The Investigation Of Traveling Wave Solutions For Several Types Discrete Reaction Diffusion Equations With Delays

As we all know,a lot of models in the fields of physical,chemical and biology can be translated into initial value issue or initial and boundary value problem of reaction diffusion equations.As a major subject of reaction diffusion equations,traveling wave solutions attract the attention of many scholars and experts.With the development of computer technology,scholars dispersed continuous reaction diffusion equations by time variation and space variation in order to provide help for numerical calculation and numerical analysis.The reaction diffusion equation dispersed by space variation was called lattice differential equation.In the paper,we investigate the existence of traveling wave solutions for a temporally discrete reaction-diffusion system with delays and the stability and uniqueness of travelling wave fronts for a delayed differential equation with the nonlocal interactions.For time discrete reaction diffusion system,we investigate the existence of traveling wave solutions for 2 species Lotka-Volterra competitive system with delays.By using upper-lower solution methods and a cross iteration scheme,we obtain that there exist at least one traveling wave solution if its wave profile have a pair of upper-lower solution satisfying some assumptions.Next,we extend the result obtained above to the case of 3 species in a properly subset equipped with exponential decay norm by Schauder's fixed point theorem.For space discrete reaction diffusion system,we investigate the stability and uniqueness(up to shift)of traveling wave fronts for a delayed differential equation with the nonlocal interactions by using the weighted energy method and the squeezing technique with the help of Gronwall's inequality.For all traveling wave fronts with the wave speedc>c_*(?),where_*c(?)>0 is the critical wave speed,we obtain that these traveling waves are asymptotically stable,when the initial perturbation around the traveling waves decay exponentially at far fields,but can be allowed arbitrarily large in other locations.This essentially improves the stability results obtained by Cheng,Li and Wang[Discrete Cont.Dyna.Syst.B,2010]with the small initial perturbation and the large wave speed,by Yu and Yuan[Osaka J.Math.,2013]with the large initial perturbation and the large wave speed,by Wu and Liu[Electronic J.Diff.Eqns,2014]with the large initial perturbation and the large wave speed.