{"id":1724,"date":"2023-11-06T18:14:15","date_gmt":"2023-11-06T23:14:15","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/?post_type=chapter&p=1724"},"modified":"2025-10-30T08:57:00","modified_gmt":"2025-10-30T12:57:00","slug":"shared-foraging-arenas","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/chapter\/shared-foraging-arenas\/","title":{"raw":"Shared foraging arenas","rendered":"Shared foraging arenas"},"content":{"raw":"
\r\n

The basic Ecosim formulation for predation interactions assumes that each non-zero consumption of a prey type i<\/em> by a predator type j<\/em> takes place in a foraging arena unique to that interaction[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 7:139-172. https:\/\/doi.org\/10.1023\/A:1018479526149<\/a>[\/footnote] [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 3: 70-83. https:\/\/doi.org\/10.1007\/s100210000011<\/a>[\/footnote] [footnote]Christensen, V. and C. J. Walters. 2004. Ecopath with Ecosim: methods, capabilities and limitations. Ecol. Model. 172:109-139 https:\/\/doi.org\/10.1016\/j.ecolmodel.2003.09.003<\/a>[\/footnote].\u00a0 The rationale for this assumption is that each arena is defined by the combined behaviors of both the prey and the predator, and possibly also by selection of particular prey sub-types, (e.g., sizes), such that multiple predators can feed on the same prey type in different ways (at different depths, times of day, spatial microhabitats) without competing directly for prey within the typically very confined space represented by each arena.<\/p>\r\n

The general foraging arena assumption that predation typically is concentrated within restricted arenas, and hence at restricted rates, has profound implications for model predictions about ecosystem stability, and the further assumption that each predator-prey interaction takes place within a unique arena has equally profound implication for the maintenance of ecosystem structure and diversity[footnote]Walters, C. J. and Martell, S. J. D., 2004. Fisheries Ecology and Management. Princeton University Press, Princeton. 399 pp. [\/footnote].\u00a0 It essentially represents the possibility of a distinct \"feeding niche\" for each of the predators that takes type i<\/em> prey, hence allowing for the possibility that multiple predators can coexist while feeding on only that prey type.\u00a0 A prototype example of this possibility is with rockfishes (Sebastes <\/em>spp.) along the Pacific coast, where a diverse collection of species all feed on euphausids, but avoid direct competition for these euphausids by feeding at different depths and times of day.\u00a0 An obvious evolutionary argument in favour of assuming such fine structure in feeding interactions is that if several predators were to feed within the same micro-scale foraging arena, the intense inter-specific competition caused by such behavior would result in very strong natural selection favouring differentiation of behaviors to avoid it, e.g., by feeding at different depths or times.<\/p>\r\n

While there are evolutionary arguments in favour of assuming a distinct foraging arena for every interaction, Aydin and Gaichas[footnote]Aydin, K. Y. and Gaichas, S. K., 2007. In defense of complexity: towards a representation of uncertainty in multispecies models. MS, SC\/58\/E, Alaska Fisheries Science Centre, NOAA, Seattle WA[\/footnote] emphasize that there are some situations where multiple predator types are likely to feed on exactly the same prey and at the same place and time.\u00a0 An example could be where the predator types represent different life history stanzas (age-size classes) of the same predator species with very similar feeding modes (times and locations).<\/p>\r\n

We represent this possibility in Ecosim by entry of base proportions of each predator type\u2019s diet that occurs in each of the possible foraging arenas defined by all non-zero predator-prey consumption linkages.\u00a0 Vulnerable prey density in each arena is then represented as varying over time in response to abundances of all predator types that feed in the arena.<\/p>\r\n

In Ecosim, we define a list a <\/em>= 1, \u2026 , Na<\/sub><\/em> of possible foraging arenas, where Na<\/sub><\/em> is the number of non-zero consumption interactions in the Ecopath diet matrix representing consumption of each prey type i<\/em> by predator type j<\/em>.\u00a0 Each of these potential arenas has a defining prey type i(a)<\/em> and defining predator type j(a)<\/em>.<\/p>\r\n

