Dynamics of a periodic-parabolic Lotka-Volterra competition-diffusion system in heterogeneous environments

The effects of spatial heterogeneity on the dynamics of reaction-diffusion models have been studied extensively. In particular, global dynamics of general spatially heterogeneous (but temporally static) Lotka-Volterra competition-diffusion systems were completely clarified by He and Ni in 2016. However, the evolutionary impacts of temporal periodicity combined with spatial heterogeneity in population ecology remain a challenging issue. In this work, we consider a population model of two competing species in a both spatially varying and temporally periodic environment, where the two species only differ in their random dispersal rates but are otherwise ecologically identical. In a pioneering 2001 work on this model by Hutson et al., by constructing various choices of resource functions and dispersal rates of the two species, the authors demonstrated that all the following three types of dynamics are possible: (i) stable coexistence of the two species; (ii) the slower diffuser invades the faster one but not vice versa; (iii) the faster diffuser invades the slower one but not vice versa. This is in drastic contrast with the spatially heterogeneous but temporally static case, where Dockery et al. showed in 1998 that the slower diffuser always wipes out the faster one. In this paper, we completely and explicitly characterize the asymptotic stability of both semitrivial periodic solutions in terms of the two dispersal rates and the resource function, when either dispersal rate is sufficiently small or large. In particular, the direction of selection on the dispersal rate during the evolution can be elucidated in these instances. Some novel analytical methods are developed to investigate asymptotic behaviors of the underlying time-periodic parabolic eigenvalue problem and its adjoint problem. We hope that these methods are of independent interest in the area of time-periodic parabolic equations.