Spatial Dynamics Of Several Classes Of Population Diffusion Models With Nolocal Delay

Reaction-diffusion equation is often used to explain and predict the problems encountered in some specific disciplines,such as the invasion of new species and the spread of infectious diseases in mathematical ecology,enzyme catalysis and low-temperature plasma flue gas desulfurization reactions in chemical reactions,the changes of electron and hole concentrations in semiconductors,the movement of fluids in physics,etc.Since individual organisms and environmental factors are interdependent,the interaction between spatial diffusion and time delay cannot be ignored in the study of mathematical ecology.On the basis of this interaction,the weighted average of the whole space and temporal delay is introduced into the nonlinear term,i.e.,the nonlocal delay reaction-diffusion equation.Compared with the traditional model,the nonlocal delay reaction-diffusion equation will bring more difficulties.At the same time,it also reveals more dynamic behaviors,many scholars have been concerned about this model and obtained some results.In this paper,we study the traveling wave solutions and asymptotic spreading speed of the following population diffusion models with nonlocal delay:In Chapter 2,we study the stability of traveling wave solutions(monotone or non-monotone)of nonlocal Fisher-KPP equation.Since the nonlinear term leads to the lack of the comparison principle,we use the idea of anti-weighted and estimate the solution of the perturbation equation through the energy estimation and some technical methods.Finally,the global stability of the traveling wave solution of the model in the case of large wave speed is proved.In Chapter 3,we investigate the global stability of traveling waves for non-monotone infinite-dimensional lattice differential equations with time delay.The boundedness estimation of the solution of the perturbation equation is established by the method of weighted energy and Fourier transform,we obtain that:for any initial perturbations around the traveling wave,the noncritical traveling waves((8>(8_*)are globally stable with the exponential convergence rate^-1/0)^-(>0 and 0<?2),and the critical traveling waves((8=(8_*)are globally stable with the algebraic convergence rate^-1/in a weighted Sobolev space.In Chapter 4,we are devoted to studying the asymptotic spreading speed of a kind of single species model with time delay.The global existence of the solutions to the initial value problem of the model is obtained first by Banach fixed point theorem and the extension method.For the research about the spreading speed,due to the different selected parameters and kernel function,the processing methods are not compatible,then different methods are applied to study the spreading speed.Firstly,for the Food-Limited model with spatio-temporal delay,the uniform boundedness of the solution is acquired by means of the explicit structure of the kernel func-tion.Through comparison principle,we establish the asymptotic spreading speed of solutions with compactly supported initial data.Secondly,for the Food-limited model with fixed time delay,the Harnack inequality is used to study the asymptotic spreading speed of the solution with compactly supported initial.Thirdly,for the Fisher-KPP model with nonlocal delay,we study the asymptotic spreading speed of the solution with compactly supported initial data by contradiction.Furthermore,the numerical simulation is demonstrated by the finite difference method,which not only verify the theoretical conclusion,but also show that the equation may produce a positive steady state which is similar to the time periodic solution when the time delay is sufficiently large.In Chapter 5,we consider the generation of interface for a class of Nichol-son's blowflies equation with distributed delay.If the birth function satisfies the quasi-monotone condition,we utilize the nonstandard bistable approximation of the monostable problem to establish a suitable lower solution,and then employ the monostable traveling wave solution to construct a suitable upper solution.It is proved that the solution converge to a propagation interface.Furthermore,for the non-quasi-monotone situation,due to the lack of monotonicity of the equa-tion,the above method is no longer applicable.We first construct two auxiliary quasi-monotone systems,then the limit behavior of the solution of equation can be obtained from the sandwich technique and the comparison principle of Cauchy problem.Whether the birth function satisfies the quasi-monotone condition or not,the propagation speed of the interface is proved to be equal to the minimum wave speed of the corresponding traveling waves,which makes it possible to observe the minimum speed of traveling waves from a new perspective.