# American Institute of Mathematical Sciences

• Previous Article
Asymmetric dispersal and evolutional selection in two-patch system
• DCDS Home
• This Issue
• Next Article
Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere
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:

show all references

##### 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
 [1] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435 [2] Daozhou Gao, Xing Liang. A competition-diffusion system with a refuge. Discrete & Continuous Dynamical Systems - B, 2007, 8 (2) : 435-454. doi: 10.3934/dcdsb.2007.8.435 [3] Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2019  doi: 10.3934/dcdss.2020228 [4] Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193 [5] Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589 [6] Jong-Shenq Guo, Ying-Chih Lin. The sign of the wave speed for the Lotka-Volterra competition-diffusion system. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2083-2090. doi: 10.3934/cpaa.2013.12.2083 [7] Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185 [8] Yukio Kan-On. Structure on the set of radially symmetric positive stationary solutions for a competition-diffusion system. Conference Publications, 2013, 2013 (special) : 427-436. doi: 10.3934/proc.2013.2013.427 [9] E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39 [10] Xiongxiong Bao, Wan-Tong Li, Zhi-Cheng Wang. Uniqueness and stability of time-periodic pyramidal fronts for a periodic competition-diffusion system. Communications on Pure & Applied Analysis, 2020, 19 (1) : 253-277. doi: 10.3934/cpaa.2020014 [11] Qian Guo, Xiaoqing He, Wei-Ming Ni. Global dynamics of a general Lotka-Volterra competition-diffusion system in heterogeneous environments. Discrete & Continuous Dynamical Systems - A, 2020, 40 (11) : 6547-6573. doi: 10.3934/dcds.2020290 [12] Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725 [13] Kazuhiro Oeda. Positive steady states for a prey-predator cross-diffusion system with a protection zone and Holling type II functional response. Conference Publications, 2013, 2013 (special) : 597-603. doi: 10.3934/proc.2013.2013.597 [14] Yuan Lou, Salomé Martínez, Wei-Ming Ni. On $3\times 3$ Lotka-Volterra competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 175-190. doi: 10.3934/dcds.2000.6.175 [15] Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367 [16] Robert Stephen Cantrell, Xinru Cao, King-Yeung Lam, Tian Xiang. A PDE model of intraguild predation with cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3653-3661. doi: 10.3934/dcdsb.2017145 [17] Michael Winkler, Dariusz Wrzosek. Preface: Analysis of cross-diffusion systems. Discrete & Continuous Dynamical Systems - S, 2020, 13 (2) : ⅰ-ⅰ. doi: 10.3934/dcdss.20202i [18] Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147 [19] Wei-Ming Ni, Masaharu Taniguchi. Traveling fronts of pyramidal shapes in competition-diffusion systems. Networks & Heterogeneous Media, 2013, 8 (1) : 379-395. doi: 10.3934/nhm.2013.8.379 [20] Kousuke Kuto, Yoshio Yamada. Coexistence states for a prey-predator model with cross-diffusion. Conference Publications, 2005, 2005 (Special) : 536-545. doi: 10.3934/proc.2005.2005.536

2019 Impact Factor: 1.338