Uniqueness of limit cycles and multiple attractors in a Gause-typepredator-prey model with nonmonotonic functional response and Allee effecton prey

Eduardo González-Olivares; Betsabé González-Yañez; Jaime Mena-Lorca; José D. Flores

doi:10.3934/mbe.2013.10.345

Article Contents

2013, Volume 10, Issue 2: 345-367. Doi: 10.3934/mbe.2013.10.345

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Uniqueness of limit cycles and multiple attractors in a Gause-type predator-prey model with nonmonotonic functional response and Allee effect on prey

1.
Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso
2.
Department of Mathematics, The University of South Dakota, Vermillion, SD 57069-2390

Received: September 30, 2011

Revised: July 31, 2012

Published: December 2012

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Abstract

The main purpose of this work is to analyze a Gause type predator-prey model in which two ecological phenomena are considered: the Allee effect affecting the prey growth function and the formation of group defence by prey in order to avoid the predation.
We prove the existence of a separatrix curves in the phase plane, determined by the stable manifold of the equilibrium point associated to the Allee effect, implying that the solutions are highly sensitive to the initial conditions.
Trajectories starting at one side of this separatrix curve have the equilibrium point $(0,0)$ as their $\omega $-limit, while trajectories starting at the other side will approach to one of the following three attractors: a stable limit cycle, a stable coexistence point or the stable equilibrium point $(K,0)$ in which the predators disappear and prey attains their carrying capacity.
We obtain conditions on the parameter values for the existence of one or two positive hyperbolic equilibrium points and the existence of a limit cycle surrounding one of them. Both ecological processes under study, namely the nonmonotonic functional response and the Allee effect on prey, exert a strong influence on the system dynamics, resulting in multiple domains of attraction.
Using Liapunov quantities we demonstrate the uniqueness of limit cycle, which constitutes one of the main differences with the model where the Allee effect is not considered. Computer simulations are also given in support of the conclusions.

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Mathematics Subject Classification: Primary: 92D25, 34C; Secondary: 58F14, 58F21.

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References

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