Theoretical Ecology

, Volume 7, Issue 1, pp 77–86

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



Taylor's law (TL), widely verified in empirical ecology, states that the variance of population density approximates a power function of the mean population density, with exponent denoted b. A model of multiplicative increments in population density, where the increments are determined by a Markovian environment, predicts TL with an explicit formula for b. We give a simple theoretical example where, unexpectedly, smooth changes in environmental autocorrelation lead to an abrupt, infinite discontinuity in b. As the daily probability of change in environmental state increases from 0 to 1, b rises from 2 slowly at first, then explodes to +∞ when the population becomes critical, drops to -∞, and rises again to 2. In this model, an exponent b of large magnitude (positive or negative) signals the proximity of a population's criticality and of a singularity in b. A comparable real-world singularity in b could adversely affect fisheries, forestry, agriculture, conservation, and public health.


Markovian environment Regime change Power spectra Population density Fluctuation scaling 


Earth's climate (Broecker 2003) and biota (Barnosky et al. 2012) have experienced what Broecker (2003, p. 1522) described as "large and abrupt changes so richly recorded in ice and sediment." Sometimes, an abrupt biotic change resulted from an abrupt environmental change, such as an asteroid impact or volcanic eruption. Sometimes, an abrupt biotic change was a non-linear response to a smooth environmental change (Scheffer et al. 2001). For example (deMenocal et al. 2000), the vegetation cover of much of the Sahara disappeared abruptly ∼5,700–5,200 years before the present, at the end of the so-called African Human Period (AHP), though the transition was more gradual in some places. The AHP ended when summer monsoons dropped off as a result of a smooth decline in summer insolation, a decline driven by the precession (wobble) of the Earth's axis. The abruptness, timing, magnitude, and spatial localization of the biotic changes in different parts of northern Africa remain under investigation (deMenocal and Tierney 2012).

Some dynamical systems exhibit recognizable precursors of dramatic shifts in state or behavior (Scheffer et al. 2009; D'Odorico et al. 2013) and some do not (Hastings and Wysham 2010). In a simplified model of the thermohaline circulation system in the North Atlantic, Kleinen (2005) showed that a change in the flux of fresh water into the ocean could push the thermohaline circulation to a saddle-node bifurcation. In this model, as the freshwater flux approaches its critical value at the bifurcation point, the power spectrum of the circulation strength shifts upward. Consequently, the total variance of circulation strength (the area under the spectral density function) increases, mainly at low frequencies (Kleinen 2005, p. 18, his Fig. 2.2). In this theoretical example, as the freshwater flux approaches its critical value, the system gives premonitory signals of increasing variance in the circulation strength. The variance increases because of increased amplitudes of oscillations of very low frequencies. Such increases in the amplitudes of low-frequency oscillations would be very difficult to detect from short-term observations. In the real world, it could be difficult to identify with assurance the precursors of dramatic shifts in the thermohaline circulation.

The causes, necessary and sufficient conditions, leading indicators, and consequences of abrupt biotic responses to smooth environmental changes remain insufficiently understood (Scheffer and Carpenter 2003; Russell et al. 2012) and new ideas continue to be proposed (Lorenzo and Ohman 2013; Doney and Sailley 2013; D'Odorico et al. 2013). In this context, we investigated theoretically the effect on Taylor's law (TL), one of the most widely verified empirical patterns in ecology, of a gradual, continuous increase in the daily probability of a change in the environment. This change, we show, is equivalent to an increase in the slope of the environmental power spectral density from the negative slope of a reddened power spectrum (dominated by low frequencies of environmental change) to the zero slope of a white (flat) power spectrum and then the positive slope of a blue power spectrum (dominated by high frequencies of environmental change).

We show here, apparently for the first time, that a dramatic and abrupt shift (a singularity) can occur in the slope b of the log-linear form of TL (which is also the exponent b of the power-law form of TL) as a consequence of a sufficiently large, gradual, and continuous increase or decrease in the temporal autocorrelation of environmental states. The singularity occurs when the population becomes critical (the average population density neither declines towards extinction nor explodes in the long run). Quantifiable precursors of population criticality and of a singularity in b are values of b of large magnitude, positive or negative.

For the sake of transparency, and to avoid an extensive mathematical apparatus that underlies the result presented here, we focus on a highly idealized, simplified abstraction of real ecological dynamics, and on a simple example of the abstraction. This strategy of exploration and exposition identifies phenomena not previously suspected, and is therefore useful. But this example is not intended to replace subsequent, more realistic, more complex modeling, data analysis, and quantitative testing. By analogy, understanding a deterministic model of exponential population growth is useful, even though no biological population can grow exponentially forever.

This work forms part of a large and long-flowing stream of research on population dynamics in randomly varying environments, which includes the early Pimm and Redfearn (1988) and Lawton (1988) and the recent Schreiber et al. (2011) and Evans et al. (2013).

Material and methods

  1. (a)

    Taylor's power law of fluctuation scaling


Taylor's law presumes the following framework of observations. Suppose we have a large number O of observations of the density (number of individuals per unit of area or volume) of a set of populations. Suppose these observations are partitioned into B blocks. (A block is simply a subset of the observations, but the term "block" is conventional in the statistical theory of experimental design.) Suppose the blocks are labeled j = 1, …, B and the number of observations in block j is nj, so that n1 + …+ nB = O. For every block j, we calculate the sample mean mj and the sample variance vj of the observations in block j. Since there are B blocks, we have B pairs (mj, vj). Suppose none of the means and none of the variances is zero, and all are finite. Then we may plot these B points on log–log coordinates.

