Stochastic population dynamics in a Markovian environment implies Taylor’s power law of fluctuation scaling

doi:10.1016/j.tpb.2014.01.001

Theoretical Population Biology

Volume 93, May 2014, Pages 30-37

https://doi.org/10.1016/j.tpb.2014.01.001 Get rights and content

Abstract

Taylor’s power law of fluctuation scaling (TL) states that for population density, population abundance, biomass density, biomass abundance, cell mass, protein copy number, or any other nonnegative-valued random variable in which the mean and the variance are positive, $variance = a {(mean)}^{b}, a > 0$ , or equivalently log $variance = log a + b \times log$ mean. Many empirical examples and practical applications of TL are known, but understanding of TL’s origins and interpretations remains incomplete. We show here that, as time becomes large, TL arises from multiplicative population growth in which successive random factors are chosen by a Markov chain. We give exact formulas for $a$ and $b$ in terms of the Markov transition matrix and the values of the successive multiplicative factors. In this model, the mean and variance asymptotically increase exponentially if and only if $b > 2$ and asymptotically decrease exponentially if and only if $b < 2$ .

Introduction

Fluctuation scaling is a name popular among physicists for a lawful relationship between the mean and variance of any random variable when the mean and variance are functions of some parameter. Among statisticians, such a relationship is often called a variance function. In population biology and ecology, Taylor’s power law of fluctuation scaling (Taylor, 1961, Taylor, 1984) states that when the mean and the variance exist and are positive functions of some parameter, they are related by a power law: $variance = a {(mean)}^{b}, a > 0$ , or equivalently $log variance = log a + b \times log mean$ .

Taylor’s law (TL) began with empirical observations of insect population densities and was verified in hundreds of biological species (Eisler et al., 2008) including, recently, bacteria (Ramsayer et al., 2011, Kaltz et al., 2012), trees (Cohen et al., 2012, Cohen et al., 2013a), and humans (Cohen et al., 2013b). TL is one of the most widely verified empirical relationships in ecology. TL has also been confirmed for cell populations within specific organs (Azevedo and Leroi, 2001), stem cell populations (Klein and Simons, 2011), counts of single nucleotide polymorphisms and genes (Kendal and Jørgensen, 2011), cases of measles and whooping cough (Keeling and Grenfell, 1999), the mass of single-celled organisms of different species (Giometto et al., 2013), and in diverse other fields (for additional references, see review by Eisler et al., 2008), including cancer metastases, single nucleotide polymorphisms and genes on chromosomes, and non-biological measurements such as precipitation, packet switching on the Internet, stock market trading, and number theory. TL has practical applications in the design of sampling plans for the control of insect pests (soybeans: Kogan et al., 1974, Bechinski and Pedigo, 1981; cotton: Wilson et al., 1989; glasshouse roses: Park and Cho, 2004).

There is little consensus about why TL is so widely observed and how its estimated parameters should be interpreted. The theoretical analysis of probability distributions in which the variance is a power-law function of the mean preceded TL (Tweedie, 1946, Tweedie, 1947) (in other words, Taylor did not invent Taylor’s law) and TL has been much studied theoretically with or without recognition of its empirical roots in ecology (e.g., Anderson et al., 1982, Tweedie, 1984, Perry and Taylor, 1985, Gillis et al., 1986, Jørgensen, 1987, Kemp, 1987, Perry, 1988, Lepš, 1993, Jørgensen, 1997, Keeling, 2000, Azevedo and Leroi, 2001, Kilpatrick and Ives, 2003, Kendal, 2004, Ballantyne and Kerkhoff, 2007, Eisler et al., 2008, Engen et al., 2008, Kendal and Jørgensen, 2011, Cohen et al., 2013a). Davidian and Carroll (1987) and Wang and Zhao (2007) emphasized the importance of modeling correctly how the variance is related to the mean if one desires statistical efficiency in estimating the mean. They considered multiple variance functions including TL. But they did not identify a power-law variance function with TL or discuss models that might explain the origin of these variance functions.

Cohen et al. (2013a) showed that the Lewontin and Cohen (1969) (no relation to the present author) stochastic multiplicative population model (a geometric random walk with independently and identically distributed [i.i.d.] multiplicative increments) implies TL. Cohen et al. (2013a) calculated $log a$ and $b$ explicitly. Here we consider a more general model in which the factors that multiply the population density at each time step are history-dependent, not independent as in the Lewontin–Cohen model. We show that a multiplicative model of change in a Markovian environment leads to TL in the limit of large time, and we calculate $log a$ and $b$ explicitly.

