Fisher's equation
Encyclopedia
In mathematics, Fisher's equation, also known as the Fisher-Kolmogorov equation and the Fisher-KPP equation, named after R. A. Fisher
and A. N. Kolmogorov
, is the partial differential equation
For every wave speed c ≥ 2, it admits travelling wave solution
s 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. The wave shape for a given wave speed is unique.
For the special wave speed
, all solutions can be found in a closed form, with
where
is arbitrary, and the above limit conditions are satisfied for 
.
This equation was originally derived for the simulation of propagation of a gene in a population. It is perhaps the simplest model problem for reaction-diffusion equations
which 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.
Proof of the existence of traveling wave solutions and analysis of their properties is often done by the phase space method
.
Ronald Fisher
Sir Ronald Aylmer Fisher FRS was an English statistician, evolutionary biologist, eugenicist and geneticist. Among other things, Fisher is well known for his contributions to statistics by creating Fisher's exact test and Fisher's equation...
and A. N. Kolmogorov
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov was a Soviet mathematician, preeminent in the 20th century, who advanced various scientific fields, among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity.-Early life:Kolmogorov was born at Tambov...
, is the partial differential equation
Partial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
For every wave speed c ≥ 2, it admits travelling wave solution
Solution (disambiguation)
Solution may refer to:*In chemistry, a solution is a homogeneous mixture where one substance is dissolved in another*A solution to a problem will, metaphorically, make it dissolve...
s of the form
where
is increasing andThat is, the solution switches from the equilibrium state u = 0 to the equilibrium state u = 1. No such solution exists for c < 2. The wave shape for a given wave speed is unique.
For the special wave speed
, all solutions can be found in a closed form, withwhere
is arbitrary, and the above limit conditions are satisfied for .This equation was originally derived for the simulation of propagation of a gene in a population. It is perhaps the simplest model problem for reaction-diffusion equations
which exhibit traveling wave solutions that switch between equilibrium states given by
. Such equations occur, e.g., in ecologyEcology
Ecology is the scientific study of the relations that living organisms have with respect to each other and their natural environment. Variables of interest to ecologists include the composition, distribution, amount , number, and changing states of organisms within and among ecosystems...
, physiology
Physiology
Physiology is the science of the function of living systems. This includes how organisms, organ systems, organs, cells, and bio-molecules carry out the chemical or physical functions that exist in a living system. The highest honor awarded in physiology is the Nobel Prize in Physiology or...
, combustion
Combustion
Combustion or burning is the sequence of exothermic chemical reactions between a fuel and an oxidant accompanied by the production of heat and conversion of chemical species. The release of heat can result in the production of light in the form of either glowing or a flame...
, crystallization
Crystallization
Crystallization is the process of formation of solid crystals precipitating from a solution, melt or more rarely deposited directly from a gas. Crystallization is also a chemical solid–liquid separation technique, in which mass transfer of a solute from the liquid solution to a pure solid...
, plasma physics, and in general phase transition
Phase transition
A phase transition is the transformation of a thermodynamic system from one phase or state of matter to another.A phase of a thermodynamic system and the states of matter have uniform physical properties....
problems.
Proof of the existence of traveling wave solutions and analysis of their properties is often done by the phase space method
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 of differential equations that are first-order in...
.
External links
- Fisher's equation on MathWorldMathWorldMathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at...
. - Fisher equation on EqWorld.