The Research Of A Ratio-dependent Predator-prey System With Holling-Ⅲ Functional Response

In the cause of dealing with biological problems by establishing the mathematical models, there have been a large number of results to the qualitative analysis for predator-prey systems. However, due to the complexity of biological phenomena, if the functional response function of the predator only depends on the density of the prey, the phenomenon of the ecosystem can not be properly explained. So, some scholars have established the ratio-dependent function based on ecological phenomena. Namely, in the case that the population of the prey is limited, the productivity of the predator should be a function of the ratio of predator to prey. Of the ratio-dependent system with Holling-â…¡type functional response function, people have widely studied. In contrast, few results are found for the ratio-dependent system with Holling-â…¢type functional response. Thus, in this paper, we study a class of ratio-dependent Holling-â…¢predator-prey models as follows:In this paper, we firstly give the qualitative analysis of this system in the autonomous case. Here, we mainly research the equilibrium, the boundedness of positive solutions, the dissipation and the existence of limit cycles of this system, etc. The theorems and methods employed are Poincare-Bendixson theorem, some analysis techniques and so on. In addition, the variations of the environment plays an important role in many ecosystems. In particular, the periodic variations of the environmental usually result in the periodic fluctuations of biological populations. Therefore, we also take into account that parameters in the system are periodic functions, i.e., the system is of periodic coefficients. In the study of the existence of periodic solutions for the system, the theoretical method used is the continuation theorem of coincidence degree theory.This article is divided into five chapters.In the first chapter, the introduction is given. The chapter is divided into two sections. In sectionâ… , we introduce the historical background, the development of Holling type predator-prey systems, and the works that many authors have done on them. In sectionâ…¡, the main research contents of this article are given.In the second chapter, some theorems that we use in this article are introduced. The chapter is divided into three sections. In sectionâ… , we list the fundamental theorem of the theory of the ordinary differential equations. In sectionâ…¡, we summerize the theory on the two-dimensional system. In sectionâ…¢, the continuation theorem is given.In the third chapter, we give the qualitative analysis of this system in autonomous case. The chapter is divided into three sections. In Sectionâ… , we give the analysis of the equilibrium points. In Sectionâ…¡, we discuss the boundedness of positive solutions for this sytem. In section â…¢, an example to illustrate our results is given.In the fourth chapter, we discuss the existence of periodic solutions of this system in nonautonomous case. The chapter is divided into two sections. In Sectionâ… , the main results are proved. In sectionâ…¡, an example is given.In the fifth chapter, we make a summary of our works.