Travelling waves in delayed reaction–diffusion equations on higher dimensional lattices

Abstract

This paper is concerned with travelling wave solutions for a class of monostable delayed reaction–diffusion equations on higher dimensional lattices. We first show that, for any fixed unit vector , there is a minimal wave speed such that a travelling front exists if and only if its speed is above this minimal speed. The exact asymptotic behaviour of the wave profiles at infinity is then established. Finally, we show that any travelling wave solution is strictly monotone and unique (up to a translation), including even the minimal wave. Of particular interest is the effects of the delay, spatial dimension and direction of wave on the minimal wave speed and we obtain some interesting phenomena for the delayed lattice dynamical system which are different from the case when the spatial variable is continuous.