ReaktionsDiffusionsgleichungen

Reaction-diffusion systems describe how the concentration of chemical species changes under the combined influence of local reactions and diffusion. In their simplest form, two interacting species u(x,t) and v(x,t) evolve according to partial differential equations of the form ∂u/∂t = Du ∇^2 u + f(u,v) and ∂v/∂t = Dv ∇^2 v + g(u,v), where Du and Dv are diffusion coefficients and f and g encode the reaction kinetics. These models can be extended to larger networks of interacting species and to three-dimensional domains.

A central concept is the diffusion-driven, or Turing, instability. Although diffusion tends to homogenize concentrations, certain reaction kinetics can render a homogeneous steady state unstable in the presence of diffusion, producing spatially patterned solutions. This requires a suitable mismatch between diffusion rates (often one species diffuses much faster than the other) and specific properties of the reaction terms. Alan Turing introduced this mechanism in 1952 as a possible explanation for biological pattern formation, such as animal coat markings.

Applications of reaction-diffusion theory span chemistry, biology, and materials science. In biology, it provides a framework for understanding morphogenesis and pigment patterning; in chemistry, it models spatial oscillations and pattern formation in excitable media; and in ecology and materials science, it aids in describing spread and organization of interacting entities.

Mathematically, these systems are studied through stability analysis, bifurcation theory, and numerical simulations using finite differences, finite elements, or spectral methods. Solutions depend on geometry and boundary conditions, with no-flux and periodic boundaries commonly used in simulations.