Fisher's equation

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NOTE: this is not the Fisher equation in financial mathematics.

In mathematics, Fisher's equation, also known as the Fisher-Kolmogorov 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. ^[1]^[2]^[3]

For the special wave speed $c=5/\sqrt{6}$ , all solutions can be found in a closed form,^[4] with

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

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

In particular, the wave shape for a given wave speed is not necessarily unique.

This equation was originally derived for the simulation of propagation of a gene in a population ^[5]. 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 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.

[edit] References

^ R. A. Fisher. "The wave of advance of advantageous genes", Ann. Eugenics 7:353–369, 1937.
^ A. Kolmogorov, I. Petrovskii, and N. Piscounov. A study of the diffusion equation with increase in the amount of substance, and its application to a biological problem. In V. M. Tikhomirov, editor, Selected Works of A. N. Kolmogorov I, pages 248--270. Kluwer 1991, ISBN 90-277-2796-1. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937
^ Peter Grindrod. The theory and applications of reaction-diffusion equations: Patterns and waves. Oxford Applied Mathematics and Computing Science Series. The Clarendon Press Oxford University Press, New York, second edition, 1996 ISBN 0-19-859676-6; ISBN 0-19-859692-8.
^ Ablowitz, Mark J. and Zeppetella, Anthony, Explicit solutions of Fisher's equation for a special wave speed, Bulletin of Mathematical Biology 41 (1979) 835-840
^ Fisher, R. A., The genetical theory of natural selection. Oxford University Press, 1930. Oxford University Press, USA, New Ed edition, 2000, ISBN 978-0198504405, variorum edition, 1999, ISBN 0-19-850440-3