{"id":38,"date":"2023-04-27T09:47:07","date_gmt":"2023-04-27T13:47:07","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/eweguide\/chapter\/vulnerability\/"},"modified":"2024-05-30T17:46:13","modified_gmt":"2024-05-30T21:46:13","slug":"vulnerability-multiplier-estimator","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/eweguide\/chapter\/vulnerability-multiplier-estimator\/","title":{"raw":"Vulnerability Multiplier Estimator","rendered":"Vulnerability Multiplier Estimator"},"content":{"raw":"Simple estimator of minimum vulnerability multiplier needed to allow a predator to rebuild by some assumed amount after removal of all fishing effort\r\n\r\nThe Ecosim vulnerability multiplier interface at Ecosim > Input > Vulnerabilities > Estimate vulnerabilities allows users to obtain an initial estimate of the average prey vulnerability multiplier needed to allow a predator population to recover to some much higher level (e.g., a much higher historical abundance than the Ecopath base abundance) if fishing mortality were removed.\u00a0 This initial estimate is derived by assuming that the predator will not deplete any prey populations when it recovers, i.e. total prey biomass Bo will remain similar to the Ecopath base value.\r\n

Given this assumption about no prey depletion, an approximation for the predator biomass dynamics is given by the simple differential equation<\/p>\r\n<\/a>[latex]dP\/dt=gQ-ZP \\tag{1}\\label{1}[\/latex]\r\n

where the food conversion efficiency g<\/em> is the ratio of two Ecopath input, g = (P\/B)\/(Q\/B) = Zo\/(Q0<\/sub>\/P0<\/sub>), <\/em>and the total food consumption rate Q<\/em> summed over prey types is predicted from vulnerable prey biomass as Q=aVP<\/em> where vulnerable prey biomass V<\/em> is given by<\/p>\r\n<\/a>[latex]V=vB\/(2v+aP) \\tag{2}\\label{2}[\/latex]\r\n

The vulnerability exchange rate v<\/em> is calculated in Ecosim as<\/p>\r\n<\/a>[latex]v=KQ_0\/B_0 \\tag{3}\\label{3}[\/latex]\r\n

where K<\/em> is the vulnerability multiplier entered by the user, Q0<\/sub><\/em> is Ecopath base predator consumption, and B0<\/sub><\/em> is Ecopath base prey biomass (Ecopath base predation rate is Q0<\/sub>\/B0<\/sub><\/em>, and the maximum predation rate v<\/em> is assumed to be a multiple (K<\/em>) of this base predation rate).\u00a0 On entry to Ecosim, the predator rate of effective search a<\/em>\u00a0is estimated from K<\/em> and base predator abundance P0<\/sub><\/em>\u00a0as<\/p>\r\n<\/a>[latex]a=2v\/[(k-1)P_0] \\tag{4}\\label{4}[\/latex]\r\n\r\nNote that v<\/em> in (3<\/a>) and a<\/em> in (4<\/a>) both depend on the vulnerability multiplier K<\/em>.\r\n

In order to derive a simple estimator for the multiplier K,<\/em> we simply set dP\/dt<\/em>=0 in (1<\/a>), while setting Z=Z0<\/sub>-F<\/em>\u00a0 (i.e. set Z<\/em> to just the natural mortality rate) and P<\/em> equal to the unfished P<\/em> or more simply to P=RP0<\/sub><\/em> where R<\/em> is the ratio of the unfished biomass that we want to achieve to the Ecopath base biomass (e.g. R<\/em>=10 means we want the predator P<\/em> to be able to grow by a factor of 10 if fishing mortality rate F is set to 0).\u00a0 Setting dP\/dt<\/em>=0 creates the algebraic condition eQ=ZP<\/em>.\u00a0 Next, we substitute (2<\/a>) to (4<\/a>) into this condition, and solve the resulting equation for K<\/em>.\u00a0\u00a0 A few tedious algebraic steps then result in the Ecosim estimator for K<\/em> used in the interface:<\/p>\r\n<\/a>[latex]K=(R-1)HP_0\/(Q_0\/B_0-HP_0) \\tag{5}\\label{5}[\/latex]\r\n\r\nwhere H=Z\/(eB0). Note again in this equation R=P\/P0, implying that we need to assume proportionally larger K values for larger values of K.\u00a0 This equation can be parameterized in other ways using Ecopath input values, e.g., by noting that Q0 is estimated in Ecopath as input consumption per biomass (Q\/B=Q0\/P0) times P0.\r\n

