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Pattern formation in a cross-diffusion system
1. | Institute for Mathematical Sciences, Renmin University of China, Haidian District, Beijing, 100872, China |
2. | Center for Partial Differential Equations, East China Normal University, Minhang, Shanghai, 200241 |
3. | Department of Applied Mathematics and Informatics, Ryukoku University, Seta, Otsu, Shiga 520-2194 |
References:
[1] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
[2] |
Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[3] |
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. |
[4] |
K. Kuto and Y. Yamada, On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2745-2769.
doi: 10.3934/dcdsb.2012.17.2745. |
[5] |
M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[6] |
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. |
[7] |
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. |
[8] |
Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[9] |
Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.
doi: 10.3934/dcds.2004.10.435. |
[10] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[11] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[12] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449. |
[13] |
W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[14] |
W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary partial differential equations. Handb. Differ. Equ., North-Holland, Amsterdam, I (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[15] |
W. M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[16] |
A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, Vol. 14, 2nd ed. Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[17] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[18] |
Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of Royal Soc. Edinburg, Sect. A, 132 (2002), 1493-1511. |
[19] |
Y. Wu, The instability of spiky steady states for a competing species model with cross-diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[20] |
Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
[21] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion, Science in China, 53 (2010), 1161-1184.
doi: 10.1007/s11425-010-0141-4. |
[22] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations, Stationary Partial Differential Equations, Edited by M. Chipot, Elsevier, Amsterdam, 6 (2008), 411-501.
doi: 10.1016/S1874-5733(08)80023-X. |
[23] |
Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, 2009.
doi: 10.1142/9789812834744_0013. |
show all references
References:
[1] |
R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Series in Mathematical and Computational Biology, John Wiley and Sons, Chichester, UK, 2003.
doi: 10.1002/0470871296. |
[2] |
Y. S. Choi, R. Lui and Y. Yamada, Existence of global solutions for the Shigesada-Kawasaki-Teramoto model with strongly coupled cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 719-730.
doi: 10.3934/dcds.2004.10.719. |
[3] |
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. |
[4] |
K. Kuto and Y. Yamada, On limit systems for some population models with cross-diffusion, Discrete Contin. Dyn. Syst.-Series B, 17 (2012), 2745-2769.
doi: 10.3934/dcdsb.2012.17.2745. |
[5] |
M. Iida, M. Mimura and H. Ninomiya, Diffusion, cross-diffusion and competitive interaction, J. Math. Biol., 53 (2006), 617-641.
doi: 10.1007/s00285-006-0013-2. |
[6] |
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. |
[7] |
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. |
[8] |
Y. Lou, W.-M. Ni and Y. Wu, On the global existence of a cross-diffusion system, Discrete Contin. Dyn. Syst., 4 (1998), 193-203.
doi: 10.3934/dcds.1998.4.193. |
[9] |
Y. Lou, W.-M. Ni and S. Yotsutani, On a limiting system in the Lotka-Volterra competition with cross-diffusion, Discrete Contin. Dyn. Syst., 10 (2004), 435-458.
doi: 10.3934/dcds.2004.10.435. |
[10] |
H. Matano and M. Mimura, Pattern formation in competition-diffusion systems in nonconvex domains, Publ. RIMS. Kyoto Univ., 19 (1983), 1049-1079.
doi: 10.2977/prims/1195182020. |
[11] |
M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics, Hiroshima Math. J., 11 (1981), 621-635. |
[12] |
M. Mimura, Y. Nishiura, A. Tesei and T. Tsujikawa, Coexistence problem for two competing species models with density-dependent diffusion, Hiroshima Math. J., 14 (1984), 425-449. |
[13] |
W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc., 45 (1998), 9-18. |
[14] |
W. M. Ni, Qualitative properties of solutions to elliptic problems, Stationary partial differential equations. Handb. Differ. Equ., North-Holland, Amsterdam, I (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[15] |
W. M. Ni, Y. Wu and Q. Xu, The existence and stability of nontrivial steady states for S-K-T competition model with cross-diffusion, Discrete Contin. Dyn. Syst., 34 (2014), 5271-5298.
doi: 10.3934/dcds.2014.34.5271. |
[16] |
A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Interdisciplinary Applied Mathematics, Vol. 14, 2nd ed. Springer, Berlin, 2001.
doi: 10.1007/978-1-4757-4978-6. |
[17] |
N. Shigesada, K. Kawasaki and E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83-99.
doi: 10.1016/0022-5193(79)90258-3. |
[18] |
Y. Wu, Existence of stationary solutions with transition layers for a class of cross-diffusion systems, Proc. of Royal Soc. Edinburg, Sect. A, 132 (2002), 1493-1511. |
[19] |
Y. Wu, The instability of spiky steady states for a competing species model with cross-diffusion, J. Differential Equations, 213 (2005), 289-340.
doi: 10.1016/j.jde.2004.08.015. |
[20] |
Y. Wu and Q. Xu, The Existence and structure of large spiky steady states for S-K-T competition system with cross-diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 367-385.
doi: 10.3934/dcds.2011.29.367. |
[21] |
Y. Wu and Y. Zhao, The existence and stability of traveling waves with transition layers for the S-K-T competition model with cross-diffusion, Science in China, 53 (2010), 1161-1184.
doi: 10.1007/s11425-010-0141-4. |
[22] |
Y. Yamada, Positive solutions for Lotka-Volterra systems with cross-diffusion, Handbook of Differential Equations, Stationary Partial Differential Equations, Edited by M. Chipot, Elsevier, Amsterdam, 6 (2008), 411-501.
doi: 10.1016/S1874-5733(08)80023-X. |
[23] |
Y. Yamada, Global solutions for the Shigesada-Kawasaki-Teramoto model with cross-diffusion, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, 282-299, World Sci. Publ. Hackensack, NJ, 2009.
doi: 10.1142/9789812834744_0013. |
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