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**Dynamic complexities for prey-dependent consumption integrated pest management models with impulsive effects.**
*(English)*
Zbl 1095.92067

Summary: We consider prey-dependent consumption predator-prey (natural enemy-pest) models with age structure for the predator; immature and mature natural enemies are released and pesticide is applied impulsively. We prove that, when the impulsive period is no longer than some threshold, the pest-eradication solution is globally asymptotically stable, or say, the pest population can be eradicated totally. But from the point of ecological balance and saving resources, we only need to control the pest population under the economic threshold level instead of eradicating it totally; so we further prove that, when the impulsive period is longer than the threshold, pest population and natural enemy population can coexist, i.e., the system is uniformly permanent.

Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term to the prey population’s intrinsic growth rate. From two aspects, i.e., when the period of the forcing term is the same as the impulsive period and when the two periods are different, we illustrate that the dynamical behaviors of the corresponding impulsive system are very complex.

Considering population communities always are imbedded in periodically varying environments, and the parameters in ecosystem models may oscillate simultaneously with the periodically varying environments, we add a forcing term to the prey population’s intrinsic growth rate. From two aspects, i.e., when the period of the forcing term is the same as the impulsive period and when the two periods are different, we illustrate that the dynamical behaviors of the corresponding impulsive system are very complex.

### MSC:

92D40 | Ecology |

34A37 | Ordinary differential equations with impulses |

34D23 | Global stability of solutions to ordinary differential equations |

93C95 | Application models in control theory |

37N25 | Dynamical systems in biology |

### Keywords:

uniform permanence
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\textit{J. Hui} and \textit{D. Zhu}, Chaos Solitons Fractals 29, No. 1, 233--251 (2006; Zbl 1095.92067)

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### References:

[1] | Bainov, D. D.; Simeonov, P. S., System with impulsive effect: stability, theory and applications (1989), John Wiley & Sons: John Wiley & Sons New York · Zbl 0676.34035 |

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[3] | Hastings, A., Delay in recruitment at different trophic levels, effects on stability, J Math Biol, 21, 35-44 (1984) · Zbl 0547.92014 |

[4] | Laksmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of impulsive differential equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002 |

[5] | Volterra, V., Variazione e fluttuazini del numerod’individui in specie animali convienti, Mem Acad Nazionale Lincei (Ser. 6), 2, 31-113 (1926) |

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