Evolution of Dispersal in Advective Homogeneous Environments: Inflow Versus Outflow

.We consider a single species model and a two species competition model in one-dimensional open advective environments featured by an inflow (resp., outflow) of individuals at the upstream (resp., downstream) end as measured by a parameter \(b_u≥ 0\) (resp., \(b_d≥ 0\)). The two species are assumed to follow the same population dynamics but have different random diffusion rates. Under certain mild conditions on \(b_u\) and \(b_d\), we first determine clearly the global dynamics of the single species model in terms of critical habitat size or critical advection speed, and then further give a complete understanding on the global dynamics of the two species competition model. Our results suggest that in an open environment with mild inflow and outflow rates and with totally unfavorable boundary effect (outflow rate greater than inflow rate), "faster diffusion can evolve," a different mechanism behind evolution of dispersal from that ("slower diffusion can evolve") observed by Tang and Chen [J. Differential Equations, 269 (2020), pp. 1465–1483]. We also discuss other scenarios with different settings of \(b_u\) and \(b_d\) and propose problems that deserve future investigation.