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Mathematical Biosciences and Engineering (MBE)
 

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

Pages: 345 - 367, Volume 10, Issue 2, April 2013      doi:10.3934/mbe.2013.10.345

 
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Eduardo González-Olivares - Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile (email)
Betsabé González-Yañez - Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile (email)
Jaime Mena-Lorca - Grupo de Ecología Matemática, Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Valparaíso, Chile (email)
José D. Flores - Department of Mathematics, The University of South Dakota, Vermillion, SD 57069-2390, United States (email)

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.

Keywords:  Predator-prey model, Allee effect, non-monotonic functional response, stability, bifurcation, limit cycle.
Mathematics Subject Classification:  Primary: 92D25, 34C; Secondary: 58F14, 58F21.

Received: October 2011;      Accepted: August 2012;      Available Online: January 2013.

 References