When only predator type j(a)<\/em> feeds in arena a<\/em>, vulnerable prey density Va<\/sub><\/em> is predicted by the basic foraging arena equation,<\/a><\/p>\r\n[latex]V_a=\\frac{v_a \\cdot B_{i(a)}}{v_a+v_a' +\\alpha_a \\cdot P_{j(a)}}\\tag{1}[\/latex]\r\n

Here, va<\/sub><\/em> and v<\/em>a<\/sub>' are vulnerability exchange rates of prey to and from arena a<\/em>, Bi(a)<\/sub><\/em> is prey biomass, Pj(a)<\/sub><\/em> is predator abundance (biomass or sum of numbers times search rates per\u00a0 predator for multi-stanza predators), and \u03b1a<\/em><\/sub> is the predator rate of effective search (volume swept per time divided by foraging arena volume).\u00a0 The predation flow rate (biomass of prey i(a)<\/em> consumed per unit of time by predator j(a)<\/em>) is then predicted as Qi(a),j(a) <\/sub><\/em>= \u03b1a<\/sub>Va<\/sub>Pj(a)<\/sub><\/em>.\u00a0 The va<\/sub><\/em> and \u03b1a <\/em><\/sub>are parameterized by having model builders define va<\/em> from maximum possible mortality rates expressed as multiples Ka<\/sub><\/em> of Ecopath base instantaneous predation rates M<\/em>ij<\/sub>(0)<\/sup> = Q<\/em>ij<\/sub>(0) <\/sup>\/ B<\/em>i<\/sub>(0)<\/sup>, simply by setting va<\/sub> = Ka<\/sub><\/em>\u00a0where the superscript (0) designates Q<\/em>\u2019s and B<\/em>\u2019s estimated as base (initial) values of abundances and flows in the Ecopath baseline model.\u00a0 The back-exchange parameter v\u2019<\/em> is set equal to v<\/em> since it cannot be estimated separately from the \u03b1a<\/em><\/sub> parameter.<\/p>\r\n

The shared-arena extension of Eq. 1<\/a> is straightforward,<\/a><\/p>\r\n[latex]V_a=\\frac{v_a \\cdot B_{i(a)}}{v_a+v_a'+ \\sum \\limits_k \\alpha_{ak} \\cdot P_k}\\tag{2}[\/latex]\r\n

Here the predator impact on Va<\/sub><\/em> is represented by a sum over all possible predators k<\/em> of arena-specific search rates \u03b1ak <\/em><\/sub>times predator abundances Pk<\/sub><\/em>. [In the software-implementation of this, we do not actually sum over all k<\/em> but instead construct a list of all non-zero \u03b1ak<\/em><\/sub> flow combinations, and sum the \u03b1ak<\/sub>Pk <\/sub><\/em>denominator terms only over the elements of that list.]<\/p>\r\n

To parameterize Eq. 2<\/a> in a relatively simple way while assuring that it predicts predation rates equal to Ecopath base rates when the system is at its Ecopath base state, we need to specify base proportions pak<\/sub><\/em> of each predator k<\/em>\u2019s diet that is taken in arena a<\/em>. These proportions are constrained to sum to Ecopath base consumption rates \u00a0over all a<\/em> for which i(a) <\/em>= i<\/em>.\u00a0 That is, we take the by-arena base flows \u00a0to be p<\/em>ak<\/sub> Q<\/em>i(a),k<\/sub>(0)<\/sup>.\u00a0 These base flows then imply a base instantaneous mortality rate M<\/em>a<\/sub>(0)<\/sup>\u00a0totaled over predators feeding in a<\/em>, for prey i(a)<\/em>,<\/p>\r\n[latex]M_a^{(0)}=\\frac{\\sum \\limits_k Q_{ak}^{(0)}}{B_{i(a)}}\\tag{3}[\/latex]\r\n

Using this input or baseline estimate of M<\/em> for each arena and an assumed vulnerability multiplier Ka<\/sub><\/em> for that arena, we simply set va <\/sub><\/em>= Ka<\/sub><\/em> (and v<\/em>a<\/sub>' = v<\/em>a<\/sub>).<\/p>\r\n

