# Fisher's equation

In mathematics, **Fisher's equation** (named after statistician and biologist Ronald Fisher) also known as the **Kolmogorov–Petrovsky–Piskunov equation** (named after Andrey Kolmogorov, Ivan Petrovsky, and N. Piskunov), **KPP equation** or **Fisher–KPP equation** is the partial differential equation:

It is a kind of Reaction–diffusion system that can be used to model population growth and wave propagation.

## DetailsEdit

Fisher's equation belongs to the class of reaction-diffusion equation: in fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term

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

Fisher proposed this equation in his 1937 paper *The wave of advance of advantageous genes* in the context of population dynamics to describe the spatial spread of an advantageous allele and explored its travelling wave solutions.^{[1]}
For every wave speed ( in dimensionless form) it admits travelling wave solutions of the form

where is increasing and

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]} 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 , all solutions can be found in a closed form,^{[4]} with

where is arbitrary, and the above limit conditions are satisfied for .

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

## Fisher–Kolmogorov equationEdit

A generalization is given by

which gives the above equation upon setting , and rescaling the coordinate by a factor of .^{[5]}^{[6]}^{[7]}

## See alsoEdit

## ReferencesEdit

- ^
^{a}^{b}Fisher, R. A. (1937). "The Wave of Advance of Advantageous Genes" (PDF).*Annals of Eugenics*.**7**(4): 353–369. doi:10.1111/j.1469-1809.1937.tb02153.x. hdl:2440/15125. **^**A. Kolmogorov, I. Petrovskii, and N. Piskunov. "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 doi:10.1007/BF02462380**^**Trefethen (August 30, 2001). "Fisher-KPP Equation" (PDF).*Fisher 2*.**^**Griffiths, Graham W.; Schiesser, William E. (2011). "Fisher–Kolmogorov Equation".*Traveling Wave Analysis of Partial Differential Equations*. Academy Press. pp. 135–146. ISBN 978-0-12-384652-5.**^**Adomian, G. (1995). "Fisher–Kolmogorov equation".*Applied Mathematics Letters*.**8**(2): 51–52. doi:10.1016/0893-9659(95)00010-N.

## External linksEdit

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