Initial Value Problems For Lotka-Volterra Competitive Systems With Nonlocal Diffusion

In this paper,we consider the propagation properties of solutions to the fol-lowing Lotka-Volterra competitive systems with nonlocal diffusion when the initial value satisfies different assumptions and the diffusion kernel functions are expo-nentially decaying#12 where t>0,x?R,0<?1,?2<1,and the other parameters are positive.The main results include the following three parts.Firstly,we obtain the existence of traveling wave solutions of the nonlocal Lotka-Volterra competitive system by constructing a pair of sub-super solutions and using a cross iteration scheme.We further get the asymptotic behavior of the traveling wave solutions by a contracting rectangle method.The initial value problem of the nonlocal Lotka-Volterra competive system with a front-like initial value is considered.We obtain the properties that the solutions travel in the form of traveling waves and the approximate propagation speed by constructing a pair of sub-super solutions that are similar to the proof of the existence of traveling wave solutions.Secondly,we consider the speed of asymptotic spreading of the solutions when the initial value has non-empty compact support.By constructing auxiliary equa-tions and using the sub-super solution method,we obtain that the competitive nonlinear term can slow the propagation speed of species with slower propagation speed when there is no competition and has no effect to species with faster speed.But ultimately,it can reduce the population density of the two species in the coexistence domainFinally,assuming that the species with slower spreading speed without com-petition has a slow decay initial value,and the initial value of another species still has a non-empty compact support.We obtain that the species with a slow decay initial value has infinite propagation speed and the competitive nonlinear term slows down the propagation speed of another species by constructing auxiliary equations and using the sub-super solution method.This reflects the impact of competition and reducibility on propagation of species.