Most predators in Ecosim models feed on multiple prey types.\u00a0 The predicted consumption rate Q(p) of each prey type p<\/em> as predator and prey biomasses vary can be expressed using Kerim Aydin\u2019s [footnote]Aydin, K.Y., 2004. Age structure or functional response? Reconciling the energetics of surplus production between single-species models and ECOSIM. African Journal of Marine Science, 26, pp.289-301. https:\/\/doi.org\/10.2989\/18142320409504062<\/a>[\/footnote]\u00a0parameterization of the Ecosim consumption rate equation [footnote]see also Whitehouse, G.A. and Aydin, K.Y., 2020. Assessing the sensitivity of three Alaska marine food webs to perturbations: an example of Ecosim simulations using Rpath. Ecological Modelling, 429, p.109074. https:\/\/doi.org\/10.1016\/j.ecolmodel.2020.109074<\/a>[\/footnote]<\/p>\r\n[latex]Q(p)=Q_0(p)K(p)[B(p)\/B_0(p)][P\/P_0]\/[K(p)-1+P\/P_0]\\tag{6}\\label{6}[\/latex]\r\n

Provided the K(p)<\/em> are similar for all prey types that contribute significantly to the predator diet and that none show substantial depletion (B(p)\/B0<\/sub>(p)<\/em> stays near 1.0 as P<\/em> grows, for all p<\/em>), total consumption rate per predator \u2211pQ(p)\/P<\/em> will behave the same (all Q(p)<\/em> will increase by the same function of P\/Po<\/em>, diet composition will not change) as if there were just one prey type as assumed in the basic derivation above.\u00a0 That is, the really critical assumption for (5<\/a>) to work is that all important prey types have similar K(p)<\/em> and are not depleted (or changed in biomass due to other factors like fishing) as the predator biomass P<\/em> recovers from its initial level P0<\/sub><\/em>.<\/p>\r\n

Note that absent dynamic changes in prey biomasses B(p)<\/em> due to factors other than changes in P\/P0<\/sub><\/em>, (5<\/a>) represents a minimum estimate of K<\/em> needed to allow an assumed growth ratio R<\/em>.\u00a0 If increases in P\/P0<\/sub><\/em>\u00a0do cause some decreases in key prey type abundances, K<\/em> will need to be set to higher values in order to reach the target R<\/em>, and there may in fact be no K<\/em> value for which some high assumed R<\/em> (Punfished<\/sub>\/P0<\/sub><\/em>) can be achieved, i.e. the modeled prey production rates may simply be incapable of supporting such a high assumed unfished predator biomass.\u00a0 This happens for example in models where prey biomasses and net production at the Ecopath base system state are substantially lower (e.g., due to fishing) than the prey biomasses that supported the historically higher Punfished<\/sub><\/em>.<\/p>\r\n

A more elaborate, iterative approach is typically needed to set by-stanza K<\/em>\u2019s for multi-stanza populations represented in Ecosim by an age-structured model rather than a simple biomass rate equation.\u00a0 Typically, fishing mortality rates for multistanza populations are substantial only for the older (larger) fish.\u00a0 K<\/em>\u2019s for at least one juvenile stanza are typically set to low values in order to represent density dependent juvenile mortality (stock-recruitment relationship).\u00a0 Biomass of older fish can then be limited either by food availability or the number of recruits.\u00a0 As a first iterative step, increases in biomass of older fish can be represented by setting high K<\/em>\u2019s for older stanzas, so that removal of fishing basically just results in increased biomass-per-recruit (due to increased longevity).\u00a0 If that biomass-per-recruit response is insufficient to predict assumed recovery of biomass of older fish, it will then be necessary as a second iterative step to increase the K\u2019s for at least one unfished juvenile stanza, in order to allow both increased numerical recruitment and maintenance of juvenile growth rates.<\/p>","rendered":"

Simple estimator of minimum vulnerability multiplier needed to allow a predator to rebuild by some assumed amount after removal of all fishing effort<\/p>\n

