Taylor's Law Research Articles

Taylor's law on abundance unravels the role of traits and environmental conditions on population dynamics.

The variance-mean scaling in population abundance or Taylor's power law (TPL) has been reported hundreds of times and is related to ecological processes such as competition, dispersal or territoriality. In this sense, the TPL was extensively validated to resume population variability and to show the action of ecological mechanisms. Baumgartner and Peláez (2024) combine databases of fish dynamics along the United States, species traits, species phylogeny and climatic conditions, estimating the TPL for 180 species along 972 populations. The observed scaling suggests that the variability of the population decreases with abundance. Notably, 68% of the variation in the variance-mean scaling was explained by species traits and environmental conditions. Specifically, the life history of the species, associated with its body size, was the main explanation for the TPL, also reporting that the variability of the population increased faster with mean abundance in the headwaters than in other river sections. Equally important, the diversity of the community in which the species were inserted did not affect the scaling. These results provide overwhelming evidence on the nature of TPL on large geographic scales and how they are affected by species biology and environmental conditions. A contribution that should motivate further empirical and theoretical analysis of the TPL and its determinants.

Adaptability of the gut microbiota of the German cockroach Blattella germanica to a periodic antibiotic treatment

High-throughput sequencing studies have shown that diet or antimicrobial treatments impact animal gut microbiota equilibrium. However, properties related to the gut microbial ecosystem stability, such as resilience, resistance, or functional redundancy, must be better understood. To shed light on these ecological processes, we combined advanced statistical methods with 16 S rRNA gene sequencing, functional prediction, and fitness analyses in the gut microbiota of the cockroach Blattella germanica subject to three periodic pulses of the antibiotic (AB) kanamycin (n=512). We first confirmed that AB did not significantly affect cockroaches' biological fitness, and gut microbiota changes were not caused by insect physiology alterations. The sex variable was examined for the first time in this species, and no statistical differences in the gut microbiota diversity or composition were found. The comparison of the gut microbiota dynamics in control and treated populations revealed that (1) AB treatment decreases diversity and completely disrupts the co-occurrence networks between bacteria, significantly altering the gut community structure. (2) Although AB also affected the genetic composition, functional redundancy would explain a smaller effect on the functional potential than on the taxonomic composition. (3) As predicted by Taylor's law, AB generally affected the most abundant taxa to a lesser extent than the less abundant taxa. (4) Taxa follow different trends in response to ABs, highlighting "resistant taxa," which could be critical for community restoration. (5) The gut microbiota recovered faster after the three AB pulses, suggesting that gut microbiota adapts to repeated treatments.

Multiscaling behavior of braided channel networks: An alternative formulation of Taylor's law variate transformations.

Braided channel networks exhibit a complex interplay between spatial and temporal dynamics. Their behavior is characterized by both simple and multiscaling patterns, and the mechanisms underlying the stochastic processes associated with this dynamics remain incompletely understood. Leveraging Taylor's pioneering work [Nature (London) 189, 732 (1961)NATUAS0028-083610.1038/189732a0], which unveiled scaling relations in a plethora of natural phenomena through what is now known as the Taylor power law (TPL), we propose a physical interpretation of braided channel systems. This interpretation utilizes a specific class of transformation functions applied to the mean of fluvial geomorphic variables measured along cross sections, namely, the number of wet channels, the average width of wet channels, and the entropic braiding index. By analyzing remotely sensed data of the Brahmaputra-Jamuna River in Bangladesh we obtain valuable insight into the spatiotemporal scaling of these geomorphological variables and gather a deeper understanding of the complexity of braided channel systems. Finally, through a direct analysis employing the TPL in conjunction with a fixed-mass multifractal algorithm, we prove that braided channel networks exhibit a multiscaling behavior.

Quantifying how much host, pathogen, and other factors affect human protective adaptive immune responses.

