Propagation phenomena for time-space periodic monotone semiflows and applications to cooperative systems in multi-dimensional media

Abstract

The current paper is concerned with propagation phenomena for time-space periodic monotone semiflows and applications to time-space periodic cooperative systems in multi-dimensional media. We first establish some abstract theory on spreading speeds and traveling waves for time-space periodic monotone semiflows in the space of vector-valued functions on RN. Among others, we prove the equivalence of spreading speeds adopted by two different approaches, several spreading properties in terms of the spreading speeds, and the existence of periodic traveling waves which extends several known results in various special cases. By applying the abstract theory, we study the spreading speeds and traveling waves of time-space periodic cooperative systems in multi-dimensional media. It is proved that such a system admits a single spreading speed (resp. asymptotic spreading ray speed and asymptotic spreading set) under certain conditions. A set of sufficient conditions are also given for the single spreading speed to be linearly determinate. Furthermore, we show that the spreading speed can be characterized as the minimal wave speed of periodic traveling waves. The obtained results are then applied to the two-species periodic competition system in multi-dimensional media.