Next, note that to be consistent with Ecopath baseline inputs, we must require that Ecosim predict Q<\/em>ak<\/sub>(0)<\/sup> when all biomasses (and p<\/em>\u2019s) are at their Ecopath base values. The Ecosim prediction of rate Q<\/em>ak<\/sub> (flow rate of prey to predator k<\/em> from feeding in arena a<\/em>) at any time is Qak <\/sub><\/em>= \u03b1ak<\/sub>Va<\/sub>Pk<\/sub><\/em>, implying we must constrain the \u03b1ak<\/em><\/sub> so that Q<\/em>ak<\/sub>(0)<\/sup> = \u03b1<\/em>ak<\/sub> V<\/em>a<\/sub>(0)<\/sup> P<\/em>k<\/sub>(0)<\/sup>, i.e. we must set \u03b1<\/em>ak <\/sub>= Q<\/em>ak<\/sub>(0)<\/sup> \/ (V<\/em>a<\/sub>(0) <\/sup>P<\/em>k<\/sub>(0) <\/sup>).\u00a0 This means that to estimate the \u03b1ak<\/em><\/sub> we must first estimate the base vulnerable abundances V<\/em>a<\/sub>(0)<\/sup>.<\/p>\r\n

This estimation turns out to be remarkably simple, when we note that the Ecopath base value of \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k <\/sub>must equal \u2211k\u00a0<\/sub>Q<\/em>ak<\/sub>(0)<\/sup> \/ V<\/em>a<\/sub>(0)<\/sup>, (simply sum Qak<\/sub><\/em> over k<\/em>, which must equal Va = \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k<\/sub>, and solve for \u2211k<\/sub> \u03b1<\/em>ak <\/sub>P<\/em>k<\/sub>). Substituting \u2211k\u00a0<\/sub>Q<\/em>ak<\/sub>(0)<\/sup> \/ V<\/em>a<\/sub>(0)<\/sup> for \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k<\/sub> in Eq. 2<\/a>, then solving for V<\/em>a<\/sub>(0)<\/sup>, we calculate the base vulnerable abundances to be simply,<\/p>\r\n[latex]V_a^{(0)}= v_a \\cdot B_{i(a)} - \\frac{\\sum \\limits_k Q_{ak}^{(0)}}{v_a+v_a'}\\tag{4}[\/latex]\r\n

The \u03b1ak<\/em><\/sub> are then calculated from these base vulnerable biomasses. Time-varying values of Qak<\/sub><\/em> are computed efficiently in Ecosim by setting up a list h <\/em>= 1<\/em>, \u2026 , Nh<\/sub><\/em> of all non-zero by-arena flows (N<\/em>h\u00a0<\/sub>\u2265 N<\/em>a<\/sub>), where for each list element we store its associated prey type i(h)<\/em>, predator type k(h)<\/em>, and arena a(h)<\/em>.<\/p>\r\nTo calculate Qak<\/sub><\/em>, we sweep down this list repetitively.\u00a0 On the first sweep, we accumulate the denominator sums \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k <\/sub>for Eq. 2<\/a>.\u00a0 We then sweep down the arena list and calculate Va<\/sub><\/em> for every a<\/i>\u00a0again using Eq. 2<\/a>.\u00a0 Then we sweep again down the h<\/em> list, calculating Qak <\/sub><\/em>= \u03b1ak<\/sub>Va<\/sub>Pk<\/sub><\/em> and accumulating predictions of total predation rates on the prey i(a)<\/em> and food consumption rates by predators k(a)<\/em>.\r\n

As an added bit of model realism, one can specify a non-zero prey handling times for predator k<\/em> (type II functional response[footnote]Holling, C.S., 1959. The components of predation as revealed by a study of small mammal predation of the European pine sawfly 91, 293\u2013320. https:\/\/doi.org\/10.4039\/Ent91293-5<\/a>[\/footnote]), and the Qak<\/sub><\/em> calculation is modified to be Qak <\/sub><\/em>= (\u03b1ak<\/sub><\/em>\/Hk<\/sub><\/em>) Va <\/sub>Pk<\/sub><\/em>, where Hk<\/sub><\/em> is the denominator of Holling's multi-species disc equation for predator k<\/em> feeding.\u00a0 This handling time correction is also applied in the bout-feeding formulation described in the next chapter.<\/p>\r\n

