{"id":1026,"date":"2023-09-29T22:04:08","date_gmt":"2023-09-30T02:04:08","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/?post_type=chapter&p=1026"},"modified":"2025-10-30T08:56:45","modified_gmt":"2025-10-30T12:56:45","slug":"predicting-consumption","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/chapter\/predicting-consumption\/","title":{"raw":"Predicting consumption","rendered":"Predicting consumption"},"content":{"raw":"This is what Ecosim (and all other dynamic ecosystem models) really is about. If the number of consumers change over time, how much do they eat? How does consumption change with population density?\r\n\r\nAll ecosystem models predict consumption (Qij<\/sub><\/em>) changes based on a variant of Lotka-Volterra dynamics (see chapter<\/a>), including Ecosim which uses simple Lotka-Volterra or \"mass action\" assumptions for prediction of consumption rates. But importantly, the assumption is modified to consider \"foraging arena\" properties so that the flow rates depend on abundance of vulnerable prey rather than total prey abundance. In the foraging arena model structure, prey can be in states that are or are not vulnerable to predation, for instance by hiding, (e.g., in crevices in reefs, inside a school, where predators don't go) when not feeding, and only being subject to predation when having left their shelter to feed. (see chapter<\/a>).\r\n\r\nIn the original Ecosim formulations[footnote]Walters, C., V. Christensen and D. Pauly. 1997. Structuring dynamic models of exploited ecosystems from trophic mass-balance assessments. Reviews in Fish Biology and Fisheries<\/a> 7:139-172.[\/footnote] \u00a0[footnote]Walters, C.J., J.F. Kitchell, V. Christensen and D. Pauly. 2000. Representing density dependent consequences of life history strategies in aquatic ecosystems: Ecosim II. Ecosystems<\/a> 3: 70-83.[\/footnote]) the foraging arena consumption rate for a given predator i<\/em> feeding on a prey j<\/em> was predicted as,\r\n

[latex]Q_{ij}=\\frac{a_{ij} \\ v_{ij} \\ B_i \\ B_j}{2v_{ij}+a_{ij} \\ B_j}\\tag{1}[\/latex]<\/p>\r\nwhere, aij<\/sub><\/em> is the effective search rate for predator j<\/em> feeding on a prey i<\/em>, vij<\/sub><\/em> base vulnerability expressing the rate with which prey move between being vulnerable and not vulnerable, Bi<\/sub><\/em> prey biomass, and Bj<\/sub><\/em> predator abundance (for multi-stanza groups, Bj<\/sub> <\/em>in this calculation is replaced by an estimate of the area swept by organisms of varying sizes, summed over ages within each stanza).\r\n

\r\n\r\nFor discussion about the relationship between top-down vs bottom-up and carrying capacity, see the Density dependence<\/a> chapter.\r\n\r\n<\/div>\r\nThe model as implemented implies that \"top-down vs. bottom-up\" control is in fact a continuum, where low v<\/em>\u2019s implies bottom-up and high v<\/em>\u2019s top-down control.\r\n\r\nExperience with Ecosim has led to a more elaborate expression to describe how consumption may vary with a variety of factors:<\/a>\r\n\r\n[latex]Q_{ij}=\\frac{v_{ij} \\ a_{ij} \\ B_i \\ B_j \\ T_i \\ T_j \\ S_{ij} M_{ij}\/D_j }{v_{ij}+v_{ij} \\ T_i \\ M_{ij}+a_{ij} \\ M_{ij} \\ B_j \\ S_{ij} \\ T_j\/D_j\/A} \\cdot f(Env_t)\\tag{2}[\/latex]\r\n\r\nwhere, Ti<\/sub><\/em> represents prey relative feeding time, Tj<\/sub><\/em> predator relative feeding time, Sij<\/sub><\/em> user-defined seasonal or long term forcing effects, Mij<\/sub><\/em> mediation forcing effects, A is foraging arena size, f(Envt<\/sub>)<\/em> is an environmental response function that impacting the size of the foraging arena to account for external drivers, which may change over time[footnote]Christensen, V, M Coll, J Steenbeek, J Buszowski, D Chagaris, and CJ Walters. 2014. Representing variable habitat quality in a spatial food web model. Ecosystems 17(8): 1397-1412. http:\/\/www.jstor.org\/stable\/43678116<\/a>[\/footnote], and Dj<\/sub><\/em> represents effects of handling time as a limit to consumption rate (1\/D<\/em>j<\/sub> is proportion of time spent feeding):\r\n

