Taylor's law

Taylor's power law is an empirical law in ecology that relates the variance of the number of individuals of a species per unit area of habitat to the corresponding mean by a power law relationship. It is named after the ecologist who first proposed it in 1961, Lionel Roy Taylor (1924–2007). Taylor's original name for this relationship was the law of the mean. The name Taylor's law was coined by Southwood in 1966. This law was originally defined for ecological systems, specifically to assess the spatial clustering of organisms. For a population count with mean and variance , Taylor's law is written where a and b are both positive constants. Taylor proposed this relationship in 1961, suggesting that the exponent b be considered a species specific index of aggregation. This power law has subsequently been confirmed for many hundreds of species. Taylor's law has also been applied to assess the time dependent changes of population distributions. Related variance to mean power laws have also been demonstrated in several non-ecological systems: cancer metastasis the numbers of houses built over the Tonami plain in Japan. measles epidemiology HIV epidemiology, the geographic clustering of childhood leukemia blood flow heterogeneity the genomic distributions of single-nucleotide polymorphisms (SNPs) gene structures in number theory with sequential values of the Mertens function and also with the distribution of prime numbers from the eigenvalue deviations of Gaussian orthogonal and unitary ensembles of random matrix theory The first use of a double log-log plot was by Reynolds in 1879 on thermal aerodynamics. Pareto used a similar plot to study the proportion of a population and their income. The term variance was coined by Fisher in 1918. Pearson in 1921 proposed the equation (also studied by Neyman) Smith in 1938 while studying crop yields proposed a relationship similar to Taylor's. This relationship was where Vx is the variance of yield for plots of x units, V1 is the variance of yield per unit area and x is the size of plots.