The effect of predator avoidance and travel time delay on the stability of predator-prey metacommunities

Abstract

The stability conditions for an isolated specialist predator-prey community are fairly well understood. The spatial coupling of several such systems through dispersal of individuals can generate new dynamic behavior that is not yet completely understood. Many factors are known to be stabilizing or neutral, e.g., random dispersal or time delays, while others may induce instabilities in some cases but not others, e.g., density-dependent movement. We study the combination of two stabilizing mechanisms in a two-patch Rosenzweig-MacArthur model with a novel density-dependent movement term. Specifically, we assume that prey move between patches according to their perceived predation risk, and we include travel time between patches as a time delay. We show that the combination of mechanisms may be destabilizing even though each mechanism by itself is stabilizing. Our results show that a detailed knowledge of mechanisms and their temporal scales is necessary to correctly predict the stability of a metacommunity.

Appendices

Appendix A: Stability of the coexistence steady state when τ=0

In this appendix, we demonstrate that the coexistence steady state for the two-patch model (2.2) with τ=0 is stable if and only if the coexistence state is stable for the single-patch model (2.3). The transformation introduced in Hauzy et al. (2010) gives a block-diagonal matrix whose stability properties can easily be determined.

We denote the vector of densities on patch i with X _i =(h _i ,p _i )^T. Then system (2.2) with τ=0 can be written as

$$\begin{array}{@{}rcl@{}} \frac{d X_{1}}{d s}&=& F(X_{1})-G(X_{1})+G(X_{2}) \\ \frac{d X_{2}}{ds}&=& F(X_{2})-G(X_{2})+G(X_{1}), \end{array} $$

(4.1)

where

$$\begin{array}{@{}rcl@{}} F(X_{i})&=&\left( \epsilon(h_{i}(1-\frac{h_{i}}{k} )-\frac{h_{i}p_{i}}{1+h_{i}}), \frac{h_{i}p_{i}}{1+h_{i}}-\mu p_{i}\right),\\ G(X_{i})&=&\left( d(\alpha \gamma\frac{p_{i}h_{i}}{1+h_{i}}+(1-\alpha)h_{i}), 0\right). \end{array} $$

The crucial transformation consists of the change of variables: U=(X ₁+X ₂)/2,V=(X ₁−X ₂)/2. Then, system (4.1) is equivalent to

$$\begin{array}{@{}rcl@{}} \frac{d U}{d s}&=& \frac{F(U+V)+F(U-V)}{2}, \\ \frac{d V}{d s}&=& \frac{F(U+V)-F(U-V)}{2}\\ &&+G(U-V)-G(U+V). \end{array} $$

(4.2)

Linearizing system (4.2) at the coexistence state U ^∗=(h ^∗,p ^∗)^T,V ^∗=(0,0)^T, we get the linear system

$$\begin{array}{@{}rcl@{}} \frac{d U}{d s}&=& DF(U^{*})U \\ \frac{d V}{d s}&=& [DF(U^{*})-2DG(U^{*})]V, \end{array} $$

(4.3)

where D F(⋅) and D G(⋅) denote the Jacobian matrix of functions F and G, respectively. This system is in block-diagonal form, and the steady state is stable if all eigenvalues of the two matrices D F(U ^∗) and D F(U ^∗)−2D G(U ^∗) have negative real parts.

We calculate both matrices.

$$\begin{array}{@{}rcl@{}} \kern4.4pc DF(U^{*})=\left( \begin{array}{ll} \epsilon(1-2h^{*}/k-p^{*}/(1+h^{*})^2)\kern0.8pc - & \kern-0.3pc \epsilon \mu \\ \kern1.7pc p^{*}/(1+h^{*})^2 &\kern-0.3pc 0 \end{array} \right),\end{array} $$

and D F(U ^∗)−2D G(U ^∗)=

$$\begin{array}{@{}rcl@{}} \ \left( \begin{array}{ll} \!\! \epsilon(1-2h^{*}/k-p^{*}/(1+h^{*})^{2})-2d(\alpha \gamma p^{*}/(1+h^{*})^{2} +(1-\alpha))\ \ \ \ -(\epsilon+2d\alpha \gamma) \mu \!\!\! \\ \kern7.9pc p^{*}/(1+h^{*})^{2} \kern12.8pc 0 \\ \end{array} \right)\!.{}\\ \end{array} $$

(4.4)

Since the determinants of both matrices are positive, stability depends on the sign of the two traces. Since the trace of the second matrix is smaller than the one of the first, the coexistence state is stable precisely when the trace of D F(U ^∗) is negative. But the matrix D F(U ^∗) is precisely the community matrix of the single-patch Rosenzweig-MacArthur model (2.3).

The same method gives the additional result that (k,0,k,0) is stable for Eq. 2.2 if and only if (k,0) is stable for the single-patch system. That is, the local stability of symmetric equilibria does not depend on the values of d and α.

Appendix B: Zeros of a transcendental polynomial

A key element in our analysis is the control of eigenvalues with a positive real part of the linearization at the coexistence state. The following lemma from (Cooke and Grossman 1982) shows that in our case, an eigenvalue with a positive real part can only appear by passing through the imaginary axis and not, for example, in a saddle-node bifurcation. For ease of referral and convenience of the reader, we state this not-so-well-known lemma here.

Lemma 4.1 (Cooke and Grossman (1982))

Let f(λ,τ)=λ ² +a ₁ λ+a ₂ λe ^−τλ +a ₃ +a ₄ e ^−λτ , where a _i and τ are real numbers and τ≥0. Then, as τ varies, the sum of the multiplicities of zeros of f in the open right half-plane can change only if a zero appears on or crosses the imaginary axis.