Reaction-diffusion processes

The cellular automaton in the main text can be viewed as a discrete idealization of a reaction-diffusion process. The notion that diffusion might be important in embryo development had been suggested in the early 1900s (see page 1004), but it was only in 1952 that Alan Turing showed how it could lead to the formation of definite patterns. Diffusion of a single chemical always tends to smooth out distributions of concentration. But Turing pointed out that with two chemicals in which each can be produced from the other it is possible for separated regions to develop. If c = {u[t, x], v[t, x]} is a vector of chemical concentrations, then for suitable values of parameters even the standard linear diffusion equation ∂_tc  d . ∂_xxc + m . c can exhibit an instability which causes disturbances with certain spatial wavelengths to grow (compare page 988). In his 1952 paper Turing used a finite difference approximation to a pair of diffusion equations to show that starting from a random distribution of concentration values dappled regions could develop in which one or the other chemical was dominant. With purely linear equations, any instability will always eventually lead to infinite concentrations, but Turing noted that this could be avoided by using realistic nonlinear chemical rate equations. In the couple of years before his death in 1954, Turing appears to have tried to simulate such nonlinear equations on an early digital computer, but my cursory efforts to understand his programs—written as they are in a 32-character machine code—were not successful.

Following Turing's work, the fact that simple reaction-diffusion equations can yield spatially inhomogeneous patterns has been rediscovered—with varying degrees of independence—many times. In the early 1970s Ilya Prigogine termed the patterns dissipative structures. And in the mid-1970s, Hermann Haken considered the phenomenon a cornerstone of what he called synergetics.

Many detailed mathematical analyses of linear reaction-diffusion equations have been done since the 1970s; numerical solutions to linear and occasionally nonlinear such equations have also often been found, and in recent years explicit pictures of patterns—rather than just curves of related functions—have commonly been generated. In the context of biological pigmentation patterns detailed studies have been done particularly by Hans Meinhardt and James Murray.