Blocking (statistics)

Numerical simulation of the Fisher–KPP equation. In colors: the solution u(t,x); in dots : slope corresponding to the theoretical velocity of the traveling wave.

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

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

Fisher proposed this equation to describe the spatial spread of an advantageous allele and explored its travelling wave solutions. For every wave speed c ≥ 2, it admits travelling wave solutions of the form

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

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 $c=\pm 5/{\sqrt {6}}$, all solutions can be found in a closed form, with

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

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

It is perhaps the simplest example of a semilinear reaction-diffusion equation

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

which can 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.

Traveling wave solutions

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See also

• List of plasma (physics) articles
• Allen–Cahn equation

References

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External links

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