American Institute of Mathematical Sciences

Advanced
February  2012, 5(1): 147-158. doi: 10.3934/dcdss.2012.5.147

A relation between cross-diffusion and reaction-diffusion

Hideki Murakawa 1,

1. 

Graduate School of Science and Engineering for Research, University of Toyama, 3190 Gofuku, Toyama 930-8555, Japan

Received  March 2009 Revised  January 2010 Published  February 2011

Reaction-diffusion system approximations to a cross-diffusion system are investigated. Iida and Ninomiya~[Recent Advances on Elliptic and Parabolic Issues, 145--164 (2006)] proposed a semilinear reaction-diffusion system with a small parameter and showed that the limit equation takes the form of a weakly coupled cross-diffusion system provided that solutions of both the reaction-diffusion and the cross-diffusion systems are sufficiently smooth. In this paper, the results are extended to a more general cross-diffusion problem involving strongly coupled systems. It is shown that a solution of the problem can be approximated by that of a semilinear reaction-diffusion system without any assumptions on the solutions. This indicates that the mechanism of cross-diffusion might be captured by reaction-diffusion interaction.
Keywords: Reaction-diffusion system, Cross-diffusion system, Approximation..
Mathematics Subject Classification: 35K55, 35K5.
Citation: Hideki Murakawa. A relation between cross-diffusion and reaction-diffusion. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 147-158. doi: 10.3934/dcdss.2012.5.147
References:
[1]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2006), 301-322. doi: 10.1137/S0036141003427798.  Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002.  Google Scholar

[3]

M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math, 32 (1974), 1-9.  Google Scholar

[4]

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.  Google Scholar

[5]

M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164. doi: 10.1142/9789812774170_0007.  Google Scholar

[6]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

[7]

E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255. doi: 10.1007/BF02476361.  Google Scholar

[8]

H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332. doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[9]

H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590. Google Scholar

[10]

H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().   Google Scholar

[11]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar

[12]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273. doi: 10.1016/j.jde.2004.01.004.  Google Scholar

[13]

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.  Google Scholar

[14]

R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001.  Google Scholar

show all references

References:
[1]

L. Chen and A. Jüngel, Analysis of a multidimensional parabolic population model with strong cross-diffusion, SIAM J. Math. Anal., 36 (2006), 301-322. doi: 10.1137/S0036141003427798.  Google Scholar

[2]

L. Chen and A. Jüngel, Analysis of a parabolic cross-diffusion population model without self-diffusion, J. Differential Equations, 224 (2006), 39-59. doi: 10.1016/j.jde.2005.08.002.  Google Scholar

[3]

M. E. Gurtin, Some mathematical models for population dynamics that lead to segregation, Quart. Appl. Math, 32 (1974), 1-9.  Google Scholar

[4]

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.  Google Scholar

[5]

M. Iida and H. Ninomiya, A reaction-diffusion approximation to a cross-diffusion system, in "Recent Advances on Elliptic and Parabolic Issues" (eds. M. Chipot and H. Ninomiya), World Scientific, (2006), 145-164. doi: 10.1142/9789812774170_0007.  Google Scholar

[6]

T. Kadota and K. Kuto, Positive steady states for a prey-predator model with some nonlinear diffusion terms, J. Math. Anal. Appl., 323 (2006), 1387-1401. doi: 10.1016/j.jmaa.2005.11.065.  Google Scholar

[7]

E. H. Kerner, Further considerations on the statistical mechanics of biological associations, Bull. Math. Biophys., 21 (1959), 217-255. doi: 10.1007/BF02476361.  Google Scholar

[8]

H. Murakawa, Reaction-diffusion system approximation to degenerate parabolic systems, Nonlinearity, 20 (2007), 2319-2332. doi: 10.1088/0951-7715/20/10/003.  Google Scholar

[9]

H. Murakawa, A solution of nonlinear diffusion problems by semilinear reaction-diffusion systems, Kybernetika, 45 (2009), 580-590. Google Scholar

[10]

H. Murakawa, Discrete-time approximation to nonlinear degenerate parabolic problems using a semilinear reaction-diffusion system,, preprint., ().   Google Scholar

[11]

A. Okubo and S. A. Levin, "Diffusion and Ecological Problems: Modern Perspectives. Second Edition," Interdisciplinary Applied Mathematics, 14, Springer-Verlag, New York, 2001.  Google Scholar

[12]

