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Journal of Mathematical Chemistry

, Volume 48, Issue 4, pp 841–855 | Cite as

Turing pattern amplitude equation for a model glycolytic reaction-diffusion system

  • A. K. DuttEmail author
Original Paper

Abstract

For a reaction-diffusion system of glycolytic oscillations containing analytical steady state solution in complicated algebraic form, Turing instability condition and the critical wavenumber at the Turing bifurcation point, have been derived by a linear stability analysis. In the framework of a weakly nonlinear theory, these relations have been subsequently used to derive an amplitude equation, which interprets the structural transitions and stability of various forms of Turing structures. Amplitude equation also conforms to the expectation that time-invariant amplitudes are independent of complexing reaction with the activator species.

Keywords

Turing patterns Reaction-Diffusion systems Amplitude equation Glycolytic oscillations 

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References

  1. 1.
    Cross M.C., Hohenberg P.C.: Rev. Mod. Phys 65, 851 (1993)CrossRefGoogle Scholar
  2. 2.
    Manneville P.: Dissipative Structures and Weak Turbulence. Academic Press, New York (1990)Google Scholar
  3. 3.
    Turing A.: Philos. Trans. R. Soc. (London) B 237, 37 (1952)CrossRefGoogle Scholar
  4. 4.
    DeKepper P., Boissonade J., Epstein I.: J. Phys. Chem. 94, 6525 (1990)CrossRefGoogle Scholar
  5. 5.
    Castets V., Dulos E., Boissonade J., DeKepper P.: Phys. Rev. Letts. 64, 2953 (1990)CrossRefGoogle Scholar
  6. 6.
    DeKepper P., Castets V., Dulos E., Boissonade J.: Physica D 49, 161 (1991)CrossRefGoogle Scholar
  7. 7.
    Walgraef D., Dewel G., Borckmans P.: Adv. Chem. Phys. 49, 311 (1982)CrossRefGoogle Scholar
  8. 8.
    Keener J.P., Tyson J.J.: Physica D 21, 307 (1986)CrossRefGoogle Scholar
  9. 9.
    Verdasca J., DeWit A., Dewel G., Borckmans P.: Phys. Letts. A. 168, 194 (1992)CrossRefGoogle Scholar
  10. 10.
    Dewel G., Borckmans P., DeWit A., Rudovics B., Perraud J.-J., Dulos E., Boissonade J., Dekepper P.: Physica A 213, 181 (1995)CrossRefGoogle Scholar
  11. 11.
    DeWit A., Lima D., Dewel G., Borckmans P.: Phys. Rev. E 54, 261 (1996)CrossRefGoogle Scholar
  12. 12.
    Dufiet V., Boissonade J.: Phys. Rev. E 53, 4883 (1996)CrossRefGoogle Scholar
  13. 13.
    Lima D., DeWit A., Dewel G., Borckmans P.: Phys. Rev. E 53, R1305 (1996)CrossRefGoogle Scholar
  14. 14.
    Hilali M.F., Metens S., Borckmans P., Dewel G.: Phys. Rev. E 51, 2046 (1995)CrossRefGoogle Scholar
  15. 15.
    Bestehorn M., Haken H.: Z. Phys. B 57, 329 (1984)CrossRefGoogle Scholar
  16. 16.
    Rabinovich M.I., Fabrikant A.L., Tsimring L.S.: Usp. Fiz. Nauk 162, 1 (1992)CrossRefGoogle Scholar
  17. 17.
    Richter P., Regmus P., Ross J.: Prog. Theor. Phys. 66, 385 (1981)CrossRefGoogle Scholar
  18. 18.
    Dutt A.K.: Chem. Phys. Letts. 208, 139 (1993)CrossRefGoogle Scholar
  19. 19.
    Sel’kov E. E.: Eur. J. Biochem. 4, 79 (1968)CrossRefGoogle Scholar
  20. 20.
    Dutt A.K.: J. Chem. Phys. 92, 3058 (1990)CrossRefGoogle Scholar
  21. 21.
    Lengyel I., Epstein I.: PNAS(USA) 89, 3977 (1992)CrossRefGoogle Scholar
  22. 22.
    Dutt A.K.: J. Phys. Chem. B 109, 17679 (2005)CrossRefGoogle Scholar
  23. 23.
    Murray J.D.: Mathematical Biology. Springer, Berlin (1989)Google Scholar
  24. 24.
    Kapral R., Showalter K.: Chemical Patterns and Waves. Kluwer, Amsterdam (1995)Google Scholar
  25. 25.
    P. Borckmans, G. Dewel, A. DeWit and D. Walgraef, in : Chemical Patterns and Waves, ref. 22Google Scholar
  26. 26.
    L.M. Pismen, in Dynamics of Nonlinear Systems, ed. by V. Hlavacec (Gordon and Breach, New York, 1986), p. 47Google Scholar
  27. 27.
    Busse F.H.: J. Fluid. Mech. 30, 625 (1967)CrossRefGoogle Scholar
  28. 28.
    Ciliberto S., Coullet P., Lega J., Pampaloni E., Perez-Garcia C.: Phys. Rev. Letts. 65, 2370 (1990)CrossRefGoogle Scholar
  29. 29.
    Busse F.H.: Rep. Prog. Phys. 41, 1929 (1978)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Faculty of Computing, Engineering and Mathematical Sciences, Du Pont BuildingUniversity of the West of EnglandBristolUK
  2. 2.HooghlyIndia

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