Kolmogorov–Petrovsky–Piskounov equation

In mathematics, Fisher's equation (named after statistician and biologist Ronald Fisher; also known as Kolmogorov–Petrovsky–Piskunov equation—named after Andrey Kolmogorov, Ivan Petrovsky, and N. Piskunov—or KPP equation or Fisher–KPP equation) is the partial differential equation:

Fisher's equation belongs to the class of reaction-diffusion equation: in fact, it is one of the simplest semilinear r.d.e. , the one which has the inhomogeneous term

which can exhibit traveling wave solutions that switch between equilibrium states given by ${\displaystyle f(u)=0}$. 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. For every wave speed ${\displaystyle c\geq 2{\sqrt {rD}}}$ (${\displaystyle c\geq 2}$ in dimensionless form) it admits travelling wave solutions of the form

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