Traveling Wave Solutions Of Discrete Time Reaction-diffusion Systems

With the development of science and technology, reaction-diffusion equations play an important role in modeling the spatial-temporal patterns, and many researchers focus on the properties of traveling wave solutions, which can explain many natural phenomena, such as species dispersal, invasion and spreading of infections diseases,therefore, it is of important theoretical and practical significance to research the travel-ing wave solutions of reaction-diffusion equations. In the real world, the evolution of a system usually depends not only on the present states but also the past states, which means that time delay and diffusion are inevitable in many evolutionary processes. In nature, many species evolve with non-overlapping generations and the changes of the number of population is discrete, so discrete time model can describe the growth of population more accurately. Therefore,in many cases,a more realistic model would be delayed reaction-diffusion equations modelling diffusive phenomena. Based on the above reasons, we shall study the existence of traveling wave solution for some discrete time reaction-diffusion equations in this thesis. This dissertation is composed of six chapters and the content are as follows:In Chapter 1, a brief introduction to the historical background and recent ad-vances in discrete time reaction-diffusion systems are presented.In Chapter 2, we study the existence of traveling wavefronts for integro-difference equations. We establish the existence of traveling wavefronts under rather weak assumptions on the growth function of my model by using comparison theorem,monotone iteration technique and upper-lower solution method. This work improves some of the previous theoretical results in this field.In chapter 3, we consider the existence of traveling wave solution for dis-crete time delayed predator-prey system. In this work,we introduce partially quasi-monotone condition (PQM) for the nonlinearity. Techniques such as upper-lower so-lutions together with cross-iteration and the Schauder's fixed point theorem are used.In addition, we apply the results to predator-prey system and obtain the existence of the traveling wave solutions connecting 0 steady state to a coexistence steady state.In Chapter 4, higher dimensional discrete time delayed competition-cooperation diffusion system was considered. Under a new mixed quasi-monotone condition,Schauder's fixed point theorem is applied to some operators to prove the existence of traveling wave solutions in a properly subset equipped with exponential decay norm. Further, we apply the results to three-dimensional discrete time delay K-type monotonous diffusion system and obtain the existence of traveling wave solutions of this system.In Chapter 5, for discrete time delayed K-type competition diffusion systems,under another mixed quasi-monotone condition, we consider the existence of trav-eling wave solution for a three-dimensional discrete time delay K-type competition diffusion system.The summary of this dissertation and the outlook for the further research work are stated in Chapter 6.