Taylor (1961, 1984) and colleagues (e.g. Taylor et al. 1980) observed that, in many species, the logarithm of the sample variance was an approximately linear function of the logarithm of the sample mean: for some a > 0, log(vj) ≈ log(a) + b × log(mj). This relationship, equivalent to the power law, vja(mj)b, was previously observed in the spatial distribution of Ribes by Fracker and Brischle (1944), but because Taylor et al. emphasized its generality and importance, it became known as Taylor's law (henceforth TL). Most empirical tests of TL have ignored the sampling error in log(mj) when fitting a linear regression by the usual least-squares formulas. For large enough numbers nj of observations per block and a sufficiently wide range of values of log(mj), the error introduced by this procedure may not be serious.

TL has multiple forms depending on how the blocks and samples within each block are defined. For example, suppose the observations are organized in a rectangular table with R rows and C columns, and suppose each row is a sampling site or quadrat and each column is a census date. In a spatial TL, each block is one census date, i.e., the observations in that block are the elements in one column of the data matrix, and the mean and variance of density are calculated over different sites (rows) at a given time; here, B = C. Alternatively, with the same array of data, in a temporal TL, each row is a block, and the mean and variance of density are calculated over different censuses (columns) at a given site; here, B = R. In a hierarchical spatial TL, each quadrat is subdivided into smaller subplots, and the mean and variance of density are calculated over the subplots within each quadrat. In a hierarchical temporal TL, each block is a longer time interval (e.g., decade) and each observation within that block refers to density observed in a shorter time interval (e.g., a year) within each decade. There are other forms of TL as well.

In its various forms, TL has been verified in hundreds of species (Eisler et al. 2008) including, recently, bacteria in laboratory experiments (Ramsayer et al. 2012; Kaltz et al. 2012) and trees in a temperate forest (Cohen et al. 2012a, 2013). TL also describes the relation of variance to mean in populations of stem cells (Klein and Simons 2011), stock market trading, precipitation, packet switching on the Internet, measles cases, and the occurrence of single nucleotide polymorphisms (see the review by Eisler et al. 2008).

Here, we focus on a spatial TL. In theoretical models of TL, the sample mean and sample variance are replaced by the population mean and population variance, respectively. A spatial TL applies exactly to the population moments, by definition, if and only if there exist real constants a > 0 and b such that the population mean E(N(t)) and the population variance Var(N(t)) of the density N(t) satisfy, at every time t,
$$ \log \kern0.5em Var\left(N(t)\right)\hbox{--} b\kern0.5em \log \kern0.5em E\left(N(t)\right)= \log \kern0.5em a. $$
Here, the approximate equality (≈) in the empirical form of TL is replaced by exact equality (=). We emphasize that E(N(t)) in Eq. (1) pertains to an average over space (ensemble average) at a given time t, and Var(N(t)) pertains to a variance over space at a given time t, not to an average or variance over time. TL applies in the limit as t gets large, by definition, if and only if there exist real constants a > 0 and b such that
$$ { \lim}_t{}_{\to \infty}\left[ \log \kern0.5em Var\left(N(t)\right)\hbox{--} b\kern0.5em \log \kern0.5em E\left(N(t)\right)\right]= \log \kern0.5em a. $$

If TL holds for any base of logarithms, it holds for all choices. For the following mathematical analysis, log = loge. The value of b remains the same for any choice of the base of logarithms and regardless of the units in which N(t) is measured (e.g., cells per cubic millimeter or cells per cubic kilometer), though the value of log a changes. Hence attention here, and in much work on TL, focuses on the slope b rather than on the intercept.

  1. (b)



In this example, we first create a model of an environment and then create a model of a population whose dynamics depend on this environment. This population has no stage-, age-, sex-, or other structure; it is "single-type." For expository convenience, we select the unit of time in this model as 1 day.

In defining the model of the environment, we distinguish between "climate" (a pattern of weather) and "weather" (the conditions on a particular day). We model a climate as a sequence of transitions among different states of weather, and we model these transitions by using a finite-state, homogeneous Markov chain (the Appendix gives details). Suppose a climate has two states of weather. State 1 represents "good weather" and state 2 represents "bad weather." Assume that the probability that tomorrow's weather will differ from today's weather is λ, so the probability that tomorrow's weather will be the same as today's is 1 - λ, where 0 ≤ λ ≤ 1. Assume that initially (at time 0), good weather occurs with probability π1 = 1/2 and bad weather occurs with probability π2 = 1/2. We summarize the transition probabilities P(λ) and the initial probability distribution π of the climate in matrix form:
$$ P\left(\lambda \right)=\left(\begin{array}{cc}\hfill 1-\lambda \hfill & \hfill \lambda \hfill \\ {}\hfill \lambda \hfill & \hfill 1-\lambda \hfill \end{array}\right),\pi =\left(\begin{array}{c}\hfill 1/2\hfill \\ {}\hfill 1/2\hfill \end{array}\right). $$

Then it is easy to check that P(λ)π = π, regardless of λ. This means that in every climate, no matter what the value of λ, π1 = 1/2 and π2 = 1/2 is the stationary probability distribution of the weather, i.e., these probabilities of good and bad weather, respectively, are the same for all times t = 0, 1, 2, … . Intuitively, that is because the transition from good to bad weather is just as likely as the reverse transition, so by symmetry, the two states of weather are equally likely at every time.

