A semilinear equation with large advection in population dynamics.

Abstract

A system of semilinear equations with large advection term arising from theoretical ecology is studied. The 2X2 system of partial differential equations models two theoretical species competing for a common resources with density m(x) that is spatially unevenly distributed. The two species are identical except for their modes of dispersal: one of them disperses completely randomly, while the other one, in addition to random diffusion, has a tendency to move up the gradient of the resource m(x). It is proved in [Cantrell et. al. Proc. R. Soc. Edinb. 137A (2007), pp. 497-518.] that under mild condition on the resource density m(x), the two species always coexist stably whenever the strength of the directed movement is large, regardless of initial conditions. In this paper we show that every equilibrium densities of the two species approaches a common limiting profile exhibiting concentration phenomena. As a result, the mechanism of coexistence of the two species are better understood and a recent conjecture of Cantrell, Cosner and Lou is resolved under mild conditions.