Traveling waves and spreading speeds for time-space periodic monotone systems

In this paper, we first establish the existence of traveling waves and spreading speeds for time-space periodic monotone systems with monostable structure via the Poincaré maps approach combined with an evolution viewpoint. Our construction of time-space periodic wave profiles also gives rise to a family of almost pulsating waves, which is a new observation in time and space periodic media. We then apply the developed theory to two species competitive reaction-advection-diffusion systems, and prove that the minimal wave speed exists and coincides with the single spreading speed for such a system no matter whether the spreading speed is linearly determinate. We further obtain a set of sufficient conditions for the linear determinacy.