| In recent years, the study on the ecology has been a hot topic to many scientists. Prey-predator system is an important branch of ecological system. It is mainly used mathematical ideas, established mathematical model, and used the theory and method of differential equation, to study the population living conditions, the relationship between population and environment, so as to reveal the change of population, promote ecological balance.This paper mainly uses the qualitative and stability theory of ordinary differen-tial equation and bifurcation theory, study the dynamic behavior of a prey-predator system with non-monotonic functional response. The thesis is divided into six chap-ters.The first chapter is the introduction. This chapter mainly introduces (1)the origin and development of the theory of bifurcation, bifurcation problem, the term bifurcation appeared for the first time. It got a lot of bifurcation phenomenon after many experiments, and the bifurcation theory is widely used in real life.(2)The development and current situation of functional response. It is mainly summarize and sum up the image of the response function for each stage, as well as the existing problems. This paper mainly research a predator-prey model with non-monotonic functional response.(3)The Allee effect, which mainly introduces strong Allee effect and weak Allee effect for the need in this paper.(4)The development and present situation of predator-prey system with functional response. Through analysis the model in the reference [1][2], we get the model of this paper, it is as follows: andThe second chapter is the preliminary knowledge. This chapter mainly intro-duces the basic concepts and lemmas needed in this paper, include:the stability theory, the theory of equilibrium bifurcation theory and Hopf bifurcation theory, the Gronwall lemma.The third chapter discusses the existence and stability of equilibrium points of the predator-prey system with non-monotonic functional response in (a,d) plane. According to the model, we know that the system may have five equilibrium points, which have three boundary equilibrium points, also there are two possible equilib-rium points. But the existence of equilibrium point depends on different parameters. In order to study the influence of strong Allee effect on predator density, the paper chooses a and d as the bifurcation parameter, and consider the existence and sta-bility of equilibrium point in the (a, d) plane. By using Gronwall lemma, we prove the existence of equilibrium point, as long as the equilibrium exists it is bounded. By using the combination of number and shape and analysis the existing conditions of equilibrium point in (a,d) plane. Then we obtain the distribution of equilibrium point. By discussing the change matrix of any equilibrium point, we obtain the types of the equilibrium points and their local stability.The fourth chapter discusses bifurcation phenomenon about the predator-prey system with non-monotonic functional response in (a,d) plane, including a saddle node bifurcation and Hopf bifurcation, by calculating the coefficient of Liapunov and proveing the supercritical Hopf curve is existence.The fifth chapter discusses limit cycle. This chapter mainly discusses the system is no limit cycle in the first quadrant as long as a hypothesis which satisfied the conditions.The sixth chapter is the summary and prospect of this paper. This chapter mainly summarized the main conclusion:First, we discuss the number of solutions in the system and then study the existence and the existence of the regional about equilibrium point in (a,d) plane. According to the analysis of the stability of equilib-rium point, we get a number of conclusions; Second, we research various bifurcation curves appear in (a, d) plane, prove that the Hopf bifurcation curve is supercritical. In the fourth chapter, we cannot calculate the expression of the Hopf bifurcation curves, therefore cannot determine whether there is a Bogdanov-Takens branch. |
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