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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.