Traveling wave phenomena of [formula omitted]-dimensional diffusive predator–prey systems

Abstract

In this work, we deal with the n-dimensional reaction-diffusive predator–prey systems with the discrete time delay, which have numerous applications in biology and ecology. By applying an asymptotic analysis, the Schauder’s fixed point theorem as well as the upper and lower solution method, we investigate the existence, non-existence, exponentially asymptotic stability and asymptotic behaviors of traveling wave solutions with the appropriate small delay when the wave speed c varies. Moreover, we derive the minimal wave speed c∗ based on the well-established properties of the Fisher equation, and prove the existence of traveling wave solutions when the wave speed c≥c∗, which provides a sharp contrast to the case of c<c∗. The obtained results extend the existing ones in the literature to arbitrary finite dimensional systems with the discrete time delay and demonstrate asymptotic behaviors of traveling wave solutions.