Individual behavioral variation in predator–prey models

Abstract

The role of individual behavioral variation in community dynamics was studied. Behavioral variation in this study does not refer to differences in average responses (e.g., average response between presence and absence of antipredator behavior). Rather it refers to the variation around the average response that is not explained by trivial experimental treatments. First, the effect of behavioral variation was examined based on Jensen’s inequality. In cases of commonly used modeling framework with type II functional response, neglecting behavioral variation (a component of encounter rate) causes overestimation of predation effects. The effect of this bias on community processes was examined by incorporating the behavioral variation in a commonly used consumer-resource model (Rosenzweig–MacArthur model). How such a consideration affects a model prediction (paradox of enrichment) was examined. The inclusion of behavioral variation can both quantitatively and qualitatively alter the model characteristics. Behavioral variation can substantially increase the stability of the community with respect to enrichment.

Appendix: Model from Genkai-Kato and Yamamura (1999)

There are two prey species R ₁ and R ₂ and predators N that consume them.

$$ \frac{{\text{d}R_{1} }} {{\text{d}t}} = r_{1} R_{1} {\left( {\frac{{K_{1} - R_{1} - \alpha _{{12}} R_{2} }} {{K_{1} }}} \right)} - \frac{{p_{1} C_{1} R_{1} N}} {{1 + p_{1} C_{1} h_{1} R_{1} + p_{2} C_{2} h_{2} R_{2} }} $$

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$$ \frac{{\text{d}R_{2} }} {{\text{d}t}} = r_{2} R_{2} {\left( {\frac{{K_{2} - \alpha _{{21}} R_{1} - R_{2} }} {{K_{2} }}} \right)} - \frac{{p_{2} C_{2} R_{2} N}} {{1 + p_{1} C_{1} h_{1} R_{1} + p_{2} C_{2} h_{2} R_{2} }} $$

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$$ \frac{{\text{d}N}} {{\text{d}t}} = \frac{{b_{1} p_{1} C_{1} R_{1} N}} {{1 + p_{1} C_{1} h_{1} R_{1} + p_{2} C_{2} h_{2} R_{2} }} + \frac{{b_{2} p_{2} C_{2} R_{2} N}} {{1 + p_{1} C_{1} h_{1} R_{1} + p_{2} C_{2} h_{2} R_{2} }} - dN $$

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Parameter description follows the basic model described in the main text. New parameters are competition coefficients, α₁₂ and α₂₁ that describe the competitive effect of prey species 2 on species 1 and vice versa, respectively. p ₁ and p ₂ are the probability that predators will attack prey species 1 and species 2, respectively. For this behavioral strategy of predators, optimal foraging behavior was assumed. Assuming that species 1 is more profitable (i.e., b ₁/h ₁ > b ₂/h ₂), predators should always attack species 1 (i.e., p ₁ = 1). The theory suggests that p ₂ is either 0 or 1 and does not hold an intermediate value (Stephens and Krebs 1986) and can be determined by the following rule,

$$ p_{2} = \left\{ {\begin{array}{*{20}c} {0} \\ {1} \\ \end{array} } \right.\begin{array}{*{20}c} {{}} \\ {{}} \\ \end{array} \begin{array}{*{20}l} {{{\text{if}}}} \\ {{{\text{otherwise}}}} \\ \end{array} \begin{array}{*{20}c} {{\frac{{b_{1} C_{1} R_{1} }} {{1 + C_{1} h_{1} R_{1} }} > \frac{{b_{2} }} {{R_{2} }}}} \\ {{}} \\ \end{array} $$

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Parameter values used in this study were r ₁ = 0.5, r ₂ = 0.25, d = 0.25, α₁₂ = 0.1, α₂₁ =  0.4, C ₁ = C ₂ = 1, b ₁ = b ₂ = 0.5, h ₁ = 1, h ₂ = 2.083. In Fig. 3b, dynamics for R ₁ and N is plotted. For the model with behavioral variance, the same parameter values for the prey species 1 were used.