Fisher–KPP Wave

About the Fisher–KPP Wave

The Fisher-KPP equation describes how a beneficial allele or invasive species spreads spatially as a traveling wave. It combines:

• Diffusion (migration): Spatial dispersal at rate D
• Selection (growth): Logistic growth at rate r

The equation: ∂u/∂t = D ∂²u/∂x² + ru(1-u)

Starting from a localized introduction, u spreads as a traveling wave with constant speed v = 2√(rD). The wave front maintains its shape, connecting the absent state (u=0) to the established state (u=1).

Applications: range expansion, biological invasions, epidemic spread, gene flow.