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Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey

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Abstract

This work aims to examine the global behavior of a Gause type predator–prey model considering two aspects: (i) the functional response is Holling type III and, (ii) the prey growth is affected by the Allee effect. We prove the origin of the system is an attractor equilibrium point for all parameter values. It has also been shown that it is the ω-limit of a wide set of trajectories of the system, due to the existence of a separatrix curve determined by the stable manifold of the equilibrium point (m,0), which is associated to the Allee effect on prey. When a weak Allee effect on the prey is assumed, an important result is obtained, involving the existence of two limit cycles surrounding a unique positive equilibrium point: the innermost cycle is unstable and the outermost stable. This property, not yet reported in models considering a sigmoid functional response, is an important aspect for ecologists to acknowledge as regards the kind of tristability shown here: (1) the origin; (2) an interior equilibrium; and (3) a limit cycle of large amplitude. These models have undoubtedly been rather sensitive to disturbances and require careful management in applied conservation and renewable resource contexts.

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  1. Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaíso, Chile

    Eduardo González-Olivares & Alejandro Rojas-Palma

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  1. Eduardo González-Olivares
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  2. Alejandro Rojas-Palma
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Correspondence to Eduardo González-Olivares.

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González-Olivares, E., Rojas-Palma, A. Multiple Limit Cycles in a Gause Type Predator–Prey Model with Holling Type III Functional Response and Allee Effect on Prey. Bull Math Biol 73, 1378–1397 (2011). https://doi.org/10.1007/s11538-010-9577-5

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  • DOI: https://doi.org/10.1007/s11538-010-9577-5

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