Section snippets

Taylor’s law

Let a family of nonnegative random variables $N (t)$ be parameterized by $t \in Θ$ , where $Θ$ is an index set. Assume that, for all $t \in Θ$ , the mean $E (N (t))$ and the variance $V ar (N (t))$ are finite and positive, so $log V ar (N (t))$ and $log E (N (t))$ are well defined. We may think of $N (t)$ as population density at time $t$ .

Definition

TL applies to $N (t)$ exactly for all $t \in Θ$ if and only if there exist real constants $a > 0$ and $b$ such that, for all $t \in Θ, V ar (N (t)) = a {(E (N (t)))}^{b}$ . Equivalently, TL applies to $N (t)$ exactly for all $t \in Θ$ if and

Scalar discrete-time Markovian multiplicative growth

Assume $N (0)$ is a fixed positive number. Suppose that $N (t) = A (t - 1) A (t - 2) \dots A (0) N (0), t = 0, 1, 2, \dots .$

Then $A (t - 1) = N (t) / N (t - 1), t = 1, 2, \dots$ represents the random factor of change from time $t - 1$ to time $t$ . Assume that each value of $A (t)$ is taken from a finite set of positive numbers ${d_{1}, \dots, d_{s}}, s > 1$ , at least two of which are distinct. Intuitively, $s$ is the number of states of the environment. By assumption, each state of the environment determines a multiplicative factor of change: if $A (t - 1) = d_{i}$ , then $N (t) = d_{i} N (t - 1)$ ,

Branching processes and birth-and-death processes

Taylor’s power law of fluctuation applies asymptotically to other Markovian population processes in addition to the example just studied.

First, the discrete-generation Galton–Watson branching process (Bartlett, 1955, 1966, Section 2.3, Eq. (3), p. 40 in 1955, p. 42 in 1966) assumes that each individual of the $t$ th generation independently has a stochastically distributed number of offspring in the next generation. If the initial number of individuals is $m_{0} > 0$ , and the mean and variance of the

Multiple models lead to Taylor’s law

A wide range of models can yield TL exactly or in the limit of large time. For example, a deterministic model of exponential clonal growth (Cohen, 2013a, Cohen et al., 2013b), the Galton–Watson branching process (with $v = 0$ or $m > 1$ ), and the birth and death process (with $λ > μ$ ) all converge (for large time) to TL with $b = 2$ . The gamma distribution satisfies TL exactly with $b = 2$ . Hence $b = 2$ in TL need not indicate deterministic population growth. Both the model of Lewontin and Cohen (1969) with i.i.d.

Acknowledgments

I thank Lee Altenberg (as colleague and self-identified reviewer), Roy Malka, Michael Plank, Shripad Tuljapurkar, Meng Xu, and an anonymous reviewer for very helpful comments, Priscilla K. Rogerson for assistance, Michael Plank and the family of William T. Golden for hospitality during this work, and the US National Science Foundation grants EF-1038337 and DMS-1225529 and the Marsden Fund of the Royal Society of New Zealand (08-UOC-034) for partial support.

References (56)

Lee Altenberg
A sharpened condition for strict log-convexity of the spectral radius via the bipartite graph
Linear Algebra Appl.
(2013)
J.E. Cohen
Taylor’s power law of fluctuation scaling and the growth-rate theorem
Theor. Popul. Biol.
(2013)
D.M. Gillis et al.
Taylor’s power law as a consequence of Fretwell’s ideal free distribution
J. Theoret. Biol.
(1986)
M.J. Keeling
Simple stochastic models and their power-law type behavior
Theor. Popul. Biol.
(2000)
W.S. Kendal
Taylor’s ecological power law as a consequence of scale invariant exponential dispersion models
Ecol. Complex.
(2004)
S.D. Tuljapurkar
Population dynamics in variable environments. II. Correlated environments, sensitivity analysis and dynamics
Theor. Popul. Biol.
(1982)
S.D. Tuljapurkar et al.
Population dynamics in variable environments. I. Long run growth rates and extinction
Theor. Popul. Biol.
(1980)
R.M. Anderson et al.
Variability in the abundance of animal and plant species
Nature
(1982)
R.B.R. Azevedo et al.
A power law for cells
Proc. Natl. Acad. Sci.
(2001)
F. Ballantyne et al.
The observed range for temporal mean–variance scaling exponents can be explained by reproductive correlation
Oikos
(2007)

M.S. Bartlett

Introduction to Stochastic Processes with Special Reference to Methods and Applications

(1955,1966)

E.J. Bechinski et al.

