# Vulnerability Multiplier Estimator

Carl J. Walters

Simple estimator of minimum vulnerability multiplier needed to allow a predator to rebuild by some assumed amount after removal of all fishing effort

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. 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.

Given this assumption about no prey depletion, an approximation for the predator biomass dynamics is given by the simple differential equation

[latex]dP/dt=gQ-ZP \tag{1}\label{1}[/latex]

where the food conversion efficiency *g* is the ratio of two Ecopath input, *g = (P/B)/(Q/B) = Zo/(Q _{0}/P_{0}), *and the total food consumption rate

*Q*summed over prey types is predicted from vulnerable prey biomass as

*Q=aVP*where vulnerable prey biomass

*V*is given by

[latex]V=vB/(2v+aP) \tag{2}\label{2}[/latex]

The vulnerability exchange rate *v* is calculated in Ecosim as

[latex]v=KQ_0/B_0 \tag{3}\label{3}[/latex]

where *K* is the vulnerability multiplier entered by the user, *Q _{0}* is Ecopath base predator consumption, and

*B*is Ecopath base prey biomass (Ecopath base predation rate is

_{0}*Q*, and the maximum predation rate

_{0}/B_{0}*v*is assumed to be a multiple (

*K*) of this base predation rate). On entry to Ecosim, the predator rate of effective search

*a*is estimated from

*K*and base predator abundance

*P*as

_{0}[latex]a=2v/[(k-1)P_0] \tag{4}\label{4}[/latex]

Note that *v* in (3) and *a* in (4) both depend on the vulnerability multiplier *K*.

In order to derive a simple estimator for the multiplier *K,* we simply set *dP/dt*=0 in (1), while setting *Z=Z _{0}-F* (i.e. set

*Z*to just the natural mortality rate) and

*P*equal to the unfished

*P*or more simply to

*P=RP*where

_{0}*R*is the ratio of the unfished biomass that we want to achieve to the Ecopath base biomass (e.g.

*R*=10 means we want the predator

*P*to be able to grow by a factor of 10 if fishing mortality rate F is set to 0). Setting

*dP/dt*=0 creates the algebraic condition

*eQ=ZP*. Next, we substitute (2) to (4) into this condition, and solve the resulting equation for

*K*. A few tedious algebraic steps then result in the Ecosim estimator for

*K*used in the interface:

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

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. 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.

Most predators in Ecosim models feed on multiple prey types. The predicted consumption rate Q(p) of each prey type *p* as predator and prey biomasses vary can be expressed using Kerim Aydin’s ^{[1]} parameterization of the Ecosim consumption rate equation ^{[2]}

[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]

Provided the *K(p)* are similar for all prey types that contribute significantly to the predator diet and that none show substantial depletion (*B(p)/B _{0}(p)* stays near 1.0 as

*P*grows, for all

*p*), total consumption rate per predator

*∑pQ(p)/P*will behave the same (all

*Q(p)*will increase by the same function of

*P/Po*, diet composition will not change) as if there were just one prey type as assumed in the basic derivation above. That is, the really critical assumption for (5) to work is that all important prey types have similar

*K(p)*and are not depleted (or changed in biomass due to other factors like fishing) as the predator biomass

*P*recovers from its initial level

*P*.

_{0}Note that absent dynamic changes in prey biomasses *B(p)* due to factors other than changes in *P/P _{0}*, (5) represents a minimum estimate of

*K*needed to allow an assumed growth ratio

*R*. If increases in

*P/P*do cause some decreases in key prey type abundances,

_{0}*K*will need to be set to higher values in order to reach the target

*R*, and there may in fact be no

*K*value for which some high assumed

*R*(

*P*) can be achieved, i.e. the modeled prey production rates may simply be incapable of supporting such a high assumed unfished predator biomass. 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

_{unfished}/P_{0}*P*.

_{unfished}A more elaborate, iterative approach is typically needed to set by-stanza *K*’s for multi-stanza populations represented in Ecosim by an age-structured model rather than a simple biomass rate equation. Typically, fishing mortality rates for multistanza populations are substantial only for the older (larger) fish. *K*’s for at least one juvenile stanza are typically set to low values in order to represent density dependent juvenile mortality (stock-recruitment relationship). Biomass of older fish can then be limited either by food availability or the number of recruits. As a first iterative step, increases in biomass of older fish can be represented by setting high *K*’s for older stanzas, so that removal of fishing basically just results in increased biomass-per-recruit (due to increased longevity). 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’s for at least one unfished juvenile stanza, in order to allow both increased numerical recruitment and maintenance of juvenile growth rates.

- 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 ↵
- 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 ↵