Turing Structures

How did the stripes on the tiger or the patterns on sea-shells form? Nature abounds with examples of beautiful patterns. An attempt to understand such patterns from a physio-chemical perspective is the Turing structure: a stationary patern formed when the propagating chemical waves of the BZ type are combined with the effects of diffusion in a medium (that is, a reaction diffusion system). Normally one would think that the effects of diffusion would be to smoothen out inhomogeneities caused by an autocatylitic process such as BZ. However suppose that the diffusion rates for the various chemicals in the reaction is different. Then diffusion would actually create inhomogeneities, since the reaction rates would then become different at different regions. If the conditions are just right then the inhomogeneities will lead to a stationary spatial structure, called a Turing structure. See pictures in Ref.[1] As expected for out-of-equilibrium structures, the particular patterns that are formed depend on the initial conditions (the various parameters in the dynamical equations). This means one can obtain a great deal of variety in one simulation. Furthermore, the patterns also depend on the geometry of the surface, and this too is necessary to reflect the diversity observed in nature. For example, the beautiful patterns seen on lady-bug beetles have been simulated in a model defined on a spherical surface, Ref.[9].