2009, 6(1): 145-172. doi: 10.3934/mbe.2009.6.145

The dynamics of a simple Laissez-Faire model with two predators

1. 

University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420, United States, United States

Received  May 2008 Revised  June 2008 Published  December 2008

In this paper, we study the dynamics of a laissez-faire predator--prey model with both a specialist and a generalist predator. We analyze the stabilities of equilibria by performing linearized stability analyses. We then reexamine the stability of the equilibrium where the prey and predator coexist by constructing a Lyapunov function. If we hold the generalist predator population constant, treating it as a bifurcation parameter, we show that our model can possess multiple (up to three) limit cycles that surround an equilibrium in the interior of the first quadrant. Our model shows rich dynamics including fold, transcritical, pitchfork, Hopf, cyclic-fold, and Bautin bifurcations as well as heteroclinic connections. If we instead vary the generalist predator population slowly across bifurcations, the model exhibits bursting behavior as it alternates between a repetitive spiking phase and a quiescent phase.
Citation: Gunog Seo, Mark Kot. The dynamics of a simple Laissez-Faire model with two predators. Mathematical Biosciences & Engineering, 2009, 6 (1) : 145-172. doi: 10.3934/mbe.2009.6.145
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