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Decaying localized structures beyond Turing space in an activator–inhibitor system

  • Dhritiman Talukdar
  • Kishore DuttaEmail author
Regular Article
  • 6 Downloads

Abstract

We perform numerical simulations beyond Turing space in an activator–inhibitor system involving quadratic and cubic nonlinearities . We show that while all the three fixed points of the system are stable nodes, it exhibits spatially stable patterns as diverse as labyrinths, worms, negatons, and combination of them. The transition among the patterns is found to be dependent on the relative strength (h) of quadratic and cubic couplings. The labyrinths and worms are formed for small values of h while stable negatons are obtained for \(0.0891\le h\le 0.1106\). The negatons start showing decaying behavior for \(h\ge 0.1135\) and, finally they vanish at random positions by emitting remnant solitary waves, yielding a pattern of stable concentric rings. The spatial extension of the concentric rings is found to depend on the initial concentration profile of the decaying negatons. The resulting concentric rings do not decay further due to limitation of numerical precision. We also find that the transient period for each pattern also depends on h.

Supplementary material

13360_2019_63_MOESM1_ESM.mp4 (686 kb)
Supplementary material 1 (mp4 686 KB)

References

  1. 1.
    D.W. Thompson, On Growth and Form (Cambridge University Press, Cambridge, 1961)Google Scholar
  2. 2.
    A.M. Turing, On the chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond. B 237, 37–72 (1952)ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    J.D. Murray, A pre-pattern formation mechanism for animal coat markings. J. Theor. Biol. 88, 161–199 (1981)MathSciNetCrossRefGoogle Scholar
  4. 4.
    J.D. Murray, On pattern formation mechanism for lepidopteran wing patterns and mammalian coat markings. Philos. trans. R. Soc. Lond. B 295, 473–496 (1981)ADSCrossRefGoogle Scholar
  5. 5.
    J.B.L. Bard, A model for generating aspects of zebra and other mammalian coat patterns. J. Theor. Biol. 93, 363–385 (1981)MathSciNetCrossRefGoogle Scholar
  6. 6.
    J.D. Murray, M.R. Myerscough, Pigmentation pattern formation on snakes. J. Theor. Biol. 149, 339–360 (1991)CrossRefGoogle Scholar
  7. 7.
    S. Kondo, R. Asai, A reaction-diffusion wave on the marine angelfish pomacanthus. Nature 376, 765–768 (1995)ADSCrossRefGoogle Scholar
  8. 8.
    R. Asai et al., Zebrafish leopard gene as a component of the putative reaction-diffusion system. Mech. Dev. 89, 87–92 (1999)CrossRefGoogle Scholar
  9. 9.
    J.D. Murray, Mathematical Biology, vol. I & II, 3rd edn. (Springer, New York, 2002)CrossRefGoogle Scholar
  10. 10.
    H. Shoji et al., Origin of directionality in the fish stripe pattern. Dev. Dyn. 226, 627–633 (2003)CrossRefGoogle Scholar
  11. 11.
    R.A. Barrio et al., Modeling the skin pattern of fishes. Phys. Rev. E 79, 031908 (2009)ADSCrossRefGoogle Scholar
  12. 12.
    V. Castets, E. Dulos, J. Boissonade, P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern. Phys. Rev. Lett. 64, 2953 (1990)ADSCrossRefGoogle Scholar
  13. 13.
    I. Lengyel, I.R. Epstein, A chemical approach to designing turing patterns in reaction-diffusion systems. Proc. Natl. Acad. Sci. USA 89, 3977 (1992)ADSCrossRefGoogle Scholar
  14. 14.
    I.R. Epstein, J.A. Pojman, An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Oxford University Press, New York, 1998)Google Scholar
  15. 15.
