Linear and Nonlinear Speed Selection for Mono-Stable Wave Propagations

Abstract

In this paper, we study the selection mechanism of the minimal wave speed for traveling waves to an abstract monotone semiflow. A necessary and sufficient condition for the nonlinear selection of the minimal wave speed is established. Based on this result, we then derive conditions under which the linear or nonlinear selection is realized by way of a comparison principle. Our results on nonlinear selection are new and novel, and they can be viewed as breakthroughs in this topic; and for the linear selection, we successfully improve previous conventional results that always require that the monotone semiflow is dominated by its linear map. The applications to various biological models are also successful. We establish a series of new results to reaction-diffusion models with delay interactions, a lattice system, a scalar integro-difference equation, and a cooperative system, which completely solve some open problems and conjectures in the related references.