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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

Yukio Kan-On

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].

Keywords: Competition-diffusion system, cross-diffusion, limiting system.
Mathematics Subject Classification: Primary: 35Q92; Secondary: 92B05.
Citation: Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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Figure 2.  Density distribution of function $ (u, v) $
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Figure 1">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
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Figure 3">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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