Invasion waves for a class of multi-species non-cooperative systems with nonlocal dispersal

This paper is concerned with the invasion waves for a class of multi-species non-cooperative systems with nonlocal dispersal. We first establish a sharp existence result of the weak traveling wave solution connected the semi-trivial equilibrium for a general multi-species nonlocal dispersal system by Schauder’s fixed-point theorem. And then we apply this result to discuss the traveling wave solutions for a disease-transmission model and a predator–prey model respectively, where we prove that the weak traveling wave solutions connect the positive equilibrium with the help of Lyapunov functional. To get the asymptotic behavior of traveling wave solutions at $+\infty$, we have to overcome the difficulties brought by the nonlocal dispersal and the non-cooperative of systems themselves.