The Ecosim vulnerability multiplier interface at Ecosim > Input > Vulnerabilities > Estimate vulnerabilities allows users to obtain an initial estimate of the average prey vulnerability multiplier needed to allow a predator population to recover to some much higher level (e.g., a much higher historical abundance than the Ecopath base abundance) if fishing mortality were removed.\u00a0 This initial estimate is derived by assuming that the predator will not deplete any prey populations when it recovers, i.e. total prey biomass Bo will remain similar to the Ecopath base value.<\/p>\n

Given this assumption about no prey depletion, an approximation for the predator biomass dynamics is given by the simple differential equation<\/p>\n

<\/a>[latex]dP\/dt=gQ-ZP \\tag{1}\\label{1}[\/latex]<\/p>\n

where the food conversion efficiency g<\/em> is the ratio of two Ecopath input, g = (P\/B)\/(Q\/B) = Zo\/(Q0<\/sub>\/P0<\/sub>), <\/em>and the total food consumption rate Q<\/em> summed over prey types is predicted from vulnerable prey biomass as Q=aVP<\/em> where vulnerable prey biomass V<\/em> is given by<\/p>\n

<\/a>[latex]V=vB\/(2v+aP) \\tag{2}\\label{2}[\/latex]<\/p>\n

The vulnerability exchange rate v<\/em> is calculated in Ecosim as<\/p>\n

<\/a>[latex]v=KQ_0\/B_0 \\tag{3}\\label{3}[\/latex]<\/p>\n

where K<\/em> is the vulnerability multiplier entered by the user, Q0<\/sub><\/em> is Ecopath base predator consumption, and B0<\/sub><\/em> is Ecopath base prey biomass (Ecopath base predation rate is Q0<\/sub>\/B0<\/sub><\/em>, and the maximum predation rate v<\/em> is assumed to be a multiple (K<\/em>) of this base predation rate).\u00a0 On entry to Ecosim, the predator rate of effective search a<\/em>\u00a0is estimated from K<\/em> and base predator abundance P0<\/sub><\/em>\u00a0as<\/p>\n

<\/a>[latex]a=2v\/[(k-1)P_0] \\tag{4}\\label{4}[\/latex]<\/p>\n

Note that v<\/em> in (3<\/a>) and a<\/em> in (4<\/a>) both depend on the vulnerability multiplier K<\/em>.<\/p>\n

In order to derive a simple estimator for the multiplier K,<\/em> we simply set dP\/dt<\/em>=0 in (1<\/a>), while setting Z=Z0<\/sub>-F<\/em>\u00a0 (i.e. set Z<\/em> to just the natural mortality rate) and P<\/em> equal to the unfished P<\/em> or more simply to P=RP0<\/sub><\/em> where R<\/em> is the ratio of the unfished biomass that we want to achieve to the Ecopath base biomass (e.g. R<\/em>=10 means we want the predator P<\/em> to be able to grow by a factor of 10 if fishing mortality rate F is set to 0).\u00a0 Setting dP\/dt<\/em>=0 creates the algebraic condition eQ=ZP<\/em>.\u00a0 Next, we substitute (2<\/a>) to (4<\/a>) into this condition, and solve the resulting equation for K<\/em>.\u00a0\u00a0 A few tedious algebraic steps then result in the Ecosim estimator for K<\/em> used in the interface:<\/p>\n

<\/a>[latex]K=(R-1)HP_0\/(Q_0\/B_0-HP_0) \\tag{5}\\label{5}[\/latex]<\/p>\n

where H=Z\/(eB0). Note again in this equation R=P\/P0, implying that we need to assume proportionally larger K values for larger values of K.\u00a0 This equation can be parameterized in other ways using Ecopath input values, e.g., by noting that Q0 is estimated in Ecopath as input consumption per biomass (Q\/B=Q0\/P0) times P0.<\/p>\n

Most predators in Ecosim models feed on multiple prey types.\u00a0 The predicted consumption rate Q(p) of each prey type p<\/em> as predator and prey biomasses vary can be expressed using Kerim Aydin\u2019s [1]<\/sup><\/a>\u00a0parameterization of the Ecosim consumption rate equation [2]<\/sup><\/a><\/p>\n

