Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder's fixed point theorem for c > c(*)(tau) and using a limiting argument for c = c(*)(tau). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara's Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for partial derivative(2)f(0,0) > 0 and the delay is independent of the minimal wave speed for partial derivative(2)f(0,0) = 0. (C) 2012 Elsevier Ltd. All rights reserved.