The Extinction And Permanent Existence Of An Impulse Lotka--Volterra System

In population ecology, the Lotka-Voterra model is a fundamental one. It can be classified three types according to ecological meaning: predator-prey, compete, cooperation.[]1—4] Especially, predator-prey model has always been a hotspot of research. In this chapter, we investigate a Lotka-Voterra system with impulsive effect. We will obtain some conclusion about solution of the systems, another word, we will use a Lyapunov function such that the systems be permanence or extinguish. But much phenomena exhibit impulsive effects. Impulsive is more natural to describe the processes which characterized by the fact that at certain moments of time. Especially, realism meaning of the systems describing the growth of species and the behavior of epidemic dynamics. Its result of the model become scientific and true.The L—V systems with impulsive have very important meaning of theory and practice. In this paper, by using a typical Lyapunov function to judge permanence and using the systems boundary solution to judge permanence and extinction. Introduce some knowledge which we will use in next paper.Chapter 1 By front knowledge to judge general L—V systems'permanence. Theorem 1.1 Assume a predator-prey system with impulsive is Assume that it has constantτ> 0, m1≤0,and0≤γk≤γsuch that△τk<τand maxfor all k ,so system's solution is uniformly quasipermanent .Chapter 2 Introduce the systems with HollingII function. Consider permanence of the systems with impulsive. We study next system:...