Permanence And Stability Of A Two Predator-two Cooperative Prey System

Population dynamics is an important part of biomathematics. It has a far-reaching significancein agriculture and fishery researching the population dynamics for its function of controllingpopulation and guiding harvesting. There are three kinds of interspecific relationship in ecosystem,namely predator-prey, competition and mutualism. But the relationships between species are notalways single. In this article, we mainly discuss a two predator-two cooperative prey model withBeddington-DeAngelis functional response for the predator species individually.This article is divided into five chapters. Time delay and impulsive effect are taken intoaccount in chapter3and chapter4respectively for the reason of actual complexity.We introduce the origin and development of biomathematics in chapter1. Some notablelandmarks in the progress are expounded, so were the background of our work and severalinterrelated definition and lemma.In chapter2, we study the persistence and stability of the two predator-two cooperative preymodel. The permanence of the system is obtained by comparison theorem. Through constructing aLiapunov function, the global asymptotic stability of the system is proved under some appropriateconditions. Further, for the periodic case, we give a set of sufficient conditions, which guaranteethe existence, uniqueness and global asymptotic stability of a positive periodic solution.Based on the model above, time delay is taken into account, we consider this new system inchapter3. Under the condition given by chapter2, we can see that permanence and stability of thesystem stay the same. The difference is the ultimate bounded region of the system is changed withthe time delay factor.We study the system with impulsive effect in chapter4. By using the Floquet theory ofimpulsive differential equation, we obtained a set of sufficient conditions, which guarantee thelocal asymptotic stability of pest-eradication periodic solution. Further, the permanence of thesystem is proved under some appropriate conditions.In the last chapter, we give a result of the work we did above, and its biological significancealso. What’s more, based on the deficiency of this article, we give direction for future research.