The Study Of Stability And Bifurcation On Some Reaction-Diffusion Systems

It is an important research subject for many mathematical modelers to explore the predatory relationship between biological populations in nature by establishing mathematical models.If we only consider the influence of a certain factor on the development of a specific system in a local range,we can study it by establishing an ordinary differential system.If we are concerned about the influence of spatial range,we need to establish a reaction-diffusion system to study the influence of space and time on the development of the system.In this dissertation,we mainly apply Poincaré-Andronov Hopf bifurcation theory,Jacobian matrix eigenvalue theory,normal form method and the center manifold theorem to study the dynamic properties of several types of reactiondiffusion systems,and discuss the stability,Turing instability and Hopf bifurcation of the systems.The first chapter of this paper is the introduction,which mainly describes the background and significance of the establishment of predator-prey system and the research status and analysis process of marine sediment model and predator-prey model with schooling behavior.The second chapter of this paper considers the dynamic properties of the reactiondiffusion minimum sediment model with homogeneous Neumann boundary conditions.Local asymptotic stability,Turing instability and the existence Hopf bifurcation of the unique constant positive equilibrium solution are obtained by analyzing the relevant eigenvalue problem.In addition,the direction and stability of spatially homogeneous Hopf bifurcation are derived by means of the normal form method and the center manifold theorem.The third chapter of this paper presents a reaction diffusion predator-prey system with schooling behavior.Firstly,the local asymptotic stability of the unique positive equilibrium of the local system is discussed,and the existence and related properties of Hopf bifurcation are obtained by using Poincaré-Andronov Hopf bifurcation theorem.Secondly,the local asymptotic stability and Turing instability of the reaction diffusion predator-prey system are studied.Finally,the existence of Hopf bifurcation for spatially homogeneous system and spatially inhomogeneous system are derived,respectively,and the direction and stability of relevant Hopf bifurcation are studied.The fourth chapter summarizes the main conclusions of this dissertation and the future work.