[latex]D_j={1+h_j\\sum_k a_{kj} V_k T_k M_{kj}}\\tag{3}[\/latex]<\/p>\r\nwhere hj<\/sub><\/em> is the predator handling time and V<\/em>k<\/sub> is the vulnerable density of prey type k<\/em> to predator j<\/em> (V<\/em>k <\/sub>is estimated numerically in the Ecosim code).\r\n

\r\n\r\nFor multi-stanza groups, Bi<\/sub><\/em> is replaced by a sum over ages of numbers at age times body weights to the \u2154 power.\r\n\r\n<\/div>\r\nThe food consumption prediction relationship in the second equation above contains two parameters that directly influence the time spent feeding and the predation risk that feeding may entail: aij<\/sub><\/em> and v\u2019ij<\/sub><\/em>. To model possible linked changes in these parameters with changes in food availability over time (t<\/em>) as measured by per biomass food intake rate ci,t<\/sub><\/em> = Qi,t<\/sub><\/em> \/ Bi,t<\/sub><\/em> we need to specify how changes in ci,t<\/sub><\/em> will influence at least relative time spent foraging.\r\n\r\nDenoting the relative time spent foraging as Ti,t<\/sub><\/em>, measured such that the rate of effective search during any model time step t<\/em> can be predicted as aji,t<\/sub><\/em> = Ti,t<\/sub><\/em> aij<\/sub><\/em>\u00a0for each prey type i<\/em> that j<\/em> eats, we may (optionally) assume that time spent vulnerable to predation, as measured by v\u2019ij<\/sub><\/em> for all predators j<\/em> on i<\/em>, is inversely related to Ti,t<\/sub><\/em>, i.e., v\u2019ij,t<\/sub><\/em> = v\u2019ij<\/sub><\/em> \/ Ti,t<\/sub><\/em>. An alternative structure that gives similar results is to leave the aij<\/sub><\/em> constant, while varying the vij<\/sub><\/em> by setting vij,t<\/sub><\/em> = Tj,t<\/sub><\/em> \u00b7 vij<\/sub><\/em> in the numerator of Eq. 2<\/a> and vij,t<\/sub><\/em> = Ti,t<\/sub><\/em> \u00b7 vij<\/sub><\/em> in the denominator.\r\n\r\nFor convenience in estimating the aij<\/sub><\/em> and v\u2019ij<\/sub><\/em> parameters, we scale Ti,t<\/sub><\/em> so that Ti,0<\/sub><\/em> = 1, and v\u2019ij<\/sub><\/em>= vij<\/sub><\/em>. Using these scaling conventions, the key issue then becomes how to functionally relate Ti,t<\/sub><\/em> to food intake rate ci,t<\/sub> so as to represent the hypothesis that animals with lots of food available will simply spend less time foraging, rather than increase food intake rates.\r\n\r\nIn Ecosim, a simple functional form for Ti,t<\/sub><\/em> is implemented that will result in near constant feeding rates, but changing time at risk to predation, in situations where rate of effective search aij<\/sub><\/em>\u00a0is the main factor limiting food consumption rather than prey behaviour as measured by vji<\/sub><\/em>. This is implemented in form of the relationship,\r\n\r\n[latex]T_{i,t}=T_{i,t-1 }(1-\u03b1+\u03b1\\frac{Q_{opt,i,t}}{Q_{i,t-1}}) \\tag{5}[\/latex]\r\n\r\nwhere, \u237a<\/em>\u00a0is a feeding time adjustment rate [0, 1]and Q<\/em>opt<\/sub> is the Ecopath base consumption rate per biomass (QB) for group j .\u00a0 This calculation is subject to a user-defined maximum relative foraging time for each predator, and the result of that upper limit is for the predator functional response to be approximately the Holling Type I (rectilinear, see Holling functional response<\/a> chapter) form with steepness proportional to the maximum relative foraging time.\r\n\r\n\"Composite<\/strong>\r\n\r\nThe relationship between foraging time, consumption and predator biomass (when adjustment rate a<\/em> is assumed nonzero) is illustrated in Figure 1.\r\n\r\nFigure 1. Relationship between relative foraging time (T<\/em>), Q\/B<\/em> and predator biomass. If Q\/B<\/em> is held constant the foraging time (and hence predation risk) is a linear function of the predator biomass (solid line). If T<\/em> is held constant the Q\/B<\/em> will decrease asymptotically with predator biomass (stippled line). The predation risk is assumed proportional to the relative foraging time.<\/strong>\r\n
Attribution: <\/strong>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.<\/div>","rendered":"