To edit the pak<\/sub><\/em> diet proportions array, we display a matrix for each prey type i<\/em> of the non-zero i-k <\/em>consumption proportions, as shown schematically in Table 1. In this table, m<\/em> is the number of non-zero flows from prey i<\/em>to predators k<\/em> where each such flow defines a potential foraging arena.\u00a0 Note that each column of the table must sum to 1.0, i.e. all of the consumption by predator kj<\/em> of prey type i<\/em> must be accounted for by feeding in one of the m<\/em> identifiable arenas for prey type i<\/em>.\u00a0 The Ecosim default proportions for this table imply that each predator takes all of its consumption of prey type i<\/em> in a unique arena, i.e. the table is an identity matrix, (with values of 1 on the shaded diagonal in Table 1).<\/p>\r\n\"\"\r\n

The opposite extreme of this default assumption would be that all consumption of prey type i<\/em> by its predators occur in only one arena or behavioral state for prey i<\/em>, as shown in Table 2. This case implies maximum possible impact of predators k<\/em> on availability of prey i<\/em> to one another, and will cause competitive exclusion of at least some predator types in Ecosim unless the predators are well-differentiated in terms of overall diet composition, i.e. where each predator \"specializes\" on a different prey type i<\/em>, which dominates the diet composition, as for instance shown by Schmidt[footnote]Schmidt, K. A., 2004. Incidental predation, enemy-free space and the coexistence of incidental prey. Oikos, 106:335-343. https:\/\/doi.org\/10.1111\/j.0030-1299.2004.13093.x<\/a> [\/footnote]. Studies rather tend to indicate resource partitioning between competing predator species, leading to non-additive mortality rates<\/a>, see e.g., Griffen and Byers[footnote]Griffen, B. D. and Byers, J. E., 2006. Partitioning mechanisms of predator interference in different habitats. Oecologia, 146:608-614. https:\/\/doi.org\/10.1007\/s00442-005-0211-4<\/a>[\/footnote]. Separation where diet compositions indicate predator overlap may also be caused by temporal exclusion of prey based on availability to the predator[footnote]Scheuerell, J. M., Schindler, D. E., Scheuerell, M. D., Fresh, K. L., Sibley, T. H., Litt, A. H. and Shepherd, J. H., 2005. Temporal dynamics in foraging behavior of a pelagic predator. Canadian Journal of Fisheries and Aquatic Sciences, 62:2494-2501. https:\/\/doi.org\/10.1139\/f05-164<\/a>[\/footnote].<\/p>\r\n\"\"\r\n

In the special case where a set of predators feeds on only one prey type in a single arena (Table 2), and where there are no complications such as multistanza population dynamics where abundance of one or more predator types may be limited by recruitment rates from younger stanzas, the above formulation implies that there is not even a unique equilibrium point for predator abundances.\u00a0 Rather, all predator abundance combinations that predict V=V<\/em>(0<\/em>) in Eq. 2<\/a> are neutral stable points provided predator mortality rates remain at Ecopath base values, such that any temporary pulse of differential mortality that causes one or more predators to decline will then be followed by persistence of the new predator abundance combination if mortality rates return to the base values.\u00a0 Any predator that suffers a persistent differential increase in mortality rate is predicted to decline toward extinction.<\/p>\r\n\r\n<\/div>\r\n

\r\n\r\nAttribution\u00a0<\/strong>This chapter was inspired by Kerim Aydin\u2019s work on a foraging arenas, and <\/span>is based on Walters and Christensen. 2007.[footnote]Walters, C and V. Christensen. 2007. Adding realism to foraging arena predictions of trophic flow rates in Ecosim ecosystem models: shared foraging arenas and bout feeding. Ecological Modelling 209:342-350. https:\/\/doi.org\/10.1016\/j.ecolmodel.2007.06.025<\/a>[\/footnote], used with permission from Elsevier, Licence Number 5663310244809.\r\n\r\n<\/div>","rendered":"
\n

