Stability And Turing Patterns Of A Predator-prey System With Diffusion

All kinds of speckle patterns exist in nature,which can be divided into those formed in thermodynamic equilibrium or those far away from thermodynamic equilibrium.Generally,we can explain the formation of the first kind of speckle pattern with the law of thermodynamic equilibrium state.However,the study of the second kind of speckle pattern has gradually developed into a branch of nonlinear science,namely,the dynamics of speckle pattern.In this paper,the dynamic behavior of predator-prey system under the influence of diffusion is studied.The conditions of Turing instability are analyzed theoretically,and the results of numerical simulation verify that various spatiotemporal models can be generated in Turing parameter domain.The specific work of this paper is as follows:In chapter 1,Taking time as the number axis,this paper describes the evolution process of species growth model horizontally,especially the research background and significance of predator-prey model.At the same time,it also analyzes the current research status at home and abroad,all of which provide help and theoretical basis for this study.In chapter 2,we study a class of predator-prey system with B-D reaction term and modified Leslie-Gower term.Firstly,we discuss the stability of the normal equilibrium state and the existence of Hopf bifurcation of the ordinary differential system.The results show that when the diffusion speed of the predator is much faster than that of the prey,Turing instability can occur in the system,In addition,by constructing appropriate Lyapunov function,it can be concluded that the positive equilibrium point of the system is globally asymptotically stable under certain conditions.In chapter 3,a kind of predator-prey system with self diffusion and cross diffusion is studied.The conditions of Hopf bifurcation and Turing instability are given through analysis.The Turing parameter domain can be determined.The nonlinear amplitude equation describing spatiotemporal dynamics is derived through weak nonlinear analysis.The theoretical results are verified by numerical simulation.In chapter 4,A kind of predator-prey system with self diffusion and nonlinear staggered diffusion is considered.The local asymptotic stability of the system and the condition of Turing instability are discussed.On the basis of theory,the spatiotemporal model of the prey population is numerically simulated.Through calculation,the global stability condition of the positive equilibrium point of the predator-prey system with staggered diffusion is obtained.The Harnack inequality and the maximum principle are used,A priori estimate of the upper and lower bounds of the positive solution of the system is given.Using the theory of Leray Schauder degree,it is verified that the system can have a non constant positive steady state.Finally,the main conclusions of the paper are summarized,and the future research directions are pointed out.