Turing patterns
A Turing pattern is a spatial pattern generated when diffusion destabilises an equilibrium that is stable in the corresponding non-spatial system.
Two-component system
\[\frac{\partial u}{\partial t}=D_u\nabla^2u+f(u,v),\]\[\frac{\partial v}{\partial t}=D_v\nabla^2v+g(u,v).\]Without diffusion, the homogeneous equilibrium may be stable. With unequal diffusion rates, some spatial perturbations can grow.
Diffusion-driven instability
The mathematical test compares the reaction Jacobian with the diffusion coefficients and determines whether a spatial mode has positive growth rate.
Biological interpretation
Turing mechanisms have been studied as possible explanations for developmental and pigmentation patterns.
Key idea. In a Turing mechanism, diffusion does not simply smooth differences; together with reaction dynamics it can create stable spatial structure.