Fisher's equation

Not to be confused with the Fisher equation in financial mathematics.
Further information: Fisher-Kolmogorov equation
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 (named after R. A. Fisher,[1] also known as Kolmogorov-Petrovsky-Piscounov equation, KPP equation or Fisher-KPP equation) is the partial differential equation

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

Fisher proposed this equation in 1937 in the context of population dynamics to describe the spatial spread of an advantageous allele and explored its travelling wave solutions.[2] For every wave speed $c \geq 2 \sqrt{r D}$ ($c \geq 2$ in dimensionless form) 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.[2][3][4] The wave shape for a given wave speed is unique. The travelling-wave solutions are stable against near-field perturbations, but not to far-field perturbations which can thicken the tail. One can prove using the comparison principle and super-solution theory that all solutions with compact initial data converge to waves with the minimum speed.

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

$v(z) = \left( 1 + C \mathrm{exp}\left(\pm{z}/{\sqrt6}\right) \right)^{-2}$

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.