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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 and Continuous Dynamical Systems, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
##### References:
 [1] A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15. [2] K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [3] K. Kuto, Limiting structure of shrinking solutions to the stationary Shigesada-Kawasaki-Teramoto model with large cross-diffusion, SIAM J. Math. Anal., 47 (2015), 3993-4024.  doi: 10.1137/140991455. [4] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [5] Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559. [6] Y. Lou, Yu an, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [7] Y. Lou, W.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589. [8] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020. [9] T. Mori, T. Suzuki and S. Yotsutani, Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation, Math. Models Methods Appl. Sci., 28 (2018), 2191-2210.  doi: 10.1142/S0218202518400122. [10] W.-M. Ni, The mathematics of diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. [11] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.

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##### References:
 [1] A. Jüngel, Diffusive and nondiffusive population models, in Mathematical Modeling of Collective Behavior in Socio-Economic and Life Sciences, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2010,397–425. doi: 10.1007/978-0-8176-4946-3_15. [2] K. Kishimoto and H. F. Weinberger, The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.  doi: 10.1016/0022-0396(85)90020-8. [3] K. Kuto, Limiting structure of shrinking solutions to the stationary Shigesada-Kawasaki-Teramoto model with large cross-diffusion, SIAM J. Math. Anal., 47 (2015), 3993-4024.  doi: 10.1137/140991455. [4] Y. Lou and W.-M. Ni, Diffusion, self-diffusion and cross-diffusion, J. Differential Equations, 131 (1996), 79-131.  doi: 10.1006/jdeq.1996.0157. [5] Y. Lou and W.-M. Ni, Diffusion vs cross-diffusion: An elliptic approach, J. Differential Equations, 154 (1999), 157-190.  doi: 10.1006/jdeq.1998.3559. [6] Y. Lou, Yu an, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.  doi: 10.3934/dcds.2004.10.435. [7] Y. Lou, W.-M. Ni and S. Yotsutani, Pattern formation in a cross-diffusion system, Discrete Contin. Dyn. Syst., 35 (2015), 1589-1607.  doi: 10.3934/dcds.2015.35.1589. [8] H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. Res. Inst. Math. Sci., 19 (1983), 1049-1079.  doi: 10.2977/prims/1195182020. [9] T. Mori, T. Suzuki and S. Yotsutani, Numerical approach to existence and stability of stationary solutions to a SKT cross-diffusion equation, Math. Models Methods Appl. Sci., 28 (2018), 2191-2210.  doi: 10.1142/S0218202518400122. [10] W.-M. Ni, The mathematics of diffusion, CBMS-NSF Regional Conference Series in Applied Mathematics, 82, SIAM, Philadelphia, PA, 2011. doi: 10.1137/1.9781611971972. [11] N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theoret. Biol., 79 (1979), 83-99.  doi: 10.1016/0022-5193(79)90258-3.
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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