Wave propagation for a nonlocal dispersal predator-prey system with single predator and multiple preys

This paper is concerned with a nonlocal dispersal predator-prey system with single predator and multiple preys. We study the invading phenomenon of an alien predator to the habitat of multiple aborigine preys by traveling wave solutions connecting the predator-free state to the co-existence state. The existence of traveling wave solutions that converge to predator-free equilibrium as the moving coordinate goes to [Formula: see text] is proved by constructing suitable upper-lower solutions and using Schauder’s fixed point theorem, when the wave speed [Formula: see text] for some positive number [Formula: see text]. Then by constructing a Lyapunov functional, we show that the traveling wave solutions converge to the co-existence state as the moving coordinate goes to [Formula: see text]. Furthermore, by using a limiting argument, we prove the existence of traveling wave solutions with speed [Formula: see text]. Finally, by using a contradictory approach with the properties of the corresponding characteristic equation to the system, we prove the non-existence of traveling wave solutions when [Formula: see text]. It turns out that [Formula: see text] is the minimum wave speed for traveling wave solutions connecting the predator-free state and the co-existence state.