# American Institute of Mathematical Sciences

November  2012, 17(8): 2745-2769. doi: 10.3934/dcdsb.2012.17.2745

## On limit systems for some population models with cross-diffusion

 1 Department of Communication Engineering and Informatics, The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu-shi, Tokyo 182-8585, Japan 2 Department of Applied Mathematics, Waseda University, 3-4-1 Ohkubo, Shinjuku-ku, Tokyo 169-8555, Japan

Received  April 2011 Revised  September 2011 Published  July 2012

This paper deals with the following reaction-diffusion system $$(SP) $$\left\{\begin{array}{11} \Delta[(1+\alpha v)u]+u(a-u-cv)=0, \\ \Delta[(1+\beta u)v]+v(b-du-v)=0, \end{array} \right.$$$$ in a bounded domain of $\Bbb{R}^N$ with homogeneous Neumann boundary conditions or Dirichlet boundary conditions. Our main purpose is to understand the structure of positive solutions of (SP) and know the effects of cross-diffusion coefficients $\alpha$ and $\beta$. For this purpose, our strategy is to study limiting behavior of positive solutions when $\alpha$ or $\beta$ goes to $\infty$ and derive the corresponding limit systems. We will obtain a priori estimates of $u$ and $v$ independently of $\beta$ (resp. $\alpha$) with small $\alpha\ge0$ (resp. $\beta\ge0$) in case $1\le N\le 3$ under Neumann boundary conditions, while we will obtain a priori estimates of $u$ and $v$ independently of $\alpha$ and $\beta$ in case $1\le N\le 5$ under Dirichlet boundary conditions. These a priori estimates allow us to investigate limiting behavior of positive solutions. When $\alpha=0$ and $\beta\to\infty$, we can derive two limit systems for Neumann conditions and one limit system for Dirichlet conditions. We will also give some results on the structure of positive solutions for such limit systems.
Citation: Kousuke Kuto, Yoshio Yamada. On limit systems for some population models with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2745-2769. doi: 10.3934/dcdsb.2012.17.2745
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