Spatial Patterns And Traveling Waves Of Some Ecological Models

The recent development of reaction diffusion systems in ecology and the traditional importance of theses systems in physics lead to extensive study in various aspects of nonlinear partial differential equations.In this dissertation we are focus on the spatial patterns and the traveling wave solutions of some reaction diffusion systems which describe the phenomena of population dynamics.Since Turing(1952) first proposed the theory of spatial pattern,the spatial pattern has been studied extensively in physics,chemistry and biology.Murray(2002) exhibited a lot of practical applications to biology.Recently,Maini et al stated that the spatial pattern plays the significant role in embryology.The theory of traveling wave solutions of parabolic differential equations is one of the fastest developing areas of modern mathematics and has attracted much attention due to its important nature in biology,chemistry,epidemiology and physics.In Chapter 0,the background of the spatial patterns are presented.Besides self-diffusion,cross-diffusion can also induce spatial patterns,this phenomenon was first proposed by Shigesada et al(1979).Considering the competitive model with self-diffusion and cross-diffusion,Lou et al(1996) showed that the cross-diffusion induces the nonconstant stationary state,which is the significant method for the spatial patterns.Farkas(1997) gave two ways to introduce the cross-diffusion in reaction-diffusion systems.We consider that the cross-diffusion induce spatial patterns.Since Fisher(1937) proposed Fisher-KPP equation,the traveling wave has always attracted many attentions in modern mathematics and physics.By the monotone iteration owing to Wu and Zou(2001),we investigate the more general reaction functions and multidimensional traveling wave.In Chapter 1,we consider the spatial patterns of the general reaction diffusion system with strongly-coupled diffusion,which involves two equations.First the necessary conditions and the sufficient conditions for the pattern formation in the whole space are given.Then in rectangular region we give the sufficient conditions for the pattern formation.Finally,to illustrate our results,the spatial patterns for a competitive model are presented.Chapter 2 is devoted to the spatial patterns of a three-species food chain model with cross-diffusion.We first present the sufficient conditions for Turing instability.Our result is that only cross-diffusion can induce Turing instability, that is to answer the question:when does the spatial pattern happen? Secondly, by using Leray-Schauder theory together with a prior estimates of positive steady state,we obtain the existence of the nonhomogeneous steady state,that is to answer the question:why does the spatial pattern happen? Moreover,by using numerical methods,we illustrat that the cross-diffusion induces spatial pattern. In this way,we tried to answer the question:what does it look like?Chapter 3 deals with the traveling solutions of the general delayed reaction diffusion system,where the reaction function possesses a quasimonotone property. By the monotone iteration method due to Pao(1992),we first construct a pair of upper and lower solutions to guarantee the existence of the traveling solution. More precisely,we reduce the existence of traveling waves to the existence of an admissible pair of quasi-upper and quasi-lower solutions.As a special example, our theorem is also applicable to the case that the reaction function is monotone. And our theorem is consistent to the previous concludes.Finally,to illustrate our results,we study the spatial patterns for a cooperative system.In Chapter 4,considering a insect breaking model as a example of reactiondiffusion equation,we study the multidimensional traveling solutions.Generalizing the method of upper and lower solutions and monotone iteration to multidimensional space,we give out the generalized upper solution and the generalized lower solution.We show that under the conditions of upper solution and generalized lower solution,if the wave speed is larger than a positive constant the model admits the multidimensional traveling wavefronts.Moreover,by using of the numerical simulations we illustrate the speed of the wavefront.We summarize our work in Chapter 5 and propose some problems,which need more attention in the future.