Taylor's Power Law

Taylor's Power Law (also known in statistical physics as fluctuation scaling) is a ubiquitous empirical scaling law in population ecology and macroecology stating that the sample variance (s²) of the abundance or population density of a species across space or time is proportional to a fractional power of its sample mean density (m). First formulated in 1961 by British entomologist Lionel Roy Taylor (1924–2007) at the Rothamsted Experimental Station and tested across hundreds of taxa by L. R. Taylor, I. P. Woiwod, and J. N. Perry (1978, 1980), the law demonstrates that biological populations rarely distribute themselves uniformly or via independent Poisson randomness. Instead, living organisms cluster into spatial and temporal aggregations whose variance expands predictably as average population density rises toward carrying capacity.

Tested across more than 400 species ranging from viruses, soil bacteria, and marine phytoplankton to agricultural insect pests, parasitic helminths, forest trees, and pelagic fishes, Taylor's Power Law spans 30 orders of magnitude in variance and 15 orders of magnitude in mean abundance. Beyond linking behavioral aggregation, metapopulation demography, and stochastic forcing, the law provides a mathematical foundation in pest management, fisheries stock assessment, and disease ecology for sample-size design and variance stabilization.

Suppose an ecologist measures the population density or count X of a focal species across a collection of N quadrats, traps, or time intervals, repeated across multiple regions, habitats, or census dates so that each set of samples yields a sample mean m and a sample variance s²:

m = (1 / N) ∑ X_i ,     s² = [ 1 / (N − 1) ] ∑ (X_i − m)²

Taylor (1961) demonstrated that across sets of samples varying in mean abundance m, the variance s² obeys the power-law equation:

s² = a · m^b

Taking the base-10 or natural logarithm of both sides transforms the power law into a linear regression relationship on a log-log coordinate plane:

log₁₀(s²) = log₁₀(a) + b · log₁₀(m)

where a (a > 0) is a sampling and scaling proportionality factor whose magnitude depends on the physical size of the sampling unit (quadrat area, soil core volume, or trap duration) and baseline environmental heterogeneity, and b is the dimensionless Taylor scaling exponent (the slope of log₁₀(s²) versus log₁₀(m)), which serves as an intrinsic index of spatial or temporal aggregation (dispersion).

Four Operational Variants of Taylor's Power Law

As systematized by Joel E. Cohen and colleagues (2012, 2013), ecologists distinguish four operational variants of Taylor's Power Law depending on how observations are grouped: (1) Spatial Taylor's Power Law, where each (m_t, s_t²) point is computed across N spatial sites at time t and fitted across census times t = 1, ..., T; (2) Temporal Taylor's Power Law, where each (m_i, s_i²) point is computed across a time series of T censuses at station i and fitted across stations i = 1, ..., N; (3) Community (Interspecific) Taylor's Power Law, where each point represents a different species s within a community; and (4) Food-Web Taylor's Power Law, computed across trophic guilds or host-parasite burden distributions within a food web.

Because the variance-to-mean ratio (s² / m = a · m^b−1) changes systematically with mean density whenever b ≠ 1, the numerical value of the Taylor exponent b diagnoses the underlying dispersion regime of a population:

In their survey of 382 animal and plant taxa (including 102 species of Lepidoptera and Homoptera across the Rothamsted Insect Survey, plus birds, fishes, and plankton), Taylor, Woiwod, and Perry (1978, 1980) found that over 85% of natural populations exhibit exponents between 1 < b < 2, with a mean of b ≈ 1.45 to 1.60.

