Effects of anisotropic diffusion in heterogeneous time-periodic environments

We study a reaction-diffusion system involving two species competing in temporally periodic and spatially heterogeneous environments. In this system, the species move horizontally and vertically with different probabilities, which can be regarded as dispersal strategies. The selection mechanisms in this case are more intricate than those observed in random diffusion scenarios. We investigate the stability of the semi-trivial periodic solutions in terms of the sign of the principal eigenvalue associated with a linear periodic eigenvalue problem. Furthermore, we provide sufficient conditions for the coexistence of two species. Additionally, numerical simulations are performed to facilitate further research and exploration.