Recognizing the "essential" factors that contribute to a clinical outcome is critical for designing appropriate therapies and prioritizing limited medical resources. Demonstrating a high correlation between a factor and an outcome does not necessarily imply an essential role of the factor to the outcome. Human protective adaptive immune responses to pathogens vary among (and perhaps within) pathogenic strains, human individual hosts, and in response to other factors. Which of these has an "essential" role? We offer three statistical approaches that predict the presence of newly contributing factor(s) and then quantify the influence of host, pathogen, and the new factors on immune responses. We illustrate these approaches using previous data from the protective adaptive immune response (cellular and humoral) by human hosts to various strains of the same pathogenic bacterial species. Taylor's law predicts the existence of other factors potentially contributing to the human protective adaptive immune response in addition to inter-individual host and intra-bacterial species inter-strain variability. A mixed linear model measures the relative contribution of the known variables, individual human hosts and bacterial strains, and estimates the summed contributions of the newly predicted but unknown factors to the combined adaptive immune response. A principal component analysis predicts the presence of sub-variables (currently not defined) within bacterial strains and individuals that may contribute to the combined immune response. These observations have statistical, biological, clinical, and therapeutic implications.

Quantifying factors that explain the slopes of the temporal Taylor's law of Hokkaido vole populations

AbstractTaylor's law (TL) describes the relationship between the variance and mean of population density: log10(variance) ≈ log10(a) + b × log10(mean), a > 0. This study analyzed the temporal TL, for which mean and variance are calculated over time, separately for each population in a collection of populations, considering the effects of the parameters of the Gompertz model (a second‐order autoregressive time‐series model) and the skewness of the density frequency distribution. Time series of 162 populations of the gray‐sided vole in Hokkaido, Japan, spanning 23–31 years, satisfied the temporal TL: log10(variancej) ≈ 0.199 + 1.687 × log10(meanj). This model explained 62% of the variation of log10(variancej). An extended model with explanatory variables log10(meanj), the density‐dependent coefficient for 1‐year lag (α1,j), that for 2‐year lag (α2,j), the density‐independent variability (σj2), and the skewness (γj), explained 93.9% of the log10(variancej) variation. In the extended model, the coefficient of log10(meanj) was 1.949, close to the null value (b = 2) of the TL slope. The standardized partial regression coefficients indicated that density‐independent effects (σj2 and γj) dominated density‐dependent effects (α1,j and α2,j) apart from log10(meanj). The negative correlations observed between σj2 and log10(meanj), and between γj and log10(meanj), played an essential role in explaining the difference between the estimated slope of TL (b = 1.687) and the null slope (b = 2). The effects of those explanatory variables on log10(variancej) were interpreted based on the theory of a second‐order autoregressive time‐series model.

Capillary flow of two immiscible liquids initially separated by a Taylor gas bubble

We investigate the capillary displacement of silicone oil columns by water in tandem with a Taylor gas bubble in horizontally aligned tubes that initially separate the two immiscible liquids. The experiments show that the gas slug's motion forces the drainage of the prefilling liquid through a thin film deposited on the tube's wall. The displaced oil is then recovered at the advancing water meniscus. The dynamics of the capillary displacement and drainage are studied as functions of the silicone oil and gas bubble volumes. We find that the Taylor bubble's length has a small influence on the capillary displacement and drainage of the prefilling liquid. The study presents an adapted model for the capillary drainage and oil recovery using Taylor's law for the deposited film thickness on the tube's wall, which collapses all experimental data onto a single curve.

Deciphering the Taylor Law: A Novel Interpretation of Variance and Mean Relations in Population Dynamics

Deciphering the Taylor Law: A Novel Interpretation of Variance and Mean Relations in Population Dynamics

Spatiotemporal dynamics of bacterial wilt in Eucalyptus

AbstractBacterial wilt (caused by Ralstonia spp.) is one of the most damaging diseases of Eucalyptus species, and is responsible for substantial losses to producers. For efficient, scientifically‐based management of this disease, it is necessary to understand the various factors involved in its development, including greater knowledge of the epidemiology of the pathogen on Eucalyptus spp. This study aimed to determine the spatialtemporal dynamics of bacterial wilt in Eucalyptus under natural infection conditions. An experiment was conducted in a commercial plantation in the municipality of Itinga, state of Maranhão, Brazil, using the clone FGCA0385 (Eucalyptus urophylla var. platyphylla). The study comprised of four plots composed of 450 plants each, subdivided into nine rows with 50 plants per row with a spacing of 3 × 3 m. Disease incidence was quantified over 1 year. The spatial dynamics of the disease was determined using the dispersion index, a modified Taylor law, and the analysis of the dynamics and structure of foci. For temporal dynamics, the curve of the disease incidence progress was plotted, and the data were analysed by simple linear regression analysis fitted to three empirical models: logistic, monomolecular, and Gompertz. The distribution pattern of Eucalyptus bacterial wilt was random, as confirmed by the Index of dispersion. Analysis of the dynamics and structure of the foci, showed that 69 disease foci occurred, 44 of which were unitary, with an average number of plants per focus of 1.63. Foci had greater length in the direction of the planting line. The epidemics were best described by the monomolecular model, with an estimated incidence of Eucalyptus bacterial wilt of 27.77% in the fourth year of the study. According to the spatiotemporal dynamics of this work, management strategies such as eliminating symptomatic plants and crop renovation can be used based on economic viability.

