Power-law species–area relationships and self-similar species distributions within finite areas

The species–area relationship (SAR) is often expressed as a power law, which indicates scale invariance. It has been claimed that the scale invariance – or self‐similarity at the community level – is not compatible with the self‐similarity at the level of spatial distribution of individual species, because the power law would only emerge if distributions for all species had identical fractal dimensions (FD). Here we show that even if species differ in their FD, the resulting SAR is approximately linear on a log–log scale because observed spatial distributions are inevitably spatially restricted – a phenomenon we term the ‘finite‐area effect’. Using distribution atlases, we demonstrate that the apparent power law of SARs for central European birds is attributable to this finite‐area effect affecting species that indeed reveal self‐similar distributions. We discuss implications of this mechanism producing the SAR.     

Population fluctuations,power laws and mixtures of lognormal distributions

A number of investigators have invoked a cascading local interaction model to account for power‐law‐distributed fluctuations in ecological variables. Invoking such a model requires that species be tightly coupled, and that local interactions among species influence ecosystem dynamics over a broad range of scales. Here we reanalyse bird population data used by Keitt & Stanley (1998, Dynamics of North American breeding bird populations. Nature, 393, 257–260) to support a cascading local interaction model. We find that the power law they report can be attributed to mixing of lognormal distributions. More tentatively, we propose that mixing of distributions accounts for other empirical power laws reported in the ecological literature.     

Self-organization and complexity in historical landscape patterns

Self‐organization describes the evolution process of complex structures where systems emerge spontaneously, driven internally by variations of the system itself. Self‐organization to the critical state is manifested by scale‐free behavior across many orders of magnitude (Bak et al. 1987, Bak 1996, Solé et al. 1999). Spatial scale‐free behavior implies fractal properties and is quantified by the fractal dimension. Temporal scale‐free behavior is evident in power spectra of fluctuations that obey power laws. Self‐organized criticality is a universal phenomenon that likely produces some of the fractals and power laws observed in nature. We investigated the historical landscape of southern Wisconsin (USA) (60,000 km²) for self‐organization and complexity. The landscape is patterned into prairies, savannas, and open and closed forests, using data from the United States General Land Office Surveys that were conducted during the 19th century, at a time prior to Euro‐American settlement. We applied a two‐dimensional cellular automaton model with one adjustable parameter. Model evolution replaces a cell that dies at random times by a cell chosen randomly from within a circular radius r, where r typically takes values between 1 (local) and 10 units (regional). Cluster probability is used to measure the degree of organization. The model landscape self‐organizes to a realistic critical state if neighborhoods of intermediate size (r=3) are chosen, indicating that (a) no particular time or space scale for the clusters is singled out, i.e. the spatial dependence is fractal, and temporal fluctuations in the cluster probability exhibit power laws; (b) a simple model suffices to replicate the landscape pattern resulting from complex spatial and temporal interactions. Measures of comparison between the observed and the simulated landscape show good agreement: fractal dimensions for simulated (1.6) and observed landscapes (1.64), cluster probabilities for simulated (32.3%) and observed (32.6%) landscapes, and algorithmic complexity for simulated (6792 bytes) and observed (6205 bytes) landscapes. The results are robust towards variation of initial and boundary conditions as well as perturbations.     

Metabolic partitioning across individuals in ecological communities

The mechanistic origin and shape of body‐size distributions within communities are of considerable interest in ecology. A recently proposed light‐limitation model provides a good fit to the distribution of tree sizes in a tropical forest plot. The maximum entropy theory of ecology (METE) also predicts size distributions, but without explicit mechanistic assumptions, and thus its predictions should hold in ecosystems generally, regardless of whether they are light limited. A comparison of the form and success of the predictions of the model and the theory can provide insight into the role that mechanisms play in shaping patterns in macroecology. The prediction by the METE of the size distribution of organisms is remarkably similar in form to that of the model: power‐law behaviour in the size range where the light‐limitation model predicts a power law, and exponential behaviour in the size range where the model predicts an exponential tail. The METE prediction matches data widely, including data in ecosystems where light is not limiting. We show examples for three disparate communities: trees in a tropical forest plot; herbaceous plants in a treeless subalpine meadow; and island arthropods. We conclude that the success of METE's predicted form across systems, including those that are clearly not light limited, enriches our capacity to predict patterns in macroecology without making explicit mechanistic assumptions and provides a unified framework that can capture ubiquitous features of those patterns across diverse ecosystems governed by a variety of mechanisms.     

