Analysis Of The Solutions Of Three Types Of Predator-prey Models

This thesis mainly considers the solutions of three predat,or-prey models.We discuss the exsitence,stability,bifurcation phenomenon of solutions by using some mathematical tools,such as degree theory,eigenvalue theory and bifurcation theory under homogeneous Neumann boundary condition.Chapter 1 is the preface.In this chapter,we introduce the development,and current,states of biological model in recent years,and bring in the three predator-prey models that this thesis will study,analyzing the research meaning of these models.In chapter 2,we add the factor of stage structure to a predator-prey model,and investigate the stability of steady states of the model.Firstly,by the comparison principle,the boundedness of solutions of system is obtained.Furthermore,we prove the local stability of nonnegative solutions of system by means of eigenvalue theory,which is compared with that of ordinary differential equations.By comparison,we find that the homogeneity of space has certain effect on the stability of solutions when some conditions are established.At last,by constructing a Lyapunov function,the conditions that positive solutions are stable globally are given.In chapter 3,we introduce a constant prey refuge and a constant harvest to a.predator-prey model,and study the existence of positive solutions of model.Firstly,a priori estimate of the positive solutions is obtained.Next,the stability of positive constant solution is given.Finally,the conditions of non-existence and existence of non-constant positive solutions are given by using the energy integral method and degree theory.In chapter 4,we investigate a toxic-phytoplankton-zooplankton model with harvesting,studying the properties of bifurcation solutions of model.In the first part,a priori estimate of positive solutions is given,and the local stability of positive solutions is obtained.In the second part,by employing the local bifurcation and stability theory,the conditions of existence and stability of the bifurcation solutions are acquired respectively.In the third part,the bifurcation solutions that we get are extended from local branch to global branch by applying global bifurcation theory.