Reaction-diffusion system

A reaction-diffusion system is a mathematical model describing how the concentration of one or more substances evolves under the combined influence of local chemical reactions and spatial diffusion. The canonical form is a set of coupled partial differential equations: $\partial_t u = D \nabla^2 u + f(u)$, where $ is a diffusion matrix and $ encodes the reaction kinetics. These systems are the primary mechanism of spontaneous pattern formation in homogeneous media.

Alan Turing's 1952 paper established that the interplay of a slowly diffusing activator and a rapidly diffusing inhibitor can destabilize a uniform state and generate stable patterns — stripes, spots, and labyrinths — from noise. This activator-inhibitor mechanism is the mathematical basis of morphogenetic fields and explains patterning in developmental biology, ecology, and chemical systems such as the Belousov-Zhabotinsky reaction. Reaction-diffusion systems are also a paradigm of symmetry breaking: the homogeneous state possesses full spatial symmetry, while the patterned solutions do not.