Taylor's law and related allometric power laws in New Zealand mountain beech forests: the roles of space,time and environment

Taylor's law says that the variance of population density of a species is proportional to a power of mean population density. Density–mass allometry says that mean population density is proportional to a power of mean biomass per individual. These power laws predict a third, variance–mass allometry: the variance of population density of a species is proportional to a power of mean biomass per individual. We tested these laws using 10 censuses of New Zealand mountain beech trees in 250 plots over 30 years at spatial scales from 5 m to kilometers. We found that: 1) a single‐species forest not disrupted by humans obeyed all three laws; 2) random sampling explained the parameters of Taylor's law at a large spatial scale in 8 of 10 censuses, but not at a fine spatial scale; 3) larger spatial scale increased the exponent of Taylor's law and decreased the exponent of variance–mass allometry (this is the first empirical demonstration that the latter exponent depends on spatial scale), but affected the exponent of density–mass allometry slightly; 4) despite varying natural disturbance, the three laws varied relatively little over the 30 years; 5) self‐thinning and recruiting plots had significantly different intercepts and slopes of density–mass allometry and variance–mass allometry, but the parameters of Taylor's law were not usually significantly affected; and 6) higher soil calcium was associated with higher variance of population density in all censuses but not with a difference in the exponent of Taylor's law, while elevation above sea level and soil carbon‐to‐nitrogen ratios had little effect on the parameters of Taylor's law. In general, the three laws were remarkably robust. When their parameters were influenced by spatial scale and environmental factors, the parameters could not be species‐specific indicators. We suggest biological mechanisms that may explain some of these findings.     

Scale criticality in estimating ecosystem carbon dynamics

Scaling is central to ecology and Earth system sciences. However, the importance of scale (i.e. resolution and extent) for understanding carbon dynamics across scales is poorly understood and quantified. We simulated carbon dynamics under a wide range of combinations of resolution (nine spatial resolutions of 250 m, 500 m, 1 km, 2 km, 5 km, 10 km, 20 km, 50 km, and 100 km) and extent (57 geospatial extents ranging from 108 to 1 247 034 km²) in the southeastern United States to explore the existence of scale dependence of the simulated regional carbon balance. Results clearly show the existence of a critical threshold resolution for estimating carbon sequestration within a given extent and an error limit. Furthermore, an invariant power law scaling relationship was found between the critical resolution and the spatial extent as the critical resolution is proportional to Aⁿ (n is a constant, and A is the extent). Scale criticality and the power law relationship might be driven by the power law probability distributions of land surface and ecological quantities including disturbances at landscape to regional scales. The current overwhelming practices without considering scale criticality might have largely contributed to difficulties in balancing carbon budgets at regional and global scales.     

Power law analysis of the human microbiome

Taylor's (1961, Nature, 189:732) power law, a power function (V = am^b) describing the scaling relationship between the mean and variance of population abundances of organisms, has been found to govern the population abundance distributions of single species in both space and time in macroecology. It is regarded as one of few generalities in ecology, and its parameter b has been widely applied to characterize spatial aggregation (i.e. heterogeneity) and temporal stability of single‐species populations. Here, we test its applicability to bacterial populations in the human microbiome using extensive data sets generated by the US‐NIH Human Microbiome Project (HMP). We further propose extending Taylor's power law from the population to the community level, and accordingly introduce four types of power‐law extensions (PLEs): type I PLE for community spatial aggregation (heterogeneity), type II PLE for community temporal aggregation (stability), type III PLE for mixed‐species population spatial aggregation (heterogeneity) and type IV PLE for mixed‐species population temporal aggregation (stability). Our results show that fittings to the four PLEs with HMP data were statistically extremely significant and their parameters are ecologically sound, hence confirming the validity of the power law at both the population and community levels. These findings not only provide a powerful tool to characterize the aggregations of population and community in both time and space, offering important insights into community heterogeneity in space and/or stability in time, but also underscore the three general properties of power laws (scale invariance, no average and universality) and their specific manifestations in our four PLEs.     

