KPP equation
The Kolmogorov-Petrovsky-Piscounov equation ( KPP equation according to Andrei Kolmogorow , Iwan Petrowski and N. Piscounov 1937) is a nonlinear partial differential equation in the form of a reaction-diffusion equation .
A special case is Fisher's equation (after Ronald Aylmer Fisher 1937) of population dynamics , a continuous variant of the logistic equation (see also logistic function ).
Bulk
The KPP equation has the form:
with a nonlinear function which satisfies: , and for (the interval [0,1] is often the interval of definition of the variable , if this indicates a concentration).
Fisher's equation is a special case of the form:
Sometimes a term is given instead of the reaction term , similar to the logistic equation.
This is a semi-linear parabolic equation of the second order. It is used to model various processes in nature, such as population dynamics or chemical reactions .
The differential equation consists of a diffusion term and a nonlinear reaction term .
If one uses a location-independent function , one obtains the ordinary differential equation
- .
At this it can be seen that the model an exponential growth is modeled, but which a saturation term includes. This stands z. B. in the population dynamics for the limited food supply or in chemical reactions for the saturation of the concentration .
Reaction fronts
If the equation is used to model a locally localized starting reaction, it is clear that a reaction front is developing. As can be shown, this has a minimal velocity of propagation.
Using the usual approach for waves
- ,
so after insertion one obtains the ordinary differential equation of the second order
- .
After linearization and assuming that the "concentration" f can only assume values between 0 and 1, the equation for the eigenvalues is obtained
- .
Since these must be real for stable waves , it must apply.
Generalizations
The Fisher equation can be generalized to:
with a positive integer .
In the case of the Fisher equation then applies .
See also
Individual evidence
- ↑ BH Gilding u. a. (Ed.), Traveling waves in nonlinear diffusion-convection equation reaction, Birkhäuser 2004, p. 2
- ↑ F. Hamel, N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math. , Vol. 52, 1999, pp. 1255-1276, doi : 10.1002 / (SICI) 1097-0312 (199910) 52:10 <1255 :: AID-CPA4> 3.0.CO; 2-W .