Pattern formation in a cross-diffusion system

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

Pattern formation in a cross-diffusion system

Yuan Lou ^1, , Wei-Ming Ni ^2, and Shoji Yotsutani ^3,

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

Abstract
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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.

Keywords: existence, competition, stability, Density-dependent diffusion, steady states..

Mathematics Subject Classification: Primary: 35B40, 35K57; Secondary: 92D25, 92D4.

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11 (1981), 621. Google Scholar
[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. Google Scholar
[13]	W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	M. Mimura, Stationary pattern of some density-dependent diffusion system with competitive dynamics,, Hiroshima Math. J., 11 (1981), 621. Google Scholar
[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. Google Scholar
[13]	W. M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states,, Notices Amer. Math. Soc., 45 (1998), 9. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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