Population dispersion and development of sampling plans for Orius insidiosus and Nabis spp. in soybeans

Environ. Entomol.

(1981)

B.H. Bharucha

On the stability of randomly varying systems

H. Caswell

Matrix Population Models: Construction, Analysis and Interpretation

(2001)

J.E. Cohen

Ergodicity of age structure in populations with Markovian vital rates, I: countable states

J. Amer. Statist. Assoc.

(1976)

J.E. Cohen

Ergodicity of age structure in populations with Markovian vital rates, II: general states

Adv. Appl. Probab.

(1977)

J.E. Cohen

Ergodicity of age structure in populations with Markovian vital rates, III: finite-state moments and growth rates; an illustration

Adv. Appl. Probab.

(1977)

J.E. Cohen

Taylor’s law and abrupt biotic change in a smoothly changing environment

Theor. Ecol.

(2013)

J.E. Cohen

Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices

Proc. Amer. Math. Soc.

(2014)

J.E. Cohen et al.

Allometric scaling of population variance with mean body size is predicted from Taylor’s law and density-mass allometry

Proc. Natl. Acad. Sci.

(2012)

J.E. Cohen et al.

Taylor’s law applies to spatial variation in a human population

Genus

(2013)

J.E. Cohen et al.

Stochastic multiplicative population growth predicts and interprets Taylor’s power law of fluctuation scaling

Proc. R. Soc. Ser. B

(2013)

M. Davidian et al.

Variance function estimation

J. Amer. Statist. Assoc.

(1987)

Z. Eisler et al.

Fluctuation scaling in complex systems: Taylor’s law and beyond

Adv. Phys.

(2008)

S. Engen et al.

A general model for analyzing Taylor’s spatial scaling laws

Ecology

(2008)

H. Furstenberg et al.

Products of random matrices

Ann. Math. Statist.

(1960)

F.R. Gantmacher

Theory of Matrices

(1960)

Cited by (37)

