Evolution of dispersal in advective homogeneous environment: The effect of boundary conditions

We consider a single species model and a two-species competition model in a one-dimensional advective homogeneous environment. One interesting feature in these models concerns the boundary condition at the downstream end, where the species can be exposed to a net loss of individuals, as tuned by a parameter b which measures the magnitude of the loss. We first determine necessary and sufficient conditions for the persistence of a single species for general value of b  , in terms of the critical habitat size and the critical advection rate. Then for the competition model, we assume that two species are identical except their random diffusion rates. We obtain complete understanding when 0≤b<10≤b<1, and our result indicates that larger diffusion rate is selected, extending an earlier work [20] (b=1b=1). However, for b>1b>1 the dynamics can be quite different, and particularly we illustrate that some intermediate diffusion rate may be selected when b>32.