Stability Of Traveling Wave Solutions For Nonlocal Reaction Diffusion Systems With Delay

As a typical parabolic equation,reaction-diffusion equation comes from a large number of mathematical models in many natural sciences.With recent researches,it is recognized that convolution operators can more accurately describe long range diffusion.However,the introduction of nonlocal operators(convolution operators)lead to many mathematical difficulty and complicated dynamics.On the other hand,the phenomenon of time delay,such as the latent period of a virus,the mature period of species,is worthy of attention.Therefore,it is interesting and valuable to study the dynamics for reaction-diffusion systems with delay or nonlocal effect.In this thesis,we study the stability of traveling wavefronts for a delayed nonlocal dispersal system and a delayed Lotka-Volterra competition systems with stage structure respectively.The main results in this thesis are as follows:For a nonlocal dispersal man-environment-man epidemic system with delay,the stability of all non-critical traveling wave solutions is studied.In the case of quasi-monotonicity,the existence and comparison theorem to the solutions for the initial value problems are firstly established by appealing to the theories of Banach fixed point and differential equations.Then,by using a comparison theorem and the weighted energy method,the global exponential stability for all non-critical traveling wave solutions under some small perturbation is proved.Finally,the influence of time delay on the minimal wave speed and stability of traveling wave solution is discussed.For a delayed Lotka-Volterra competition system with age structures,the stability of traveling wave solutions with large wave speed is studied.Firstly,the original competition system can be transformed into an equivalent cooperation system by the linear transformation.Then,by using the theories of analytic semigroup and differential equation,we establish the existence and comparison principle to the solutions for the initial value problem on R under quasi-monotone condition.Further,based on the comparison principle and weighed energy method,the energy estimation is established.Finally,combined with the embedding theorem and the squeezing method,the globally exponential stability for the traveling wave solutions connecting two semi-trivial equilibriums is proved.