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June  2020, 40(6): 3561-3570. doi: 10.3934/dcds.2020161

## On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates

 Department of Mathematics, Faculty of Education, Ehime University, 790-8577, Japan

Received  February 2019 Revised  January 2020 Published  March 2020

Fund Project: The work was supported by JSPS KAKENHI Grant Number JP16K05233

In 1979, Shigesada, Kawasaki and Teramoto [11] proposed a mathematical model with nonlinear diffusion, to study the segregation phenomenon in a two competing species community. In this paper, we discuss limiting systems of the model as the cross-diffusion rates included in the nonlinear diffusion tend to infinity. By formal calculation without rigorous proof, we obtain one limiting system which is a little different from that established in Lou and Ni [5].

Citation: Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
##### References:

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##### References:
Density distribution of radially symmetric solution $(w, z)(x, t)$ for (1.2) with $\Omega = \{ \, x \in \mathbb{R}^2 \, | \, | \, x \, | < \pi \, \}$ for the case where $a = 1.04$, $b = 1.1$, $c = 1.1$, $d = 15.0$, $\varepsilon = 0.005$, $\alpha = 1200.0$ and $\beta = 2400.0$. The horizontal axis and the vertical axis indicate the distance $r = | \, x \, |$ and the time $t$, respectively
Density distribution of function $(u, v)$
Density distribution of solution ${U}^* = (U^*, V^*)(x, t)$ for (2.6), where $a$, $b$, $c$, $d$ and $\varepsilon$ are the same as in Figure 1
Density distribution of $(W^*, Z^*) = (\Psi_w, \Psi_z)({U}^*)$ with ${U}^* = {U}^*(x, t)$ shown in Figure 3
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