Evolution of Dispersal in Open Advective Patchy Environments

A Lotka–Volterra competitive patch model in advective homogeneous environments is investigated, where two species are supposed to differ only in their diffusion rates and the environment is assumed to be open so that there may be an inflow (resp. outflow) of individuals at the upstream (resp. downstream) patch. Under certain conditions on the inflow and outflow rates, a complete understanding on the global dynamics is obtained, which, biologically, suggests that in open patchy environments with mild inflow and outflow rates, faster diffusion can evolve, extending two existing results obtained by Chen et al. (Stud. Appl. Math., 149: 762-797, 2022) and (J. Nonlinear Sci., 33: Paper No. 40, 35 pp, 2023) to more general biological situations. Moreover, our main result does not depend on the size relation between the inflow and outflow rates, different from the corresponding space-continuous case treated recently by Wang et al. (SIAM J. Math. Anal., 56: 1643-1671, 2024).