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Stability and Bifurcation Analysis of a Delayed Leslie‐Gower Predator‐Prey System with Nonmonotonic Functional Response

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  • Jiao Jiang
  • Yongli Song

Abstract

A delayed Leslie‐Gower predator‐prey model with nonmonotonic functional response is studied. The existence and local stability of the positive equilibrium of the system with or without delay are completely determined in the parameter plane. Using the method of upper and lower solutions and monotone iterative scheme, a sufficient condition independent of delay for the global stability of the positive equilibrium is obtained. Hopf bifurcations induced by the ratio of the intrinsic growth rates of the predator and prey and by delay, respectively, are found. Employing the normal form theory, the direction and stability of Hopf bifurcations can be explicitly determined by the parameters of the system. Some numerical simulations are given to support and extend our theoretical results. Two limit cycles enclosing an equilibrium, one limit cycle enclosing three equilibria and different types of heteroclinic orbits such as connecting two equilibria and connecting a limit cycle and an equilibrium are also found by using analytic and numerical methods.

Suggested Citation

  • Jiao Jiang & Yongli Song, 2013. "Stability and Bifurcation Analysis of a Delayed Leslie‐Gower Predator‐Prey System with Nonmonotonic Functional Response," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:152459
    DOI: 10.1155/2013/152459
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    References listed on IDEAS

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    1. Li, Yilong & Xiao, Dongmei, 2007. "Bifurcations of a predator–prey system of Holling and Leslie types," Chaos, Solitons & Fractals, Elsevier, vol. 34(2), pages 606-620.
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    Cited by:

    1. Mei Yan & Yongfeng Li & Zhongyi Xiang, 2014. "Time Delayed Stage‐Structured Predator‐Prey Model with Birth Pulse and Pest Control Tactics," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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