Limiting profiles of semilinear elliptic equations with large advection in population dynamics

Limiting profiles of solutions to a 2$\times$2 Lotka-Volterra competition-diffusion-advection system, when the strength of the advection tends to infinity, are determined. The two species, competing in a heterogeneous environment, are identical except for their dispersal strategies: One is just random diffusion while the other is "smarter" - a combination of random diffusion and a directed movement up the environmental gradient. With important progress made, it has been conjectured in [2] and [3] that for large advection the "smarter" species will concentrate near a selected subset of positive local maximum points of the environment function. In this paper, we establish this conjecture in one space dimension, with the peaks located and the limiting profiles determined, under mild hypotheses on the environment function.