Now we add to this model of a Markovian environment a highly simplified model of multiplicative population dynamics. Assume that a population starts with some positive initial population density N(0) > 0. Since we will be interested only in long-run growth rates, the initial population density has no effect, provided it is positive, so no generality is lost if we assume N(0) = 1. Assume that in good weather, the population density increases by a factor of A(t) = d1 > 1. Assume that in bad weather, the population density falls by a factor of A(t) = d2 < 1. We chose d1 = 2, so that the population density doubles in good weather, and d2 = 1/4, so that the population is reduced to 1/4 its previous density in bad weather.

The choices d1 = 2 and d2 = 1/4 guarantee that the average multiplicative growth factor is E(A(t)) = (2 + 1/4)/2 = 9/8 > 1 but that E(log A(t)) = (ln 2 + ln(1/4))/2 = (ln 2–2 ln 2)/2 = (−ln 2)/2 < 0.

An elementary (but long) calculation shows that, regardless of the values of d1 and d2, the first-order autocorrelation of the time series {A(t)} is exactly p11(λ)–p12(λ) = 1–2λ, which is also the second eigenvalue of P(λ) (the first eigenvalue being 1). When λ ≈ 0, the autocorrelation is near 1 and the power spectrum of {A(t)} is red. The climate has long runs (sojourn times) of good weather (the population grows by a factor of A(t) = d1 > 1 on each such day), and the climate also has long sojourn times of bad weather (the population falls by a factor of d2 on each such day), and the weather changes rarely from good to bad or vice versa. When λ ≈ 1, the autocorrelation is near -1 and the power spectrum is blue. Two days in a row rarely have the same weather.

When λ = 1/2, the autocorrelation is 0, successive values of A(t) are independent, and the power spectrum is white. This special case of serial independence of successive values of A(t) is assumed in the model of Lewontin and Cohen (1969) [no relation to the present author]. The Lewontin–Cohen model cannot be used to investigate how differences in environmental autocorrelation affect TL because its environmental autocorrelation is always 0. From Eq. (3),
$$ { \lim}_t{}_{\to \infty}\left(1/t\right)\ \log \kern0.5em E\left(N(t)\right)={ \lim}_t{}_{\to \infty}\left(1/t\right)\ \left[ \log \kern0.5em E\left(N(0)\right)+ \log \kern0.5em E\left(A(0)\right)+\cdots + \log \kern0.5em E\left(A\left(t-1\right)\right)\right]= \log \kern0.5em E\left(A(0)\right)>0. $$
The mean population density (averaged over an infinite ensemble of infinitely precise simulations or bacterial dishes of infinite capacity) at time t diverges to +∞ with increasing time t, because some of those realizations will have long runs of good weather. But
$$ { \lim}_t{}_{\to \infty}\left(1/t\right)E\left( \log \kern0.5em N(t)\right)={ \lim}_t{}_{\to \infty}\left(1/t\right)\ \left[E\left( \log \kern0.5em N(0)\right)+E\left( \log \kern0.5em A(0)\right)+\cdots +E\left( \log \kern0.5em A\left(t-1\right)\right)\right]=E\left( \log \kern0.5em A(0)\right)<0, $$
so almost every realization of the process (trajectory of population density) approaches (but never reaches) 0 with increasing time t. This apparently paradoxical situation was discussed extensively by Lewontin and Cohen (1969).

For each fixed value of λ from 0 to 1, we calculated an exact, explicit formula for the log-linear TL slope b = b(λ). This method is known as "comparative statics" in mechanics and economics. It assumes λ is permanently fixed at different values in different climates and compares the TL parameter b = b(λ) in different climates. When we speak of the effects of changes in λ, we are comparing the outcomes in the limit of large time t in different climates in which λ is constant at different values. This method approximates usefully what should be expected if λ, rather than being assumed constant, changed much more slowly than (1/t) log Var(N(t)) converges to a linear function of (1/t) log E(N(t)). We summarize the results in the following section and give detail in the Appendix.

Results: a singularity in the slope of Taylor's law

In terms of sample paths (or infinitely precise simulations with no underflow and no overflow) of the population density N(t) in this model generally (under broad conditions) and in this example, the spatial mean density E(N(t)) grows or declines exponentially or is constant with increasing time t. Thus log E(N(t)) is a linear function of t and log r1(λ) is the slope of that linear function. Likewise, the spatial variance of density Var(N(t)) grows or declines exponentially or is constant with increasing time t. Thus log Var(N(t)) is a linear function of t and log r2(λ) is the slope of that linear function.

For each possible climate with a fixed probability λ of a change in the weather from 1 day to the next, Fig. 1a shows the limiting exponential growth rate per unit time of the mean density, r1(λ), and the limiting exponential growth rate per unit time of the variance of population density, r2(λ). These quantities are defined mathematically in (9) (Appendix) and are calculated explicitly there for this example. Both r1(λ) and r2(λ) are positive for all 0 ≤ λ ≤ 1, but are greater than 1 for λ near 0 and less than 1 for λ near 1. Consequently, the slopes of both log E(N(t)) and log Var(N(t)) as functions of t are positive for λ near 0 and negative for λ near 1. Since the slopes are continuous functions of λ, they must pass through 0 for some value of λ between 0 and 1. The surprising singularity happens precisely where the slope of log E(N(t)) passes through 0, as now described in further detail.
Fig. 1

a The long-run rate of growth r1 of mean population density (solid curve) and the long-run rate of growth r2 of the variance of population density (dotted curve) and b the slope b(λ) of Taylor's law (solid curves) depend strongly on the daily probability λ of change in the weather from good to bad or from bad to good. a The larger the probability λ, the smaller r1 and r2. b As λ increases, the slope b = b(λ) of TL increases slowly at first, then more rapidly, and then explodes when the mean population density becomes critical at λ = 3/5, i.e., r1(3/5) = 1 or log r1(3/5) = 0, so the denominator of Eq. (8) is 0 when λ = 3/5. When λ > 3/5, b(λ) increases rapidly through negative values and more slowly through positive values toward 2

