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**Dynamic complexities of a Holling I predator-prey model concerning periodic biological and chemical control.**
*(English)*
Zbl 1058.92047

Summary: We investigate the dynamic behavior of a Holling I predator-prey model with impulsive effect concerning biological and chemical control strategies – periodic releasing natural enemies and spraying pesticides at different fixed time. By using Floquet theorem and small amplitude perturbation method, we prove that there exists an asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value.

The condition for the permanence of the system is given. It is shown that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Further, the effects of impulsive perturbations on the unforced continuous system is studied. We find that the system we considered has more complex dynamic behavior and is dominated by periodic, quasi-periodic and chaotic solutions. We also find that our impulsive forced system may have different dynamic behaviors with different range of initial values, with which the solutions of the unforced system tend either to the inherent stable limit cycle or to a stable positive equilibrium.

The condition for the permanence of the system is given. It is shown that our impulsive control strategy is more effective than the classical one if we take chemical control efficiently. Further, the effects of impulsive perturbations on the unforced continuous system is studied. We find that the system we considered has more complex dynamic behavior and is dominated by periodic, quasi-periodic and chaotic solutions. We also find that our impulsive forced system may have different dynamic behaviors with different range of initial values, with which the solutions of the unforced system tend either to the inherent stable limit cycle or to a stable positive equilibrium.

### MSC:

92D40 | Ecology |

37N25 | Dynamical systems in biology |

49N25 | Impulsive optimal control problems |

34A37 | Ordinary differential equations with impulses |

34C25 | Periodic solutions to ordinary differential equations |

34C60 | Qualitative investigation and simulation of ordinary differential equation models |

49N90 | Applications of optimal control and differential games |

34D05 | Asymptotic properties of solutions to ordinary differential equations |

### Keywords:

impulsive perturbation analysis
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\textit{B. Liu} et al., Chaos Solitons Fractals 22, No. 1, 123--134 (2004; Zbl 1058.92047)

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