Dynamics Of Several Discontinuous Predator-prey Models

Predator-prey system has always been one of the hot issues in mathematics and ecology,and prey shelters have an important influence on the coexistence of predators and prey populations.Combining with the practice,we can also see the necessity of shelters,which can protect the prey when the number of prey is relatively small and avoid the extinction of prey.Therefore,with the study of the classic prey-prey model before,the predator-prey model with shelters emerges.On the basis of predecessors,this paper studies several kinds of predator-prey models with discontinuous strategies.The local and global dynamics of these two models are studied by using the stability theory of plane equations and the right-hand discontinuous differential equation theory,and abundant dynamic behaviors are obtained.The full text includes the following four parts.In chapter 1,introduces some research background and significance of the predatorprey model,the research background and significance of this topic,the development of the predator-prey model,and the structural arrangement of this paper.In chapter 2,introduces some basic mathematical theories involved in studying the model in this paper.In chapter 3,we study a discontinuous predator-prey model with functional response functions of the type I,and decide whether to use refuge according to the density of the prey.By constructing Lyapunov function,we illustrate the stability of the equilibrium in the subsystem.For different parameters,we can get the global dynamics of three cases.In chapter 4,based on the classic predator-prey model,this paper adds the Holing-Ⅰfunction response function,and when the number of predators exceeds the threshold,it takes measures to capture them.The qualitative theory of Filippov is used to analyze the global stability of various equilibrium in this model.In chapter 5,on the basis of the chapter 4,we take measures to capture predators when the ratio of prey to predators is less than the threshold,and use the qualitative theory of Filippov to analyze the global stability of various equilibrium in this model.Finally,in addition to summarizing the research work of this paper,another type of prey shelter is also summarized,and the further prospect of this model is put forward.