality (M2_{i}=∑B_{j} DC_{ji}) | year^{-1} |
Parameter estimation
Not all parameters used to construct a model need to be entered. The Ecopath model “links” the production of each group with the consumption of all groups, and uses the linkages to estimate missing parameters, based on the mass-balance requirement of the second Ecopath Master Equation Eq. 2 and Eq. 4, that production from any of the groups has to end somewhere else in the system. Ecopath balances the system using one production equation for each group in the system. For a system with n groups, n production equations as in Eq. 4 are used,
[latex]\begin{eqnarray} B_1(\frac PB)_1 EE_1-B_1 (\frac QB)_1 DC_{11}-B_2(\frac QB)_2 DC_{21} \ldots -B_n (\frac QB)_n DC_{n1} - Y_1 - E_1 - BA_1=0 \\ B_2(\frac PB)_2 EE_2-B_1 (\frac QB)_1 DC_{12}-B_2(\frac QB)_2 DC_{22} \ldots -B_n (\frac QB)_n DC_{n2} - Y_2 - E_2 - BA_2=0 \\ B_3(\frac PB)_3 EE_3-B_1 (\frac QB)_1 DC_{13}-B_2(\frac QB)_2 DC_{23} \ldots -B_n (\frac QB)_n DC_{n3} - Y_3 - E_3 - BA_3=0 \\ \vdots \\ \vdots \\ B_n(\frac PB)_n EE_n-B_1 (\frac QB)_1 DC_{1n}-B_2(\frac QB)_2 DC_{2n} \ldots -B_n (\frac QB)_n DC_{nn} - Y_n - E_n - BA_n=0 \end{eqnarray}\tag{7}[/latex]
where the parameters are as in Tables 1 and 2. A system of linear equation as in Eq. 7 can be solved using standard matrix algebra – you may have learned that in precalculus or algebra classes. If, however, the determinant of a matrix is zero or if the matrix is not square, it has no ordinary inverse. Still, a generalized inverse can be found in most cases. For Ecopath, we have adopted an approach described by McKay^{[5]} to estimate the generalized inverse. If the system of linear equations is overdetermined (more equations than unknowns), and the equations are not mutually consistent, the generalized inverse method provides least square estimates to minimize discrepancies^{[6]}. While the generalized inverse in principle is a great way of solving a system of linear equations, it is in practice not used much in the Ecopath mass-balance routine. By iteration through the system, it is usually possible to solve many of the equations. Those equations are eliminated and the inversion is only used where and if needed.
An important implication of the mass-balance equation Eq. 7 is that information about predator consumption rates and diets concerning a given prey can be used to estimate the predation mortality term for the group, or, alternatively, that if the predation mortality for a given prey is known the equation can be used to estimate the consumption rates for one or more predators instead.
The gross food conversion efficiency, g_{i}, is estimated using
[latex]g_i=\frac{(P/B)_i}{(Q/B)_i}\tag{8}[/latex]
while Q/B are attempted solved by inverting the same equation. The P/B ratio is then estimated (if possible) from
[latex](\frac PB)_i=\frac{\sum \limits_{j=1}^{n} Q_j \cdot DC_{ji}+ C_i+E_i+BA_i}{B_i \cdot EE_i} \tag{9}[/latex]
This expression can be solved if both the catch, biomass and ecotrophic efficiency of group i, and the biomasses and consumption rates of all predators on group i are known (including group i if a zero order cycle, i.e., “cannibalism” exists). The catch, net migration and biomass accumulation rates are required input, and hence always known;
The EE is estimated from
[latex]EE_i=\frac{M2_i\cdot B_i+C_i+E_i+BA_i}{P_i}\tag{10}[/latex]
where the predation mortality M2_{i} is estimated as in Table 3 (= the first term of the numerator in Eq. 9.
In cases where all input parameters have been estimated for all prey for a given predator group it is possible to estimate both the biomass and consumption/biomass ratio for such a predator. The details of this are described in the original Ecopath II User Guide Appendix 4, Algorithm 3.
If for a group the total predation can be estimated it is possible to calculate the biomass for the group as described in detail in the original Ecopath II User Guide, Appendix 4, Algorithm 4.