Figure 1b shows b(λ), the limiting value at large time of the log-linear TL slope b in Eq. (2). It is derived from the two curves in Fig. 1a as the ratio of their logarithms (see Eq. (5)), b(λ) = log r2(λ)/log r1(λ). This formula shows that b(λ) is a ratio of the two slopes described in the previous paragraph. We now describe the behavior of r1(λ) and r2(λ) and b(λ) from left to right as λ increases from 0 to 1.

When λ = 0, the weather never changes so the only randomness is in the initial conditions: does day 0 have good weather or bad? In this case, the long-run rate of growth of the mean density is r1(0) = 2, meaning that, in the long run, the mean density doubles every day. Half of all realizations, or sample paths, are stuck initially and permanently with bad weather. For them, the density is multiplied by d2 = 1/4 every day. These realizations approach 0 but never vanish. The other half of all realizations have good weather initially and permanently, and the density in those realizations doubles at every time step because d1 = 2. So the mean density (over all realizations, including the half of all realizations that approach zero) also asymptotically doubles at every time step, and that mean is just half of the density in the realizations with permanent good weather. Hence the absolute difference between the mean and the minimum density (as well as the absolute difference between the mean and the maximum density) asymptotically doubles at every time step. So the variance, which is the average of the squared deviations from the mean, asymptotically quadruples with each time step.

When 0 < λ < 1, both r1(λ) and r2(λ) decrease with increasing λ (Fig. 1a). A plot of log E(N(t)) as a function of t has slope 2 when λ = 0, and this slope, log r1(λ), decreases as λ increases. When λ = 3/5, the slope of log E(N(t)) as a function of t is exactly 0. Hence, for large t, changes in log E(N(t)) are negligible compared to t, and a plot of log E(N(t)) as a function of t is asymptotically flat when λ = 3/5.

Meanwhile, a plot of log Var(N(t)) as a function of t has slope 4 when λ = 0, and this slope, log r2(λ), also decreases as λ increases. But when λ = 3/5, the slope log r2(λ) of log Var(N(t)) as a function of t is still positive, meaning that the variance of N(t) is still growing exponentially (or log Var(N(t)) is still growing linearly) while log E(N(t)) is changing negligibly compared to t. In this situation, when λ = 3/5, if you try to regress a linearly changing quantity log Var(N(t)) as a function of an essentially unchanging quantity log E(N(t)), you get an infinite slope. This explains why the slope b(λ) of TL becomes singular at λ = 3/5. As λ → 3/5 from below, b(λ) diverges to +∞ because log r1(λ), the denominator of b(λ), approaches 0, and not because the variance explodes.

In the vicinity of λ = 3/5, a very small difference in λ (which implies a very small difference in the autocorrelation) may alter b(λ) dramatically and discontinuously. A very slight deviation of λ below 3/5 makes the regression slope b(λ) extremely positive. A very slight deviation of λ above 3/5 makes the regression slope b(λ) extremely negative.

In the interval 0 ≤ λ < 3/5, the increase of b(λ) with increasing λ means that any given fractional increase in mean density is associated (in the long term) with an ever greater fractional increase in the variance of density as λ increases.

The singularity in b(λ) and the large absolute values of b(λ) show (Fig. 1b) that the relation of variance to mean population density described by TL can be exquisitely sensitive to smooth changes in the pattern of environmental variability, even when the mean and variance of density separately are well behaved and changing smoothly (Fig. 1a). The sharp singularity at λ = 3/5 looks just as sharp on a plot of log b(λ) as on the linear scale in Fig. 1b.

In climates with 3/5 < λ < 15/19, b(λ) increases rapidly from -∞ through negative values. A negative value b(λ) < 0 means that the log variance of density decreases with increasing log mean density. Then as λ increases from 15/19 toward 1, b(λ) increases slowly toward 2.

When λ = 1, the weather alternates deterministically between good and bad. Again, the only randomness is in the initial conditions: half the realizations start out with good weather, half with bad, but all realizations alternate weather daily. Therefore, over any two successive days, the mean density grows deterministically for all realizations by a factor of d1d2 = 2 × (1/4) = 1/2, so r1(λ) = limt→∞ [E(N(t))]1/t = (1/2)1/2 < 1, so log r1(λ) < 0, i.e., log E(N(t)) has a negative slope as a function of t, and the population is subcritical. In both extreme cases, λ = 0 and λ = 1, the TL slope is b(0) = b(1) = 2, i.e., the variance is proportional to (mean)2, or the coefficient of variation (standard deviation divided by mean) of population density is constant. This behavior is expected for deterministic dynamics (Ballantyne 2005; Cohen 2013a).


