FisherKPP

The Fisher-KPP equation, named after Ronald Fisher and Kolmogorov, Petrovsky, and Piskunov, is a reaction-diffusion model used to describe the spread of a population or allele through space. It combines local growth with spatial diffusion and is a canonical example of a monostable reaction-diffusion system.

Mathematically, it is written as ∂u/∂t = D ∂^2u/∂x^2 + f(u), where u(x,t) represents a population density or gene frequency, D is the diffusion coefficient, and f(u) encodes local growth. In the classical Fisher form, f(u) = r u(1 − u/K), with r the intrinsic growth rate and K the carrying capacity. A common non-dimensional form uses f(u) = u(1 − u). The equation was independently introduced by Fisher (1937) and by Kolmogorov, Petrovsky, and Piskunov (1937) to study the invasion of advantageous genes and the spread of populations.

A key feature of Fisher-KPP is the existence of traveling wave solutions, u(x,t) = U(x − ct), describing a front moving with speed c. For monostable nonlinearities with f′(0) > 0, there is a minimal wave speed c* = 2√(D f′(0)). In the canonical Fisher case, c* = 2√(Dr). The wave typically connects the stable equilibrium u = K (or 1, in nondimensional form) to the unstable equilibrium u = 0, with fronts that are often termed pulled waves due to their dependence on the leading edge.

Applications span ecology, genetics, and epidemiology, modeling invasive species, spread of advantageous alleles, or epidemic fronts under diffusion and local growth. The Fisher-KPP framework has influenced theoretical studies of pattern formation, nonlinear dynamics, and reaction-diffusion phenomena, with numerous extensions to heterogeneous media and higher dimensions.