The basic Ecosim formulation for predation interactions assumes that each non-zero consumption of a prey type i<\/em> by a predator type j<\/em> takes place in a foraging arena unique to that interaction[1]<\/sup><\/a> [2]<\/sup><\/a> [3]<\/sup><\/a>.\u00a0 The rationale for this assumption is that each arena is defined by the combined behaviors of both the prey and the predator, and possibly also by selection of particular prey sub-types, (e.g., sizes), such that multiple predators can feed on the same prey type in different ways (at different depths, times of day, spatial microhabitats) without competing directly for prey within the typically very confined space represented by each arena.<\/p>\n

The general foraging arena assumption that predation typically is concentrated within restricted arenas, and hence at restricted rates, has profound implications for model predictions about ecosystem stability, and the further assumption that each predator-prey interaction takes place within a unique arena has equally profound implication for the maintenance of ecosystem structure and diversity[4]<\/sup><\/a>.\u00a0 It essentially represents the possibility of a distinct “feeding niche” for each of the predators that takes type i<\/em> prey, hence allowing for the possibility that multiple predators can coexist while feeding on only that prey type.\u00a0 A prototype example of this possibility is with rockfishes (Sebastes <\/em>spp.) along the Pacific coast, where a diverse collection of species all feed on euphausids, but avoid direct competition for these euphausids by feeding at different depths and times of day.\u00a0 An obvious evolutionary argument in favour of assuming such fine structure in feeding interactions is that if several predators were to feed within the same micro-scale foraging arena, the intense inter-specific competition caused by such behavior would result in very strong natural selection favouring differentiation of behaviors to avoid it, e.g., by feeding at different depths or times.<\/p>\n

While there are evolutionary arguments in favour of assuming a distinct foraging arena for every interaction, Aydin and Gaichas[5]<\/sup><\/a> emphasize that there are some situations where multiple predator types are likely to feed on exactly the same prey and at the same place and time.\u00a0 An example could be where the predator types represent different life history stanzas (age-size classes) of the same predator species with very similar feeding modes (times and locations).<\/p>\n

We represent this possibility in Ecosim by entry of base proportions of each predator type\u2019s diet that occurs in each of the possible foraging arenas defined by all non-zero predator-prey consumption linkages.\u00a0 Vulnerable prey density in each arena is then represented as varying over time in response to abundances of all predator types that feed in the arena.<\/p>\n

In Ecosim, we define a list a <\/em>= 1, \u2026 , Na<\/sub><\/em> of possible foraging arenas, where Na<\/sub><\/em> is the number of non-zero consumption interactions in the Ecopath diet matrix representing consumption of each prey type i<\/em> by predator type j<\/em>.\u00a0 Each of these potential arenas has a defining prey type i(a)<\/em> and defining predator type j(a)<\/em>.<\/p>\n

When only predator type j(a)<\/em> feeds in arena a<\/em>, vulnerable prey density Va<\/sub><\/em> is predicted by the basic foraging arena equation,<\/a><\/p>\n

[latex]V_a=\\frac{v_a \\cdot B_{i(a)}}{v_a+v_a' +\\alpha_a \\cdot P_{j(a)}}\\tag{1}[\/latex]<\/p>\n

Here, va<\/sub><\/em> and v<\/em>a<\/sub>‘ are vulnerability exchange rates of prey to and from arena a<\/em>, Bi(a)<\/sub><\/em> is prey biomass, Pj(a)<\/sub><\/em> is predator abundance (biomass or sum of numbers times search rates per\u00a0 predator for multi-stanza predators), and \u03b1a<\/em><\/sub> is the predator rate of effective search (volume swept per time divided by foraging arena volume).\u00a0 The predation flow rate (biomass of prey i(a)<\/em> consumed per unit of time by predator j(a)<\/em>) is then predicted as Qi(a),j(a) <\/sub><\/em>= \u03b1a<\/sub>Va<\/sub>Pj(a)<\/sub><\/em>.\u00a0 The va<\/sub><\/em> and \u03b1a <\/em><\/sub>are parameterized by having model builders define va<\/em> from maximum possible mortality rates expressed as multiples Ka<\/sub><\/em> of Ecopath base instantaneous predation rates M<\/em>ij<\/sub>(0)<\/sup> = Q<\/em>ij<\/sub>(0) <\/sup>\/ B<\/em>i<\/sub>(0)<\/sup>, simply by setting va<\/sub> = Ka<\/sub><\/em>\u00a0where the superscript (0) designates Q<\/em>\u2019s and B<\/em>\u2019s estimated as base (initial) values of abundances and flows in the Ecopath baseline model.\u00a0 The back-exchange parameter v\u2019<\/em> is set equal to v<\/em> since it cannot be estimated separately from the \u03b1a<\/em><\/sub> parameter.<\/p>\n