  1. (a)

    Principal findings


Under certain broad conditions specified in the Appendix, but not under all conditions, a simple model of multiplicative population growth in a Markovian environment implies TL in the limit of large time, and the limiting slope b of TL (Eq. 8) can be any real number, positive or negative (Cohen 2013c). The variation in population growth may be driven by environmental variation and/or demographic stochasticity. In the model, b depends on the values of the multiplicative factors in different environments (summarized in this particular example by d1 and d2) and on the sequential dependence in the occurrence of different environmental states (summarized by the transition probabilities in the matrix P(λ) in this example), even when the long-run relative frequency of different environmental states (summarized by the stationary probability vector π) is held constant.

TL has been studied theoretically from many different perspectives (e.g., Anderson et al. 1982; Tweedie 1984; Perry and Taylor 1985; Gillis et al. 1986; Jørgensen 1987; Kemp 1987; Perry 1988; 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; Fronczak and Fronczak 2010; Cohen et al. 2012b, 2013; Cohen 2013a, c). Probability distributions in which the variance is a power-law function of the mean were studied (Tweedie 1946, 1947) even before the "law" was named for Taylor. Nevertheless, there is little consensus about why TL is so widely observed and how its estimated parameters should be interpreted.

Here, in a simple example, we compared hypothetical climates with differing rates of change from good weather to bad and vice versa, holding constant the probability of good or bad weather on an average day. As the daily probability λ of a change in weather increased from 0 to a threshold value of 3/5 in this example, the slope of TL rose from b = 2 with increasing rapidity and then diverged to +∞ when the mean population density became critical (Fig. 1b), i.e., when log(r1(λ)) = 0 or r1(λ) = 1. This critical case includes, but is not limited to, constant mean population density. Variations in mean population density that are sufficiently small compared to large time are not excluded. With further increase in λ beyond the threshold 3/5, the slope of TL dropped instantaneously to -∞ and then rose, first rapidly then more slowly, approaching b = 2 from below as λ approached 1. In this example, a slope b of very large magnitude, positive or negative, was a premonitory signal of a population's approach to criticality (the dividing line between ultimate extinction and exponential growth in mean population density).

This discovery of a singularity in b resulting from smooth change in λ in a simple example was unexpected. It may stimulate and contribute to a theory of abrupt biotic change.
  1. (b)

    Generality and significance


From Fig. 1, it is clear that a singularity in b would occur in any dynamical model with three characteristics. First, the long-run exponential rate of growth r1 of mean population density and the long-run exponential rate of growth r2 of the variance of population density are both functions of some parameter θ. (In our example, θ = λ.) Second, there exist two distinct values of θ, say, θ1 and θ2, such that r1 is positive at one of these values and negative at the other, i.e., r1(θ1)r1(θ2) < 0, and r1 is continuous in the closed interval from θ1 to θ2. Then there exists at least one intermediate value, say θ*, such that r1(θ*) = 0, by the intermediate value theorem of the calculus. Third, r2(θ*) ≠ 1. Under these three conditions, b(θ) = log  r2(θ)/log  r1(θ) is singular at θ = θ*.

These conditions seem likely to be satisfied by a very wide variety of models. The condition that r1(θ) be positive at one value of θ and negative at another requires that the parameter θ be good for population growth at some value and bad for population growth at some other value. Temperature, salinity, precipitation, pH, and a host of other environmental factors all satisfy such a condition.

The example given here could be generalized immediately to models of population dynamics in Markovian environments with s > 2 environmental states, general multiplicative growth factors d1, …, ds, general transition probabilities, and an age- or stage-structured population instead of a single-type population, provided only that the three conditions above be satisfied.

The model of multiplicative population dynamics in a Markovian environment (Appendix (a)), and not only the example above, allows, as a special case, independent multiplicative increments, as assumed in the Lewontin and Cohen (1969) model. The Lewontin–Cohen model satisfies TL (Cohen et al. 2013, p. 3). In the Lewontin–Cohen model, for a given fixed positive variance in the multiplicative factors A(t), the TL slope b is less than 2 if and only if E(A(t)) < 1 (the mean population density approaches zero), and b is greater than 2 if and only if E(A(t)) > 1 (the mean population density explodes). In the present model where the successive multiplicative factors A(t) are selected by a Markov chain, under broad conditions, and not merely for the specific example given here, similar relationships hold. Specifically (Cohen 2013c), b(λ) > 2 if and only if r1(λ) > 1 and b(λ) < 2 if and only if r1(λ) < 1.

The probability λ of a change in weather from 1 day to the next in the model is reflected in the shape of the power spectrum of environmental variation. Caswell and Cohen (1995, p. 308) showed that the nth order autocovariance function of an environmental function of a Markov chain specified by Eq. (3) decays exponentially with increasing n. The spectral density function is dominated by low frequencies (is said to be "red," as the spectral density function decreases with increasing frequency) when 0 < λ < 1/2, is dominated by high frequencies (is "blue," as the spectral density function increases with increasing frequency) when 1/2 < λ < 1, and is white (the spectral density function is flat, or the same for every frequency) when λ = 1/2. Caswell and Cohen (1995, p. 303, their Fig. 1) graphed illustrative sample paths of a function of the environment with two states and λ = 1/10, 1/2, and 9/10.

