American Institute of Mathematical Sciences

Advanced
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 & 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

show all references

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

[1]

Kaigang Huang, Yongli Cai, Feng Rao, Shengmao Fu, Weiming Wang. Positive steady states of a density-dependent predator-prey model with diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3087-3107. doi: 10.3934/dcdsb.2017209
[2]

Wei-Ming Ni, Yaping Wu, Qian Xu. The existence and stability of nontrivial steady states for S-K-T competition model with cross diffusion. Discrete & Continuous Dynamical Systems - A, 2014, 34 (12) : 5271-5298. doi: 10.3934/dcds.2014.34.5271
[3]

Yaping Wu, Qian Xu. The existence and structure of large spiky steady states for S-K-T competition systems with cross-diffusion. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 367-385. doi: 10.3934/dcds.2011.29.367
[4]

Qing Li, Yaping Wu. Existence and instability of some nontrivial steady states for the SKT competition model with large cross diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 0 (0) : 0-0. doi: 10.3934/dcds.2020051
[5]

Tracy L. Stepien, Erica M. Rutter, Yang Kuang. A data-motivated density-dependent diffusion model of in vitro glioblastoma growth. Mathematical Biosciences & Engineering, 2015, 12 (6) : 1157-1172. doi: 10.3934/mbe.2015.12.1157
[6]

Weiping Yan. Existence of weak solutions to the three-dimensional density-dependent generalized incompressible magnetohydrodynamic flows. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1359-1385. doi: 10.3934/dcds.2015.35.1359
[7]

Jishan Fan, Tohru Ozawa. An approximation model for the density-dependent magnetohydrodynamic equations. Conference Publications, 2013, 2013 (special) : 207-216. doi: 10.3934/proc.2013.2013.207
[8]

Jacques A. L. Silva, Flávia T. Giordani. Density-dependent dispersal in multiple species metapopulations. Mathematical Biosciences & Engineering, 2008, 5 (4) : 843-857. doi: 10.3934/mbe.2008.5.843
[9]

Wenshu Zhou, Hongxing Zhao, Xiaodan Wei, Guokai Xu. Existence of positive steady states for a predator-prey model with diffusion. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2189-2201. doi: 10.3934/cpaa.2013.12.2189
[10]

Pierre Degond, Silke Henkes, Hui Yu. Self-organized hydrodynamics with density-dependent velocity. Kinetic & Related Models, 2017, 10 (1) : 193-213. doi: 10.3934/krm.2017008
[11]

J. X. Velasco-Hernández, M. Núñez-López, G. Ramírez-Santiago, M. Hernández-Rosales. On carrying-capacity construction, metapopulations and density-dependent mortality. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1099-1110. doi: 10.3934/dcdsb.2017054
[12]

Baojun Song, Wen Du, Jie Lou. Different types of backward bifurcations due to density-dependent treatments. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1651-1668. doi: 10.3934/mbe.2013.10.1651
[13]

Quansen Jiu, Zhouping Xin. The Cauchy problem for 1D compressible flows with density-dependent viscosity coefficients. Kinetic & Related Models, 2008, 1 (2) : 313-330. doi: 10.3934/krm.2008.1.313
[14]

Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20
[15]

Azmy S. Ackleh, Linda J. S. Allen. Competitive exclusion in SIS and SIR epidemic models with total cross immunity and density-dependent host mortality. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 175-188. doi: 10.3934/dcdsb.2005.5.175
[16]

Xulong Qin, Zheng-An Yao, Hongxing Zhao. One dimensional compressible Navier-Stokes equations with density-dependent viscosity and free boundaries. Communications on Pure & Applied Analysis, 2008, 7 (2) : 373-381. doi: 10.3934/cpaa.2008.7.373
[17]

Jishan Fan, Tohru Ozawa. Global Cauchy problem of an ideal density-dependent MHD-$\alpha$ model. Conference Publications, 2011, 2011 (Special) : 400-409. doi: 10.3934/proc.2011.2011.400
[18]

Jishan Fan, Tohru Ozawa. A regularity criterion for 3D density-dependent MHD system with zero viscosity. Conference Publications, 2015, 2015 (special) : 395-399. doi: 10.3934/proc.2015.0395
[19]

Xulong Qin, Zheng-An Yao. Global solutions of the free boundary problem for the compressible Navier-Stokes equations with density-dependent viscosity. Communications on Pure & Applied Analysis, 2010, 9 (4) : 1041-1052. doi: 10.3934/cpaa.2010.9.1041
[20]

Mei Wang, Zilai Li, Zhenhua Guo. Global weak solution to 3D compressible flows with density-dependent viscosity and free boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 1-24. doi: 10.3934/cpaa.2017001

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (46)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences