diffusionreaction

Diffusion-reaction models describe how the concentrations of chemical species evolve in space and time under the combined influence of diffusion and local chemical reactions. Diffusion tends to smooth spatial gradients, while reaction kinetics can create or consume species, leading to complex spatiotemporal behavior.

For a system of N species with concentrations u_i(x,t), the evolution is typically written as ∂u_i/∂t = D_i ∇^2 u_i + f_i(u_1,...,u_N), where D_i are diffusion coefficients and f_i are nonlinear reaction terms. The equations are completed with boundary conditions (for example no-flux or fixed concentrations) and initial conditions.

Because diffusion and nonlinear reactions interact, the system can exhibit steady states, oscillations, traveling waves, and, under certain conditions, stable spatial patterns. A notable mechanism is diffusion-driven or Turing instability, in which diffusion destabilizes a homogeneous steady state and generates patterns such as stripes or spots in two- or three-dimensional domains.

Applications span chemistry, biology, and beyond. They include modeling chemical waves in reacting media, morphogenesis and pattern formation in developing organisms, growth and spread of populations in ecology, and materials science phenomena such as phase separation with reaction terms.

Analytical approaches include linear stability analysis and weakly nonlinear theory; computational methods rely on finite difference or finite element discretizations and, for stiff systems, implicit solvers or operator-splitting schemes. Diffusion-reaction models are a foundational tool for understanding spatiotemporal pattern formation.