Zipf's law organizes a psychiatric ward.

We developed a simple mathematical model based on power law fitting for describing the interactions among patients from a psychiatric ward. First we defined a protocol in order to evaluate in a quantative way the state of the patient, measuring sociability/restlessness through a daily analysis of the behavior and attributing a grade for both parameters, per patient. The grades were checked by two different specialists and a table of incidence was constructed. This table generated power laws for the grades and their variations. We concluded that power laws, like Zipf's law, may be good to explain the data, showing a self-organizing process that indicates a strong interaction component determining the whole behavior. We would like to see more data being collected, in other centers and among normal populations, trying to quantify complex collective behavioral phenomena using self-organizing criticality laws.     

Cell cycle coordination and regulation of bacterial chromosome segregation dynamics by polarly localized proteins

What regulates chromosome segregation dynamics in bacteria is largely unknown. Here, we show in Caulobacter crescentus that the polarity factor TipN regulates the directional motion and overall translocation speed of the parS/ParB partition complex by interacting with ParA at the new pole. In the absence of TipN, ParA structures can regenerate behind the partition complex, leading to stalls and back‐and‐forth motions of parS/ParB, reminiscent of plasmid behaviour. This extrinsic regulation of the parS/ParB/ParA system directly affects not only division site selection, but also cell growth. Other mechanisms, including the pole‐organizing protein PopZ, compensate for the defect in segregation regulation in ΔtipN cells. Accordingly, synthetic lethality of PopZ and TipN is caused by severe chromosome segregation and cell division defects. Our data suggest a mechanistic framework for adapting a self‐organizing oscillator to create motion suitable for chromosome segregation.     

Self-organised criticality in a genetic model of species-species interaction

Summary Genetics incorporated into a two trophic level species-species interaction model allows the populations of species to evolve. The system enters a regime where the extinction of species follows an erratic pattern in time and appears chaotic to the eye. Quantitative measurements suggest that the dynamics of the model exhibits self organised criticality. Fourier Transform analysis of the time series and autocorrelation function for the population data, as well as the distribution of lineage sizes and longevity of lineages, all show power law behavior. The lineages are the analogue of avalanches in other models of self organised criticality     

Broadband Criticality of Human Brain Network Synchronization

Self-organized criticality is an attractive model for human brain dynamics, but there has been little direct evidence for its existence in large-scale systems measured by neuroimaging. In general, critical systems are associated with fractal or power law scaling, long-range correlations in space and time, and rapid reconfiguration in response to external inputs. Here, we consider two measures of phase synchronization: the phase-lock interval, or duration of coupling between a pair of (neurophysiological) processes, and the lability of global synchronization of a (brain functional) network. Using computational simulations of two mechanistically distinct systems displaying complex dynamics, the Ising model and the Kuramoto model, we show that both synchronization metrics have power law probability distributions specifically when these systems are in a critical state. We then demonstrate power law scaling of both pairwise and global synchronization metrics in functional MRI and magnetoencephalographic data recorded from normal volunteers under resting conditions. These results strongly suggest that human brain functional systems exist in an endogenous state of dynamical criticality, characterized by a greater than random probability of both prolonged periods of phase-locking and occurrence of large rapid changes in the state of global synchronization, analogous to the neuronal “avalanches” previously described in cellular systems. Moreover, evidence for critical dynamics was identified consistently in neurophysiological systems operating at frequency intervals ranging from 0.05–0.11 to 62.5–125 Hz, confirming that criticality is a property of human brain functional network organization at all frequency intervals in the brain's physiological bandwidth.     

