Skip to main content
SpringerLink
Log in

Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion

  • Original Paper
  • Published:
Journal of Mathematical Chemistry Aims and scope Submit manuscript

Abstract

Auto-wave solutions in nonlinear time-fractional reaction–diffusion systems are investigated. It is shown that stability of steady-state solutions and their subsequent evolution are mainly determined by the eigenvalue spectrum of a linearized system and level of anomalous diffusion (orders of fractional derivatives). The results of linear stability analysis are confirmed by computer simulations. To illustrate the influence of anomalous diffusion on stability properties and possible dynamics in fractional reaction–diffusion systems, we generalized two classical activator–inhibitor nonlinear models: FitzHugh–Nagumo and Brusselator. Based on them a common picture of typical nonlinear solutions in nonlinear incommensurate time-fractional activator–inhibitor systems is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

References

  1. S. Abad, B. Yuste, K. Lindenberg, Reaction–subdiffusion and reaction–superdiffusion equations for evanescent particles performing continuous-time random walks. Phys. Rev. E 81, 031115 (2010)

    Article  CAS  Google Scholar 

  2. F. Amblard, A. Maggs, B. Yurke, A. Pargellis, S. Leibler, Subdiffusion and anomalous local viscoelasticity in acting networks. Phys. Rev. Lett. 77, 4470–3 (1996)

    Article  CAS  Google Scholar 

  3. B.P. Belousov, A periodic reaction and its mechanism, in Oscillations and Traveling Waves in Chemical Systems, ed. by R.J. Field, M. Burger (Wiley, New York, 1985)

    Google Scholar 

  4. M. Cross, P. Hohenberg, Pattern formation out of equilibrium. Rev. Mod. Phys. 65, 851–1112 (1993)

    Article  CAS  Google Scholar 

  5. B. Datsko, V. Gafiychuk, I. Podlubny, Solitary travelling auto-waves in fractional reaction–diffusion systems. Commun. Nonlinear Sci. Numer. Simul. 23, 378–387 (2015)

    Article  Google Scholar 

  6. B. Datsko, V. Gafiychuk, Complex spatio-temporal solutions in fractional reaction–diffusion systems near a bifurcation point. Fract. Calc. Appl. Anal. 21, 237–253 (2018)

    Article  Google Scholar 

  7. R.J. Field, R.M. Noyes, Oscillations in chemical systems IV. Limit cycle behavior in a model of a real chemical reaction. J. Chem. Phys. 60, 1877–1884 (1974)

    Article  CAS  Google Scholar 

  8. S. Fomin, V. Chugunov, T. Hashida, Mathematical modeling of anomalous diffusion in porous media. Fract. Differ. Calc. 1, 1–28 (2011)

    Google Scholar 

  9. S. Fomin, V. Chugunov, T. Hashida, Non-Fickian mass transport in fractured porous media. Adv. Water Resour. 34, 205–214 (2011)

    Article  Google Scholar 

  10. V. Gafiychuk, B. Datsko, Pattern formation in a fractional reaction–diffusion system. Physica A 365, 300–306 (2006)

    Article  CAS  Google Scholar 

  11. V. Gafiychuk, B. Datsko, Stability analysis and oscillatory structures in time-fractional reaction–diffusion systems. Phys. Rev. E. 75, R 055201-1-4 (2007)

  12. V. Gafiychuk, B. Datsko, Spatiotemporal pattern formation in fractional reaction–diffusion systems with indices of different order. Phys. Rev. E. 77, 066210-1-9 (2008)

  13. V. Gafiychuk, B. Datsko, V. Meleshko, Mathematical modeling of time fractional reaction–diffusion systems. J. Comput. Appl. Math. 372, 215–225 (2008)

    Article  Google Scholar 

  14. V. Gafiychuk, I. Lubashevsky, B. Datsko, Fast heat propagation in living tissue caused by branching artery network. Phys. Rev. E 72, 051920 (2005)

    Article  CAS  Google Scholar 

  15. A. Golovin, B. Matkowsky, V. Volpert, Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69, 251–272 (2008)

    Article  Google Scholar 

  16. M. Harris-White, S. Zanotti, S. Frautschy, A. Charles, Spiral intercellular calcium waves in hippocampal slice cultures. J. Neurophysiol. 79, 1045–1052 (1998)

    Article  CAS  Google Scholar 

  17. B. Henry, T. Langlands, S. Wearne, Fractional cable models for spiny neuronal dendrites. Phys. Rev. Lett. 100, 128103 (2008)

    Article  CAS  Google Scholar 

  18. G. Hornung, B. Berkowitz, N. Barkai, Morphogen gradient formation in a complex environment: an anomalous diffusion model. Phys. Rev. E. 72, 041916-1-10 (2005)

  19. A. Iomin, Toy model of fractional transport of cancer cells due to self-entrapping, Phys. Rev. E. 73, 061918-1-5 (2006)

  20. A. Kaminaga, V. Vanag, I. Epstein, A reaction–diffusion memory device. Angew. Chem. Int. Ed. 45, 3087–3089 (2006)

    Article  CAS  Google Scholar 

  21. A. Kindzelskii, H. Petty, From the cover: apparent role of traveling metabolic waves in oxidant release by living neutrophils. Proc. Natl. Acad. Sci. USA 99, 9207–9212 (2002)

