14.1 Reaction–diffusion equations

A reaction–diffusion equation is \[ u_t = D\Delta u + f(u) . \] Without diffusion it is the ordinary differential equation \(u' = f(u)\), whose equilibria are the zeros of \(f\); without reaction it is the heat equation. The two standard choices of \(f\) in one dimension, after scaling \(D\) and the rate to \(1\), are

• the Fisher–KPP equation \(u_t = u_{xx} + u(1 - u)\): logistic growth, with \(u = 0\) unstable and \(u = 1\) stable;
• the bistable (Nagumo, or Allen–Cahn) equation \(u_t = u_{xx} + u(1 - u)(u - a)\) with \(0 < a < 1\): both \(0\) and \(1\) are stable, and \(a\) is a threshold.

The scaling \(x \mapsto x\sqrt {r/D}\), \(t \mapsto rt\) removes \(D\) and the growth rate \(r\), so a speed \(c\) in scaled units is \(c\sqrt {rD}\) in physical units.

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