### Mathematical Biosciences and Engineering

2009, Issue 1: 145-172. doi: 10.3934/mbe.2009.6.145

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

• Received: 01 May 2008 Accepted: 29 June 2018 Published: 01 December 2008
• MSC : 34D05, 34D20, 92D25

• 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[J]. Mathematical Biosciences and Engineering, 2009, 6(1): 145-172. doi: 10.3934/mbe.2009.6.145

### Related Papers:

• 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.

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• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

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