Abstract
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.
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