Dynamics of Predator–Prey Metapopulations with Allee Effects

Abstract

Allee effects increasingly are recognized as influential determinants of population dynamics, especially in disturbed landscapes. We developed a predator–prey metapopulation model to study the impact of an Allee effect on predator–prey. The model incorporates habitat destruction and predators with imperfect information about prey distribution. Criteria are established for the existence and stability of equilibria, and the possible existence of a limit cycle is discussed. Numerical bifurcation analysis of the model is carried out to examine the impact of Allee effects as well as other key processes on trophic dynamics. Inclusion of Allee effects produces a richer array of dynamics than earlier models in which it was absent. When prey interacts with generalist predators, Allee effects operate synergistically to depress prey populations. Allee effects are more likely to depress occupancy levels when destruction of habitat patches is moderate; at severe levels of destruction, Allee effects are swamped by demographic effects of habitat loss. Stronger Allee effects correspond to lower thresholds of predator colonization rates at which prey become extinct. We discuss implications of our model for conservation of rare species as well as pest management via biocontrol.

Appendices

Appendix 1: The Proof of Lemma 2.1

Since $E^{*}(x^{*},y^{*})$ is an interior equilibrium of (6), one has

$$\begin{aligned} y^{*}=\displaystyle \frac{c_{x}}{\mu p}(p-x^{*})(x^{*}-a), \quad y^{*}=\displaystyle \frac{\psi }{c_{y}}x^{*}+q. \end{aligned}$$

(11)

Then

$$\begin{aligned} J(E^{*}(x^{*},y^{*}))=\left( \begin{array}{cc} \displaystyle \frac{c_{x}}{p}x^{*}(a+p-2x^{*})&{}-x^{*}\mu \\ y^{*}\psi &{}-c_{y}y^{*} \end{array}\right) . \end{aligned}$$

Hence, if

$$\begin{aligned} \det (J(E^{*}(x^{*},y^{*})))= & {} x^*y^*\left[ -\displaystyle \frac{c_{x}c_{y}}{p}(a+p-2x^{*})+\mu \psi \right] >0,\nonumber \\ \mathrm{Tr}(J(E^{*}(x^{*},y^{*})))= & {} \displaystyle \frac{c_{x}}{p}x^{*}(a+p-2x^{*})-c_{y}y^{*}<0, \end{aligned}$$

(12)

then $E^{*}$ is l.a.s.

Note that, if $x^{*}\ge (a+p)/2$, then (12) is satisfied and hence $E^{*}$ is l.a.s.

Now we deal with the case of $x^{*}<(a+p)/2$. If

$$\begin{aligned} x^{*}>\displaystyle \frac{a+p}{2}-\displaystyle \frac{p\mu \psi }{2c_{x}c_{y}}, \end{aligned}$$

then $\det (J(E^{*}(x^{*},y^{*})))>0$. Moreover, if

$$\begin{aligned} \displaystyle \frac{2c_{x}}{p}x^{*}+\displaystyle \frac{c_{y}q}{x^{*}}-\displaystyle \frac{c_{x}(a+p)}{p}+\psi >0, \end{aligned}$$

then $\mathrm{Tr}(J(E^{*}(x^{*},y^{*})))<0$. Therefore, $E^{*}$ is l.a.s. In addition, if

$$\begin{aligned} \displaystyle \frac{2c_{x}}{p}x^{*}+\displaystyle \frac{c_{y}q}{x^{*}}-\displaystyle \frac{c_{x}(a+p)}{p}+\psi <0, \end{aligned}$$

then $\mathrm{Tr}(J(E^{*}(x^{*},y^{*})))>0$. Hence, $E^*$ is an unstable node or focus. If

$$\begin{aligned} x^{*}<\displaystyle \frac{a+p}{2}-\displaystyle \frac{p\mu \psi }{2c_{x}c_{y}}, \end{aligned}$$

then $\det (J(E^{*}(x^{*},y^{*})))<0$. Therefore, $E^{*}$ is a saddle. This completes the proof. $\square$

Appendix 2: Existence of Limit Cycle

If (6) admits a unique non-saddle interior equilibrium $E^*$ being unstable (e.g., (10) is satisfied), then (6) has one limit cycle in $\Omega$ with $E^*$ lying in its interior.

Proof

Define

$$\begin{aligned} \Omega =\{(x,y)\in {\mathbb {R}}^2: 0<x<p, 0<y<y_0\}, \end{aligned}$$

where $y_0>\max \left\{ q+{\psi p}/{c_{y}}, {c_{x}(a-p)^{2}}/{(4\mu p)}\right\}$. We prove the existence of a limit cycle by the Poincaré–Bendixson Theorem and show that $\Omega$ is the desired trapping region. Let us now consider the behavior of the vector field on the boundary of $\Omega$. On the straight line $x=p$, one has

$$\begin{aligned} \displaystyle \frac{\hbox {d}x}{\hbox {d}t}\Big |_{x=p}=-\mu py<0,\quad y>0, \end{aligned}$$

so that the vector field points right-to-left into the interior of $\Omega$. On $y=y_0$ with $x<p$, we have

$$\begin{aligned} \displaystyle \frac{\hbox {d}y}{\hbox {d}t}\Big |_{y=y_{0}}= & {} c_{y}y_{0}(1-y_{0}-D)-e_{y}y_{0}-\psi y_{0}(1-x)\nonumber \\= & {} y_{0}(c_{y}q-c_{y}y_{0}+\psi x)<0, \end{aligned}$$

so that the vector field points top-down into the interior of $\Omega$. In addition, $x=0$ and $y=0$ are trajectories of (6). Therefore, $x=p, y=y_0$, positive x-axis and positive y-axis constitute the outer boundary curve. Since the unique interior equilibrium is unstable, by the Poincaré–Bendixson theorem for annular regions, (6) has at least one stable limit cycle in $\Omega$ containing $E^*$ in its interior. $\square$