Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay.

*(English)*Zbl 1172.92038Summary: Continuous and impulsive harvesting policies are considered in a predator-prey system with stage-structure. In the case where continuous harvesting is used, it is shown that the mature predator becomes extinct under appropriate conditions. In the case where impulsive harvesting is used, using the discrete dynamical system determined by the stroboscopic map, we obtain the mature predator-eradication periodic solution of the system which is globally attractive. The conditions of permanence are established by the method of comparison involving multiple Lyapunov functions and auxiliary functions.

These results indicate that: a short period of pulse harvest is a sufficient condition for the eradication of the population; the impulsive harvest policy is more effective than the continuous one from the eradicating predator point of view. Our results offer a more economical and safe strategy in controlling pest in contrast with biological control and chemical control. Furthermore, we give a summary of the dynamic behavior when the impulsive period takes values in different intervals. Finally, numerical results show that the impulsive system we considered has more complex dynamics including quasi-periodic oscillation and chaos.

These results indicate that: a short period of pulse harvest is a sufficient condition for the eradication of the population; the impulsive harvest policy is more effective than the continuous one from the eradicating predator point of view. Our results offer a more economical and safe strategy in controlling pest in contrast with biological control and chemical control. Furthermore, we give a summary of the dynamic behavior when the impulsive period takes values in different intervals. Finally, numerical results show that the impulsive system we considered has more complex dynamics including quasi-periodic oscillation and chaos.

##### MSC:

92D40 | Ecology |

34K45 | Functional-differential equations with impulses |

34K20 | Stability theory of functional-differential equations |

37N25 | Dynamical systems in biology |

93C95 | Application models in control theory |

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

65L12 | Finite difference and finite volume methods for ordinary differential equations |

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\textit{Y. Pei} et al., Math. Comput. Simul. 79, No. 10, 2994--3008 (2009; Zbl 1172.92038)

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