The shared-arena extension of Eq. 1<\/a> is straightforward,<\/a><\/p>\n

[latex]V_a=\\frac{v_a \\cdot B_{i(a)}}{v_a+v_a'+ \\sum \\limits_k \\alpha_{ak} \\cdot P_k}\\tag{2}[\/latex]<\/p>\n

Here the predator impact on Va<\/sub><\/em> is represented by a sum over all possible predators k<\/em> of arena-specific search rates \u03b1ak <\/em><\/sub>times predator abundances Pk<\/sub><\/em>. [In the software-implementation of this, we do not actually sum over all k<\/em> but instead construct a list of all non-zero \u03b1ak<\/em><\/sub> flow combinations, and sum the \u03b1ak<\/sub>Pk <\/sub><\/em>denominator terms only over the elements of that list.]<\/p>\n

To parameterize Eq. 2<\/a> in a relatively simple way while assuring that it predicts predation rates equal to Ecopath base rates when the system is at its Ecopath base state, we need to specify base proportions pak<\/sub><\/em> of each predator k<\/em>\u2019s diet that is taken in arena a<\/em>. These proportions are constrained to sum to Ecopath base consumption rates \u00a0over all a<\/em> for which i(a) <\/em>= i<\/em>.\u00a0 That is, we take the by-arena base flows \u00a0to be p<\/em>ak<\/sub> Q<\/em>i(a),k<\/sub>(0)<\/sup>.\u00a0 These base flows then imply a base instantaneous mortality rate M<\/em>a<\/sub>(0)<\/sup>\u00a0totaled over predators feeding in a<\/em>, for prey i(a)<\/em>,<\/p>\n

[latex]M_a^{(0)}=\\frac{\\sum \\limits_k Q_{ak}^{(0)}}{B_{i(a)}}\\tag{3}[\/latex]<\/p>\n

Using this input or baseline estimate of M<\/em> for each arena and an assumed vulnerability multiplier Ka<\/sub><\/em> for that arena, we simply set va <\/sub><\/em>= Ka<\/sub><\/em> (and v<\/em>a<\/sub>‘ = v<\/em>a<\/sub>).<\/p>\n

Next, note that to be consistent with Ecopath baseline inputs, we must require that Ecosim predict Q<\/em>ak<\/sub>(0)<\/sup> when all biomasses (and p<\/em>\u2019s) are at their Ecopath base values. The Ecosim prediction of rate Q<\/em>ak<\/sub> (flow rate of prey to predator k<\/em> from feeding in arena a<\/em>) at any time is Qak <\/sub><\/em>= \u03b1ak<\/sub>Va<\/sub>Pk<\/sub><\/em>, implying we must constrain the \u03b1ak<\/em><\/sub> so that Q<\/em>ak<\/sub>(0)<\/sup> = \u03b1<\/em>ak<\/sub> V<\/em>a<\/sub>(0)<\/sup> P<\/em>k<\/sub>(0)<\/sup>, i.e. we must set \u03b1<\/em>ak <\/sub>= Q<\/em>ak<\/sub>(0)<\/sup> \/ (V<\/em>a<\/sub>(0) <\/sup>P<\/em>k<\/sub>(0) <\/sup>).\u00a0 This means that to estimate the \u03b1ak<\/em><\/sub> we must first estimate the base vulnerable abundances V<\/em>a<\/sub>(0)<\/sup>.<\/p>\n

This estimation turns out to be remarkably simple, when we note that the Ecopath base value of \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k <\/sub>must equal \u2211k\u00a0<\/sub>Q<\/em>ak<\/sub>(0)<\/sup> \/ V<\/em>a<\/sub>(0)<\/sup>, (simply sum Qak<\/sub><\/em> over k<\/em>, which must equal Va = \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k<\/sub>, and solve for \u2211k<\/sub> \u03b1<\/em>ak <\/sub>P<\/em>k<\/sub>). Substituting \u2211k\u00a0<\/sub>Q<\/em>ak<\/sub>(0)<\/sup> \/ V<\/em>a<\/sub>(0)<\/sup> for \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k<\/sub> in Eq. 2<\/a>, then solving for V<\/em>a<\/sub>(0)<\/sup>, we calculate the base vulnerable abundances to be simply,<\/p>\n