1. Taylor's Behavioral Δ-Model vs. Stochastic Birth-Death Processes

L. R. Taylor and R. A. J. Taylor (1977) initially proposed that the exponent b is a species-specific evolutionary trait governed by a behavioral balance between congregation (gregarious attraction toward conspecifics) and migration (dispersal away from crowded patches, compare the ideal free distribution). However, in 1982, Roy M. Anderson, D. M. Gordon, M. J. Crawley, and Michael P. Hassell demonstrated mathematically that Taylor's Power Law with 1 < b < 2 emerges naturally in both mobile animals and sessile plants from stochastic birth-death-immigration-emigration (BDIE) Markov processes. When subpopulations occupy a heterogeneous landscape, demographic stochasticity (generating s² ∝ m¹ at low densities) interacts with environmental multiplicative variance (generating s² ∝ m² at high densities), producing an intermediate log-log slope b between 1.3 and 1.9.

2. Tweedie Dispersion Distributions and Multiplicative Growth

In probability theory, Byron Jørgensen (1987, 1997) proved that Taylor's Power Law is the defining variance function of the Tweedie family of dispersion distributions, Var(Y) = φ · E[Y]^p. For 1 < p < 2, the resulting distribution is a compound Poisson-Gamma distribution—the sum of a Poisson-distributed number of clusters whose sizes follow a continuous Gamma distribution. Furthermore, Joel E. Cohen, Meng Xu, and William S. F. Schuster (2012, 2013) proved using random matrix products (Lewontin-Cohen multiplicative models) that spatial synchrony across subpopulations (the Moran effect) determines whether b approaches or exceeds 2.

1. Minimum Sample Size and Variance-Stabilizing Transformations

When an entomologist or fisheries biologist censuses a stock with mean density m and standard error SE = √(s² / n), estimating the population mean within a fixed relative precision D = SE / m (typically D = 0.10 or 0.25) requires the optimal sample size (n_opt):

n_opt = ( s / (D · m) )² = a · m^b−2 / D²

Because b < 2 for almost all agricultural insect pests, the required number of field samples n_opt decreases as pest infestation density m rises, forming the mathematical basis of sequential pest sampling plans (Green, 1970; Wilson & Room, 1983). Moreover, Taylor (1961) proved that transforming raw count data X via the power transformation Y = X^{1 − b/2} (or Y = log(X + 1) when b = 2) decouples the variance from the mean for parametric analysis of variance (yielding √X for Poisson b = 1, and X^0.25 for b = 1.5).

2. Macroparasite Aggregation in Disease Ecology

In disease ecology, Roy M. Anderson and Robert M. May (1978, 1985) applied Taylor's Power Law to the distribution of macroparasites (nematodes, schistosomes, and ticks) across vertebrate hosts. Across wild and human host populations, parasite burden variance scales with mean worm burden at 1.5 < b < 2.1, reflecting the "20/80 rule" in which roughly 20% of host individuals harbor 80% of the total parasite population. Incorporating the Taylor aggregation relationship into the negative binomial clumping parameter (k = m² / (s² − m)) allows epidemiologists to model density-dependent parasite regulation and target mass drug administration (MDA) campaigns.

1. Taylor, L. R. (1961). Aggregation, variance and the mean. Nature, 189(4766), 732–735. https://doi.org/10.1038/189732a0
2. Taylor, L. R., Woiwod, I. P., & Perry, J. N. (1978). The density-dependence of spatial behaviour and the rarity of randomness. Journal of Animal Ecology, 47(2), 383–406. https://doi.org/10.2307/3790
3. Anderson, R. M., Gordon, D. M., Crawley, M. J., & Hassell, M. P. (1982). Variability in the abundance of animal and plant species. Nature, 296(5854), 245–248. https://doi.org/10.1038/296245a0
4. Cohen, J. E., Xu, M., & Schuster, W. S. F. (2012). Allometric scaling of population variance with mean body size is predicted from Taylor's law and density-mass allometry. Proceedings of the National Academy of Sciences, 109(39), 15829–15834. https://doi.org/10.1073/pnas.1212883109
5. Eisler, Z., Bartos, I., & Kertész, J. (2008). Fluctuation scaling in complex systems: Taylor's law and beyond. Advances in Physics, 57(1), 89–142. https://doi.org/10.1080/00018730801893043