The Fisher equation is an economic formula stating that the nominal interest rate equals the real interest rate plus the expected inflation rate.In 1937, the great statistician and geneticist Ronald Fisher wondered: at what speed would a beneficial mutation take over a population? He derived an equation that combined genetics and diffusion mathematics. It turned out that the gene spread front moves at a constant speed v = 2√(D·s), where D is the diffusion coefficient (a measure of mobility), and s is the selective advantage. This was the first mathematical model of waves in biology. Later it was discovered that similar equations describe combustion, tumor growth, and even the dynamics of language change. Fisher's equation (often called the Fisher–Kolmogorov–Petrovsky–Piskunov equation) became a classic of mathematical biology.
Ronald Fisher, a British statistician and geneticist, published the equation in 1937 in the paper 'The wave of advance of advantageous genes'. Independently, in the same year, Soviet mathematicians Andrei Kolmogorov, Ivan Petrovsky, and Nikolai Piskunov derived a similar equation while studying combustion. Hence, the equation is often called the Fisher–KPP equation. It became the foundation for the theory of population waves and spatial ecology.
The equation is used to model species invasions, epidemic spread, tumor growth, and even cultural trends. It predicts that wave speed is proportional to the square root of the product of mobility and selective advantage. Limitations: the equation assumes a homogeneous environment and constant parameters, which are rare in nature. It also ignores genetic drift and population discreteness.
The equation describes how quickly species spread across new territory. Epidemiologists use it to predict the speed of infection spread.
The equation predicts that wave speed does not depend on the initial number of mutants. A single carrier of a beneficial gene can trigger a wave that sweeps across an entire continent within a few thousand generations.
| \Delta\theta | variable | Standard deviation of the estimator for parameter theta — standard deviation |
| \mathcal{F}(\rho_\th | variable | Quantum Fisher information of the state rho_theta — quantum fisher information |
