This paper is concerned with the traveling waves for a class of delayed reaction-diffusion equations with crossingmonostability. In the previous papers, we established the existence and uniqueness of traveling waves which may not be monotone. However, the stability of such traveling waves remains open. In this paper, by means of the (technical) weighted energy method, we prove that the traveling wave is exponentially stable, when the initial perturbation around the wave is relatively small in a weighted norm. As applications, we consider the delayed diffusive Nicholson's blowflies equation in population dynamics and Mackey-Glass model in physiology. (2000). 35K57 · 35R10 · 35B40 · 92D25. Mathematics Subject Classification

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This paper is concerned with the traveling fronts of a diffusive food-limited population model with spatiotemporal delay. Sufficient conditions are established for the existence of traveling wave fronts by choosing different kinds of delay kernels. The approach used here is the upper-lower solution method and monotone iteration technique. Our work extends and/or covers some previous results.