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Article Contents

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

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].

Mathematics Subject Classification: Primary: 35Q92; Secondary: 92B05.

 Citation:

• Figure 1.  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

Figure 2.  Density distribution of function $(u, v)$

Figure 3.  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

Figure 4.  Density distribution of $(W^*, Z^*) = (\Psi_w, \Psi_z)({U}^*)$ with ${U}^* = {U}^*(x, t)$ shown in Figure 3

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