Stability And Turing Pattern Of Several Kinds Of Prey-predator Systems With Diffusion

The spatiotemporal dynamics between prey and predator has long been one of the main research topics in the ecosystem.The prey-predator model is an important tool for describing the relationship between populations.This paper mainly studies three kinds of prey-predator models with diffusion and cross-diffusion,and the stability and Turing pattern of these systems are discussed.The main work is as follows:In chapter 1,we mainly introduce the background and significance of the prey-predator systems,the research of status at home and abroad,and generalize the work in this paper.In Chapter 2,the stability and Turing pattern of a diffusive prey-predator system with modified Leslie-Gower term are is investigated.The local asymptotic stability of the system and the existence of Hopf bifurcation under certain conditions are proved.And the conditions of Turing instability are given.It is proved that the Turing instability of system can occur only when the predator's diffusion speed is greater than the diffusion rate of the prey.And our con-clusions are proved through the specific numerical examples.Then through the weak nonlinear analysis of system,the amplitude equation of the pattern for the system at the threshold of the Turing bifurcation are derived,based on the method of standard multiple scales.In Chapter 3,The steady state of a ratio-dependent prey-predator systems with diffusion and cross-diffusion is studied.Using the maximal principle and Harnack inequality,a priori estimate of the upper and lower bounds of the system are obtained.Then the Leray-Schauder degree theory is used to prove the existence of the non-constant positive steady state.It indicates that the system has no non-constant positive steady state in the absence of cross-diffusion,while the system has at least one non-constant positive steady state under certain assumptions,when cross-diffusion occurs.In Chapter 4,the stability and Turing pattern of prey-predator system with diffusion and cross-diffusion are studied.By analyzing the stability of the positive solution for the system,the local asymptotic stability of system under certain conditions is proved.When cross-diffusion occurs,the Turing instability emerge and the corresponding Turing parameter space is obtained.Through theoretical analysis and legends,it is proved that the cross-diffusion coefficient d12 suppresses the formation of Turing pattern,and the d21 promotes the pattern generation.Then we verified our theoretical conclusions by numerical simulation,and the spatial distribution regular of the system is given.In the last chapter,we summarize the research contents of this paper,and give some questions that can be further studied.