Minimal wave speed for a class of non-cooperative diffusion–reaction system

In this paper, we consider a class of non-cooperative diffusion–reaction systems, which include prey–predator models and disease-transmission models. The concept of weak traveling wave solutions is proposed. The necessary and sufficient conditions for the existence of such solutions are obtained by the Schauder's fixed-point theorem and persistence theory. The introduction of persistence theory is very technical and crucial. The LaSalle's invariance principle is applied to show that traveling wave solutions connect two equilibria. The nonexistence of traveling wave solutions is proved by introducing a negative one-sided Laplace transform. The results are applied to a prey–predator model and a disease-transmission model with specific interaction functions. We find that the profile of traveling wave solutions may depend on different eigenvalues according to the corresponding condition, which is a new phenomenon.