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Extinction and permanence of a two-prey one-predator system with impulsive effect. (English) Zbl 1046.92051

Summary: We investigate a two-prey one-predator system with impulsive effects on the predator at fixed moments. By using Floquet’s theorem and small-amplitude perturbation skills, we show that there exists a globally asymptotically stable two-pest eradication periodic solution when the impulsive period is less than some critical value. Further, we prove that the system is permanent if the impulsive period is larger than some critical value, and meanwhile the conditions for the extinction of one of the two prey and permanence of the remaining two species are given. Finally, numerical simulation shows that there exists a stable positive periodic solution with a maximum value no larger than a given level. Therefore, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels.

MSC:

92D40 Ecology
34A37 Ordinary differential equations with impulses
34C60 Qualitative investigation and simulation of ordinary differential equation models
92D25 Population dynamics (general)
34D99 Stability theory for ordinary differential equations
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