KPP–Fisher equation

In mathematics, Fisher's equation, also known as the Fisher-Kolmogorov equation and the Fisher-KPP equation, named after R. A. Fisher and A. N. Kolmogorov, is the partial differential equation

${\frac {\partial u}{\partial t}}=u(1-u)+{\frac {\partial ^{2}u}{\partial x^{2}}}.\,$

For every wave speed c ≥ 2, it admits travelling wave solutions of the form

$u(x,t)=v(\pm x+ct),\,$

where $\textstyle v$ is increasing and

$\lim _{z\rightarrow -\infty }v\left(z\right)=0,\quad \lim _{z\rightarrow \infty }v\left(z\right)=1.$

That is, the solution switches from the equilibrium state u = 0 to the equilibrium state u = 1. No such solution exists for c < 2. The wave shape for a given wave speed is unique.

For the special wave speed $c=5/{\sqrt {6}}$, all solutions can be found in a closed form, with

$v(z)=\left(1+C\mathrm {exp} \left(-{z}/{\sqrt {6}}\right)\right)^{-2}$

where $C$ is arbitrary, and the above limit conditions are satisfied for $C>0$.

This equation was originally derived for the simulation of propagation of a gene in a population . It is perhaps the simplest model problem for reaction-diffusion equations

${\frac {\partial u}{\partial t}}=\Delta u+f\left(u\right),$

which exhibit traveling wave solutions that switch between equilibrium states given by f(u) = 0. Such equations occur, e.g., in ecology, physiology, combustion, crystallization, plasma physics, and in general phase transition problems.

Proof of the existence of traveling wave solutions and analysis of their properties is often done by the phase space method.

External links

• Fisher's equation on MathWorld.
• Fisher equation on EqWorld.

See also

• Allen-Cahn equation