[latex]V_a^{(0)}= v_a \\cdot B_{i(a)} - \\frac{\\sum \\limits_k Q_{ak}^{(0)}}{v_a+v_a'}\\tag{4}[\/latex]<\/p>\n

The \u03b1ak<\/em><\/sub> are then calculated from these base vulnerable biomasses. Time-varying values of Qak<\/sub><\/em> are computed efficiently in Ecosim by setting up a list h <\/em>= 1<\/em>, \u2026 , Nh<\/sub><\/em> of all non-zero by-arena flows (N<\/em>h\u00a0<\/sub>\u2265 N<\/em>a<\/sub>), where for each list element we store its associated prey type i(h)<\/em>, predator type k(h)<\/em>, and arena a(h)<\/em>.<\/p>\n

To calculate Qak<\/sub><\/em>, we sweep down this list repetitively.\u00a0 On the first sweep, we accumulate the denominator sums \u2211k<\/sub> \u03b1<\/em>ak<\/sub> P<\/em>k <\/sub>for Eq. 2<\/a>.\u00a0 We then sweep down the arena list and calculate Va<\/sub><\/em> for every a<\/i>\u00a0again using Eq. 2<\/a>.\u00a0 Then we sweep again down the h<\/em> list, calculating Qak <\/sub><\/em>= \u03b1ak<\/sub>Va<\/sub>Pk<\/sub><\/em> and accumulating predictions of total predation rates on the prey i(a)<\/em> and food consumption rates by predators k(a)<\/em>.<\/p>\n

As an added bit of model realism, one can specify a non-zero prey handling times for predator k<\/em> (type II functional response[6]<\/sup><\/a>), and the Qak<\/sub><\/em> calculation is modified to be Qak <\/sub><\/em>= (\u03b1ak<\/sub><\/em>\/Hk<\/sub><\/em>) Va <\/sub>Pk<\/sub><\/em>, where Hk<\/sub><\/em> is the denominator of Holling’s multi-species disc equation for predator k<\/em> feeding.\u00a0 This handling time correction is also applied in the bout-feeding formulation described in the next chapter.<\/p>\n

To edit the pak<\/sub><\/em> diet proportions array, we display a matrix for each prey type i<\/em> of the non-zero i-k <\/em>consumption proportions, as shown schematically in Table 1. In this table, m<\/em> is the number of non-zero flows from prey i<\/em>to predators k<\/em> where each such flow defines a potential foraging arena.\u00a0 Note that each column of the table must sum to 1.0, i.e. all of the consumption by predator kj<\/em> of prey type i<\/em> must be accounted for by feeding in one of the m<\/em> identifiable arenas for prey type i<\/em>.\u00a0 The Ecosim default proportions for this table imply that each predator takes all of its consumption of prey type i<\/em> in a unique arena, i.e. the table is an identity matrix, (with values of 1 on the shaded diagonal in Table 1).<\/p>\n

\"\"<\/p>\n

The opposite extreme of this default assumption would be that all consumption of prey type i<\/em> by its predators occur in only one arena or behavioral state for prey i<\/em>, as shown in Table 2. This case implies maximum possible impact of predators k<\/em> on availability of prey i<\/em> to one another, and will cause competitive exclusion of at least some predator types in Ecosim unless the predators are well-differentiated in terms of overall diet composition, i.e. where each predator “specializes” on a different prey type i<\/em>, which dominates the diet composition, as for instance shown by Schmidt[7]<\/sup><\/a>. Studies rather tend to indicate resource partitioning between competing predator species, leading to non-additive mortality rates<\/a>, see e.g., Griffen and Byers[8]<\/sup><\/a>. Separation where diet compositions indicate predator overlap may also be caused by temporal exclusion of prey based on availability to the predator[9]<\/sup><\/a>.<\/p>\n

