Existence And Stability Of Traveling Wave Solutions For Discrete Diffusion Systems

In recent decades,discrete diffusion systems have attracted more and more attention from many researchers.The traveling wave solution is an important topic in the study of discrete diffusion systems,since they can explain the propagations with finite speed and oscillatory phenomena in nature.Firstly,we study the stability of traveling wavefronts for a class of three-component Lotka-Volterra competition system on a lattice.By using the comparison principle and the weighted energy method,we obtain the exponentially stability of the traveling wavefronts with large speed.Secondly,we investigate the existence of invasion traveling waves for a class of ratio-dependent predator-prey system on a lattice.By using Schauder's fixed point theorem combined with the sub-and supersolutions method and a limiting argument,we prove the existence of traveling wave solutions,when the speed wave c?c*.Then by the asymptotic spreading theory,the non-existence of traveling wave solutions with speed c<c*is obtained.Finally,we study the global stability of non-critical traveling waves for a class of non-monotone discrete diffusion equation.With the help of comparison princi-ple,the weighted energy method and Fourier s transform,the global exponential stability of all non-critical traveling wave solutions is established,when the initial perturbations in weighted Sobolev space are arbitrary large.