Testing Taylor’s Law in Urban Population Dynamics Worldwide with Simultaneous Equation Models

Knowledge of long-term population trends is still incomplete at the global scale. In this perspective, human and animal ecology has intensively studied the relationship between the Mean (M) size and the Variance (V) of specific attributes of subpopulations within a given regional system. One of the best-known relationships between these two attributes suitable to describe long-term population trends is governed by Taylor’s law (TL). The present article contributes to the recent literature on population trends worldwide by testing the long-term relationship (1950–2015) between the overall variance and mean in the total population of 1857 metropolitan agglomerations in 155 countries classified into 9 world macro-regions. To estimate the unknown parameter(s) of the V–M relation we made use of a simultaneous equation system using both linear (classical TL) and quadratic specifications, with the aim of ascertaining a wide range of simplified (or more complex) association rules between the two dimensions of demographic change. The empirical results show that TL is verified in all nine cases, although a quadratic relationship provides slightly better results than the classical, linear relationship. More specifically, similar estimates for both linear and quadratic relationships were characteristic of ‘new’ demographic continents with more recent and intense urbanization processes (the Americas, and African and Asian countries). The predominance of quadratic relationships characterized regions with long-established urbanization processes, such as Europe, Russia, and, partly, China and the Middle East. The relevance of the TL for a refined understanding of urbanization mechanisms worldwide, and the importance of a quadratic term for distinguishing metropolitan systems that have experienced different development paths, were finally discussed.

COVID-19 cases and deaths in the United States follow Taylor’s law for heavy-tailed distributions with infinite variance

The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor's law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state's variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor's law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction.

Temporal and Spatial Taylor's Law: Application to Japanese Subnational Mortality Rates

AbstractTaylor's law is a widely observed empirical pattern that relates the variances to the means of population densities. We present four extensions of the classical Taylor's law (TL): (1) a cubic extension of the linear TL describes the mean–variance relationship of human mortality at subnational levels well; (2) in a time series, long-run variance measures not only variance but also autocovariance, and it is a more suitable measure than variance alone to capture temporal/spatial correlation; (3) an extension of the classical equally weighted spatial variance takes account of synchrony and proximity; (4) robust linear regression estimators of TL parameters reduce vulnerability to outliers. Applying the proposed methods to age-specific Japanese subnational death rates from 1975 to 2018, we study temporal and spatial variations, compare different coefficient estimators, and interpret the implications. We apply a clustering algorithm to the estimated TL coefficients and find that cluster memberships are strongly related to prefectural gross domestic product. The time series of spatial TL coefficients has a decreasing trend that confirms the narrowing gap between rural and urban mortality in Japan.

Variance in Landscape Connectivity Shifts Microbial Population Scaling.

Understanding mechanisms shaping distributions and interactions of soil microbes is essential for determining their impact on large scale ecosystem services, such as carbon sequestration, climate regulation, waste decomposition, and nutrient cycling. As the functional unit of soil ecosystems, we focus our attention on the spatial structure of soil macroaggregates. Emulating this complex physico-chemical environment as a patchy habitat landscape we investigate on-chip the effect of changing the connectivity features of this landscape as Escherichia coli forms a metapopulation. We analyze the distributions of E. coli occupancy using Taylor's law, an empirical law in ecology which asserts that the fluctuations in populations is a power law function of the mean. We provide experimental evidence that bacterial metapopulations in patchy habitat landscapes on microchips follow this law. Furthermore, we find that increased variance of patch-corridor connectivity leads to a qualitative transition in the fluctuation scaling. We discuss these results in the context of the spatial ecology of microbes in soil.

Protein concentration fluctuations in the high expression regime: Taylor's law and its mechanistic origin.

