Journal of Mathematical Chemistry

, Volume 57, Issue 2, pp 638–654 | Cite as

Complex pattern formation arising from wave instabilities in a three-agent chemical system with superdiffusion

  • Alain MvogoEmail author
  • Jorge E. Macías-Díaz
Original Paper


This work investigates analytically and numerically the wave instabilities in a three-agent chemical system undergoing anomalous diffusion. Anomalous diffusion is modeled using higher-dimensional space-fractional operators. The model considers one activator and two inhibitors, and it is a superdiffusive extension of some reaction-diffusion systems in the literature (Phys D Nonlinear Phenom 199(1–2):264–277, 2004). The model under investigation is presented in generic form as a three-species reaction-diffusion system, so that it has broad applicability to a wide range of chemical and biological systems. A weakly nonlinear analysis is performed, dispersive curves of eigenvalues are plotted and their behavior is analyzed. This analysis reveals that the critical value of the wave number for wave instability increases when the superdiffusive exponent decreases. The numerical scheme is performed and some numerical simulations are conducted as evidence of the analytical predictions. We show how the evolution of spatio-temporal patterns is related to meaningful parameters. In particular, we demonstrate that the system exhibits the coexistence of regular and irregular structures, forming complex patterns that are not familiar in standard reaction-diffusion systems.


Wave instability Chemical system Anomalous diffusion Space-fractional operators Superdiffusive exponent Reaction-diffusion systems Pattern formation Regular and irregular structures 



One of us (AM) Acknowledges fruitful discussions with Professor K. Showalter of the University of West Virginia, USA, during the event “Hands-on Research in Complex Systems School” at the Abdus Salam International Centre for Theoretical Physics (ICTP). He also Acknowledges the invitation and Financial support from the ICTP.