\"\"<\/p>\n

In the special case where a set of predators feeds on only one prey type in a single arena (Table 2), and where there are no complications such as multistanza population dynamics where abundance of one or more predator types may be limited by recruitment rates from younger stanzas, the above formulation implies that there is not even a unique equilibrium point for predator abundances.\u00a0 Rather, all predator abundance combinations that predict V=V<\/em>(0<\/em>) in Eq. 2<\/a> are neutral stable points provided predator mortality rates remain at Ecopath base values, such that any temporary pulse of differential mortality that causes one or more predators to decline will then be followed by persistence of the new predator abundance combination if mortality rates return to the base values.\u00a0 Any predator that suffers a persistent differential increase in mortality rate is predicted to decline toward extinction.<\/p>\n<\/div>\n

\n

Attribution\u00a0<\/strong>This chapter was inspired by Kerim Aydin\u2019s work on a foraging arenas, and <\/span>is based on Walters and Christensen. 2007.[10]<\/sup><\/a>, used with permission from Elsevier, Licence Number 5663310244809.<\/p>\n<\/div>\n

  1. 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 7:139-172. https:\/\/doi.org\/10.1023\/A:1018479526149<\/a> ↵<\/a><\/li>
  2. 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 3: 70-83. https:\/\/doi.org\/10.1007\/s100210000011<\/a> ↵<\/a><\/li>
  3. Christensen, V. and C. J. Walters. 2004. Ecopath with Ecosim: methods, capabilities and limitations. Ecol. Model. 172:109-139 https:\/\/doi.org\/10.1016\/j.ecolmodel.2003.09.003<\/a> ↵<\/a><\/li>
  4. Walters, C. J. and Martell, S. J. D., 2004. Fisheries Ecology and Management. Princeton University Press, Princeton. 399 pp. ↵<\/a><\/li>
  5. Aydin, K. Y. and Gaichas, S. K., 2007. In defense of complexity: towards a representation of uncertainty in multispecies models. MS, SC\/58\/E, Alaska Fisheries Science Centre, NOAA, Seattle WA ↵<\/a><\/li>
  6. Holling, C.S., 1959. The components of predation as revealed by a study of small mammal predation of the European pine sawfly 91, 293\u2013320. https:\/\/doi.org\/10.4039\/Ent91293-5<\/a> ↵<\/a><\/li>
  7. Schmidt, K. A., 2004. Incidental predation, enemy-free space and the coexistence of incidental prey. Oikos, 106:335-343. https:\/\/doi.org\/10.1111\/j.0030-1299.2004.13093.x<\/a> ↵<\/a><\/li>
  8. Griffen, B. D. and Byers, J. E., 2006. Partitioning mechanisms of predator interference in different habitats. Oecologia, 146:608-614. https:\/\/doi.org\/10.1007\/s00442-005-0211-4<\/a> ↵<\/a><\/li>
  9. Scheuerell, J. M., Schindler, D. E., Scheuerell, M. D., Fresh, K. L., Sibley, T. H., Litt, A. H. and Shepherd, J. H., 2005. Temporal dynamics in foraging behavior of a pelagic predator. Canadian Journal of Fisheries and Aquatic Sciences, 62:2494-2501. https:\/\/doi.org\/10.1139\/f05-164<\/a> ↵<\/a><\/li>
  10. Walters, C and V. Christensen. 2007. Adding realism to foraging arena predictions of trophic flow rates in Ecosim ecosystem models: shared foraging arenas and bout feeding. Ecological Modelling 209:342-350. https:\/\/doi.org\/10.1016\/j.ecolmodel.2007.06.025<\/a> ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":1909,"menu_order":19,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1724","chapter","type-chapter","status-publish","hentry"],"part":411,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapters\/1724","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/wp\/v2\/users\/1909"}],"version-history":[{"count":16,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapters\/1724\/revisions"}],"predecessor-version":[{"id":3699,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapters\/1724\/revisions\/3699"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/parts\/411"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapters\/1724\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/wp\/v2\/media?parent=1724"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/pressbooks\/v2\/chapter-type?post=1724"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/wp\/v2\/contributor?post=1724"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ewemodel\/wp-json\/wp\/v2\/license?post=1724"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}