Population density, Distribution (Probability theory)
Taylor’s law (TL), a widely verified quantitative pattern in ecology and other sciences, describes the variance in a species’ population density (or other nonnegative quantity) as a power-law function of the mean density (or other nonnegative quantity): Approximately, variance = a(mean)b, a > 0. Multiple mechanisms have been proposed to explain and interpret TL. Here, we show analytically that observations randomly sampled in blocks from any skewed frequency distribution with four finite moments give rise to TL. We do not claim this is the only way TL arises. We give approximate formulae for the TL parameters and their uncertainty. In computer simulations and an empirical example using basal area densities of red oak trees from Black Rock Forest, our formulae agree with the estimates obtained by least-squares regression. Our results show that the correlated sampling variation of the mean and variance of skewed distributions is statistically sufficient to explain TL under random sampling, without the intervention of any biological or behavioral mechanisms. This finding connects TL with the underlying distribution of population density (or other nonnegative quantity) and provides a baseline against which more complex mechanisms of TL can be compared.
Cohen, Joel E. and Xu, Meng, "Random Sampling of Skewed Distributions Implies Taylor’s Power Law of Fluctuation Scaling" (2015). Mathematics Faculty Publications. 2.
Cohen JE, Xu M (2015). Random sampling of skewed distributions implies Taylor's power law of fluctuation scaling. Proceedings of the National Academy of Sciences, U.S.A. 112 (25): 7749-7754, June 23, 2015.doi: 10.1073/pnas.1503824112.