Research On The Dynamics Of Reaction-diffusion Systems With Non-local Terms

A variety of reaction-diffusion equations have been widely used to describe the natural phenomena arising in bioscience,material science,chemistry and physics etc.In order to model the behavior of species which is diffusing,aggregating,reproducing and competing for space and resources,mathematicians proposed the nonlocal reaction-diffusion equations.Recently,the study on the nonlocal reaction-diffusion equations focuses on the traveling wave solutions and global solutions of the corresponding problems in a bounded domain in one dimensional case or high dimensional case.In the real world,the species will have a tendency to emigrate from the boundary to obtain their habitats,i.e.,they will move outward along the unknown cure as time increases,people consider the unknown cure as a free boundary.The difficulty of the problem lies in the lack of comparsion principle for the nonlocal reaction-diffusion equations.Then many classical methods cannot be used directly.This motivates us to find new ideas and techniques.Therefore,the research on nonlocal reaction-diffusion equations has important theoretical and practical significance.In this thesis,we investigate the free boundary problems of reaction-diffusion systems with nonlocal effects.The main objective is to understand the influence of the nonlocal term in the form of an integral convolution on the dynamics of the species,and the dynamic behavior of this model is studied.Due to the existence of nonlocal term,when spreading happens,the equilibrium solution structure of this problem is very complicated.We provide some sufficient conditions to guarantee the species stabilizes at a positive equilibrium state.The organization of this thesis is as follows.In Chapter 1,we make a introduction of the background and the development about the research topic.Chapter 2 investigates a free boundary problem of reaction-diffusion model with a nonlocal effect.We derive a spreading-vanishing dichotomy for the invasive population of free boundary problem.Specially,when the species successfully spreads into infinity as time goes to infinity,we proved that the species stabilizes at a positive equilibrium state under rather mild conditions.We also give the sharp criteria governing spreading and vanishing.When spreading happens,we present the estimates of asymptotic spreading speeds.In Chapter 3,we consider a free boundary problem for the reaction-diffusion competition systems with a nonlocal effect.For the weak competition case,when spreading occurs,we provide some sufficient conditions to prove that two competing species stabilize at a positive constant equilibrium state.Furthermore,for the case of successful spreading,we give estimates of the spreading speed.At last,we summarize the results in this thesis.Views and suggestions are put forward for future research work.