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Journal of Mathematical Chemistry

, Volume 55, Issue 3, pp 818–831 | Cite as

Turing pattern formation in anisotropic medium

  • Debojyoti Das
Original Paper

Abstract

Diffusion of reacting species in chemical and biochemical systems in anisotropic medium is markedly different from those occurring in isotropic medium, therefore approximating diffusion coefficients as constants may not be desirable as this has dynamical consequences. This paper is devoted to the analytical and numerical investigation of the development of spatial patterns in such systems. To this end we consider a general reaction–diffusion system with concentration-dependent diffusion and formulate a scheme to derive the general form of envelope equation for such systems. The theory is applied to the chlorite–iodide–malonic acid system, a standard paradigm for activator–inhibitor mechanism, to derive the instability condition in terms of the anisotropy parameters (\(\kappa _{i}, i = u, v\) that impart concentration-dependence to the diffusion coefficients) and identify the supercritical and subcritical Turing regions in the bifurcation diagram. The theoretical predictions are in good agreement with the numerical simulations.

Keywords

Concentration/position-dependent diffusion Pattern formation Turing amplitude equation 

Mathematics Subject Classification

35B36 35K57 

Notes

Acknowledgements

D. D. would like to thank Professor Deb Shankar Ray for insightful and detailed discussions and thorough reading of the manuscript. Thanks are due to the Council of Scientific and Industrial Research, Government of India, for partial financial support (File No: 09/080(0742)/2010-EMR-I).