This is what Ecosim (and all other dynamic ecosystem models) really is about. If the number of consumers change over time, how much do they eat? How does consumption change with population density?<\/p>\n

All ecosystem models predict consumption (Qij<\/sub><\/em>) changes based on a variant of Lotka-Volterra dynamics (see chapter<\/a>), including Ecosim which uses simple Lotka-Volterra or “mass action” assumptions for prediction of consumption rates. But importantly, the assumption is modified to consider “foraging arena” properties so that the flow rates depend on abundance of vulnerable prey rather than total prey abundance. In the foraging arena model structure, prey can be in states that are or are not vulnerable to predation, for instance by hiding, (e.g., in crevices in reefs, inside a school, where predators don’t go) when not feeding, and only being subject to predation when having left their shelter to feed. (see chapter<\/a>).<\/p>\n

In the original Ecosim formulations[1]<\/sup><\/a> \u00a0[2]<\/sup><\/a>) the foraging arena consumption rate for a given predator i<\/em> feeding on a prey j<\/em> was predicted as,<\/p>\n

[latex]Q_{ij}=\\frac{a_{ij} \\ v_{ij} \\ B_i \\ B_j}{2v_{ij}+a_{ij} \\ B_j}\\tag{1}[\/latex]<\/p>\n

where, aij<\/sub><\/em> is the effective search rate for predator j<\/em> feeding on a prey i<\/em>, vij<\/sub><\/em> base vulnerability expressing the rate with which prey move between being vulnerable and not vulnerable, Bi<\/sub><\/em> prey biomass, and Bj<\/sub><\/em> predator abundance (for multi-stanza groups, Bj<\/sub> <\/em>in this calculation is replaced by an estimate of the area swept by organisms of varying sizes, summed over ages within each stanza).<\/p>\n

\n

For discussion about the relationship between top-down vs bottom-up and carrying capacity, see the Density dependence<\/a> chapter.<\/p>\n<\/div>\n

The model as implemented implies that “top-down vs. bottom-up” control is in fact a continuum, where low v<\/em>\u2019s implies bottom-up and high v<\/em>\u2019s top-down control.<\/p>\n

Experience with Ecosim has led to a more elaborate expression to describe how consumption may vary with a variety of factors:<\/a><\/p>\n

[latex]Q_{ij}=\\frac{v_{ij} \\ a_{ij} \\ B_i \\ B_j \\ T_i \\ T_j \\ S_{ij} M_{ij}\/D_j }{v_{ij}+v_{ij} \\ T_i \\ M_{ij}+a_{ij} \\ M_{ij} \\ B_j \\ S_{ij} \\ T_j\/D_j\/A} \\cdot f(Env_t)\\tag{2}[\/latex]<\/p>\n

where, Ti<\/sub><\/em> represents prey relative feeding time, Tj<\/sub><\/em> predator relative feeding time, Sij<\/sub><\/em> user-defined seasonal or long term forcing effects, Mij<\/sub><\/em> mediation forcing effects, A is foraging arena size, f(Envt<\/sub>)<\/em> is an environmental response function that impacting the size of the foraging arena to account for external drivers, which may change over time[3]<\/sup><\/a>, and Dj<\/sub><\/em> represents effects of handling time as a limit to consumption rate (1\/D<\/em>j<\/sub> is proportion of time spent feeding):<\/p>\n

[latex]D_j={1+h_j\\sum_k a_{kj} V_k T_k M_{kj}}\\tag{3}[\/latex]<\/p>\n

where hj<\/sub><\/em> is the predator handling time and V<\/em>k<\/sub> is the vulnerable density of prey type k<\/em> to predator j<\/em> (V<\/em>k <\/sub>is estimated numerically in the Ecosim code).<\/p>\n