Protein concentration in a living cell fluctuates over time due to noise in growth and division processes. In the high expression regime, variance of the protein concentration in a cell was found to scale with the square of the mean, which belongs to a general phenomenon called Taylor's law (TL). To understand the origin for these fluctuations, we measured protein concentration dynamics in single E. coli cells from a set of strains with a variable expression of fluorescent proteins. The protein expression is controlled by a set of constitutive promoters with different strength, which allows to change the expression level over 2 orders of magnitude without introducing noise from fluctuations in transcription regulators. Our data confirms the square TL, but the prefactor A has a cell-to-cell variation independent of the promoter strength. Distributions of the normalized protein concentration for different promoters are found to collapse onto the same curve. To explain these observations, we used a minimal mechanistic model to describe the stochastic growth and division processes in a single cell with a feedback mechanism for regulating cell division. In the high expression regime where extrinsic noise dominates, the model reproduces our experimental results quantitatively. By using a mean-field approximation in the minimal model, we showed that the stochastic dynamics of protein concentration is described by a Langevin equation with multiplicative noise. The Langevin equation has a scale invariance which is responsible for the square TL. By solving the Langevin equation, we obtained an analytical solution for the protein concentration distribution function that agrees with experiments. The solution shows explicitly how the prefactor A depends on strength of different noise sources, which explains its cell-to-cell variability. By using this approach to analyze our single-cell data, we found that the noise in production rate dominates the noise from cell division. The deviation from the square TL in the low expression regime can also be captured in our model by including intrinsic noise in the production rate.

Disordered beta thinned ensemble with applications.

It recently has been found that methods of the statistical theories of spectra can be a useful tool in the analysis of spectra far from levels of Hamiltonian systems. The purpose of the present study is to deepen this kind of approach by performing a more comprehensive spectral analysis that measures both the local- and long-range statistics. We have found that, as a common feature, spectra of this kind can exhibit a situation in which local statistics are relatively quenched while the long-range ones show large fluctuations. By combining three extensions of the standard random matrix theory (RMT) and considering long spectra, we demonstrate that this phenomenon occurs when disorder and level incompleteness are introduced in an RMT spectrum. Consequently, the long-range statistics follow Taylor's law, suggesting the presence of a fluctuation scaling (FS) mechanism in this kind of spectra. Applications of the combined ensemble are then presented for spectra originate from several very diverse areas, including complex networks, COVID-19 time series, and quantitative linguistics, which demonstrate that short- and long-range statistics reflect the rigid and elastic characteristics of a given spectrum, respectively. These observations may shed some light on spectral data classification.

Taylor’s law of fluctuation scaling for semivariances and higher moments of heavy-tailed data

We generalize Taylor's law for the variance of light-tailed distributions to many sample statistics of heavy-tailed distributions with tail index α in (0, 1), which have infinite mean. We show that, as the sample size increases, the sample upper and lower semivariances, the sample higher moments, the skewness, and the kurtosis of a random sample from such a law increase asymptotically in direct proportion to a power of the sample mean. Specifically, the lower sample semivariance asymptotically scales in proportion to the sample mean raised to the power 2, while the upper sample semivariance asymptotically scales in proportion to the sample mean raised to the power [Formula: see text] The local upper sample semivariance (counting only observations that exceed the sample mean) asymptotically scales in proportion to the sample mean raised to the power [Formula: see text] These and additional scaling laws characterize the asymptotic behavior of commonly used measures of the risk-adjusted performance of investments, such as the Sortino ratio, the Sharpe ratio, the Omega index, the upside potential ratio, and the Farinelli-Tibiletti ratio, when returns follow a heavy-tailed nonnegative distribution. Such power-law scaling relationships are known in ecology as Taylor's law and in physics as fluctuation scaling. We find the asymptotic distribution and moments of the number of observations exceeding the sample mean. We propose estimators of α based on these scaling laws and the number of observations exceeding the sample mean and compare these estimators with some prior estimators of α.

Integrating metabolic scaling variation into the maximum entropy theory of ecology explains Taylor's law for individual metabolic rate in tropical forests

Individual trait variation has important ecological implications for species populations and communities. In particular, individual variation of metabolic rate links directly with the energy use estimation and scaling patterns in community ecology. Here, we examine the mean-variance relationship of individual metabolic rate by testing Taylor's law (a power-function relationship between mean and variance) for individual rescaled metabolic rate across tree communities in tropical forests. We use a constraint-based model called maximum entropy theory of ecology (METE) to estimate and predict the parameters of Taylor's law. Our results show that, when assuming a universal metabolic scaling between metabolic rate and aboveground biomass, the METE generates the form of Taylor's law but fails to predict its slope. When setting the metabolic scaling exponent as a community-level parameter in the METE model, the estimated and predicted slopes of Taylor's law agree with each other. Our parameterized METE model reveals the positive effect of number of individuals on the metabolic scaling exponent. These results suggest that fluctuation scaling of individual metabolic rate can be explained solely by the macroecological constraints of communities, without relying on the physiological or genetic characters of individual organisms.

