Stability And Hopf Branching Of Reaction-diffusion Models With Time Delays

In this thesis,we study a tumor immune reaction-diffusion model with time delay under homogeneous Neumann boundary conditions and a plankton reactiondiffusion model with time delay,the stability of equilibriums,Hopf bifurcation,persistence and other dynamic properties of the two models are discussed separately by using the Maximum principle,Upper and lower solutions,and Hopf bifurcation theory,and the appropriate biological explanations are given.The main contents in the thesis are as follows:In Chapter 1,we briefly introduce the research background and development status of tumor immune models and plankton models,and introduce two models which will be studied in this article.In Chapter 2,we consider the stability and Hopf bifurcation of the tumor immune reaction-diffusion model with time delay.Firstly,the stability of the boundary equilibrium and the positive equilibrium and the existence of the Hopf bifurcation for model are studied,the results show that the time delay can destroy the stability of the positive equilibrium and affect the existence of the Hopf bifurcation.Secondly,the direction and stability of the Hopf bifurcation is discussed by using the method proposed by Hassard.Finally,numerical simulation is used to verify the correctness of the theoretical results.In Chapter 3,we discuss the dynamic behavior of the phytoplankton ecological reaction-diffusion model.Firstly,the stability and persistence of the boundary equilibrium solution are proved by using the Maximum principle.Secondly,the sufficient condition for the existence and uniqueness of the positive equilibrium is given,and the stability of the equilibrium under this condition is discussed;Then,the existence and properties of the Hopf bifurcation are discussed by using the method proposed by Jianhong Wu,and compared with the second chapter,the method used in this chapter can specifically determine the first four coefficients of the canonical type,so that the formula for judging the properties of Hopf bifurcation can be accurately given.Finally,the correctness of the theoretical results is verified by numerical simulation.