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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 |
In 1979, Shigesada, Kawasaki and Teramoto [
References:
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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. |
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The spatial homogeneity of stable equilibria of some reaction-diffusion systems on convex domains, J. Differential Equations, 58 (1985), 15-21.
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[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. |
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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. |
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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. |
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. |
[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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