    Article  CAS  Google Scholar 

  22. B. Kerner, V. Osipov, Autosolitons (Kluwer, Dordrecht, 1994)

    Book  Google Scholar 

  23. Y. Kuramoto, Chemical Oscillations, Waves, and Turbulence (Springer, Berlin, 1984)

    Book  Google Scholar 

  24. T. Langlands, B. Henry, S. Wearne, Anomalous subdiffusion with multispecies linear reaction dynamics. Phys. Rev. E. 77, 021111-1-9 (2008)

  25. J. Macias-Diaz, A. Hendy, Numerical simulation of Turing patterns in fractional hyperbolic reaction–diffusion model with Grunwald differences. Eur. Phys. J. Plus 134, 324 (2019)

    Article  Google Scholar 

  26. J. Macias-Diaz, An efficient and fully explicit model to simulate delayed activator–inhibitor systems with anomalous diffusion. J. Math. Chem. 57, 1902–1923 (2019)

    Article  CAS  Google Scholar 

  27. R. Metzler, J.H. Jeon, A.G. Cherstvy, Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128 (2014)

    Article  CAS  Google Scholar 

  28. R. Metzler, J.H. Jeon, A.G. Cherstvy, E. Barkai, Non-Brownian diffusion in lipid membranes: experiments and simulations. Biochim. Biophys. Acta 1858, 2451–2467 (2016)

    Article  CAS  Google Scholar 

  29. V. Mendez, S. Fedotov, W. Horsthemke, Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities (Springer, New York, 2009)

    Google Scholar 

  30. A. Mvogo, J. Macias-Diaz, T. Kofane, Diffusive instabilities in a hyperbolic activator–inhibitor system with superdiffusion. Phys. Rev. E 97(3), 032129 (2018)

    Article  CAS  Google Scholar 

  31. G. Nicolis, I. Prigogine, Self-organization in Non-equilibrium Systems (Wiley, New York, 1997)

    Google Scholar 

  32. F.A. Oliveira, R.M.S. Ferreira, L.C. Lapas, M.H. Vainstein, Anomalous diffusion: a basic mechanism for the evolution of inhomogeneous systems. Front. Phys. 19, 00018 (2019)

    Article  Google Scholar 

  33. I. Podlubny, Fractional Differential Equations (Acad. Press, San Diego, 1999)

    Google Scholar 

  34. Y. Povstenko, Linear Fractional Diffusion-Wave Equation for Scientists and Engineers (Birkhäuser, New York, 2015)

    Book  Google Scholar 

  35. Y. Povstenko, Fractional Thermoelasticity (Springer, New York, 2015)

    Book  Google Scholar 

  36. S. Samko, A. Kilbas, O. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, San Diego, 1993)

    Google Scholar 

  37. R. Torabi, Z. Rezaei, Instability in reaction–superdiffusion systems. Phys. Rev. E 94, 052202 (2005)

    Article  Google Scholar 

  38. V. Uchaikin, Fractional Derivatives for Physicists and Engineers (Springer, Berlin, 2013)

    Book  Google Scholar 

  39. V. Uchaikin, R. Sibatov, Fractional theory for transport in disorder semiconductors. Commun. Nonlinear Sci. Numer. Simul. 13, 715–27 (2008)

    Article  Google Scholar 

  40. F. Valdes-Parada, J. Ochoa-Tapia, J. Alvarez-Ramirez, Effective medium equation for fractional Cattaneo’s diffusion and heterogeneous reaction in disordered porous media. Physica A 369, 318–328 (2006)

    Article  CAS  Google Scholar 

  41. V. Vanag, Waves and patterns in reaction-diffusion systems. Belousov–Zhabotinsky reaction in water-in-oil microemulsions. Phys. Usp. 47(9), 923–943 (2004)

    Article  CAS  Google Scholar 

  42. V. Vasiliev, Yu. Romanovskii, D. Chernavskii, V. Yakhno, Autowave Processes in Kinetic Systems: Spatial and Temporal Self-organization in Physics, Chemistry, Biology, and Medicine (Kluwer, Dordrecht, 1987)

    Book  Google Scholar 

  43. L. Zelenyi, A. Milovanov, Fractal topology and strange kinetics: from percolation theory to problems in cosmic electrodynamics. Phys. Usp. 47(8), 809–852 (2004)

    Article  Google Scholar 

  44. A. Zhokh, P. Strizhak, Non-Fickian diffusion of methanol in mesoporous media: geometrical restrictions or adsorption-induced? J. Chem. Phys. 146, 124704 (2017)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rzeszow University of Technoogy, Rzeszow, Poland

    B. Datsko, M. Kutniv & A. Włoch

  2. Institute for Applied Problems of Mechanics and Mathematics, Lviv, Ukraine

    M. Kutniv

Authors
  1. B. Datsko
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Kutniv
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Włoch
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to B. Datsko.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Datsko, B., Kutniv, M. & Włoch, A. Mathematical modelling of pattern formation in activator–inhibitor reaction–diffusion systems with anomalous diffusion. J Math Chem 58, 612–631 (2020). https://doi.org/10.1007/s10910-019-01089-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10910-019-01089-y

Keywords

Mathematics Subject Classification

Search

Navigation