Reaction-diffusion

Reaction-diffusion systems are mathematical models that couple local chemical reactions with the spatial diffusion of reactants. First studied systematically by Alan Turing in his 1952 paper on morphogenesis, these systems demonstrate that the interaction of reaction kinetics and diffusive transport can produce stable spatial patterns — Turing patterns — from initially homogeneous conditions. The governing equations are partial differential equations of the form ∂u/∂t = D_u ∇²u + f(u,v), where the reaction term f encodes the local chemistry and the Laplacian ∇² encodes diffusion. Despite their apparent simplicity, reaction-diffusion systems exhibit a rich phenomenology including pattern formation, traveling waves, and spatiotemporal chaos. They remain the primary mathematical framework for understanding morphogenesis, chemical oscillations in space, and ecological patterning such as vegetation banding.

The systems insight is that reaction-diffusion dynamics are not chemistry plus geometry. They are a new class of behavior that emerges only from the coupling: neither reaction alone nor diffusion alone produces patterns, but their interaction does. This makes reaction-diffusion systems the canonical example of emergence through coupling — a principle that extends far beyond chemistry into developmental biology, ecology, and even the theory of cities and economic geography.