Universal features in the genome-level evolution of protein domains

Background  

Protein domains can be used to study proteome evolution at a coarse scale. In particular, they are found on genomes with notable statistical distributions. It is known that the distribution of domains with a given topology follows a power law. We focus on a further aspect: these distributions, and the number of distinct topologies, follow collective trends, or scaling laws, depending on the total number of domains only, and not on genome-specific features.     

Universal scaling of the C--G distribution of genes

Using our previous result that the C--G distribution in genomes is very broad, varying as a power law of the size of the block of genome considered, we examine the C--G distribution in genes themselves. We show that the widths of the C--G distributions for the genes of several simple organisms also vary as power laws. This suggests that the power law behavior gives a universal scaling whereby the distributions for the C--G content of the genes from all species are mapped onto a single function.     

Spatial patterns of the ant Crematogaster scutellaris in a model ecosystem

1. The spatial arrangement of individuals and populations may have deep influences on all the biotic interactions within a community. 2. The spatial distribution of nests of the ant Crematogaster scutellaris Olivier was analysed in an olive orchard in central Italy. As this species nests inside tree trunks, the regular structure of this simplified ecosystem may help to reduce the confounding effect of habitat heterogeneity on the spatial distribution of nests. In total, 531 trees were mapped and their shape (size and structure of the trunk) recorded. The presence of C. scutellaris nests in each tree was assessed in spring–summer and autumn 2006 and 2007. 3. The number of occupied trees changed in time, from a maximum of 129 (summer 2006) to a minimum of 60 (autumn 2007). Occupancy of tree was related to its shape, with larger trees being more frequently and more steadily occupied than smaller ones. 4. Nests were spatially aggregated, forming well‐defined clusters, but aggregation was not explained by a corresponding clumping of larger trees. Nests belonging to the same cluster were usually not aggressive to each other, whereas aggression was more common between nests belonging to different clusters. The dynamic nature of the system coupled with the clustered distribution of nests, is consistent with a hypothesis of seasonal polydomy, and suggest that whereas some trees are steadily occupied (core) others are opportunistically colonised when new resources are discovered (satellites). 5. Clusters size distribution was shown to follow a truncated power‐law, a finding consistent with the idea that clusters are self‐organised structures dependent on local interactions. These results suggest that spatial self‐organisation in ant colonies may in principle be more common than previously thought, although the mechanisms generating these patterns still need to be clarified.     

Truncated power laws: a tool for understanding aggregation patterns in animals?

Statistical distributions like the negative binomial distribution are commonly used to describe aggregation patterns in animals. However, recently it has been suggested that truncated power laws (TPLs) may also be used for this kind of analysis. A TPL consists of two power functions separated by a cut-off size ( C *). The cut-off size and the slope of power function one (β₁) for the smallest group sizes have been suggested to have a biological explanatory value.
We applied TPLs to aggregation data of tephritid seed predators on a composite plant, aphids on willows and grey seals on a haulout site. β₁ varied between 0.60 and and −0.72, which is higher than predicted. In addition, resource distribution and animal density influenced β₁ and C *. This indicates that environmental dimensionality suggested to affect β₁ is masked by ecological factors. We conclude that TPLs are useful due to their simplicity and, in comparison with traditional methods, provide additional biologically relevant information. Truncated power laws can therefore prove to be useful in studies of animal behaviour and population dynamics.     

Pollinator directionality as a response to nectar gradient: promoting outcrossing while avoiding geitonogamy

Plants with multiple flowers could be prone to autonomous self‐pollination and insect‐mediated geitonogamy, but physiological and ecological features have evolved preventing costs related to autogamy. We studied the rare perennial herb Dictamnus albus as a model plant, with the aim of describing the plant–pollinator system from both plant and pollinator perspectives and analysing features that promote outcrossing in an entomophilous species. The breeding system and reproductive success of D. albus were investigated in experimental and natural conditions, showing that it is potentially self‐compatible, but only intra‐inflorescence insect‐mediated selfing is possible. Nectar analysis showed gender‐biased production towards the female phase, which follows the male phase, and during flowering, full blooming is found in flowers at the bottom of the raceme. Among a wide spectrum of insect visitors, three genera (Bombus, Apis, Megachile) were found to be principal pollinators. A study of insect behaviour showed a tendency towards bottom‐to‐top flights for the most important pollinators Bombus spp. and Apis mellifera: upward movements on the racemes could be explained by foraging behaviour, from more to less rewarding flowers. In accordance with the ‘declining reward hypothesis’, bumblebees and honeybees leave the plant when gain of reward is low, after which few flowers are visited, reducing the chance of self‐pollen transfer among flowers. Intra‐flower self‐pollination is prevented in D. albus by protandry and herkogamy, while the nectar‐induced sequential pattern of pollinator visits avoids geitonogamy and tends to maximise pollen export, promoting outcrossing. All these features for preventing selfing benefit plant fitness and population genetic structure.     

