Traveling waves for competitive Lotka–Volterra systems with spatial diffusions and spatio-temporal delays

This paper is concerned with the existence of traveling waves for reaction–diffusion equations with spatio-temporal delays. Our approach is to use the cross-iteration technique and Schauder’s fixed point theorem. The iterative technique can be established for a class of integral operators when the reaction terms satisfy new monotonicity conditions, which is different from those in previous works. The results can be well applied to a two species competitive Lotka–Volterra system with more general kernel functions. In order to overcome the difficulty of spatio-temporal delays, we need to use a basic result from a PDE textbook. Existence of traveling waves which connect the trivial equilibrium and the positive equilibrium indicates that there is a transition zone moving the steady state with no species to the steady state with the coexistence of two species. Our results can extend some existing ones for two species diffusion competitive Lotka–Volterra systems.