Application of genetic algorithm to land use optimization for non-point source pollution control based on CLUE-S and SWAT
2018, Journal of Hydrology
The genetic algorithm (GA) was combined with the Conversion of Land Use and its Effect at Small regional extent (CLUE-S) model to obtain an optimized land use pattern for controlling non-point source (NPS) pollution. The performance of the combination was evaluated. The effect of the optimized land use pattern on the NPS pollution control was estimated by the Soil and Water Assessment Tool (SWAT) model and an assistant map was drawn to support the land use plan for the future. The Xiangxi River watershed was selected as the study area. Two scenarios were used to simulate the land use change. Under the historical trend scenario (Markov chain prediction), the forest area decreased by 2035.06 ha, and was mainly converted into paddy and dryland area. In contrast, under the optimized scenario (genetic algorithm (GA) prediction), up to 3370 ha of dryland area was converted into forest area. Spatially, the conversion of paddy and dryland into forest occurred mainly in the northwest and southeast of the watershed, where the slope land occupied a large proportion. The organic and inorganic phosphorus loads decreased by 3.6% and 3.7%, respectively, in the optimized scenario compared to those in the historical trend scenario. GA showed a better performance in optimized land use prediction. A comparison of the land use patterns in 2010 under the real situation and in 2020 under the optimized situation showed that Shennongjia and Shuiyuesi should convert 1201.76 ha and 1115.33 ha of dryland into forest areas, respectively, which represented the greatest changes in all regions in the watershed. The results of this study indicated that GA and the CLUE-S model can be used to optimize the land use patterns in the future and that SWAT can be used to evaluate the effect of land use optimization on non-point source pollution control. These methods may provide support for land use plan of an area.
Heterogeneous ‘proportionality constants’ – A challenge to Taylor's Power Law for temporal fluctuations in abundance
2016, Journal of Theoretical Biology
Citation Excerpt :
This many-to-one mapping further implies that b may be a poor correlate of the ecological and life-historical processes that affect population dynamics; even when sampling issues do not mask the underlying dynamics (Kalyuzhny et al., 2014; Giometto et al., 2015). The interpretability of b has been questioned previously with the demonstration that different models can give rise to TL of similar exponents (Cohen, 2014). A second, related implication is that negative covariance between ln(a) and ln(M) is sufficient to generate values of b<2 (Fig. 2A, Eq. (1)).
Taylor's Power Law for the temporal fluctuation in population size (TL) posits that the variance in abundance scales according to aM ^b; where M is the mean abundance and a and b are the ‘proportionality’ and ‘scaling’ coefficients. As one of the few empirical rules in population ecology, TL has attracted substantial theoretical and empirical attention. Much of this attention focused on the scaling coefficient; particularly its ubiquitous deviation from the null value of 2. Here we present a line of reasoning that challenges the power-law interpretation of the empirical log-linear relationship between the mean and variance of population size. At the core of our reasoning is the proposition that populations vary not only with respect to M but also with respect to a; which leaves the log-linear relationship intact but forfeits its power-law interpretation. Using the stochastic logistic-growth model as an example, we show that ignoring among-population variation in a is akin to ignoring the variation in the intrinsic rate of growth (r). Accordingly, we show that the slope of the log-linear relationship (b) is a function of the among-population (co)variation in r and the carrying-capacity. We further demonstrate that local environmental stochasticity is sufficient to generate the full range of observed values of b, and that b can in fact be insensitive to substantial differences in the balance between variance-generating and stabilizing processes.
Waveform Proportionality and Taylor's Law Induced by Synchronization of Periodic and Chaotic Oscillators
2023, arXiv
Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models
2023, Economies
COVID-19 cases and deaths in the United States follow Taylor’s law for heavy-tailed distributions with infinite variance
2022, Proceedings of the National Academy of Sciences of the United States of America
Taylor’s Law for Some Infinitely Divisible Probability Distributions from Population Models
2022, Journal of Statistical Physics

View all citing articles on Scopus

View full text

Stochastic population dynamics in a Markovian environment implies Taylor’s power law of fluctuation scaling

Abstract

Introduction

Section snippets

Taylor’s law

Scalar discrete-time Markovian multiplicative growth

Branching processes and birth-and-death processes

Multiple models lead to Taylor’s law

Acknowledgments

Linear Algebra Appl.

Theor. Popul. Biol.

J. Theoret. Biol.

Theor. Popul. Biol.

Ecol. Complex.

Theor. Popul. Biol.

Theor. Popul. Biol.

Variability in the abundance of animal and plant species

Nature

A power law for cells

Proc. Natl. Acad. Sci.

The observed range for temporal mean–variance scaling exponents can be explained by reproductive correlation

Oikos

Introduction to Stochastic Processes with Special Reference to Methods and Applications

Population dispersion and development of sampling plans for Orius insidiosus and Nabis spp. in soybeans

Environ. Entomol.

On the stability of randomly varying systems

Matrix Population Models: Construction, Analysis and Interpretation

Ergodicity of age structure in populations with Markovian vital rates, I: countable states

J. Amer. Statist. Assoc.

Ergodicity of age structure in populations with Markovian vital rates, II: general states

Adv. Appl. Probab.

Ergodicity of age structure in populations with Markovian vital rates, III: finite-state moments and growth rates; an illustration

Adv. Appl. Probab.

Taylor’s law and abrupt biotic change in a smoothly changing environment

Theor. Ecol.

Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices

Proc. Amer. Math. Soc.

Allometric scaling of population variance with mean body size is predicted from Taylor’s law and density-mass allometry

Proc. Natl. Acad. Sci.

Taylor’s law applies to spatial variation in a human population

Genus

Stochastic multiplicative population growth predicts and interprets Taylor’s power law of fluctuation scaling

Proc. R. Soc. Ser. B

Variance function estimation

J. Amer. Statist. Assoc.

Fluctuation scaling in complex systems: Taylor’s law and beyond

Adv. Phys.

A general model for analyzing Taylor’s spatial scaling laws

Ecology

Products of random matrices

Ann. Math. Statist.

Theory of Matrices