Qualitative Analysis Of Two Predator-Prey Systems With HollingⅡ Function

With the development of science and technology, the ecological problems become more and more intimate with human. To describe the laws of biology, many models described by difference equations and differential equations are built. Making researches on dynamical property of these models is valuable both in theory and application. Recently, because of the broad application of the predator-prey model, many mathematicians and biologists have paid a great of attention on it. In this paper, we mainly discuss the existence of periodic solutions for a class of ratio-dependent three species mixed model, and the stability and the permanence for a class of ratio-dependent discrete system. The paper has four parts:In the first chapter, we introduce the background and some related researching situations of the biological dynamical system.In the second chapter, by using the coincidence degree theory, which is the common way to research the existence of periodic solutions, we obtain sufficient conditions for the exictence of periodic solutions for a class of ratio-dependent mixed model of three species. There are two preys which are reciprocal, however, the predator and preys are related by the Hollingâ…¡function.In the third chapter, for a class of ratio-dependent discrete system, by making use of some known results and skills of analyzing, we get sufficient conditions for stability of periodic solutions by constructing appropriate Lyapunov function.In the forth chapter, by employing the comparison theorem in theory of difference equations together with pure analysis, we get sufficient conditions for permanence of a class of ratio-dependent discrete system.