On traveling wave solutions in general reaction-diffusion systems with time delays

We study the existence of traveling wave solutions in a general class of mixed quasi-monotone reaction-diffusion systems with time delays. First, by applying the Schauder Fixed Point Theorem, we prove the existence of a traveling wave solution between classically defined upper and lower solutions. For better applications of the upper-lower solution method on various real-life models, the existence result is further extended under weak form or piecewise smooth upper-lower solutions. In several reaction-diffusion systems with time delays (single-species logistic growth, N-species competition, and ratio-dependent predator-prey with Gompertz growth), we apply our main result to establish the existence of traveling wave solutions flowing towards the positive or coexistent states under reasonable conditions on ecological parameters.