Power law versus exponential state transition dynamics: application to sleep-wake architecture

Background

Despite the common experience that interrupted sleep has a negative impact on waking function, the features of human sleep-wake architecture that best distinguish sleep continuity versus fragmentation remain elusive. In this regard, there is growing interest in characterizing sleep architecture using models of the temporal dynamics of sleep-wake stage transitions. In humans and other mammals, the state transitions defining sleep and wake bout durations have been described with exponential and power law models, respectively. However, sleep-wake stage distributions are often complex, and distinguishing between exponential and power law processes is not always straightforward. Although mono-exponential distributions are distinct from power law distributions, multi-exponential distributions may in fact resemble power laws by appearing linear on a log-log plot.

Methodology/Principal Findings

To characterize the parameters that may allow these distributions to mimic one another, we systematically fitted multi-exponential-generated distributions with a power law model, and power law-generated distributions with multi-exponential models. We used the Kolmogorov-Smirnov method to investigate goodness of fit for the “incorrect” model over a range of parameters. The “zone of mimicry” of parameters that increased the risk of mistakenly accepting power law fitting resembled empiric time constants obtained in human sleep and wake bout distributions.

Conclusions/Significance

Recognizing this uncertainty in model distinction impacts interpretation of transition dynamics (self-organizing versus probabilistic), and the generation of predictive models for clinical classification of normal and pathological sleep architecture.     

Taylor's law and body size in exploited marine ecosystems

Taylor's law (TL), which states that variance in population density is related to mean density via a power law, and density‐mass allometry, which states that mean density is related to body mass via a power law, are two of the most widely observed patterns in ecology. Combining these two laws predicts that the variance in density is related to body mass via a power law (variance‐mass allometry). Marine size spectra are known to exhibit density‐mass allometry, but variance‐mass allometry has not been investigated. We show that variance and body mass in unexploited size spectrum models are related by a power law, and that this leads to TL with an exponent slightly <2. These simulated relationships are disrupted less by balanced harvesting, in which fishing effort is spread across a wide range of body sizes, than by size‐at‐entry fishing, in which only fish above a certain size may legally be caught.     

Self‐medication in insects: current evidence and future perspectives

1. Self‐medication is an ability to consume or otherwise contact biologically active organic compounds specifically for the purpose of helping to clear a (parasitic) infection or reduce its symptoms. Consumption of these compounds may either take place before the infection is contracted (prophylactic consumption) or after the infection is contracted (therapeutic consumption). 2. An important insight is that self‐medication is a form of adaptive plasticity, and as such, consumption of the medicinal substance when uninfected must impose a fitness cost (otherwise the substance would be universally consumed). This distinguishes self‐medication from several closely related phenomena such as microbiome effects or compensatory diet choice. 3. A number of recent studies have convincingly demonstrated self‐medication within several different, distantly‐related, insect taxa. Here I review evidence of self‐medication in the wooly bear caterpillar Grammia incorrupta Edwards, the armyworm Spodoptera Guenée, the fruit fly Drosophila melanogaster Meigen, the monarch butterfly Danaus plexippus Kluk, and the honey bee Apis mellifera Linnaeus. 4. These studies show not only that self‐medication is possible, but that the target of the medication behaviour may in some cases be kin rather than self. They also reveal very few general patterns. I therefore end by discussing future prospects within the field of insect self‐medication.