Traveling waves in a convolution model with infinite distributed delay and non-monotonicity

In this paper, we study a population model with nonlocal diffusion and a non-monotonic reaction term with infinite distributed delay. Some existence results of traveling wavefronts for the system with a monotonic reaction term are firstly obtained by the construction of upper-lower solutions and the application of Schauder’s fixed point theorem. Then by constructing a couple of auxiliary equations with monotonicity and using the comparison method, we prove the existence of traveling waves for the system without monotonicity. We also give some discussion on the asymptotic behavior of the traveling waves as ξ=x+ct→−∞ξ=x+ct→−∞.