Spatial Dynamics In Nonlocal Reaction-Diffusion Equations

Since the nonlocal reaction-diffusion equation can describe the natural phenomena in physics,chemistry and ecology accurately,it has attracted a great deal of researcher's interests.But the introduction of nonlocal delay makes many research methods on the reaction-diffusion models to be challenged.Meanwhile,nonlocal delay can give rise to many prominent changes on the dynamical behavior.The current study on the traveling wave solution of the nonlocal delayed reaction-diffusion equations focus on such models which are involved with much weaker nonlocal delay,or quasi-monotonic,exponentially quasi-monotonic,weakly quasi-monotonic and weakly exponentially quasi-monotonic reaction term.However,there are few studies on traveling waves of the nonlocal delayed reaction-diffusion equations without those restriction.In addition,it is hard to adequately reveal many significant properties of the nonlocal delayed reaction-diffusion equations.On the other hand,the related investigations on the initial value problem and spatial pattern are also limited.The aforementioned projects are quite important in research field on reaction-diffusion equation.This thesis is devoted to the traveling wave solution,the initial problem and pattern dynamics for some classes of nonlocal reaction-diffusion equations.The main results are divided into five parts.Firstly,we study the traveling waves for a single population nonlocal reactiondiffusion model with Allee effect.Since the comparison principle for the current problem is no longer valid,many classical methods based on the comparison principle,such as super and sub-solution or sliding method are out of work.Thus,by using Leray-Schauder degree theory,we prove that the model admits positive traveling wave solutions connecting the equilibrium 0 to some unknown positive steady state if and only if the wave speed c ? 2r^1/2,where r > 0 is the intrinsic rate of the species.For the sufficiently large wave speed c,we further show that the unknown steady state is the unique positive equilibrium by variation of constant and CauchySchwartz inequality.For two types of convolution kernel functions,we investigate the change of the wave profile as the nonlocality increases and illustrate that the unknown steady state may be a positive periodic solution.Secondly,we investigated the traveling waves for a nonlocal nonlocal reactiondiffusion equation with aggregation term.Due to the emergence of aggregation term,that solutions of the model can not be controlled by the solutions of the linearized equation of this model at the zero equilibrium.By using the auxiliary equations to construct suitable upper solutions,we show that nonlocal reaction-diffusion equation with aggregation term exists traveling wave solutions connecting 0 to some unknown positive steady state.For sufficiently large speed,we also proved that the unknown positive steady state just is the equilibrium.In addition,on the basis of the lower and upper solutions method,we establish the existence of a monotone traveling wave front connecting the zero equilibrium to the positive equilibrium.Finally,for a given specific kernel function,we show numerically that the traveling wave solutions may connect the zero equilibrium to a periodic steady state as the nonlocality increased.Furthermore,by the stability analysis we explain why and when a periodic steady state can appear.Thirdly,we are interested in an initial value problem of a predator-prey system with integral term.By giving a new definition of super and sub-solution for the problem and using some auxiliary functions,we establish comparison principle and construct monotone sequences,then by them we show the existence and uniqueness for the solution.With the aid of some auxiliary equation,we further present the uniform boundedness of solutions.Finally,some conditions of occurring Turing bifurcation are given and numerical simulations are carried out to illustrate them.Fourthly,we explore the traveling wave solutions and asymptotic spreading of a nonlocal Lotka-Volterra competition system.By means of two-point boundary valued problem and the Schauder fixed point theorem,we prove the system admits traveling wave solutions connecting equilibrium?0,0?to some unkonwn positive steady state for c > c*= max{2,2dr^1/2},where d and r respectively corresponds tothe diffusion coefficients and intrinsic rate in scaled system.At the same time,we show that there is no such traveling wave solutions for speed c < c*.Finally,for a specific kernel function,some numerical simulations are established and imply that the traveling wave solutions may connects the zero equilibrium to a periodic steady state as the nonlocality increased.Finally,we are concerned with the dynamical behavior of a Lotka-Volterra competition system with nonlocal term.By stability analyses,we establish the conditions of Turing bifurcation occurring in the system.According to them and by using multiple scale method,the amplitude equations about the different Turing Patterns are obtained.Then,by analyzing stability of the amplitude equations,we get the conditions of the different patterns?including spots pattern and stripes pattern?arising in the Lotka-Volterra competition system.Finally,some numerical simulations are given to verify our theoretical analysis.