Travelling waves of a predator–prey model with a spatiotemporal delay

Abstract

This paper is concerned with the existence of traveling waves to a predator-prey model with a spatiotemporal delay. By analyzing the corresponding characteristic equations, the local stability of a positive steady state and each of boundary steady states are established, and the existence of Hopf bifurcation at the positive steady state is also discussed. By constructing a pair of upper-lower solutions and by using the cross-iteration method as well as the Schauder's fixed-point theorem, the existence of a traveling wave solution connecting the semi-trivial steady state and the positive steady state is proved. Numerical simulations are carried out to illustrate the main theoretical results.