# Turing Pattern in a Reaction-Diffusion System

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In the last few decades, developmental biologists have extensively used the reaction-diffusion model to explain pattern formation in living organisms. The original model was proposed by Alan Turing in 1952 [1]. The model is based on the idea that pattern formation results from two fundamental mechanisms: (1) coupled catalytic and autocatalytic reactions in a space element between two chemical species, an activator and an inhibitor, and (2) transfer of the interacting species to and from the neighboring space elements through a diffusional transport mechanism. Under appropriate reaction and diffusion conditions, a periodic pattern is formed from an initially homogeneous spatial distribution of activator and inhibitor [2, 3]. Examples of pattern formation can be found in biology, chemistry (the famous Belousov–Zhabotinskii reaction), physics, and mathematics [4, 5].

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Contributed by: Housam Binous and Ahmed Bellagi (August 2015)

Open content licensed under CC BY-NC-SA

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References

[1] A. M. Turing, "The Chemical Basis of Morphogenesis," *Philosophical Transactions of the Royal Society*, 237(641), 1952 pp. 37–72.

[2] H. Meinhardt, *The Algorithmic Beauty of Sea Shells*, New York: Springer-Verlag, 1995.

[3] T. Miura and P. K. Maini, "Periodic Pattern Formation in Reaction-Diffusion Systems: An Introduction for Numerical Simulation," *Anatomical Science International*, 79(3), 2004 pp. 112–123.

[4] J. D. Murray, *Mathematical Biology: I. An Introduction*, 3rd ed., New York: Springer, 2002.

[5] J. D. Murray, *Mathematical Biology II: Spatial Models and Biomedical Applications*, 3rd ed., New York: Springer, 2003.

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