Fisher-KPP Epidemic Wavefront

Fisher-KPP equation: ∂u/∂t = D ∂²u/∂x² + r·u^β·(1−u)

Originally proposed by Fisher (1937) and Kolmogorov-Petrovsky-Piskunov (1937) to model wave-like spread of a favored gene — now central to epidemic modeling, ecology, and combustion.

Speed selection: For β=1 (KPP class), any initial condition with compact support converges to the minimal speed c* = 2√(rD) — the "pulled" front, driven by linear growth at the leading edge. For β>1, the front is "pushed" by nonlinear terms and may travel faster.

The spacetime plot below shows the u=0.5 contour (white line) — its slope gives the measured wave speed. Steeper initial conditions produce faster transient speeds before settling to c*.