The comparison of hypothetical climates with different values of λ and therefore with different spectral colors has some relevance to the real world, though there is a large gap between reality and the idealized model analyzed here. García-Carreras and Reuman (2011, p. 1046) demonstrated a statistically significant global shift towards bluer power spectral densities of mean summer temperature time series from 1911–1950 to 1951–1990. Not all regions experienced the same shift. The spectra of North and South America and Europe became bluer, those of Asia and Australasia redder, and that of Africa was not significantly changed. Had values of b in TL been estimated during 1911–1950 and 1951–1990, and had all of those values been positive (as is commonly the case today), the example analyzed here would predict that the values of b should be larger in the second period than in the first where spectra became bluer (North and South America and Europe), and should be smaller in the second period than in the first where spectra became redder (Asia and Australasia), and should not be changed much where spectra did not shift significantly (Africa). It is not clear whether the observed shifts in the power spectra would have been enough to cause a singularity in b. See also García-Carreras and Reuman (2013).

In fisheries, forestry, and agriculture, efforts to increase yields or harvests may sometimes be associated with efforts to increase mean population densities of harvested species. TL applies in all these fields (for fisheries, Cohen et al. 2012b; for forestry, Fracker and Brischle 1944; Cohen et al. 2012a, 2013; for agriculture, Smith 1938; Wilson et al. 1989). TL implies that increased mean population densities should be associated with increases (if b > 0) or decreases (if b < 0) in the variance of population density. For positive b, the greater b is, the greater the increase in variance for a given increase in mean population density. Increases in the variance of the population density of the species that produce food and fiber could increase the fluctuations in the supplies available for human consumption, increase the variability of prices of these products, and increase the probability of demographic collapse (with genetic bottlenecks) or extinction of species. Increases in the variance of the population density of the species that carry, transmit, or cause human, animal, and zoonotic infections could increase the risk of outbreaks of diseases for humans and their domestic and game animals. At extremes of population density, density-dependence (e.g., positive Allee effects when density is low, negative effects of competition when density is high) may weaken the applicability of TL.
  1. (c)

    Experimental testing and further theoretical analysis


This example's predictions could be tested experimentally using, for example, bacterial microcosms (Ramsayer et al. 2012; Kaltz et al. 2012) in incubators with differing probabilities λ of daily (or even hourly) change in temperature or other environmental factors that promote or reduce population growth.

This model has no density dependence. In future empirical work, it would be desirable to examine whether populations of the flour beetle Tribolium castaneum, which are known to be density-dependent, satisfy TL. A parallel theoretical question is whether empirically validated density-dependent models of Tribolium population dynamics such as the Larvae–Pupae–Adult family of models (Dennis et al. 1997, 2001; Reuman et al. 2006, 2008) satisfy TL. If these data and models do satisfy TL, they can be used to investigate how the TL slope b depends on the parameters of the laboratory experiments and models. If these data and models do not satisfy TL, they can be used to shed light on the conditions under which TL succeeds or fails.



The author would like to thank Lee Altenberg, Carl Boettiger, Stephen R. Carpenter, Peter B. deMenocal, Alan Hastings, Michael Hochberg, Marcel Holyoak, Oliver Kaltz, Daniel C. Reuman, Meng Xu, and anonymous referees for helpful comments, US National Science Foundation grants EF-1038337 and DMS-1225529 for support, Priscilla K. Rogerson for assistance, and Michael Plank and the family of William T. Golden for their hospitality during this work.


