Fisher's equation

Fisher's equation

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 extstyle v is increasing and

: lim_{z ightarrow-infty}vleft( z ight) =0,quadlim_{z ightarrowinfty }vleft( z ight) =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. [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.]

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

: v(z) = left( 1 + C mathrm{exp}left(-{z}/{sqrt6} ight) ight)^{-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 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

] . It is perhaps the simplest model problem for reaction-diffusion equations

: frac{partial u}{partial t}=Delta u+fleft( u ight) ,

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.


External links

* [ Fisher's equation] on MathWorld.
* [ Fisher equation] on EqWorld.

ee also

*Allen-Cahn equation

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

  • FISHER (I.) — FISHER IRVING (1867 1947) Considéré, en son temps, comme le plus grand économiste américain, Irving Fisher a enseigné à l’université Yale. Statisticien et économiste, Fisher a élaboré une œuvre abondante d’où émergent, outre une importante… …   Encyclopédie Universelle

  • Fisher equation — The Fisher equation in financial mathematics and economics estimates the relationship between nominal and real interest rates under inflation.It is named after Irving Fisher who was famous for his works on the theory of interest. In finance, the… …   Wikipedia

  • Fisher's fundamental theorem of natural selection — In population genetics, R. A. Fisher s fundamental theorem of natural selection was originally stated as:: The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time. Fisher, R.A. (1930)… …   Wikipedia

  • Équation de Fisher — Irving Fisher Pour les articles homonymes, voir Fisher. Irving Fisher Naissance …   Wikipédia en Français

  • Équation différentielle stochastique — Une équation différentielle stochastique (EDS) est une généralisation de la notion d équation différentielle prenant en compte un terme de bruit blanc. Les EDS permettent de modéliser des trajectoires aléatoires, tels des cours de bourse ou les… …   Wikipédia en Français

  • Fisher hypothesis — The Fisher hypothesis is the proposition by Irving Fisher that the real interest rate is independent of monetary measures, especially the nominal interest rate. The Fisher equation is:r r = r n pi^e.This means, the real interest rate (r r) equals …   Wikipedia

  • Fisher information — In statistics and information theory, the Fisher information (denoted mathcal{I}( heta)) is the variance of the score. It is named in honor of its inventor, the statistician R.A. Fisher.DefinitionThe Fisher information is a way of measuring the… …   Wikipedia

  • fisher — 1) a person participating in a fishery (gender neutral, in preference to the previously used term fisherman ). Once archaic and now resurrected. See also angler 2) disciples in the Christian Bible 3) victualler 4) a large arboreal mammal (Martes… …   Dictionary of ichthyology

  • Fisher information metric — In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, i.e., a smooth manifold whose points are probability measures defined on a common probability space.… …   Wikipedia

  • Fisher Equation — interest rates are composed of the real rate of interest plus the expected rate of inflation (named for economist Irving Fisher) …   Eponyms, nicknames, and geographical games

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”