Reaction-Diffusion Equations

Mathematical Modeling · Axiom Academy

LESSON Reaction-Diffusion Equations Understanding how chemical reactions and spatial diffusion combine to create complex patterns in nature Reaction-diffusion equations model systems where substances undergo both Reaction (local transformation) and Diffusion (spatial spreading). Let's understand each component: Chemical reactions create or destroy substances at each point in space. These are typically nonlinear interactions between concentrations. Substances spread from high to low concentration regions. This smoothing process follows Fick's law and creates spatial coupling. 2. The General Reaction-Diffusion Equation For a single species with concentration u(x,t), the reaction-diffusion equation combines diffusion with a reaction term: Here, D is the diffusion coefficient (how fast the substance spreads), and f(u) is the reaction term describing how the substance transforms locally. 3. Two-Species Systems: The Activator-Inhibitor Model The most interesting patterns emerge when two chemical species interact. In the classic activator-inhibitor system: Promotes its own production and that of the inhibitor. Diffuses slowly. Suppresses the activator. Diffuses quickly, creating "long-range inhibition." Alan Turing discovered that even when a uniform state is stable without diffusion, adding diffusion can cause it to become unstable, leading to spontaneous pattern formation. This is called the Turing instability or diffusion-driven instability .

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