Chaos and Hopf bifurcation analysis for a two species predator-prey system with prey refuge and diffusion.

*(English)*Zbl 1222.34099The authors consider a delayed predator-prey model with Holling type II functional response incorporating a constant prey refuge and diffusion. By analyzing the characteristic equation of the linearized system corresponding to the model, the authors study the local asymptotic stability of the positive equilibrium of the system. By choosing the time delay due to gestation as a bifurcation parameter, the existence of Hopf bifurcations at the positive equilibrium is established. By applying the normal form and the center manifold theory, an explicit algorithm to determine the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is derived. Finally, using a numerical method, the influence of impulsive perturbations on the dynamics of the system is also investigated.

Reviewer: Xinyu Song (Xinyang)

##### MSC:

34K60 | Qualitative investigation and simulation of models involving functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

34K23 | Complex (chaotic) behavior of solutions to functional-differential equations |

92D25 | Population dynamics (general) |

34K20 | Stability theory of functional-differential equations |

34K17 | Transformation and reduction of functional-differential equations and systems, normal forms |

34K19 | Invariant manifolds of functional-differential equations |

34K45 | Functional-differential equations with impulses |

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\textit{X. Liu} and \textit{M. Han}, Nonlinear Anal., Real World Appl. 12, No. 2, 1047--1061 (2011; Zbl 1222.34099)

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##### References:

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