In cases where for a given predator j the P/B, B, and EE are known for all prey, and where all predation on these prey apart from that caused by predator j is known the B or Q/B for the predator may be estimated directly.
In cases where for a given prey the P/B, B, EE are known and where the only unknown predation is due to one predator whose B or Q/B is unknown, it may be possible to estimate the B or Q/B of the prey in question.
Once the loop no longer results in estimate of any missing parameters a set of linear equations is set up including the groups for which parameters are still missing. The set of linear equations is then solved using a generalized inverse method for matrix inversion described by Mackay^{[7]}. It is usually possible to estimate P/B and EE values for groups without resorting to including such groups in the set of linear equations.
The loop above serves to minimize the computations associated with establishing mass-balance in Ecopath. The desired situation is, however, that the biomasses, production/biomass and consumption/biomass ratios are entered for all groups and that only the ecotrophic efficiency is estimated, given that no procedure exists for its field estimation.
Indeed, the central point in this is that the system of linear equations in Eq. 7 can be solved for one unknown parameters for each equation. So, the advice is to leave one input parameter unknown for each group in the model, and that one parameters is preferably EE, unless no biomass estimated is available. More about that next.
Guidelines for parameter estimation
The parameters in Table 2, i.e. the diets (DC), the unassimilated part (U) and the catches (C) must always be entered as Ecopath input along with one of the six parameters in Table 1, i.e. biomass (B), production/consumption ratio (P/B), consumption/biomass ratio (Q/B), ecotrophic efficiency (EE), biomass accumulation (BA), and net migration (E). When running the Ecopath parameterization, the program will if all four basic input parameters, (B, P/B, Q/B, and EE) are entered, ask if you want to estimate biomass accumulation (BA)? If you answer no, it will ask if you want to estimate net migration (E)?
While the matrix inversion used for solving for missing parameters in Eq. 7 is flexible, it is a flexibility that should be used carefully. so a few guidelines.
Guidelines
Unless you have reason for doing it differently, leave the biomass accumulation and net migration at the default value (0).
We have a good idea of Q/B ratios for basically all kinds of organisms, so don’t let the program estimate Q/B
P/B values (year^{-1}) relates to the average longevity (B/P, year) and to standard assessment outputs (Z, year^{-1}), so should not need to be estimated.
If biomass estimates are available, use them and estimate EE.
If you don’t have biomass estimates, guess a reasonable EE value.
Note that it is generally not possible to estimate B or P/B for apex predators from which there are no predators or catches. The tutorial about mass balance can give you some hands-on experience to get started.
Attribution
This chapter is in part adapted from the unpublished EwE User Guide: Christensen V, C Walters, D Pauly, R Forrest. Ecopath with Ecosim. User Guide. November 2008.
- This figure was made in the early 1990s, and we haven't updated it for sentimental reasons (even though it would look much better with current technology). It tells the story to be told. ↵
- Winberg, G. G., 1956. Ratę of metabolism and food requirements of fishes. Nauchnye Trudy Belorusskogo Gosudarst- vennogo Universiteta. Mińsk., 253 pp. (Transl. from Russian by J. Fish. Res. Bd Can. Transl. Ser. 194, 1960). https://waves-vagues.dfo-mpo.gc.ca/library-bibliotheque/38248.pdf ↵
- Essington TE. 2007. Evaluating the sensitivity of a trophic mass balance model (Ecopath) to imprecise data inputs. CJFAS 64: 628-637 https://doi.org/10.1139/f07-04 ↵
- Polovina, J.J., 1984. Model of a coral reef ecosystem. Coral Reefs 3, 1–11. https://doi.org/10.1007/BF00306135 ↵
- Mackay A. 1981. The generalized inverse. Practical Computing, September p. 108-110 ↵
- Christensen, V., Pauly, D., 1992. ECOPATH II — a software for balancing steady-state ecosystem models and calculating network characteristics. Ecological Modelling 61, 169–185. href="https://doi.org/10.1016/0304-3800(92)90016-8">https://doi.org/10.1016/0304-3800(92)90016-8 ↵
- Mackay, op. cit. ↵