Population dynamics of two competing species in an advective heterogeneous two-patch environment

We investigate the population dynamics in an advective heterogeneous two-patch environment with general boundary conditions. We give a criterion for the persistence of a single species in such an environment in terms of the dispersal rate $ d $, the drift rate $ q $, the upstream-end flow rate $ \alpha $, and the downstream-end flow rate $ \beta $. For two competing species which have distinct dispersal rates $ d $ and $ D $ but are identical in all the other respects, we study the global dynamics of their populations under $ \alpha = 0 $. The evolution outcome of the two species depends not only on these parameters but also on the ratio of the carrying capacities of the two patches. It turns out that a coexistent equilibrium must be globally asymptotically stable if $ 0<\beta<q $. However, it could be either globally asymptotically stable or unstable if $ \beta>q $, and in the latter case, there are two stable semi-trivial equilibria. For $ \beta = q $, we give a complete and explicit classification of the dynamics, provide a formula of an evolutionarily stable dispersal strategy, and answer some open questions proposed by Lou (J. Nonlinear Modeling and Analysis 1 (2019), 151-166) regarding the competition between the faster and the slower diffusers.