P. Y. H. Pang and M. X. Wang, Strategy and stationary pattern in a three-species predator-prey model, J. Differential Equations, 200 (2004), 245-273. doi: 10.1016/j.jde.2004.01.004.  Google Scholar

[13]

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.  Google Scholar

[14]

R. Temam, "Navier-Stokes Equation Theory and Numerical Analysis," AMS Chelsea Publishing, Providence, RI., 2001.  Google Scholar

[1]

Yuan Lou, Wei-Ming Ni, Yaping Wu. On the global existence of a cross-diffusion system. Discrete & Continuous Dynamical Systems, 1998, 4 (2) : 193-203. doi: 10.3934/dcds.1998.4.193
[2]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[3]

Hirofumi Izuhara, Shunsuke Kobayashi. Spatio-temporal coexistence in the cross-diffusion competition system. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 919-933. doi: 10.3934/dcdss.2020228
[4]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. On a limiting system in the Lotka--Volterra competition with cross-diffusion. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 435-458. doi: 10.3934/dcds.2004.10.435
[5]

Yi Li, Chunshan Zhao. Global existence of solutions to a cross-diffusion system in higher dimensional domains. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 185-192. doi: 10.3934/dcds.2005.12.185
[6]

Anotida Madzvamuse, Hussaini Ndakwo, Raquel Barreira. Stability analysis of reaction-diffusion models on evolving domains: The effects of cross-diffusion. Discrete & Continuous Dynamical Systems, 2016, 36 (4) : 2133-2170. doi: 10.3934/dcds.2016.36.2133
[7]

Anotida Madzvamuse, Raquel Barreira. Domain-growth-induced patterning for reaction-diffusion systems with linear cross-diffusion. Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : 2775-2801. doi: 10.3934/dcdsb.2018163
[8]

Thomas I. Seidman. Interface conditions for a singular reaction-diffusion system. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 631-643. doi: 10.3934/dcdss.2009.2.631
[9]

Vladimir V. Chepyzhov, Mark I. Vishik. Trajectory attractor for reaction-diffusion system with diffusion coefficient vanishing in time. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1493-1509. doi: 10.3934/dcds.2010.27.1493
[10]

Salomé Martínez, Wei-Ming Ni. Periodic solutions for a 3x 3 competitive system with cross-diffusion. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 725-746. doi: 10.3934/dcds.2006.15.725
[11]

Yukio Kan-On. On the limiting system in the Shigesada, Kawasaki and Teramoto model with large cross-diffusion rates. Discrete & Continuous Dynamical Systems, 2020, 40 (6) : 3561-3570. doi: 10.3934/dcds.2020161
[12]

Jifa Jiang, Junping Shi. Dynamics of a reaction-diffusion system of autocatalytic chemical reaction. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 245-258. doi: 10.3934/dcds.2008.21.245
[13]

Sze-Bi Hsu, Junping Shi, Feng-Bin Wang. Further studies of a reaction-diffusion system for an unstirred chemostat with internal storage. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3169-3189. doi: 10.3934/dcdsb.2014.19.3169
[14]

Vo Van Au, Mokhtar Kirane, Nguyen Huy Tuan. Determination of initial data for a reaction-diffusion system with variable coefficients. Discrete & Continuous Dynamical Systems, 2019, 39 (2) : 771-801. doi: 10.3934/dcds.2019032
[15]

Nicolas Bacaër, Cheikh Sokhna. A reaction-diffusion system modeling the spread of resistance to an antimalarial drug. Mathematical Biosciences & Engineering, 2005, 2 (2) : 227-238. doi: 10.3934/mbe.2005.2.227
[16]

José-Francisco Rodrigues, Lisa Santos. On a constrained reaction-diffusion system related to multiphase problems. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 299-319. doi: 10.3934/dcds.2009.25.299
[17]

W. E. Fitzgibbon, M. Langlais, J.J. Morgan. A reaction-diffusion system modeling direct and indirect transmission of diseases. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 893-910. doi: 10.3934/dcdsb.2004.4.893
[18]

Haomin Huang, Mingxin Wang. The reaction-diffusion system for an SIR epidemic model with a free boundary. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2039-2050. doi: 10.3934/dcdsb.2015.20.2039
[19]

Sebastian Aniţa, Vincenzo Capasso. Stabilization of a reaction-diffusion system modelling malaria transmission. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1673-1684. doi: 10.3934/dcdsb.2012.17.1673
[20]

Michaël Bages, Patrick Martinez. Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 817-869. doi: 10.3934/dcdsb.2010.14.817

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (68)
  • HTML views (0)
  • Cited by (9)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences