Fisher's equation

Kolmogorov Petrovsky Piscounov equation ( KPP equation or Fisher's equation) is a partial differential equation of the form:

She is a semilinear parabolic equation of second order. The equation is used to model different processes in nature. It is for example used in the population dynamics and the description of chemical reactions.

The differential equation is composed of a diffusion term and a non-linear response term.

If one uses a non-local function, we obtain the ordinary differential equation

At this it can be seen that with the model of exponential growth is modeled but including a saturation term that is in chemical reactions, the saturation concentration or in the population dynamics for the limited food supply, for example.

Reaction fronts

If one uses the equation to model a spatially localized starting reaction, it is clear that forming a reaction front. This has, as can be shown, a minimum rate of spread.

To the usual approach used for shafts

Is obtained by inserting the ordinary differential equation of second order

After linearization, and assuming that the " concentration " f can assume only values ​​between 0 and 1, one obtains the equation of the eigenvalues ​​of

Since this must be real for stable waves, must apply.

Generalizations

The equation can be generalized to:

To a positive integer m.