Fisher-KPP Pulled Front

∂u/∂t = D·∂²u/∂x² + r·u·(1 − u) — asymptotic wave speed selection

PDE Parameters

Growth rate r 1.0

Diffusion D 0.5

Allee threshold α 0.00

Initial Condition

Step width 30

State

c* (KPP): 1.41

Measured c: —

t = 0

Fisher-KPP Equation

Proposed independently by Fisher (1937) and Kolmogorov-Petrovsky-Piskunov (1937), this PDE models the spread of a favourable allele or invasive species. The minimal wave speed is c* = 2√(D·r) — the "pulled" front propagates at this speed when initialised with a step or compact support.

With an Allee effect (negative growth for small u), the front becomes "pushed": the minimal speed exceeds c*, and a sharp threshold governs whether invasion succeeds. This transition is ecologically important for understanding biological invasions and gene surfing.

The speed plot tracks the instantaneous front position (u = 0.5 level set) and fits convergence to c*.