Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

Abstract

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay $$\frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty }^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right)\!.$$ 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*, 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.