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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

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 and Continuous Dynamical Systems, 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, 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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