Traveling fronts in monostable equations with nonlocal delayed effects

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction-diffusion equation with delay partial derivative u (x, t)/partial derivative t = d Delta u (x, t) + f (u(x, t), integral(infinity)(-infinity) h(x -y) u (y, t - tau) dy). Under the monostable assumption, we show that there exists a minimal wave speed c* > 0, such that the equation has no traveling wave front for 0 < c < c* and a traveling wave front for each c >= c*. Furthermore, we show that for c > c*, such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/ or improve a number of existing ones. In particular, our results show that (i) if. 2 f (0, 0) > 0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if. 2 f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.