Traveling wave solutions for a diffusive predator–prey model with predator saturation and competition

The purpose of this paper is to study the traveling wave solutions of a diffusive predator–prey model with predator saturation and competition functional response. The system admits three equilibria: a zero equilibrium [Formula: see text], a boundary equilibrium [Formula: see text] and a positive equilibrium [Formula: see text] under some conditions. We establish the existence of two types of traveling wave solutions which connect [Formula: see text] and [Formula: see text] and [Formula: see text] and [Formula: see text], respectively. Our main arguments are based on a simplified shooting method, a sandwich method and constructions of appropriate Lyapunov functions. Our particular interest is to investigate the oscillation of both types of traveling wave solutions when they approach the positive equilibrium.