# American Institute of Mathematical Sciences

2015, 35(4): 1589-1607. doi: 10.3934/dcds.2015.35.1589

## 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

Received  August 2013 Revised  June 2014 Published  November 2014

In this paper we study the Shigesada-Kawasaki-Teramoto model [17] for two competing species with cross-diffusion. We prove the existence of spectrally stable non-constant positive steady states for high-dimensional domains when one of the cross-diffusion coefficients is sufficiently large while the other is equal to zero.
Citation: 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
##### References:
 [1] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Series in Mathematical and Computational Biology, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. [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. [13] W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. [14] W. M. Ni, Qualitative properties of solutions to elliptic problems,, Stationary partial differential equations. Handb. Differ. Equ., I (2004), 157. 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. doi: 10.3934/dcds.2014.34.5271. [16] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Interdisciplinary Applied Mathematics, (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. 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, 132 (2002), 1493. [19] Y. Wu, The instability of spiky steady states for a competing species model with cross-diffusion,, J. Differential Equations, 213 (2005), 289. 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. 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. 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, 6 (2008), 411. 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, (2009), 282. doi: 10.1142/9789812834744_0013.

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##### References:
 [1] R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, Series in Mathematical and Computational Biology, (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. 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. 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. 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. 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. 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. 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. 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. 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. 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. [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. [13] W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. [14] W. M. Ni, Qualitative properties of solutions to elliptic problems,, Stationary partial differential equations. Handb. Differ. Equ., I (2004), 157. 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. doi: 10.3934/dcds.2014.34.5271. [16] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives,, Interdisciplinary Applied Mathematics, (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. 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, 132 (2002), 1493. [19] Y. Wu, The instability of spiky steady states for a competing species model with cross-diffusion,, J. Differential Equations, 213 (2005), 289. 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. 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. 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, 6 (2008), 411. 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, (2009), 282. doi: 10.1142/9789812834744_0013.
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