  1. Altenberg L (2013) A sharpened condition for strict convexity of the spectral radius via the bipartite graph. Linear Algebra Appl. Accessed 14 Feb. 2013
  2. Anderson RM, Gordon DM, Crawley MJ, Hassell MP (1982) Variability in the abundance of animal and plant species. Nature 296:245–248CrossRefGoogle Scholar
  3. Azevedo RBR, Leroi AM (2001) A power law for cells. Proc Natl Acad Sci U S A 98:5699–5704PubMedCentralPubMedCrossRefGoogle Scholar
  4. Ballantyne F IV (2005) The upper limit for the exponent of Taylor's power law is a consequence of deterministic population growth. Evol Ecol Res 7:1213–1220Google Scholar
  5. Ballantyne F IV, Kerkhoff AJ (2007) The observed range for temporal mean-variance scaling exponents can be explained by reproductive correlation. Oikos 116:174–180CrossRefGoogle Scholar
  6. Barnosky AD, Hadly EA, Bascompte J, Berlow EL, Brown JH, Fortelius M et al (2012) Approaching a state shift in Earth's biosphere. Nature 486:52–58PubMedCrossRefGoogle Scholar
  7. Broecker WS (2003) Does the trigger for abrupt climate change reside in the ocean or in the atmosphere? Science 300:1519–1522PubMedCrossRefGoogle Scholar
  8. Caswell H (2001) Matrix population models: construction, analysis and interpretation, 2nd edn. Sinauer Associates, Inc., Sunderland, MAGoogle Scholar
  9. Caswell H, Cohen JE (1995) Red, white and blue: environmental variance spectra and coexistence in metapopulations. J Theor Biol 176:301–316CrossRefGoogle Scholar
  10. Cohen JE (1976) Ergodicity of age structure in populations with Markovian vital rates, I: countable states. J Am Stat Assoc 71:335–339CrossRefGoogle Scholar
  11. Cohen JE (2013a) Taylor's power law of fluctuation scaling and the growth-rate theorem. Theor Popul Biol.
  12. Cohen JE (2013b) Cauchy inequalities for the spectral radius of products of diagonal and nonnegative matrices. Proc Am Math Soc, in press (accepted)Google Scholar
  13. Cohen JE (2013c) Multiplicative dynamics in a Markovian environment implies Taylor's power law of fluctuation scaling. SubmittedGoogle Scholar
  14. Cohen JE, Xu M, Schuster WSF (2012a) Allometric scaling of population variance with mean body size is predicted from Taylor's law and density-mass allometry. Proc Natl Acad Sci U S A 109:15829–15834. doi:10.1073/pnas.1212883109 PubMedCentralPubMedCrossRefGoogle Scholar
  15. Cohen JE, Plank MJ, Law R (2012b) Taylor's law and body size in exploited marine ecosystems. Ecol Evol. doi:10.1002/ece3.418 PubMedCentralPubMedGoogle Scholar
  16. Cohen JE, Xu M, Schuster WSF (2013) Stochastic multiplicative population growth predicts and interprets Taylor's power law of fluctuation scaling. Proc R Soc B 280:20122955. doi:10.1098/rspb.2012.2955 PubMedCrossRefGoogle Scholar
  17. D'Odorico P, Ridolfi L, Laio F (2013) Precursors of state transitions in stochastic systems with delay. Theor Ecol. doi:10.1007/s12080-013-0188-2 Google Scholar
  18. deMenocal PB, Tierney JE (2012) Green Sahara: African humid periods paced by Earth's orbital changes. Nat Educ Knowl 3(10):12Google Scholar
  19. deMenocal PB, Ortiz J, Guilderson T, Adkins J, Sarnthein M, Baker L et al (2000) Abrupt onset and termination of the African Humid Period: rapid climate responses to gradual insolation forcing. Quat Sci Rev 19:347–361CrossRefGoogle Scholar
  20. Dennis B, Desharnais RA, Cushing JM, Costantino RF (1997) Nonlinear demographic dynamics: mathematical models, statistical methods, and biological experiments. Ecol Monogr 65:261–281CrossRefGoogle Scholar
  21. Dennis B, Desharnais RA, Cushing JM, Henson SM, Costantino RF (2001) Estimating chaos and complex dynamics in an insect population. Ecol Monogr 71:277–303CrossRefGoogle Scholar
  22. Doney SC, Sailley SF (2013) When an ecological regime shift is really just stochastic noise. Proc Natl Acad Sci U S A 110:2438–2439. doi:10.1073/pnas.1222736110 PubMedCentralPubMedCrossRefGoogle Scholar
  23. Eisler Z, Bartos I, Kertész J (2008) Fluctuation scaling in complex systems: Taylor's law and beyond. Adv Physiol 57:89–142CrossRefGoogle Scholar
  24. Engen S, Lande R, Saether B-E (2008) A general model for analyzing Taylor's spatial scaling laws. Ecology 89:2612–2622PubMedCrossRefGoogle Scholar
  25. Evans SN, Ralph PL, Schreiber SJ, Sen A (2013) Stochastic population growth in spatially heterogeneous environments. J Math Biol 66:423–476. doi:10.1007/s00285-012-0514-0 PubMedCrossRefGoogle Scholar
  26. Fracker SB, Brischle HA (1944) Measuring the local distribution of Ribes. Ecology 25(3):283–303, July. Stable URL: Accessed 25 Aug 2013Google Scholar
  27. Fronczak A, Fronczak P (2010) Origins of Taylor's power law for fluctuation scaling in complex systems. Phys Rev E 81(6)Google Scholar
  28. García-Carreras B, Reuman DC (2011) An empirical link between the spectral colour of climate and the spectral colour of field populations in the context of climate change. J Anim Ecol 80:1042–1048PubMedCrossRefGoogle Scholar
  29. García-Carreras B, Reuman DC (2013) Are changes in the mean or variability of climate signals more important for long-term stochastic growth rate? PLoS One 8(5):e63974. doi:10.1371/journal.pone.0063974, Accessed May 14, 2013PubMedCentralPubMedCrossRefGoogle Scholar
  30. Gillis DM, Kramer DL, Bell G (1986) Taylor's power law as a consequence of Fretwell's ideal free distribution. J Theor Biol 123:281–287CrossRefGoogle Scholar
  31. Hastings A, Wysham DB (2010) Regime shifts in ecological systems can occur with no warning. Ecol Lett 13:464–472PubMedCrossRefGoogle Scholar
  32. Jørgensen B (1987) Exponential dispersion models. J Roy Stat Soc Ser B 49:127–162Google Scholar
  33. Jørgensen B (1997) The theory of dispersion models. Chapman & Hall, LondonGoogle Scholar
