Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On a limiting system in the Lotka--Volterra competition with cross-diffusion

Pages: 435 - 458, Volume 10, Issue 1/2, January/February 2004      doi:10.3934/dcds.2004.10.435

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Yuan Lou - Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210, United States (email)
Wei-Ming Ni - School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, United States (email)
Shoji Yotsutani - Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, 520-2194, Japan (email)

Abstract: In this paper we investigate a limiting system that arises from the study of steady-states of the Lotka-Volterra competition model with cross-diffusion. The main purpose here is to understand all possible solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done without the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.

Keywords:  cross-diffusion, existence, asymptotic behavior, parameter representation.
Mathematics Subject Classification:  35J55.

Received: January 2002;      Revised: March 2003;      Available Online: October 2003.