References

  1. 1.
    A.M. Turing, Philos. Trans. R. Soc. B 237, 37 (1952)CrossRefGoogle Scholar
  2. 2.
    V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Phys. Rev. Lett. 64, 2953 (1990)CrossRefGoogle Scholar
  3. 3.
    F.T. Arecchi, F. Lisi, Phys. Rev. Lett. 49, 94 (1982)CrossRefGoogle Scholar
  4. 4.
    K.M. Cuomo, A.V. Oppenheim, Phys. Rev. Lett. 71, 65 (1993)CrossRefGoogle Scholar
  5. 5.
    B. van der Pol, J. van der Mark, Nature 120, 363 (1927)CrossRefGoogle Scholar
  6. 6.
    I. Lengyel, G. Rabai, I.R. Epstein, J. Am. Chem. Soc. 112, 9104 (1990)CrossRefGoogle Scholar
  7. 7.
    I. Lengyel, I.R. Epstein, Science 251, 650 (1991)CrossRefGoogle Scholar
  8. 8.
    V. Dufiet, J. Boissonade, Phys. Rev. E 53, 4883 (1996)CrossRefGoogle Scholar
  9. 9.
    P.C. Fife, J. Chem. Phys. 64, 554 (1976)CrossRefGoogle Scholar
  10. 10.
    A. Hunding, G.D. Billing, J. Chem. Phys. 75, 486 (1981)CrossRefGoogle Scholar
  11. 11.
    J.E. Pearson, W.J. Bruno, Chaos 2, 513 (1992)CrossRefGoogle Scholar
  12. 12.
    S.S. Riaz, S. Kar, D.S. Ray, J. Chem. Phys. 121, 5395 (2004)CrossRefGoogle Scholar
  13. 13.
    D.A. Kessler, H. Levine, Phys. Rev. E 48, 4801 (1993)CrossRefGoogle Scholar
  14. 14.
    M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)CrossRefGoogle Scholar
  15. 15.
    A.J. Koch, H. Meinhardt, Rev. Mod. Phys. 66, 1481 (1994)CrossRefGoogle Scholar
  16. 16.
    S. Dutta, D.S. Ray, Phys. Rev. E 75, 016205 (2007)CrossRefGoogle Scholar
  17. 17.
    I.R. Epstein, J.A. Pojman, An Introduction to Nonlinear Chemical Dynamics (Oxford University Press, New York, 1998)Google Scholar
  18. 18.
    P.S. Hagan, Spiral waves in reaction–diffusion equations. SIAM J. Appl. Math. 42, 762 (1982)CrossRefGoogle Scholar
  19. 19.
    G. Kshirsagar, Z. Noszticzius, W. McCormick, H.L. Swinney, Phys. D 49, 5 (1991)CrossRefGoogle Scholar
  20. 20.
    R. Kapral, K. Showalter, Chemical Waves and Patterns (Kluwer, Dordrecht, 1995)CrossRefGoogle Scholar
  21. 21.
    D. Walgraef, Spatio-Temporal Pattern Formation (Springer, Berlin, 1997)CrossRefGoogle Scholar
  22. 22.
    M. Hildebrand, A.S. Mikhailov, G. Ertl, Phys. Rev. Lett. 81, 2602 (1998)CrossRefGoogle Scholar
  23. 23.
    A.M. Zhabotinsky, M. Dolnik, I.R. Epstein, J. Chem. Phys. 103, 10306 (1995)CrossRefGoogle Scholar
  24. 24.
    M. Dolnik, A.M. Zhabotinsky, A.B. Rovinsky, I.R. Epstein, J. Phys. Chem. A 103, 38 (1999)CrossRefGoogle Scholar
  25. 25.
    M. Dolnik, A.B. Rovinsky, A.M. Zhabotinsky, I.R. Epstein, Chem. Eng. Sci. 55, 223 (2000)CrossRefGoogle Scholar
  26. 26.
    R. Kapral, J. Math. Chem. 6, 113 (1991)CrossRefGoogle Scholar
  27. 27.
    S. Kar, J.K. Bhattacharjee, D.S. Ray, Eur. Phys. J. B 43, 109 (2005)CrossRefGoogle Scholar
  28. 28.
    B.K. Agarwalla, S. Galhotra, J.K. Bhattacharjee, J. Math. Chem. 52, 188 (2014)CrossRefGoogle Scholar
  29. 29.
    A.B. Rovinsky, M. Menzinger, Phys. Rev. Lett. 69, 1193 (1992)CrossRefGoogle Scholar
  30. 30.
    V.K. Vanag, A.M. Zhabotinsky, I.R. Epstein, Phys. Rev. Lett. 86, 552 (2001)CrossRefGoogle Scholar
  31. 31.
    I. Berenstein, L. Yang, M. Dolnik, A.M. Zhabotinsky, I.R. Epstein, J. Phys. Chem. A 109, 5382 (2005)CrossRefGoogle Scholar
  32. 32.
    P. Ghosh, S. Sen, S.S. Riaz, D.S. Ray, Phys. Rev. E 79, 056216 (2009)CrossRefGoogle Scholar
  33. 33.
    P. De Kepper, V. Castets, E. Dulos, J. Boissonade, Phys. D 49, 161 (1991)CrossRefGoogle Scholar
  34. 34.
    L. Yang, M. Dolnik, A.M. Zhabotinsky, I.R. Epstein, Chaos 16, 037114 (2006)CrossRefGoogle Scholar
  35. 35.
    P. Jung, G.M. Kress, Phys. Rev. Lett. 74, 2130 (1995)CrossRefGoogle Scholar
