Fisher's equation
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Fisher's equation (named after
statistician A statistician is a person who works with theoretical or applied statistics. The profession exists in both the private and public sectors. It is common to combine statistical knowledge with expertise in other subjects, and statisticians may wor ...
and
biologist A biologist is a scientist who conducts research in biology. Biologists are interested in studying life on Earth, whether it is an individual Cell (biology), cell, a multicellular organism, or a Community (ecology), community of Biological inter ...
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
) also known as the Kolmogorov–Petrovsky–Piskunov equation (named after
Andrey Kolmogorov Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Sovi ...
, Ivan Petrovsky, and
Nikolai Piskunov Nikolai Semenovich Piskunov (russian: Пискунов, Николай Семенович) (9 May 1908 – 1977)Bacaër N. (2011) The diffusion of genes (1937). In: A Short History of Mathematical Population Dynamics. Springer, London. https://d ...
), KPP equation or Fisher–KPP equation is the
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function. The function is often thought of as an "unknown" to be sol ...
: : \frac - D \frac = r u(1-u).\, It is a kind of
reaction–diffusion system Reaction–diffusion systems are mathematical models which correspond to several physical phenomena. The most common is the change in space and time of the concentration of one or more chemical substances: local chemical reactions in which the s ...
that can be used to model population growth and wave propagation.


Details

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 : f(u,x,t) = r u (1-u),\, which can exhibit traveling wave solutions that switch between equilibrium states given by f(u) = 0. Such equations occur, e.g., in
ecology Ecology () is the study of the relationships between living organisms, including humans, and their physical environment. Ecology considers organisms at the individual, population, community, ecosystem, and biosphere level. Ecology overlaps wi ...
,
physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemical ...
,
combustion Combustion, or burning, is a high-temperature exothermic redox chemical reaction between a fuel (the reductant) and an oxidant, usually atmospheric oxygen, that produces oxidized, often gaseous products, in a mixture termed as smoke. Combusti ...
,
crystallization Crystallization is the process by which solid forms, where the atoms or molecules are highly organized into a structure known as a crystal. Some ways by which crystals form are precipitating from a solution, freezing, or more rarely deposi ...
,
plasma physics Plasma ()πλάσμα
, Henry George Liddell, R ...
, and in general
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
problems. Fisher proposed this equation in his 1937 paper ''The wave of advance of advantageous genes'' in the context of
population dynamics Population dynamics is the type of mathematics used to model and study the size and age composition of populations as dynamical systems. History Population dynamics has traditionally been the dominant branch of mathematical biology, which has ...
to describe the spatial spread of an advantageous
allele An allele (, ; ; modern formation from Greek ἄλλος ''állos'', "other") is a variation of the same sequence of nucleotides at the same place on a long DNA molecule, as described in leading textbooks on genetics and evolution. ::"The chro ...
and explored its travelling wave solutions. For every wave speed c \geq 2 \sqrt ( c \geq 2 in dimensionless form) it admits travelling
wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
solutions of the form : u(x,t)=v(x \pm ct)\equiv v(z),\, where \textstyle v is increasing and : \lim_v\left( z\right) =0,\quad\lim_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.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, . Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937Peter 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 ; . 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, all solutions can be found in a closed form, with : v(z) = \left( 1 + C \mathrm\left(\mp/\right) \right)^ where C is arbitrary, and the above limit conditions are satisfied for C>0. Proof of the existence of travelling wave solutions and analysis of their properties is often done by the
phase space method In applied mathematics, the phase space method is a technique for constructing and analyzing solutions of dynamical systems, that is, solving time-dependent differential equations. The method consists of first rewriting the equations as a system o ...
.


KPP equation

In the same year (1937) as Fisher, Kolmogorov, Petrovsky and Piskunov introduced the more general reaction-diffusion equation :\frac-\frac=F(u) where F is a sufficiently smooth function with the properties that F(0)=F(1)=0, F'(0)=r>0 and F(v)>0, F'(v) for all 0. This too has the travelling wave solutions discussed above. Fisher's equation is obtained upon setting F(u)=ru(1-u) and rescaling the x coordinate by a factor of \sqrt. A more general example is given by F(u)=r u(1-u^q) with q>0. Kolmogorov, Petrovsky and Piskunov discussed the example with q=2 in the context of
population genetics Population genetics is a subfield of genetics that deals with genetic differences within and between populations, and is a part of evolutionary biology. Studies in this branch of biology examine such phenomena as adaptation, speciation, and pop ...
. The minimum speed of a KPP-type traveling wave is given by :2\sqrt which differs from other type of waves, see for example ZFK-type waves.


See also

* ZFK equation *
List of plasma (physics) articles This is a list of plasma physics topics. A * Ablation * Abradable coating * Abraham–Lorentz force * Absorption band * Accretion disk * Active galactic nucleus * Adiabatic invariant * ADITYA (tokamak) * Aeronomy * Afterglow plasma * Airg ...
* Allen–Cahn equation


References


External links


Fisher's equation
on
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Fisher equation
on EqWorld. {{DEFAULTSORT:Fisher's Equation Partial differential equations Population ecology