KPP–Fisher equation#Fisher–Kolmogorov equation

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

:$f(u,x,t)\; =\; r\; u\; (1-u),\backslash ,$

which can exhibit traveling wave solutions that switch between equilibrium states given by $f(u)\; =\; 0$. 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.{{cite journal |first=R. A. |last=Fisher |title=The Wave of Advance of Advantageous Genes |journal=Annals of Eugenics |volume=7 |issue=4 |pages=353–369 |year=1937 |doi=10.1111/j.1469-1809.1937.tb02153.x |url=https://digital.library.adelaide.edu.au/dspace/bitstream/2440/15125/1/152.pdf |hdl=2440/15125 |doi-access=free }}

For every wave speed $c\; \backslash geq\; 2\; \backslash sqrt\{r\; D\}$ ($c\; \backslash geq\; 2$ in dimensionless form) it admits travelling wave solutions of the form

:$u(x,t)=v(x\; \backslash pm\; ct)\backslash equiv\; v(z),\backslash ,$

where $\backslash textstyle\; v$ is increasing and

:$\backslash lim\_\{z\backslash rightarrow-\backslash infty\}v\backslash left(\; z\backslash right)\; =0,\backslash quad\backslash lim\_\{z\backslash rightarrow\backslash infty\; \}v\backslash left(\; z\backslash right)\; =1.$

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, {{ISBN|90-277-2796-1}}. 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 {{ISBN|0-19-859676-6}}; {{ISBN|0-19-859692-8}}. 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.{{Citation needed|reason=Also, what does compact initial data mean in this context?|date=June 2025}}

For the special wave speed $c=\backslash pm\; 5/\backslash sqrt\{6\}$, 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 {{doi|10.1007/BF02462380}} with

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

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 the same year (1937) as Fisher, Kolmogorov, Petrovsky and Piskunov introduced the more general reaction-diffusion equation

:$\backslash frac\{\backslash partial\; u\}\{\backslash partial\; t\}-\backslash frac\{\backslash partial^2\; u\}\{\backslash partial\; x^2\}=F(u)$

where $F$ is a sufficiently smooth function with the properties that

$F(0)=F(1)=0,\; F\text{'}(0)=r>0$ and

Kolmogorov, Petrovsky and Piskunov discussed the example with $q=2$ in the context of population genetics.

The minimum speed of a KPP-type traveling wave is given by

:$2\backslash sqrt\{\backslash left.\backslash frac\{dF\}\{du\}\backslash right\; |\_\{u=0\}\}$

which differs from other type of waves, see for example ZFK-type waves.

• ZFK equation
• Allen–Cahn equation

{{Reflist}}

• [https://mathworld.wolfram.com/FishersEquation.html Fisher's equation] on MathWorld.
• [https://eqworld.ipmnet.ru/en/solutions/npde/npde1101.pdf Fisher equation] on EqWorld.

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Category:Partial differential equations

Category:Population ecology

Category:Ronald Fisher