  34. Kaltz O, Escobar-Páramo P, Hochberg ME, Cohen JE (2012) Bacterial microcosms obey Taylor's law: effects of abiotic and biotic stress and genetics on mean and variance of population density. Ecol Process 1:5CrossRefGoogle Scholar
  35. Keeling MJ (2000) Simple stochastic models and their power-law type behavior. Theor Popul Biol 58:21–31PubMedCrossRefGoogle Scholar
  36. Kemp AW (1987) Families of discrete distributions satisfying Taylor's power law. Biometrics 43:693–699CrossRefGoogle Scholar
  37. Kendal WS (2004) Taylor's ecological power law as a consequence of scale invariant exponential dispersion models. Ecol Complex 1:193–209CrossRefGoogle Scholar
  38. Kilpatrick AM, Ives AR (2003) Species interactions can explain Taylor's power law for ecological time series. Nature 422:65–68PubMedCrossRefGoogle Scholar
  39. Klein AM, Simons BD (2011) Universal patterns of stem cell fate in cycling adult tissues. Development 138:3103–3111PubMedCrossRefGoogle Scholar
  40. Kleinen TC (2005) Stochastic Information in the Assessment of Climate Change. Doctoral Dissertation. University of Potsdam, February 2005. 123 pp. Accessed 25 Aug 2013
  41. Lawton JH (1988) More time means more variation. Nature 334:563CrossRefGoogle Scholar
  42. Lewontin RC, Cohen D (1969) On population growth in a randomly varying environment. Proc Natl Acad Sci U S A 62:1056–1060PubMedCentralPubMedCrossRefGoogle Scholar
  43. Lorenzo ED, Ohman MD (2013) A double-integration hypothesis to explain ocean ecosystem response to climate forcing. Proc Natl Acad Sci U S A 110:2496–2499. doi:10.1073/pnas.1218022110 PubMedCentralPubMedCrossRefGoogle Scholar
  44. Perry JN (1988) Some models for spatial variability of animal species. Oikos 51:124–130CrossRefGoogle Scholar
  45. Perry JN, Taylor LR (1985) Ades: New ecological families of species-specific frequency distributions that describe repeated spatial samples with an intrinsic power-law variance-mean property. J Anim Ecol 54:931–953CrossRefGoogle Scholar
  46. Pimm SL, Redfearn A (1988) The variability of population densities. Nature 334:613–614CrossRefGoogle Scholar
  47. Ramsayer J, Fellous S, Cohen JE, Hochberg ME (2012) Taylor's law holds in experimental bacterial populations but competition does not influence the slope. Biol Lett 8:316–319. doi:10.1098/rsbl.2011.0895 PubMedCentralPubMedCrossRefGoogle Scholar
  48. Reuman DC, Desharnais RA, Costantino RF, Ahmad OS, Cohen JE (2006) Power spectra reveal the influence of stochasticity on nonlinear population dynamics. Proc Natl Acad Sci U S A 103:18860–18865. doi:10.1073/pnas.0608571103, and supplementary online materialPubMedCentralPubMedCrossRefGoogle Scholar
  49. Reuman DC, Costantino R, Desharnais R, Cohen JE (2008) Color of environmental noise affects the nonlinear dynamics of cycling, stage-structured populations. Ecol Lett 11(8):820–830. doi:10.1111/j.1461-0248.2008.01194.x PubMedCrossRefGoogle Scholar
  50. Russell BD, Harley CDG, Wernberg T, Mieszkowska N, Widdicombe S, Hall-Spencer JM et al (2012) Predicting ecosystem shifts requires new approaches that integrate the effects of climate change across entire systems. Biol Lett 8:164–166. doi:10.1098/rsbl.2011.0779 PubMedCentralPubMedCrossRefGoogle Scholar
  51. Scheffer M, Carpenter SR (2003) Catastrophic regime shifts in ecosystems: linking theory to observation. Trends Ecol Evol 18:648–656CrossRefGoogle Scholar
  52. Scheffer M, Carpenter S, Foley JA, Folke C, Walker B (2001) Catastrophic shifts in ecosystems. Nature 413:591–596PubMedCrossRefGoogle Scholar
  53. Scheffer M, Bascompte J, Brock WA, Brovkin V, Carpenter SR, Dakos V et al (2009) Early-warning signals for critical transitions. Nature 461:53–59PubMedCrossRefGoogle Scholar
  54. Schreiber SJ, Benaïm M, Atchadé KAS (2011) Persistence in fluctuating environments. J Math Biol 62:655–683. doi:10.1007/s00285-010-0349-5 PubMedCrossRefGoogle Scholar
  55. Smith HF (1938) An empirical law describing heterogeneity in the yields of agricultural crops. J Agric Sci 28:1–23. doi:10.1017/S0021859600050516 CrossRefGoogle Scholar
  56. Taylor LR (1961) Aggregation, variance and the mean. Nature 189:732–735CrossRefGoogle Scholar
  57. Taylor LR (1984) Assessing and interpreting the spatial distributions of insect populations. Annu Rev Entomol 29:321–357CrossRefGoogle Scholar
  58. Taylor LR, Woiwod IP, Perry JN (1980) Variance and the large scale spatial stability of aphids, moths and birds. J Anim Ecol 49:831–854CrossRefGoogle Scholar
  59. Tuljapurkar SD (1990) Population dynamics in variable environments, vol 85, Lecture Notes in Biomathematics. Springer, Berlin, New YorkCrossRefGoogle Scholar
  60. Tweedie MCK (1946) The regression of the sample variance on the sample mean. J Lond Math Soc 21:22–28CrossRefGoogle Scholar
  61. Tweedie MCK (1947) Functions of a statistical variate with given means, with special reference to Laplacian distributions. Proc Camb Philos Soc 43:41–49CrossRefGoogle Scholar
  62. Tweedie MCK (1984) An index which distinguishes between some important exponential families. In Ghosh JK, Roy J (eds.) Statistics: applications and new directions. Proceedings of the Indian Statistical Institute Golden Jubilee International Conference. Indian Statistical Institute, Calcutta. pp 579–604Google Scholar
  63. Wilson LT, Sterling WL, Rummel DR, DeVay JE (1989) Quantitive sampling principles in cotton. In: Frisbie RE, El-zik KM, Wilson LT (eds) Integrated pest management systems and cotton production. Wiley, Somerset, pp 85–120Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Laboratory of PopulationsThe Rockefeller University and Columbia UniversityNew YorkUSA

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