A wide variety of reaction-diffusion systems are found in nature. A candle flame is one classic example, where evaporating wax forms an excitable medium, dynamic structures being produced through its reaction with and diffusion into the surrounding air.

Alan Turing was the first to formulate reaction-diffusion equations as such, in his 1952 landmark paper on morphogenesis - the process by which living things derive their shape, structure and function. He proposed that many varieties of spots, stripes and other markings found in nature, among other things, could be explained by reaction-diffusion equations, with a set of interacting chemicals diffusing at different rates.

In the same year Andrew Huxley and Alan Hodgkin published a system of equations describing the propagation of action potentials, the central nervous system's chief method of propagating signals electrically. Based on empirical observation of in vitro giant squid axons, their reaction-diffusion model proved seminal, winning them the Nobel prize for Medicine, forming the basis of most neurobiological models for four decades and inspiring similar models of electrical activity in the heart.

Perhaps the single phenomenon best known as a reaction-diffusion system is the Belousov-Zhabotinsky reaction. First discovered by Boris Pavlovitch Belousov in the middle years of the twentieth century, the surprising nature of the reaction - oscillating periodically between one colour and another - was initially met with incredulity by the scientific establishment. At least one editor rejected Belousov's manuscript outright on the grounds that it was clearly impossible, even before the identification of its best-known and most remarkable features - spontaneous self-organisation into a beautiful range of stripes, rings and spiral waves. When Zhabotinsky continued exploring the reaction, years later, he finally perservered through the entrenched scientific scepticism and published a series of papers on the subject; in the decades since it has attracted huge amounts of scientific interest.

Reaction-diffusion systems provide one example of dissipative structures in far-from-equilibrium systems, for which the simple laws of traditional thermodynamics prove quite inadequate. Systems of this sort frequently give rise to complexity which must be described using the language of nonlinear dynamics; seldom amenable to precise analytical solutions, it is only with the rise of computer models that their in-depth study has become possible.

*References*:

*Encyclopedia of Nonlinear Science*, edited by Alwyn Scott*Frontiers of Complexity*, by Peter Coveney and Roger Highfield

*Links*

- Some fun reaction-diffusion animations: http://cnls.lanl.gov/~aric/Simulations/Simulations.html
- A delightful interactive simulator: http://texturegarden.com/java/rd/
- A page on the Belousov-Zhabotinsky (BZ) reaction: http://www.ux.uis.no/~ruoff/BZ_Phenomenology.html
- My academic blog, where a version of this writeup first appeared: http://mimeomai.f2s.com/oolong/ac/