Permanence in an Intraguild Predation Model with Prey Switching

Abstract

Intraguild predation, a form of omnivory that can occur in simple food webs when one species preys on and competes for limiting resources with another species, can have either a stabilizing effect (McCann and Hastings in Proc. R. Soc. Lond. B 264:1249–1254, 1997) or a destabilizing effect (Holt and Polis in Am. Nat. 149:745–764, 1997), depending on the assumptions of the system. Another type of behavior that has been observed in simple food web experiments (Murdoch in Ecol. Monogr. 39:335–354, 1969) is prey switching. Prey switching can occur when the predator prefers the most abundant prey. This has also been shown to be capable of having either a stabilizing effect or a destabilizing effect and even possibly lead to predator extinction (VanLeeuwen et al. in Ecology 88:1571–1581, 2007). Therefore, it is clear that incorporating prey switching into an intraguild predation model could lead to unexpected consequences. In this paper, we propose and explore such a model.

Appendices

Appendix A: Global Asymptotic Stability of Equilibrium

Consider the predator–prey system

(27)

The general assumptions on G(x) and H(x) are:

• (a) G∈C ¹([0,∞),ℝ), G(0)>0 and there exits K>0 such that G(K)=0 and (x−K)G(x)<0 for x≠K. (Note: In our case, we can let G(x)=r(1−x/K).)

• (b) H∈C ¹([0,∞),ℝ), H(0)=0 and H′(x)>0 for all x>0. (Note: In our case, we can let H(x)=bx/(a+x).)

• (c) There exists a locally stable equilibrium (x ^∗,y ^∗) such that mH(x ^∗)−d=0 and x ^∗ G(x ^∗)−y ^∗ H(x ^∗)=0 with 0<x ^∗<K, y ^∗>0, F′(x)<0 for all x ^∗≤x≤K, where $F(x)=\frac{xG(x)}{H(x)}$.

• (d) F(2x ^∗−x)≤F(x) for all x satisfying max{0,2x ^∗−K}≤x≤x ^∗.

• (e) $\frac{d}{H(x)}-m>m-\frac{d}{H(2x^{\ast}-x)}$ for all x satisfying x ^∗≤x<min{2x ^∗,K}.

Theorem 4

(Liou and Cheng 1988)

Under the assumptions (a)–(e), (x ^∗,y ^∗) is globally asymptotically stable for system (27) in the interior of the first quadrant.

Appendix B: Uniqueness and Global Asymptotic Stability of Limit Cycle

Consider the system

(28)

where G, P, and Q are sufficiently smooth so that existence, uniqueness, and continuability for all positive time are satisfied.

Theorem 5

(Kuang and Freedman 1987)

Suppose in the system (28) that

$$\frac{d}{dx} \biggl(\frac{xG'(x)+G(x)-xG(x)\frac{P'(x)}{P(x)}}{-\gamma+Q(x)} \biggr)\le 0 \\ $$

in 0≤x<x ^∗ and x ^∗<x≤K. Then the system has exactly one limit cycle which is globally asymptotically stable with respect to the set {(x,y)∣x>0,y>0}∖{(x ^∗,y ^∗)}.