Extinction and permanence of a two-prey one-predator system with impulsive effect

In this paper, we investigate a two-prey one-predator system with impulsive effect on the predator of fixed moment. 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.