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Allee Effect in Gause Type Predator-Prey Models: Existence ofMultiple Attractors, Limit cycles and Separatrix Curves. A Brief Review

Published online by Cambridge University Press:  28 November 2013

E. González-Olivares  and
A. Rojas-Palma
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Abstract

This work deals with the consequences on structural stability of Gause type predator-preymodels, when are considered three standard functional responses and the prey growth rateis subject to an Allee effect.

An important consequence of this ecological phenomenon is the existence of a separatrixcurve dividing the behavior of trajectories in the phase plane. The origin is an attractorfor any set of parameters and the existence of heteroclinic curves can be also shown.

Conditions on the parameter values are established to ensure the existence of a uniquepositive equilibrium, which can be either an attractor or a repellor surrounded by one ormore limit cycles.

The influence of the Allee effect on the number of limit cycles is analyzed and theresults are compared with analogous models without this phenomenon, and which mainfeatures have been given in various above works. Ecological interpretations of theseresults are also given.

Keywords

Allee effectfunctional responsepredator-prey modelslimit cyclebifurcationsseparatrix curve.

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Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 6: Ecology , 2013 , pp. 143 - 164
Copyright
© EDP Sciences, 2013

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