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**Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge.**
*(English)*
Zbl 1258.34161

Summary: This paper describes a prey-predator model with Holling type II functional response incorporating prey refuge. The equilibria of the proposed system are determined and the behavior of the system is investigated around equilibria. Density-dependent mortality rate for the predator is considered as bifurcation parameter to examine the occurrence of Hopf bifurcation in the neighborhood of the co-existing equilibrium point. Discrete-type gestational delay of predators is also incorporated on the system. The dynamics of the delay induced prey-predator system is analyzed. Delay preserving stability and direction of the system is studied. Global stability of the delay preserving system is shown. Finally, some numerical simulations are given to verify the analytical results, and the system is analyzed through graphical illustrations.

### MSC:

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

92D25 | Population dynamics (general) |

34K20 | Stability theory of functional-differential equations |

34K18 | Bifurcation theory of functional-differential equations |

93C15 | Control/observation systems governed by ordinary differential equations |

37L10 | Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

Full Text:
DOI

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