    V.K. Vanag, I.R. Epstein, Inwardly rotating spiral waves in a reaction-diffusion system. Science 294, 835 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    J. Temmyo, R. Notzel, T. Tamamura, Semiconductor nanostructures formed by the Turing instability. Appl. Phys. Lett. 71, 1086 (1997)ADSCrossRefGoogle Scholar
  17. 17.
    E. Ammelt, Y.A. Astrov, H.G. Purwins, Hexagon structures in a two-dimensional dc-driven gas discharge system. Phys. Rev. E 58, 7109 (1998)ADSCrossRefGoogle Scholar
  18. 18.
    D. Walgraef, N.M. Ghoniem, Effects of glissile interstitial clusters on microstructure self-organization in irradiated materials. Phys. Rev. B 67, 064103 (2003)ADSCrossRefGoogle Scholar
  19. 19.
    R.A. Barrio, C. Varea, J.L. Araǵon, P.K. Maini, A two-dimensional numerical study of spatial pattern formation in interacting Turing systems. Bull. Math. Biol. 61, 483–505 (1999)CrossRefGoogle Scholar
  20. 20.
    T. Leppnen, M. Karttunen, R.A. Barrio, K. Kaski, Morphological transitions and bistability in Turing systems. Phys. Rev. E 70, 066202 (2004)ADSCrossRefGoogle Scholar
  21. 21.
    R.A. Barrio et al., Size-dependent symmetry breaking in models for morphogenesis. Phys. D 168, 61 (2002)MathSciNetCrossRefGoogle Scholar
  22. 22.
    J.L. Aragón et al., Turing patterns with pentagonal symmetry. Phys. Rev. E 65, 051913 (2002)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    R.T. Liu, S.S. Liaw, P.K. Maini, Two-stage Turing model for generating pigment patterns on the Leopard and the Jaguar. Phys. Rev. E 74, 011914 (2006)ADSCrossRefGoogle Scholar
  24. 24.
    J.T. Schneider, Perfect stripes from a general Turing model in different geometries, M.Sc. Thesis, Boise State University (2012)Google Scholar
  25. 25.
    T.E. Woolley et al., Analysis of stationary droplets in a generic Turing reaction-diffusion system. Phys. Rev. E 82, 051929 (2010)ADSCrossRefGoogle Scholar
  26. 26.
    A. L-Dur\(\acute{\text{a}}\)n, et al., The interplay between phenotypic and ontogenetic plasticities can be assessed using reaction-diffusion models. J. Biol. Phys. 43, 247 (2017)Google Scholar
  27. 27.
    D. Hernández et al., Self-similar Turing patterns: An anomalous diffusion consequence. Phys. Rev. E 95, 022210 (2017)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    D. Talukdar, K. Dutta, Transition of spatial patterns in an interacting Turing system. J. Stat. Phys. 174, 351 (2019)ADSMathSciNetCrossRefGoogle Scholar
  29. 29.
    C. Varea, D. Hernández, R.A. Barrio, Soliton behaviour in a bistable reaction diffusion model. J. Math. Biol. 54, 797 (2007)MathSciNetCrossRefGoogle Scholar
  30. 30.
    K. Kytta, K. Kasaki, R.A. Barrio, Complex Turing patterns in non-linearly coupled systems. Phys. A 385, 105 (2007)CrossRefGoogle Scholar
  31. 31.
    M. Dolnik, I. Berenstein, A.M. Zhabotinsky, I.R. Epstein, Spatial periodic forcing of Turing structures. Phys. Rev. Lett. 87, 238301 (2001)ADSCrossRefGoogle Scholar
  32. 32.
    M. Karttunen, N. Provatas, T. Ala-Nissila, M. Grant, Nucleation, growth, and scaling in slow combustion. J. Stat. Phys. 90, 1401 (1998)ADSCrossRefGoogle Scholar
  33. 33.
    L. Yang, M. Dolnik, A.M. Zhabotinsky, I.R. Epstein, Spatial resonances and superposition patterns in a reaction-diffusion model with interacting Turing modes. Phys. Rev. Lett. 88, 208303 (2002)ADSCrossRefGoogle Scholar
  34. 34.
    V.K. Vanag, I.R. Epstein, Pattern formation mechanisms in reaction-diffusion systems. Int. J. Dev. Biol. 53, 673 (2009)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica (SIF) and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Authors and Affiliations

  1. 1.Department of PhysicsHandique Girls’ CollegeGuwahatiIndia

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