Fisher-Kolmogorov equation

Wave processes described by the Fisher-Kolmogorov equation will be considered first. We shall analyse solutions to this equation of the travelling wave form... 

This is the Fisher-Kolmogorov equation, and whereas no analytical solution is known for it, a simple phase-plane analysis allows a determination of the minimal front propagation velocity. 

The roots Ali2(1,2) in this model are identical with the eigenvalues, for a = D = 1, in the Fisher-Kolmogorov model. As the second pair of the roots does not lead to a generation of the sensitive states, the nature of a catastrophe for the Oregonator with diffusion is the same as for the Fisher-Kolmogorov model. In other words, the waves with the velocities v centre manifold theorem the sensitive state is associated only with the zlt z2 variables. 

The kinetic term of this form is commonly referred to as KPP kinetics and is used in the Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) equation, see Chap. 4. The equation has two steady states, the trivial one, Px = 0, and p2 =. Since... 

A front corresponds to a traveling wave solution, which maintains its shape, travels with a constant velocity v, p x, t) = p(x - v t), and joins two steady states of the system. The latter are uniform stationary states, p(x, t) = p, where Ffp) = 0. For the logistic kinetics, the steady states are = 0 and jo2 = 1- While the logistic kinetics has only two steady states, three or more stationary states can exist for a broad class of systems in nonlinear chemistry and population dynamics with Alice effect, but a front can only connect two of them. To determine the propagation direction of the front, we need to evaluate the stability of the stationary states, see Sect. 1.2. The steady state jo is stable if P (fp) < 0 and unstable if F (jo) > 0. Let the initial particle density p x,0) be such that on a certain finite interval, p x,0) is different from 0 and 1, and to the left of this interval p(x,0) = 1, while to the right p x, 0) = 0. In this case, the initial condition is said to have compact support. Kolmogorov et al. [232] showed for Fisher s equation that due to the combined effects of diffusion and reaction, the region of density close to 1 expands to the... 

It was introduced in 1937 in the seminal contributions of R. A. Fisher [132] and A. N. Kolmogorov, together with 1. G. Petrovskii and N. S. Piskunov [232] as a model for the spreading of an advantageous gene. Consequently, we will refer to (4.1) also as the FKPP equation. It is the simplest and most well-known equation that has traveling wave solutions. 

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