35:
656: < 2. 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.
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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,
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Oxford
Applied Mathematics and Computing Science Series. The Clarendon Press Oxford University Press, New York, second edition, 1996
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In the same year (1937) as Fisher, Kolmogorov, Petrovsky and
Piskunov introduced the more general reaction-diffusion equation
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221:: in fact, it is one of the simplest semilinear reaction-diffusion equations, the one which has the inhomogeneous term
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634:{\displaystyle \lim _{z\rightarrow -\infty }v\left(z\right)=0,\quad \lim _{z\rightarrow \infty }v\left(z\right)=1.}
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1159:. This too has the travelling wave solutions discussed above. Fisher's equation is obtained upon setting
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Proof of the existence of travelling wave solutions and analysis of their properties is often done by the
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195:{\displaystyle {\frac {\partial u}{\partial t}}-D{\frac {\partial ^{2}u}{\partial x^{2}}}=ru(1-u).\,}
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which can exhibit traveling wave solutions that switch between equilibrium states given by
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798:{\displaystyle v(z)=\left(1+C\mathrm {exp} \left(\mp {z}/{\sqrt {6}}\right)\right)^{-2}}
50:); in dots : slope corresponding to the theoretical velocity of the traveling wave.
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Griffiths, Graham W.; Schiesser, William E. (2011). "Fisher–Kolmogorov
Equation".
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The theory and applications of reaction-diffusion equations: Patterns and waves.
1501:. Translated by V. M. Volosov from Bull. Moscow Univ., Math. Mech. 1, 1–25, 1937
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Numerical simulation of the Fisher–KPP equation. In colors: the solution
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1342:. Kolmogorov, Petrovsky and Piskunov discussed the example with
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and explored its travelling wave solutions. For every wave speed
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that can be used to model population growth and wave propagation.
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Explicit solutions of Fisher's equation for a special wave speed
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is arbitrary, and the above limit conditions are satisfied for
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1435:{\displaystyle 2{\sqrt {\left.{\frac {dF}{du}}\right|_{u=0}}}}
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The minimum speed of a KPP-type traveling wave is given by
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is a sufficiently smooth function with the properties that
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That is, the solution switches from the equilibrium state
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Traveling Wave
Analysis of Partial Differential Equations
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which differs from other type of waves, see for example
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1594:, Bulletin of Mathematical Biology 41 (1979) 835–840
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510:{\displaystyle u(x,t)=v(x\pm ct)\equiv v(z),\,}
432:in dimensionless form) it admits travelling
217:KPP–Fisher equation belongs to the class of
1590:Ablowitz, Mark J. and Zeppetella, Anthony,
1518:"The Wave of Advance of Advantageous Genes"
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1059:{\displaystyle F(0)=F(1)=0,F'(0)=r>0}
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18:Kolmogorov–Petrovsky–Piskunov equation
1256:. A more general example is given by
1491:Selected Works of A. N. Kolmogorov I
1120:{\displaystyle F(v)>0,F'(v)<r}
1637:. Academy Press. pp. 135–146.
693:{\displaystyle c=\pm 5/{\sqrt {6}}}
399:{\displaystyle c\geq 2{\sqrt {rD}}}
286:{\displaystyle f(u,x,t)=ru(1-u),\,}
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652:= 1. No such solution exists for
331:. Such equations occur, e.g., in
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1721:Partial differential equations
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1493:, pages 248–270. Kluwer 1991,
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92:partial differential equation
1677:10.1016/0893-9659(95)00010-N
1660:"Fisher–Kolmogorov equation"
1205:{\displaystyle F(u)=ru(1-u)}
535:{\displaystyle \textstyle v}
219:reaction-diffusion equations
27:Not to be confused with the
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1664:Applied Mathematics Letters
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1249:{\displaystyle {\sqrt {D}}}
1152:{\displaystyle 0<v<1}
659:For the special wave speed
10:
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1232:coordinate by a factor of
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207:reaction–diffusion system
31:in financial mathematics.
425:{\displaystyle c\geq 2}
1516:Fisher, R. A. (1937).
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1613:"Fisher-KPP Equation"
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78:) also known as the
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1361:{\displaystyle q=2}
364:population dynamics
88:Fisher–KPP equation
60:KPP–Fisher equation
1726:Population ecology
1600:10.1007/BF02462380
1526:Annals of Eugenics
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1695:Fisher's equation
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64:Andrey Kolmogorov
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1670:(2): 51–52.
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1707:on EqWorld.
56:mathematics
1715:Categories
1549:2440/15125
1470:References
355:problems.
341:combustion
337:physiology
1699:MathWorld
1291:−
1194:−
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788:−
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149:∂
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1620:Fisher 2
1453:See also
1127:for all
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333:ecology
213:Details
90:is the
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964:where
808:where
520:where
368:allele
1616:(PDF)
1521:(PDF)
1316:with
1639:ISBN
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1495:ISBN
1327:>
1144:<
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