FisherGleichung

FisherGleichung, in English commonly known as the Fisher equation or the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation, is a nonlinear reaction–diffusion partial differential equation used in population genetics and ecology to model the spatial spread of a gene, allele, or phenotype in a moving population. It combines local growth or selection with random dispersal represented by diffusion.

A standard form for an allele frequency p(x,t) is

∂p/∂t = D ∂^2p/∂x^2 + s p(1−p),

where D > 0 is the diffusion coefficient describing dispersal, and s is the selection coefficient favoring the allele. The variable p(x,t) takes values between 0 and 1, with equilibria p = 0 and p = 1 representing the absence or fixation of the allele. A related and widely used form is the Fisher–KPP equation for a normalized population density u(x,t):

∂u/∂t = D ∂^2u/∂x^2 + r u(1−u),

where r > 0 is the intrinsic growth rate.

Key properties include the existence of traveling wave solutions, u(x,t) = U(x − ct), that describe a wave of invasion moving with speed c. For the Fisher–KPP form, the minimal wave speed is c_min = 2√(Dr). The front connects the stable equilibrium u = 1 to the unstable equilibrium u = 0, and the wave is typically pulled by dynamics at the leading edge.

Historically, the equation was introduced by Ronald Fisher in 1937 and independently by Kolmogorov, Petrovsky, and Piskunov in the same year. It remains a foundational tool for modelling gene spread, ecological invasions, and related spreading processes, with numerous extensions for heterogeneous environments and nonlinearities.