Qualitative analysis of Beddington-DeAngelis type impulsive predator-prey models.

*(English)*Zbl 1204.34062This paper discusses the dynamics of a predator-prey model which is subject to periodic impulsive perturbations, understood to describe the effect of predator immigration or stocking and of pesticide spraying. Regarding the characteristics of the predation process, it is assumed that the functional response of the predator is of Beddington-deAngelis type.

First, the model is shown to be biologically well posed, the local stability of the prey-free periodic solution is then discussed by using the Floquet theory of impulsively perturbed systems. It is proved that, if a certain inequality holds, then the prey-free periodic solution is locally stable, while if the converse inequality holds, then the model is shown to be permanent. In the limit case (that is, if the corresponding equality holds instead of the previously mentioned inequalities), a nontrivial periodic solution is shown to bifurcate from the prey-free periodic solution.

A slightly more general model, in which the “proportional” and the “constant” impulsive perturbations occur at different times, but with the same periodicity, has earlier been considered from a similar viewpoint in [H. Zhang, P. Georgescu and L. Chen, On the impulsive controllability and bifurcation of a predator-pest model of IPM, Biosystems 93, No. 3, 151–171 (2009)], where the model discussed by the author is obtained for \(\tilde{l}=1\). The present paper puts more emphasis on the practical consequences of the bifurcation result, it presents a somewhat different approach towards the proof of the permanence of the model.

First, the model is shown to be biologically well posed, the local stability of the prey-free periodic solution is then discussed by using the Floquet theory of impulsively perturbed systems. It is proved that, if a certain inequality holds, then the prey-free periodic solution is locally stable, while if the converse inequality holds, then the model is shown to be permanent. In the limit case (that is, if the corresponding equality holds instead of the previously mentioned inequalities), a nontrivial periodic solution is shown to bifurcate from the prey-free periodic solution.

A slightly more general model, in which the “proportional” and the “constant” impulsive perturbations occur at different times, but with the same periodicity, has earlier been considered from a similar viewpoint in [H. Zhang, P. Georgescu and L. Chen, On the impulsive controllability and bifurcation of a predator-pest model of IPM, Biosystems 93, No. 3, 151–171 (2009)], where the model discussed by the author is obtained for \(\tilde{l}=1\). The present paper puts more emphasis on the practical consequences of the bifurcation result, it presents a somewhat different approach towards the proof of the permanence of the model.

Reviewer: Paul Georgescu (Iaşi)

##### MSC:

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

34A37 | Ordinary differential equations with impulses |

92D25 | Population dynamics (general) |

34C25 | Periodic solutions to ordinary differential equations |

34D20 | Stability of solutions to ordinary differential equations |

##### Keywords:

predator-prey model; impulsive differential equation; Beddington-DeAngelis functional response; Floquet theory; bifurcation theory
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\textit{H. K. Baek}, Nonlinear Anal., Real World Appl. 11, No. 3, 1312--1322 (2010; Zbl 1204.34062)

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##### References:

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