[latex]Q(p)=Q_0(p)K(p)[B(p)\/B_0(p)][P\/P_0]\/[K(p)-1+P\/P_0]\\tag{6}\\label{6}[\/latex]<\/p>\n

Provided the K(p)<\/em> are similar for all prey types that contribute significantly to the predator diet and that none show substantial depletion (B(p)\/B0<\/sub>(p)<\/em> stays near 1.0 as P<\/em> grows, for all p<\/em>), total consumption rate per predator \u2211pQ(p)\/P<\/em> will behave the same (all Q(p)<\/em> will increase by the same function of P\/Po<\/em>, diet composition will not change) as if there were just one prey type as assumed in the basic derivation above.\u00a0 That is, the really critical assumption for (5<\/a>) to work is that all important prey types have similar K(p)<\/em> and are not depleted (or changed in biomass due to other factors like fishing) as the predator biomass P<\/em> recovers from its initial level P0<\/sub><\/em>.<\/p>\n

Note that absent dynamic changes in prey biomasses B(p)<\/em> due to factors other than changes in P\/P0<\/sub><\/em>, (5<\/a>) represents a minimum estimate of K<\/em> needed to allow an assumed growth ratio R<\/em>.\u00a0 If increases in P\/P0<\/sub><\/em>\u00a0do cause some decreases in key prey type abundances, K<\/em> will need to be set to higher values in order to reach the target R<\/em>, and there may in fact be no K<\/em> value for which some high assumed R<\/em> (Punfished<\/sub>\/P0<\/sub><\/em>) can be achieved, i.e. the modeled prey production rates may simply be incapable of supporting such a high assumed unfished predator biomass.\u00a0 This happens for example in models where prey biomasses and net production at the Ecopath base system state are substantially lower (e.g., due to fishing) than the prey biomasses that supported the historically higher Punfished<\/sub><\/em>.<\/p>\n

A more elaborate, iterative approach is typically needed to set by-stanza K<\/em>\u2019s for multi-stanza populations represented in Ecosim by an age-structured model rather than a simple biomass rate equation.\u00a0 Typically, fishing mortality rates for multistanza populations are substantial only for the older (larger) fish.\u00a0 K<\/em>\u2019s for at least one juvenile stanza are typically set to low values in order to represent density dependent juvenile mortality (stock-recruitment relationship).\u00a0 Biomass of older fish can then be limited either by food availability or the number of recruits.\u00a0 As a first iterative step, increases in biomass of older fish can be represented by setting high K<\/em>\u2019s for older stanzas, so that removal of fishing basically just results in increased biomass-per-recruit (due to increased longevity).\u00a0 If that biomass-per-recruit response is insufficient to predict assumed recovery of biomass of older fish, it will then be necessary as a second iterative step to increase the K\u2019s for at least one unfished juvenile stanza, in order to allow both increased numerical recruitment and maintenance of juvenile growth rates.<\/p>\n

  1. Aydin, K.Y., 2004. Age structure or functional response? Reconciling the energetics of surplus production between single-species models and ECOSIM. African Journal of Marine Science, 26, pp.289-301. https:\/\/doi.org\/10.2989\/18142320409504062<\/a> ↵<\/a><\/li>
  2. see also Whitehouse, G.A. and Aydin, K.Y., 2020. Assessing the sensitivity of three Alaska marine food webs to perturbations: an example of Ecosim simulations using Rpath. Ecological Modelling, 429, p.109074. https:\/\/doi.org\/10.1016\/j.ecolmodel.2020.109074<\/a> ↵<\/a><\/li><\/ol><\/div>","protected":false},"author":1909,"menu_order":4,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["carl-j-walters"],"pb_section_license":""},"chapter-type":[],"contributor":[61],"license":[],"class_list":["post-38","chapter","type-chapter","status-publish","hentry","contributor-carl-j-walters"],"part":441,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapters\/38","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/wp\/v2\/users\/1909"}],"version-history":[{"count":14,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapters\/38\/revisions"}],"predecessor-version":[{"id":1631,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapters\/38\/revisions\/1631"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/parts\/441"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapters\/38\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/pressbooks\/v2\/chapter-type?post=38"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/wp\/v2\/contributor?post=38"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/eweguide\/wp-json\/wp\/v2\/license?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}