\n

For multi-stanza groups, Bi<\/sub><\/em> is replaced by a sum over ages of numbers at age times body weights to the \u2154 power.<\/p>\n<\/div>\n

The food consumption prediction relationship in the second equation above contains two parameters that directly influence the time spent feeding and the predation risk that feeding may entail: aij<\/sub><\/em> and v\u2019ij<\/sub><\/em>. To model possible linked changes in these parameters with changes in food availability over time (t<\/em>) as measured by per biomass food intake rate ci,t<\/sub><\/em> = Qi,t<\/sub><\/em> \/ Bi,t<\/sub><\/em> we need to specify how changes in ci,t<\/sub><\/em> will influence at least relative time spent foraging.<\/p>\n

Denoting the relative time spent foraging as Ti,t<\/sub><\/em>, measured such that the rate of effective search during any model time step t<\/em> can be predicted as aji,t<\/sub><\/em> = Ti,t<\/sub><\/em> aij<\/sub><\/em>\u00a0for each prey type i<\/em> that j<\/em> eats, we may (optionally) assume that time spent vulnerable to predation, as measured by v\u2019ij<\/sub><\/em> for all predators j<\/em> on i<\/em>, is inversely related to Ti,t<\/sub><\/em>, i.e., v\u2019ij,t<\/sub><\/em> = v\u2019ij<\/sub><\/em> \/ Ti,t<\/sub><\/em>. An alternative structure that gives similar results is to leave the aij<\/sub><\/em> constant, while varying the vij<\/sub><\/em> by setting vij,t<\/sub><\/em> = Tj,t<\/sub><\/em> \u00b7 vij<\/sub><\/em> in the numerator of Eq. 2<\/a> and vij,t<\/sub><\/em> = Ti,t<\/sub><\/em> \u00b7 vij<\/sub><\/em> in the denominator.<\/p>\n

For convenience in estimating the aij<\/sub><\/em> and v\u2019ij<\/sub><\/em> parameters, we scale Ti,t<\/sub><\/em> so that Ti,0<\/sub><\/em> = 1, and v\u2019ij<\/sub><\/em>= vij<\/sub><\/em>. Using these scaling conventions, the key issue then becomes how to functionally relate Ti,t<\/sub><\/em> to food intake rate ci,t<\/sub> so as to represent the hypothesis that animals with lots of food available will simply spend less time foraging, rather than increase food intake rates.<\/p>\n

In Ecosim, a simple functional form for Ti,t<\/sub><\/em> is implemented that will result in near constant feeding rates, but changing time at risk to predation, in situations where rate of effective search aij<\/sub><\/em>\u00a0is the main factor limiting food consumption rather than prey behaviour as measured by vji<\/sub><\/em>. This is implemented in form of the relationship,<\/p>\n

[latex]T_{i,t}=T_{i,t-1 }(1-\u03b1+\u03b1\\frac{Q_{opt,i,t}}{Q_{i,t-1}}) \\tag{5}[\/latex]<\/p>\n

where, \u237a<\/em>\u00a0is a feeding time adjustment rate [0, 1]and Q<\/em>opt<\/sub> is the Ecopath base consumption rate per biomass (QB) for group j .\u00a0 This calculation is subject to a user-defined maximum relative foraging time for each predator, and the result of that upper limit is for the predator functional response to be approximately the Holling Type I (rectilinear, see Holling functional response<\/a> chapter) form with steepness proportional to the maximum relative foraging time.<\/p>\n

\"Composite<\/strong><\/p>\n

The relationship between foraging time, consumption and predator biomass (when adjustment rate a<\/em> is assumed nonzero) is illustrated in Figure 1.<\/p>\n

Figure 1. Relationship between relative foraging time (T<\/em>), Q\/B<\/em> and predator biomass. If Q\/B<\/em> is held constant the foraging time (and hence predation risk) is a linear function of the predator biomass (solid line). If T<\/em> is held constant the Q\/B<\/em> will decrease asymptotically with predator biomass (stippled line). The predation risk is assumed proportional to the relative foraging time.<\/strong><\/p>\n

Attribution: <\/strong>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.<\/div>\n

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