Traveling Wave Solutions Of A Predator-Prey Model With Nonlocal Dispersal

Nonlocal operator is more helpful to accurately describe the nonlocal interaction of the population in the space than the classical Laplace operator,and more and more nonlocal diffusive equations is used to model the dispersal of the biological models.Since traveling waves can represent a large number of finite velocity propagation problems and the oscillation phenomenons in nature.In recent years,the research on traveling wave solutions of nonlocal diffusion biological model has attracted wide attention.In this paper,we mainly discuss traveling wave solutions for a class of nonlocal dispersal predator-prey model.Firstly,we shall consider the existence and non-existence of traveling wave solutions with different diffusion strategies of predator and prey.When the prey adopts the random diffusion and the predator adopts the nonlocal diffusion,the existence of traveling wave solutions as c>c*is proved by using the upper-lower solution technique and the Schaud-er's fixed point theorem(where c*is critical wave speed).At the same time,through the basic analytical theory,we discuss the boundary asymptotic behavior of traveling wave solutions and the existence of traveling wave solutions with c = c*.Furthermore,we linearize the equation of the predator satisfied,and prove the non-existence of traveling wave solutions by the eigenvalue theory.Secondly,the existence and non-existence of traveling wave solutions for the non local diffusion strategy of predator and prey are considered.In current paper,the existence of traveling wave solutions is proved mainly by adopting the truncation method,combined with the Schauder's fixed point theorem.By careful analysis,we get the existence of traveling wave solutions of the critical wave speed and the non-existence of traveling wave solutions with speed less than the critical speed.Finally,by comparing the two different diffusion strategies.We conclude that the diffusion form of prey does not affect the existence and non-existence of traveling wave solutions.