Qualitative Analysis On The Predator-prey Systems With Undercrowding Effect

In this paper, by using the qualitative theories of differential equation, the coincidence degree theory and Lyapunov function method, aiming at three kinds of predator-prey systems with undercrowding effect, the qualitative behavior of them is discussed by us. Main results are as follows:In the first part, considering the effect of undercrowding effect on the population, a kind of predator-prey system of Holling typeâ…¡function response with undercrowding effect is investigated. And the existence conditions of equilibrium points are given by using the qualitative analysis methods of differential equation. The center and focus is judged by the formal series method. We obtain the sufficient conditions of the global stability of positive equilibrium point by using Dulac criterion and the boundedness of solutions. The existence and uniqueness of the limit cycle is proved by using Bendixson's annular region theorem and the uniqueness theorem of Zhang Zhifen. With utilizing the relationship between the root and coefficient of algebraic equation, and the result of focus quantity we get the sufficient conditions of the existence of Hopf bifurcation. Under the help of Matlab software, the numerical simulation of the system is presented.In the second part, considering the effect of stocking on the system, we study a kind of predator-prey system with undercrowding effect and constant rate stocking of prey species. In virtue of the geometric property of the isoclinic lines, analyze the properties of equilibrium points of the system, and then obtain the sufficient conditions of existence of positive equilibrium point. With utilizing qualitative analysis methods of differential equation, we obtain the sufficient conditions of the global stability of positive equilibrium point and the existence and uniqueness of the limit cycle. The appropriate biological significance of the system is also given. With utilizing the relationship between the root and coefficient of algebraic equation, and the result of focus quantity we get the sufficient conditions of the existence of Hopf bifurcation. By the Matlab software, the numerical simulation of the system is presented.In the third part, taking into account of the impact of the change, which in many acts and properties of the population along with the time, we study a nonautonomous predator-prey system with undercrowding effect. By applying comparison principle, we obtain the sufficient conditions for the persistence of the system. For the nonautonomous periodic system, the sufficient conditions of the existence of the positive periodic solution of the periodic system are obtained through the continuation theorem of coincidence degree theory. At last, the global stability of the positive periodic solution is studied by constructing a suitable Lyapunov function.The research about stability of ecology system can be used to guide people to reasonably and persistently make full use of biological resources and scientifically manage biological resources. Those will have very important theoretical and practical significance to maintain the diversity of the biology in ecosystems and the sustainable development of the ecological environment.