  1. 1.
    F. Amblard, A.C. Maggs, B. Yurke, A.N. Pargellis, S. Leibler, Subdiffusion and anomalous local viscoelasticity in actin networks. Phys. Rev. Lett. 77(21), 4470 (1996)CrossRefGoogle Scholar
  2. 2.
    V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64(24), 2953 (1990)CrossRefGoogle Scholar
  3. 3.
    M. Cross, P. Hohenberg, For an introduction to the dynamics of excitable systems and their applications. Rev. Mod. Phys. 65, 851 (1993)Google Scholar
  4. 4.
    G. Drazer, D.H. Zanette, Experimental evidence of power-law trapping-time distributions in porous media. Phys. Rev. E 60(5), 5858 (1999)CrossRefGoogle Scholar
  5. 5.
    G. Gambino, M. Lombardo, M. Sammartino, V. Sciacca, Turing pattern formation in the brusselator system with nonlinear diffusion. Phys. Rev. E 88(4), 042925 (2013)CrossRefGoogle Scholar
  6. 6.
    S. Ghorai, S. Poria, Pattern formation in a system involving prey-predation, competition and commensalism. Nonlinear Dyn. 89(2), 1309–1326 (2017)CrossRefGoogle Scholar
  7. 7.
    A.A. Golovin, B.J. Matkowsky, V.A. Volpert, Turing pattern formation in the Brusselator model with superdiffusion. SIAM J. Appl. Math. 69(1), 251–272 (2008)CrossRefGoogle Scholar
  8. 8.
    P. Gray, S.K. Scott, Chemical Oscillations and Instabilities: Non-linear Chemical Kinetics (Oxford University Press, Oxford, 1990)Google Scholar
  9. 9.
    A.E. Hansen, D. Marteau, P. Tabeling, Two-dimensional turbulence and dispersion in a freely decaying system. Phys. Rev. E 58(6), 7261 (1998)CrossRefGoogle Scholar
  10. 10.
    S. Hata, H. Nakao, A.S. Mikhailov, Sufficient conditions for wave instability in three-component reaction-diffusion systems. Prog. Theor. Exp. Phys. 2014(1), 013A01 (2014)CrossRefGoogle Scholar
  11. 11.
    R. Kapral, K. Showalter, Chemical Waves and Patterns, vol. 10 (Springer, Berlin, 2012)Google Scholar
  12. 12.
    S. Kondo, R. Asai, A reaction-diffusion wave on the skin of the marine angelfish Pomacanthus. Nature 376(6543), 765 (1995)CrossRefGoogle Scholar
  13. 13.
    E. Lobanova, F. Ataullakhanov, Running pulses of complex shape in a reaction-diffusion model. Phys. Rev. Lett. 93(9), 098303 (2004)CrossRefGoogle Scholar
  14. 14.
    E.S. Lobanova, F.I. Ataullakhanov, Unstable trigger waves induce various intricate dynamic regimes in a reaction-diffusion system of blood clotting. Phys. Rev. Lett. 91(13), 138301 (2003)CrossRefGoogle Scholar
  15. 15.
    H. Meinhardt, Out-of-phase oscillations and traveling waves with unusual properties: the use of three-component systems in biology. Phys. D Nonlinear Phenom. 199(1–2), 264–277 (2004)CrossRefGoogle Scholar
  16. 16.
    H. Meinhardt, A. Gierer, Pattern formation by local self-activation and lateral inhibition. Bioessays 22(8), 753–760 (2000)CrossRefGoogle Scholar
  17. 17.
    R. Metzler, J. Klafter, 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(31), R161 (2004)CrossRefGoogle Scholar
  18. 18.
    K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)Google Scholar
  19. 19.
    J. Murray, Mathematical Biology ii: Spatial Models and Biochemical Applications, vol ii (2003)Google Scholar
  20. 20.
    A. Mvogo, J.E. Macías-Díaz, T.C. Kofané, Diffusive instabilities in a hyperbolic activator-inhibitor system with superdiffusion. Phys. Rev. E 97(3), 032129 (2018)CrossRefGoogle Scholar
  21. 21.
    G. Nicollis, I. Prigogine, Self Organization in Non-equilibrium Systems (Wiley, New York, 1977)Google Scholar
  22. 22.
    Q. Ouyang, H.L. Swinney, Transition from a uniform state to hexagonal and striped Turing patterns. Nature 352(6336), 610 (1991)CrossRefGoogle Scholar
  23. 23.
    I. Podlubny, Fractional Differential Equations (Academic Press, London, 1999)Google Scholar
  24. 24.
    J.M. Sancho, A. Lacasta, K. Lindenberg, I.M. Sokolov, A. Romero, Diffusion on a solid surface: anomalous is normal. Phys. Rev. Lett. 92(25), 250601 (2004)CrossRefGoogle Scholar
  25. 25.
    T. Solomon, E.R. Weeks, H.L. Swinney, Observation of anomalous diffusion and Lévy flights in a two-dimensional rotating flow. Phys. Rev. Lett. 71(24), 3975 (1993)CrossRefGoogle Scholar
  26. 26.
    A. Turing, Philosophical the royal biological transqfctions society sciences. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)CrossRefGoogle Scholar
  27. 27.
    V.K. Vanag, I.R. Epstein, Stationary and oscillatory localized patterns, and subcritical bifurcations. Phys. Rev. Lett. 92(12), 128301 (2004)CrossRefGoogle Scholar
  28. 28.
    V.K. Vanag, L. Yang, M. Dolnik, A.M. Zhabotinsky, I.R. Epstein, Coupled and forced patterns in reaction–diffusion systems. Nature (London) 406, 389 (2000)Google Scholar
  29. 29.
    V.K. Vanag, A.M. Zhabotinsky, I.R. Epstein, Oscillatory clusters in the periodically illuminated, spatially extended Belousov-Zhabotinsky reaction. Phys. Rev. Lett. 86(3), 552 (2001)CrossRefGoogle Scholar
  30. 30.
    G.M. Viswanathan, Ecology: Fish in lévy-flight foraging. Nature 465(7301), 1018 (2010)CrossRefGoogle Scholar
  31. 31.
    G.M. Viswanathan, V. Afanasyev, S. Buldyrev, E. Murphy, P. Prince, H.E. Stanley, Lévy flight search patterns of wandering albatrosses. Nature 381(6581), 413 (1996)CrossRefGoogle Scholar
  32. 32.
    X. Wang, F. Liu, X. Chen, Novel second-order accurate implicit numerical methods for the Riesz space distributed-order advection-dispersion equations. Adv. Math. Phys. 2015, 590435 (2015)Google Scholar
  33. 33.
    M. Weiss, Stabilizing Turing patterns with subdiffusion in systems with low particle numbers. Phys. Rev. E 68(3), 036213 (2003)CrossRefGoogle Scholar
  34. 34.
    M. Weiss, H. Hashimoto, T. Nilsson, Anomalous protein diffusion in living cells as seen by fluorescence correlation spectroscopy. Biophys. J. 84(6), 4043–4052 (2003)CrossRefGoogle Scholar
  35. 35.
    K.A.J. White, C.A. Gilligan, Spatial heterogeneity in three species, plant-parasite-hyperparasite, systems. Philos. Trans. R. Soc. Lond. B: Biol. Sci. 353(1368), 543–557 (1998)CrossRefGoogle Scholar
  36. 36.
    A.M. Zhabotinsky, M. Dolnik, I.R. Epstein, Pattern formation arising from wave instability in a simple reaction-diffusion system. J. Chem. Phys. 103(23), 10306–10314 (1995)CrossRefGoogle Scholar
  37. 37.
    L. Zhang, C. Tian, Turing pattern dynamics in an activator-inhibitor system with superdiffusion. Phys. Rev. E 90(6), 062915 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Laboratory of Biophysics, Department of Physics Faculty of ScienceUniversity of Yaounde ICameroonSouth Africa
  2. 2.The Abdus Salam International Center For Theoretical PhysicsTriesteItaly
  3. 3.Departamento de Matemáticas y FísicaUniversidad Autónoma de AguascalientesAguascalientesMexico

Personalised recommendations