Elsevier

Ecological Complexity

Volume 1, Issue 3, September 2004, Pages 193-209
Ecological Complexity

Review
Taylor’s ecological power law as a consequence of scale invariant exponential dispersion models

https://doi.org/10.1016/j.ecocom.2004.05.001Get rights and content

Abstract

A power function relationship between the variance and the mean number of organisms per quadrat of habitat is commonly called Taylor’s power law. Taylor’s law also manifests with other seemingly disparate processes such as the transmission of infectious diseases, human sexual behavior, childhood leukemia, cancer metastases, blood flow heterogeneity, as well as with the genomic distributions of single nucleotide polymorphisms and gene structures. The theory of errors provides, through the scale invariant exponential dispersion models, a number of statistical distributions that are characterized by Taylor’s law. One of these models, the scale invariant Poisson gamma (PG) distribution, has its power function exponent constrained to range between 1 and 2, as observed with many ecological systems. The PG model can be interpreted such that each quadrat would contain a random (Poisson-distributed) number of clusters that, on average, would themselves contain gamma-distributed number of individuals presumably determined by stochastic birth, death and immigration processes. Scale invariant exponential dispersion models also serve as limiting distributions for a wide range of other general linearized models, a property which could explain the manifestation of the variance to mean power function in so many diverse and complex natural processes and simulations.

Introduction

Ecologists have long been fascinated with a power function relationship between the variance and mean number of organisms that reflects the spatial heterogeneity of a population within its habitat (Taylor, 1961). By this relationship, if the habitat is divided into a number of equal-sized regions or quadrats and the number of organisms Z is enumerated for each quadrat, then the variance var(Z) and the mean E(Z)number of organisms per region should obeyvar(Z)=aE(Z)p,where a and p are both constants. The relationship has come to be known as Taylor’s power law (Taylor et al., 1983) and has found application to seemingly unrelated phenomena like human sexual pairing (Anderson and May, 1988), measles epidemics (Rhodes and Anderson, 1996), the clustering of childhood leukemia (Philippe, 1999), human hematogenous cancer metastases (Kendal, 2002a), regional organ blood flow (Kendal, 2001), as well as to the genomic clustering of single nucleotide polymorphisms (SNPs) (Kendal, 2003) and gene structures (Kendal, 2004). Given such broad applicability, one might ask whether some general principle might be at the basis of all these processes. Here it will be proposed that the theory of errors, through a class of probabilistic models called the scale invariant exponential dispersion models, provides just such a unifying description. Before developing this hypothesis it would be useful to review some of the history of Taylor’s law (Taylor, 1961).

Section snippets

Taylor’s ecological power law

Taylor’s empirical relationship was based upon an initial analysis of 24 data sets, derived from the spatial distributions of viral plaques, zooplankton, worms, symphylids, mites, ticks, insects, and fish (Taylor, 1961). He proposed that the parameters derived from the power law were measures characteristic of the population being studied. The power law exponent was particularly of interest as it served as an index of aggregation: if p → 0 this implied a nearly regular distribution; for p = 1

Phenomenology of Taylor’s law

Much has been written about the range of values for the power law exponent p, and how this influences the validity of the individual models. Fig. 1 presents a compilation of the values of p from a number of published data sets (Taylor, 1961, Taylor and Taylor, 1977, Taylor et al., 1983, Downing, 1986, Slone and Croft, 1998, Persaud and Yan, 2001, Blank et al., 2000, Cho et al., 2000, Elliott et al., 2003, Clarke et al., 1997). Here the estimates for each assessment of the exponent p were

Scale invariance

Taylor’s law (Eq. (1)) possesses an inherent mathematical symmetry called scale invariance. What this means is that if the measurement scale, at which the mean and variance are assessed, is changed by a factor c thenacE(Z)p=(acp)E(Z)p,and the transformed relationship thus retains the form of a power function with exponent p. Indeed, one can show that Taylor’s law is the only scale invariant relationship possible between the variance and the mean (Jørgensen, 1997).

We know that scale invariance

Exponential dispersion models

Rather than to attempt to explain the manifestations of Taylor’s law by mechanisms specific to each individual situation, it would seem desirable to find some model that might be applicable to all. Possibly the theory of errors might provide such a model. One might stipulate that the model be as general as possible. Nelder and Wedderburn’s (1972) generalized linear models provide a means to analyze a wide variety of data, and they serve as the basis for an even more general group of statistical

A population dynamic model

We know that the PG model represents the scale invariant sum of a random (Poisson-distributed) number of gamma distributions. One might then postulate that the quadrats used to enumerate a particular species within its habitat should each contain a Poisson distributed number of cluster sites. This assumption would seem quite plausible since the Poisson distribution is the usual model for such randomness, and each cluster site might be considered to represent the position of a single individual

Convergence theorems

One well-known convergence theorem, The central limit theorem, provides insight into why the Gaussian distribution is an appropriate model for the error distributions associated with many different biological and physical systems. This theorem stipulates that the sum of multiple identical and independently distributed random variables (with finite means and variances) will converge to a Gaussian form. Quite remarkably, a Gaussian distribution should result regardless of shape of the summed

Conclusions

The Gaussian distribution provides an accepted description for statistical errors associated with complex processes, and the central limit theorem provides us with the justification for why this should be so. Gauss developed his theory of errors for the specific needs of astronomers and surveyors, but in doing so he ignored the topological framework of the measurement systems (Fisher, 1953). With larger statistical errors this topology needs to be taken into account. Exponential dispersion

Acknowledgements

The author acknowledges the support provided by the Beattie Library and the Department of Radiation Oncology, both of the Ottawa Regional Cancer Centre.

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