  36. 36.
    S. Kadar, J. Wang, K. Showalter, Nature 391, 770 (1998)CrossRefGoogle Scholar
  37. 37.
    P. Jung, A. Cornell-Bell, F. Moss, S. Kadar, J. Wang, K. Showalter, Chaos 8, 567 (1998)CrossRefGoogle Scholar
  38. 38.
    D. Hochberg, M.-P. Zorzano, F. Morán, J. Chem. Phys. 122, 214701 (2005)CrossRefGoogle Scholar
  39. 39.
    S. Dutta, S.S. Riaz, D.S. Ray, Phys. Rev. E 71, 036216 (2005)CrossRefGoogle Scholar
  40. 40.
    S.S. Riaz, R. Sharma, S.P. Bhattacharyya, D.S. Ray, J. Chem. Phys. 127, 064503 (2007)CrossRefGoogle Scholar
  41. 41.
    J. Buceta, K. Lindenberg, Phys. Rev. E 68, 011103 (2003)CrossRefGoogle Scholar
  42. 42.
    D. Das, D.S. Ray, Phys. Rev. E 87, 062924 (2013)CrossRefGoogle Scholar
  43. 43.
    I. Lengyel, I.R. Epstein, Proc. Natl. Acad. Sci. USA 89, 3977 (1992)CrossRefGoogle Scholar
  44. 44.
    Y. Kuramoto, T. Tsuzuki, Prog. Theor. Phys. 54, 687 (1975)CrossRefGoogle Scholar
  45. 45.
    R.J. Gelten, A.P.J. Jansen, R.A. van Santen, J.J. Lukkien, J.P.L. Segers, P.A.J. Hilbers, J. Chem. Phys. 108, 5921 (1998)CrossRefGoogle Scholar
  46. 46.
    A. Eldar, D. Rosin, B.Z. Shilo, N. Barkai, Dev. Cell 5, 635 (2003)CrossRefGoogle Scholar
  47. 47.
    S.B. Yuste, E. Abad, K. Lindenberg, Phys. Rev. E 82, 061123 (2010)CrossRefGoogle Scholar
  48. 48.
    P.V. Gordon, V. Peter, C. Sample, A.M. Berezhkovskii, C.B. Muratov, S.Y. Shvartsman, Proc. Natl. Acad. Sci. USA 108, 6157 (2011)CrossRefGoogle Scholar
  49. 49.
    H.G. Othmer, E. Pate, Proc. Natl. Acad. Sci. 77, 4180 (1980)CrossRefGoogle Scholar
  50. 50.
    W.F. Loomis, in Development Biology: Pattern Formation, Gene Regulation, ed. by D. McMahon, C.F. Fox (Benjamin, Reading, MA), pp. 109–128 (1975)Google Scholar
  51. 51.
    N. Gottschalk, F. Mertens, M. Bär, M. Eiswirth, R. Imbihl, Phys. Rev. Lett. 73, 3483 (1994)CrossRefGoogle Scholar
  52. 52.
    H. Malchow, J. Theor. Biol. 135, 371 (1988)CrossRefGoogle Scholar
  53. 53.
    M.R. Roussel, J. Wang, Phys. Rev. Lett. 87, 188302 (2001)CrossRefGoogle Scholar
  54. 54.
    M.R. Roussel, J. Wang, J. Chem. Phys. 120, 8079 (2004)CrossRefGoogle Scholar
  55. 55.
    W.-S. Li, W.-Y. Hu, Y.-C. Pang, T.-R. Liu, W.-R. Zhong, Y.-Z. Shao, Phys. Rev. E 85, 066132 (2012)CrossRefGoogle Scholar
  56. 56.
    C.J. Roussel, M.R. Roussel, Prog. Biophys. Mol. Biol. 86, 113 (2004)CrossRefGoogle Scholar
  57. 57.
    H. Shoji, Y. Iwasa, A. Mochizuki, S. Kondo, J. Theor. Biol. 214, 549 (2002)CrossRefGoogle Scholar
  58. 58.
    H. Shoji, Y. Iwasa, S. Kondo, J. Theor. Biol. 224, 339 (2003)CrossRefGoogle Scholar
  59. 59.
    T. Aegerter-Wilmsen, C.M. Aegerter, T. Bisseling, J. Theor. Biol. 234, 13 (2005)CrossRefGoogle Scholar
  60. 60.
    H. Shoji, A. Mochizuki, Y. Iwasa, M. Hirata, T. Watanabe, S. Hioki, S. Kondo, Dev. Dyn. 226, 627 (2003)CrossRefGoogle Scholar
  61. 61.
    P.K. Maini, K.J. Painter, H.N.P. Chau, J. Chem. Soc. Faraday Trans. 93, 3601 (1997)CrossRefGoogle Scholar
  62. 62.
    Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)CrossRefGoogle Scholar
  63. 63.
    M. Ipsen, L. Kramer, P.G. Sørensen, Phys. Rep. 337, 193 (2000)CrossRefGoogle Scholar
  64. 64.
    S.H. Strogatz, Phys. D 143, 1 (2000)CrossRefGoogle Scholar
  65. 65.
    E.P. Zemskov, V.K. Vanag, I.R. Epstein, Phys. Rev. E. 84, 036216 (2011)CrossRefGoogle Scholar
  66. 66.
    R.B. Hoyle, Pattern Formation: An Introduction to Methods (Cambridge University Press, Cambridge, 2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Indian Association for the Cultivation of ScienceJadavpur, KolkataIndia

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