Stability And Bifurcation Analysis Of A Class Modified Leslie-Gower Type Predator-prey System

The development for biological populations will be affected by a variety of factors, where the predator-prey relation is one of the most important factors that affect the development of species. On the other hand, human's activities have a also direct effect on the ecological change and can lead to the extinction of species. So the study for the properties of predator-prey system is of great significance. In order to further exploit the interaction relationship between predators and preys, this paper is to study the stability and bifurcation of a kind of modified Leslie-Gower type predator-prey system with a single discrete delay. The structure of this paper is arranged as follows:The first chapter summarizes the research background, research status and significance of modified Leslie-Gower type predator-prey systems, and also put forward the main results in this paper.The second chapter introduces mainly the basic concepts and theorems used in this paper.The third chapter discusses the existence of equilibrium of the system investi-gated in this paper.The fourth chapter studies mainly the local asymptotic stability, global asymp-totic stability, instability and Hopf bifurcation of the positive equilibrium when the system considered has no delay.The fifth chapter analyzes the stability of all feasible equilibria of the system with delay. Under the condition that the other parameters in the system are fixed, by choosing the delay as the bifurcation parameter, the stability and Hopf bifurca-tion of the positive equilibrium are obtained. In addition, by means of the normal form method and the center manifold theorem of delayed differential equations, the explicit formula determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also obtained.