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

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

^⋆ Corresponding author. E-mail: ejgonzal@ucv.cl

Abstract

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

An important consequence of this ecological phenomenon is the existence of a separatrix curve dividing the behavior of trajectories in the phase plane. The origin is an attractor for 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 unique positive equilibrium, which can be either an attractor or a repellor surrounded by one or more limit cycles.

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