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July  2010, 28(3): 1051-1067. doi: 10.3934/dcds.2010.28.1051

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

King-Yeung Lam 1, and  Wei-Ming Ni 2,

1. 

School of Mathematics, University of Minnesota, Minneapolis, MN 55455, United States
2. 

School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Received  March 2010 Revised  March 2010 Published  April 2010

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.
Keywords: asymptotic behavior, ecology, large advection., Semilinear equations.
Mathematics Subject Classification: Primary: 35J57, 35B40; Secondary: 92D4.
Citation: King-Yeung Lam, Wei-Ming Ni. Limiting profiles of semilinear elliptic equations with large advection in population dynamics. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1051-1067. doi: 10.3934/dcds.2010.28.1051
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