ReviewTaylor’s ecological power law as a consequence of scale invariant exponential dispersion models
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 obeywhere 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 thenand 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.
References (77)
- et al.
Sampling distributions and sequential sampling for third instars of black Turfgrass ataenius, Ataenius spretulus (Haldeman), in control trials
Pedobiologia
(1995) Simple stochastic models and their power-law type behaviour
Theor. Pop. Biol.
(2000)A probabilistic model for the variance to mean power law in ecology
Ecol. Model.
(1995)A frequency distribution for the number of hematogenous organ metastases
J. Theor. Biol.
(2002)Spatial aggregation of the Colorado potato beetle described by an exponential dispersion model
Ecol. Model.
(2002)The scale-invariant spatial clustering of leukemia in San Francisco
J. Theor. Biol.
(1999)On the number of segregating sites in genetical models without recombination
Theor. Popul. Biol.
(1975)- et al.
Spatial distribution of the phytonematode community in Egyptian berseem clover (Trifolium alexandrinum) fields
Fund. Appl. Nematol.
(1995) - et al.
Spatial distribution of chronomid larvae (Diptera, chironomidae) in two central Florida lakes
Environ. Entomol.
(1998) - et al.
Epidemiological parameters of HIV transmission
Nature
(1988)
Variability in the abundance of animal and plant species
Nature
Emergence of scaling in random networks
Science
Fractal nature of regional myocardial blood flow heterogeneity
Circ. Res.
Methods of estimating the population of insects in the field
Biometrika
Behavioural dynamics and the negative binomial distribution
Oikos
Enumerative and binomial sampling plans for armored scale (Homoptera: Diaspididae) on kiwifruit leaves
J. Econ. Entomol.
Spatial distribution and sampling plans with fixed levels of precision for cereal aphids (Homoptera, Aphididae) infesting spring wheat
Can. Entomol.
Spatial distribution and sampling plans for Thrips palmi (Thysanoptera: Thripidae) infesting fall potato in Korea
J. Econ. Entomol.
Local dispersion of the eucalyptus leaf beetle Chrysophtharta bimaculata (Coleoptera, Chrysomelidae), and implications for forest protection
J. Appl. Ecol.
Spatial heterogeneity: evolved behaviour or mathematical artefact?
Nature
Sequential sampling for adult coccinellids in wheat
Entomol. Exp. Appl.
Fixed precision sequential sampling plans for the greenbug and bird cherry-oat aphid (Homoptera: Aphididae) in winter wheat
J. Econ. Entomol.
Metastasis results from preexisting variant cells within a malignant tumor
Science
Dispersion on a sphere
Proc. Roy. Soc. Lond. A
Scale invariance in biology: coincidence or footprint of a universal mechanism?
Biol. Rev. Camb. Philos. Soc.
Applications of fractal analysis to physiology
J. Appl. Physiol.
A sampling system for estimating population levels of the citrus thrips, Scirtothrips aurantii Faure (Thysanoptera: Thripidae), in mango orchards
Afr. Entomol.
Spatial patterns and movements in coprophagous beetles
Oikos
Mean-related stochasticity and population variability
Oikos
A statistical derivation of the significant-digit law
Stat. Sci.
Stationary time series models with exponential dispersion model margins
J. Appl. Probab.
Asymptotic behaviour of the variance function
Scand. J. Statist.
Stochastic dynamics and a power law for measles variability
Phil. Trans. Roy. Soc. Lond. Ser. B: Biol. Sci.
Cited by (68)
A wavelet-based approach to revealing the Tweedie distribution type in sparse data
2020, Physica A: Statistical Mechanics and its ApplicationsStochastic effects in mean-field population growth: The quasi-Gaussian approximation to the case of a Taylor's law-distributed substrate
2018, Physica A: Statistical Mechanics and its ApplicationsCitation Excerpt :One of the main goals of our study is to develop an analytical approximation to the community-averaged population dynamics and to determine the limits of such approximations. The statistical analysis of the parameter variability in ecological systems often shows a power law relationship between the variance and the mean of these the ecological variables, so-called Taylor’s law, see [15,16]. This fact inspired a number of recent studies, e.g. [17,18] showed that the power law scaling emerges as a result of summation of the multiplicatively growing components, each of those is weighted according to its relative input into the total mixture.
Environmental variability and density dependence in the temporal Taylor's law
2018, Ecological ModellingFluctuation scaling of color variability in automotive metallic add-on parts
2017, Progress in Organic CoatingsCitation Excerpt :Differences in ϵ between L* and the corresponding values in the (a*, b*)-plane indicate stronger correlations for color-flop variations with respect to lightness flop variations due in part to the presence of interference flakes. There are many theoretical approaches the lead to temporal fluctuation scaling in complex systems (i.e., the statistics are calculated over time) but a unified framework remains unresolved [29,28,37,38,35,32,31]. Here we have analyzed time series of CIELAB color coordinates in a different way, i.e., by calculating the moments of color coordinate distributions over different aspecular angles δ and ignoring the temporal order in the production line [21].
Heterogeneous ‘proportionality constants’ – A challenge to Taylor's Power Law for temporal fluctuations in abundance
2016, Journal of Theoretical BiologyTaylor's law, via ratios, for some distributions with infinite mean
2017, Journal of Applied Probability