On periodic nonlocal dispersal competition systems in heterogeneous shifting environments: Survival exchange and gap phenomena

This is a continuation of our work [26] to investigate the joint influences of seasonal succession, climate change and long-distance free diffusion on the competitive dynamics, where we study the scenario that the growth rates of two competing species shift in the same trend with a periodically fluctuating speed. Since the effects of climate change on the habitats of the two competing species may be not synchronized, in this paper we consider the scenario where the two growth rates shift in opposite trends with a periodically fluctuating speed. Based on the monotone iterative technique, semigroup theory, sliding and squeezing skill as well as the upper- and lower-solution method, we establish the existence, uniqueness and exponential asymptotic stability of time-periodic and survival exchange type forced wave connecting the two semi-trivial periodic solutions associated to the corresponding limiting systems when the average shifting speed falls into a finite interval. On the contrary, when the average shifting speed is beyond this interval, we find that the gap phenomena will occur by using some comparison arguments.