The wave speed for a time-periodic bistable 3-species lattice competition system

In this paper, we consider propagation direction, which can be used to predict which species will occupy the habitat or win the competition eventually, of bistable wave for a 3-species time-periodic lattice competition system with bistable nonlinearity, aiming to address an open problem proposed in [J.-S. Guo et al, The sign of traveling wave speed in bistable dynamics, Discret. Contin. Dyn. Syst. , 40 (2020), 3451]. As a first step, by transforming the competition system to a cooperative one, we study the asymptotic behavior for the bistable wave profile and then prove the uniqueness of the bistable wave speed. Secondly, we utilize comparison principle and build up two couples of upper and lower solutions to judge the sign of the bistable wave speed which provides partially the answer to the open problem. As an application, we reduce the time-periodic system to a space-time homogeneous system, we obtain the corresponding criteria and carry out numerical simulations to illustrate the availability of our results. Moreover, an interesting phenomenon we found is that the two weak competitors can wipe out the strong competitor under some circumstances.