Bulletin of Mathematical Biology

, Volume 73, Issue 6, pp 1378–1397

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

  • Eduardo González-Olivares
  • Alejandro Rojas-Palma
Original Article

DOI: 10.1007/s11538-010-9577-5

Cite this article as:
González-Olivares, E. & Rojas-Palma, A. Bull Math Biol (2011) 73: 1378. doi:10.1007/s11538-010-9577-5

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.

Keywords

Allee effectSigmoid functional responsePredator–prey modelsLimit cycleBifurcationsSeparatrix curve

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Eduardo González-Olivares
    • 1
  • Alejandro Rojas-Palma
    • 1
  1. 1.Grupo de Ecología Matemática, Instituto de MatemáticasPontificia Universidad Católica de ValparaísoValparaísoChile