reactiondiffusion

Reaction-diffusion refers to a class of mathematical models that describe how the concentration of chemical substances changes in space and time due to local reactions and diffusion. In a typical two-species system, the unknowns u(x,t) and v(x,t) represent the concentrations of two interacting chemicals or morphogens, distributed over a spatial domain. The evolution is governed by partial differential equations that couple reaction kinetics with diffusion.

A common form of the model is

∂u/∂t = Du ∇^2 u + f(u,v)

∂v/∂t = Dv ∇^2 v + g(u,v)

where Du and Dv are the diffusion coefficients, ∇^2 is the Laplacian operator describing spatial spread, and f and g encode the local chemical reactions. Boundary and initial conditions complete the specification. Variants with more species or nonlinear diffusion exist and are studied in different contexts.

The idea of diffusion-driven pattern formation, or the Turing mechanism, explains how stable homogeneous states can become spatially inhomogeneous when diffusion acts unevenly on interacting species. Conditions typically require a stable steady state in the absence of diffusion but an instability when diffusion is present, often involving at least one activator and one inhibitor with different diffusion rates. This can yield a range of patterns, including spots, stripes, and labyrinths, depending on parameters and domain geometry.

Applications include developmental biology and animal coat patterning, chemical experiments such as the Belousov–Zhabotinsky reaction, materials science, and ecological or ecological modeling. Reaction-diffusion remains a foundational concept for understanding how simple local rules can generate complex spatial organization.