Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-05T15:15:03.323Z Has data issue: false hasContentIssue false
  • English
  • Français

Turing instability of anomalous reaction–anomalous diffusion systems

Published online by Cambridge University Press:  01 June 2008

Y. NEC  and
A. A. NEPOMNYASHCHY
Y. NEC
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israel email: [email protected], [email protected]
A. A. NEPOMNYASHCHY
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa, Israel email: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Linear stability theory is developed for an activator–inhibitor model where fractional derivative operators of generally different exponents act both on diffusion and reaction terms. It is shown that in the short wave limit the growth rate is a power law of the wave number with decoupled time scales for distinct anomaly exponents of the different species. With equal anomaly exponents an exact formula for the anomalous critical value of reactants diffusion coefficients' ratio is obtained.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 19 , Issue 3 , June 2008 , pp. 329 - 349
Copyright
Copyright © Cambridge University Press 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amblard, F., Maggs, A. C., Yurke, B., Pargellis, A. N. & Leibler, S. (1996) Sub-diffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett. 77 (21), 4470–4473.CrossRefGoogle Scholar
[2]Ben-Avraham, D. & Havlin, Sh. (2000) Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, Cambridge, UK.Google Scholar
[3]Drazer, G. & Zanette, D. H. (1999) Experimental evidence of power law trapping time distributions in porous media. Phys. Rev. 60, 5858–5864.Google Scholar
[4]Fedotov, S. & Iomin, A. (2007) Migration and proliferation dichotomy in tumor cell invasion. Phys. Rev. Lett. 98, 118101.CrossRefGoogle ScholarPubMed
[5]Georgiou, G., Bahra, S. S., Mackie, A. R., Wolfe, C. A., O'Shea, P., Ladha, S., Fernandez, N. & Cherry, R. J. (2002) Measurement of lateral diffusion of human MHC class I molecules on HeLa cells by fluorescence recovery after photobleaching using a phycoerythrin probe. Biophys. J. 82, 1828–1834.CrossRefGoogle ScholarPubMed
[6]Henry, B. I., Langlands, T. A. M. & Wearne, S. L. (2005) Turing pattern formation in fractional activator–inhibitor systems. Phys. Rev. . 72, 026101.Google ScholarPubMed
[7]Henry, B. I. & Wearne, S. L. (2000) Fractional reaction–diffusion. Physica. 276, 448–455.CrossRefGoogle Scholar
[8]Henry, B. I. & Wearne, S. L. (2002) Existence of Turing instabilities in a two-species fractional reaction–diffusion system. SIAM J. Appl. Mat. 62 (3), 870–887.CrossRefGoogle Scholar
[9]Kosztołowicz, T., Dworecki, K. & Mrówczyński, S. (2005) How to measure sub-diffusion parameters. Phys. Rev. Lett. 94, 170602.CrossRefGoogle Scholar
[10]Kosztołowicz, T. & Levandowska, K. D. (2006) Time evolution of the reaction front in the system with one static and one sub-diffusive reactant. Acta Physica Polonica. 37 (5), 1571–1578.Google Scholar
[11]Langlands, T. A. M., Henry, B. I. & Wearne, S. L. (2007) Turing pattern formation with fractional diffusion and fractional reactions. J. Phys.: Condens. Matte. 19, 065115.Google Scholar
[12]Metzler, R. & Klafter, J. (2000) The random walk's guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1–77.CrossRefGoogle Scholar
[13]Metzler, R. & Klafter, J. (2004) The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics. J. Phys. A: Math. Gen. 37, R161–R208.CrossRefGoogle Scholar
[14]Murray, J. D. (1989) Mathematical Biology, Springer-Verlag, New York.Google Scholar
[15]Nec, Y. & Nepomnyashchy, A. A. (2007) Linear stability of fractional reaction–diffusion systems. Math. Model. Nat. Phenom. 2 (2), 4.CrossRefGoogle Scholar
[16]Oldham, K. B. & Spanier, J. (1974) The Fractional Calculus, Academic Press, New York.Google Scholar
[17]Podlubny, I. (1994) The Laplace Transform Method for Linear Differential Equations of the Fractional Order, Košice, Slovenská Republika.Google Scholar
[18]Ritchie, K., Shan, X. Y., Kondo, J., Iwasawa, K., Fujiwara, T. & Kusumi, A. (2005) Detection in non-Brownian diffusion in the cell membrane in single molecule tracking. Biophys. J. 88, 2266–2277.Google Scholar
[19]Seki, K., Wojcik, M. & Tachiya, M. (2003) Fractional reaction–diffusion equation. J. Chem. Phys. 119 (4), 2165–2170.CrossRefGoogle Scholar
[20]Seki, K., Wojcik, M. & Tachiya, M. (2003) Recombination kinetics in sub-diffusive media. J. Chem. Phys. 119 (14), 7525–7533.CrossRefGoogle Scholar
[21]Sokolov, I. M., Schmidt, M. G. W. & Sagués, F. (2006) Reaction–sub-diffusion equation. Phys. Rev. 73, 031102.Google Scholar
[22]Vigil, R. D., Ouyang, Q. & Swinney, H. L. (1992) Turing patterns in a simple gel reactor. Physica. 188, 17–25.CrossRefGoogle Scholar
[23]Weeks, E. R. & Weitz, D. A. (2002) Sub-diffusion and the cage effect studied near the colloidal glass transition. Chem. Phys. 284, 361–367.Google Scholar
[24]Weiss, M., Hashimoto, H. & Nilsson, T. (2003) Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. Biophys. J. 84, 4043–4052.Google Scholar
[25]Yuste, S. B., Acedo, L. & Lindenberg, K. (2004) Reaction front in an A+B C reaction–subdiffusion process. Phys. Rev. 69, 036126.Google Scholar