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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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Reaction-diffusion Dynamics and Biological Pattern Formation

Journal of Applied Nonlinear Dynamics 6(4) (2017) 547--564 | DOI:10.5890/JAND.2017.12.009

Kishore Dutta

Department of Physics, Handique Girls’ College, Guwahati, India

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The spontaneous formation of a wide variety of natural patterns with different shapes and symmetries in many physical and biological systems is one of the deep mysteries in science. This article describes the physical principles underlying the formation of various intriguing spatio-temporal patterns in Nature with special emphasis on some biological structures. We discuss how the spontaneous symmetry breaking due to diffusion driven instability in the reaction dynamics lead to the emergence of such complicated natural patterns. The mechanism of the formation of various animal coat patterns is explained via the Turing-type reaction-diffusion models.


  1. [1]  Bard, J.B.L. (1981), A model for generating aspects of zebra and other mammalian coat patterns, Journal of Theoretical Biology, 93, 363-385.
  2. [2]  Murray, J.D. (2003), Mathematical Biology, Vol. II, Springer-Verlag, Berlin.
  3. [3]  Murray, J.D. andMyerscough,M.R. (1991), Pigmentation pattern formation on snakes, Journal of Theoretical Biology, 149, 339-360.
  4. [4]  Murray, J.D. (1981), On pattern formation mechanism for lepidopteran wing patterns and mammalian coat markings, Philos. trans. R. Soc. London, Ser. B, 295, 473-496.
  5. [5]  Nijhout, H.F. (1990), A comprehensive model for color pattern formation in butterflies, Proc. R. Soc. London B, 239, 81-113.
  6. [6]  Sekimura T., Madzvamuse A., Wathen A. J., and Maini P. K. (2000), A model for color pattern formation in the butterfly wing of Papilio dardanus, Proc. R. Soc. Lond. B, 267, 851-859.
  7. [7]  Murray, J.D. (1981), A pre-pattern formation mechanism for animal coat markings, Journal of Theoretical Biology, 88, 161-199.
  8. [8]  Liu, R.T., Liaw, S.S., and Maini, P.K. (2006), Two-stage Turing model for generating pigment patterns on the leopard and the jaguar, Phys. Rev. E, 74, 011914.
  9. [9]  Kondo, S. and Asai, R. (1995), A Reaction-diffusion wave on the marine angelfish Pomacanthus, Nature, 376, 765-768.
  10. [10]  Asai, R., et al. (1999), Zebrafish Leopard gene as a component of the putative reaction-diffusion system, Mechanisms of Development, 89, 87-92.
  11. [11]  Barrio, R.A., et al. (2009), Modeling the skin pattern of fishes, Phys. Rev. E, 79, 031908.
  12. [12]  Walgraef, D. (1997), Spatio-Temporal Pattern Formation, Springer, New York.
  13. [13]  Harrison, L.G. (2011), The Shaping of Life, Cambridge University Press, New York.
  14. [14]  Cross, M. and Greenside H. (2009), Pattern Formation and Dynamics in Nonequilibrium Systems, Cambridge University Press, Cambridge.
  15. [15]  Walgraef, D. (1996), Spatiotemporal Pattern Formation, With Examples in Physics, Chemistry and Materials Science, Springer.
  16. [16]  Thompson, W. D’Arcy (1961), On Growth and Form, Cambridge University Press, Cambridge (U. K.).
  17. [17]  Bagnold, R.A. (1941), The Physics of Blown Sand and Desert Dunes, Chapman and Hall, London.
  18. [18]  Forrest, S.B. and Haff, P.K. (1992), Mechanics of wind ripple stratigraphy, Science, 255, 1240.
  19. [19]  Andreotti, B. (2004),The song of dunes as a wave-particle mode locking, Physical Review Letter 92, 238001.
  20. [20]  Ball, P. (1999),The Self-Made Tapestry: Pattern Formation in Nature, Oxford University Press, Oxford.
  21. [21]  Jacob, E.B. and Levine, H. (2001), The artistry of nature, Nature, 409, 985-986.
  22. [22]  Dutta, K. (2010), How birds fly together: the dynamics of flocking, Resonance, 15, 1097-1110.
  23. [23]  Turing, A.M. (1952), On the chemical basis of morphogenesis, Philosophical Transactions of the Royal Society of London, Series B, Biological Sciences, 237, 37-72.
  24. [24]  Nicolis, G. (1995), Introduction to nonlinear science, Cambridge University Press, Cambridge.
  25. [25]  Koch, A.J. and Meinhardt H. (1994), Biological pattern formation: from basic mechanisms to complex structures, Rev. Modern Phys., 66, 1481-1507.
  26. [26]  Sick, S., et al. (2006),WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism, Science, 314, 1447-1450.
  27. [27]  Gierer, A. and Meinhardt, H. (1972), A theory of biological pattern formation, Kybernetik, 12, 30-39.
  28. [28]  Meinhardt, H. (2003), The Algorithmic Beauty of Sea Shells, Third Edition, Springer, Berlin.
  29. [29]  Meinhardt, H. (1982), Models of Biological Pattern Formation, Academic Press, London.
  30. [30]  Yamaguchi, M., Yoshimoto, E., and Kondo, S. (2007), Pattern regulation in the stripe of zebrafish suggests an underlying dynamic and autonomous mechanism, PNAS, 104, 4790-4793.
  31. [31]  Shoji, H. et al. (2003), Origin of directionality in the fish stripe pattern, Developmental Dynamics, 226, 627-633.
  32. [32]  Venkataraman, C., et al. (2011), Modeling parr-mark pattern formation during the early development of Amago trout, Phys. Rev. E., 84, 041923.
  33. [33]  Barrio, R.A., Varea, C., Aragón, J.L., and Maini, P.K. (1999), A two-dimensional numerical study of spatial pattern formation in interacting Turing systems, Bulletin of Mathematical Biology, 61, 483-505.