Direct Scaling of Measure on Vortex Shedding through a Flapping Flag Device in the Open Channel around a Cylinder at Re∼103: Taylor's Law Approach.

The problem of vortex shedding, which occurs when an obstacle is placed in a regular flow, is governed by Reynolds and Strouhal numbers, known by dimensional analysis. The present work aims to propose a thin films-based device, consisting of an elastic piezoelectric flapping flag clamped at one end, in order to determine the frequency of vortex shedding downstream an obstacle for a flow field at Reynolds number in the open channel. For these values, Strouhal number obtained in such way is in accordance with the results known in literature. Moreover, the development of the voltage over time, generated by the flapping flag under the load due to flow field, shows a highly fluctuating behavior and satisfies Taylor’s law, observed in several complex systems. This provided useful information about the flow field through the constitutive law of the device.

Spatial and temporal Taylor’s law in 1D chaotic maps

By using low-dimensional chaotic maps, the power-law relationship established between the sample mean and variance called Taylor's Law (TL) is studied. In particular, we aim to clarify the relationship between TL from the spatial ensemble (STL) and the temporal ensemble (TTL). Since the spatial ensemble corresponds to independent sampling from a stationary distribution, we confirm that STL is explained by the skewness of the distribution. The difference between TTL and STL is shown to be originated in the temporal correlation of a dynamics. In case of logistic and tent maps, the quadratic relationship in the sample mean and variance, called Bartlett's law, is found analytically. On the other hand, TTL in the Hassell model can be well explained by the chunk structure of the trajectory, whereas the TTL of the Ricker model has a different mechanism originated from the specific form of the map.

Spatial and temporal autocorrelations affect Taylor's law for US county populations: Descriptive and predictive models.

Understanding the spatial and temporal distributions and fluctuations of living populations is a central goal in ecology and demography. A scaling pattern called Taylor's law has been used to quantify the distributions of populations. Taylor's law asserts a linear relationship between the logarithm of the mean and the logarithm of the variance of population size. Here, extending previous work, we use generalized least-squares models to describe three types of Taylor's law. These models incorporate the temporal and spatial autocorrelations in the mean-variance data. Moreover, we analyze three purely statistical models to predict the form and slope of Taylor's law. We apply these descriptive and predictive models of Taylor's law to the county population counts of the United States decennial censuses (1790–2010). We find that the temporal and spatial autocorrelations strongly affect estimates of the slope of Taylor's law, and generalized least-squares models that take account of these autocorrelations are often superior to ordinary least-squares models. Temporal and spatial autocorrelations combine with demographic factors (e.g., population growth and historical events) to influence Taylor's law for human population data. Our results show that the assumptions of a descriptive model must be carefully evaluated when it is used to estimate and interpret the slope of Taylor's law.

Heavy-tailed distributions, correlations, kurtosis and Taylor's Law of fluctuation scaling.

Pillai & Meng (Pillai & Meng 2016 Ann. Stat. 44, 2089-2097; p. 2091) speculated that 'the dependence among [random variables, rvs] can be overwhelmed by the heaviness of their marginal tails ·· ·'. We give examples of statistical models that support this speculation. While under natural conditions the sample correlation of regularly varying (RV) rvs converges to a generally random limit, this limit is zero when the rvs are the reciprocals of powers greater than one of arbitrarily (but imperfectly) positively or negatively correlated normals. Surprisingly, the sample correlation of these RV rvs multiplied by the sample size has a limiting distribution on the negative half-line. We show that the asymptotic scaling of Taylor's Law (a power-law variance function) for RV rvs is, up to a constant, the same for independent and identically distributed observations as for reciprocals of powers greater than one of arbitrarily (but imperfectly) positively correlated normals, whether those powers are the same or different. The correlations and heterogeneity do not affect the asymptotic scaling. We analyse the sample kurtosis of heavy-tailed data similarly. We show that the least-squares estimator of the slope in a linear model with heavy-tailed predictor and noise unexpectedly converges much faster than when they have finite variances.