Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions

We consider the ecological and evolutionary dynamics of a reactiondi fiusion-advection model for populations residing in a one-dimensional advective homogeneous environment, with emphasis on the effects of boundary conditions and domain size. We assume that there is a net loss of individuals at the downstream end with rate b C 0, while the no-ux condition is imposed on the upstream end. For the single species model, it is shown that the critical patch size is a decreasing function of the dispersal rate when b B 3/2; whereas it first decreases and then increases when b > 3/2. For the two-species competition model, we show that the infinite dispersal rate is evolutionarily stable for b < 3/2 and, when dispersal rates of both species are large, the population with larger dispersal rate always displaces the population with the smaller rate. For certain specific population loss rate b < 3/2, it is also shown that there can be up to three evolutionarily stable strategies. For b > 3/2, it is proved that the infinite random dispersal rate is not evolutionarily stable, and that, for some specific b > 3/2, a finite dispersal rate is evolutionarily stable. Furthermore, for the intermediate domain size, this dispersal rate is optimal in the sense that the species adopting this rate is able to displace its competitor with a similar but different rate. Finally, nine qualitatively different pairwise invasibility plots are obtained by varying the parameter b and the domain size.