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

Pages: 145 - 172, Volume 6, Issue 1, January 2009

doi:10.3934/mbe.2009.6.145       Abstract        Full Text (1578.2K)       Related Articles

Gunog Seo - University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420, United States (email)
Mark Kot - University of Washington, Applied Mathematics, Box 352420, Seattle, WA 98195-2420, United States (email)

Abstract: 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.

Keywords:  laissez-faire predator--predator--prey model, Bautin bifurcation, heteroclinic connection, bursting
Mathematics Subject Classification:  34D05, 34D20, 92D25

Received: May 2008;      Accepted: June 2008;      Published: December 2008.