Scaling in the growth of geographically subdivided populations: invariant patterns from a continent-wide biological survey

Scale invariance in the spatial-dynamics of biological invasions

Despite the enormous negative consequences of biological invasions, we have a limited understanding of how spatial demography during invasions creates population patterns observed at different spatial scales. Early stages of invasions, arrival and establishment, are considered distinct from the later stage of spread, but the processes of population growth and dispersal underlie all invasion phases. Here, we argue that the spread of invading species, to a first approximation, exhibits scale invariant spatial-dynamic patterns that transcend multiple spatial scales. Dispersal from a source population creates smaller satellite colonies, which in turn act as sources for secondary invasions; the scale invariant pattern of coalescing colonies can be seen at multiple scales. This self-similar pattern is referred to as “stratified diffusion” at landscape scales and the “bridgehead effect” at the global scale. The extent to which invasions exhibit such scale-invariant spatial dynamics may be limited by the form of the organisms’ dispersal kernel and by the connectivity of the habitat. Recognition of this self-similar pattern suggests that certain concepts for understanding and managing invasions might be widely transferable across spatial scales.

Macroecological dynamics of gut microbiota

The gut microbiota is now widely recognized as a dynamic ecosystem that plays an important role in health and disease. Although current sequencing technologies make it possible to explore how relative abundances of host-associated bacteria change over time, the biological processes governing microbial dynamics remain poorly understood. Therefore, as in other ecological systems, it is important to identify quantitative relationships describing various aspects of gut microbiota dynamics. In the present study, we use multiple high-resolution time series data obtained from humans and mice to demonstrate that, despite their inherent complexity, gut microbiota dynamics can be characterized by several robust scaling relationships. Interestingly, the observed patterns are highly similar to those previously identified across diverse ecological communities and economic systems, including the temporal fluctuations of animal and plant populations and the performance of publicly traded companies. Specifically, we find power-law relationships describing short- and long-term changes in gut microbiota abundances, species residence and return times, and the correlation between the mean and the temporal variance of species abundances. The observed scaling laws are altered in mice receiving different diets and are affected by context-specific perturbations in humans. We use the macroecological relationships to reveal specific bacterial taxa, the dynamics of which are substantially perturbed by dietary and environmental changes. Overall, our results suggest that a quantitative macroecological framework will be important for characterizing and understanding the complex dynamics of diverse microbial communities.

Climate-driven ecological stability as a globally shared cause of Late Quaternary megafaunal extinctions: the Plaids and Stripes Hypothesis

Controversy persists about why so many large-bodied mammal species went extinct around the end of the last ice age. Resolving this is important for understanding extinction processes in general, for assessing the ecological roles of humans, and for conserving remaining megafaunal species, many of which are endangered today. Here we explore an integrative hypothesis that asserts that an underlying cause of Late Quaternary megafaunal extinctions was a fundamental shift in the spatio-temporal fabric of ecosystems worldwide. This shift was triggered by the loss of the millennial-scale climate fluctuations that were characteristic of the ice age but ceased approximately 11700 years ago on most continents. Under ice-age conditions, which prevailed for much of the preceding 2.6 Ma, these radical and rapid climate changes prevented many ecosystems from fully equilibrating with their contemporary climates. Instead of today's 'striped' world in which species' ranges have equilibrated with gradients of temperature, moisture, and seasonality, the ice-age world was a disequilibrial 'plaid' in which species' ranges shifted rapidly and repeatedly over time and space, rarely catching up with contemporary climate. In the transient ecosystems that resulted, certain physiological, anatomical, and ecological attributes shared by megafaunal species pre-adapted them for success. These traits included greater metabolic and locomotory efficiency, increased resistance to starvation, longer life spans, greater sensory ranges, and the ability to be nomadic or migratory. When the plaid world of the ice age ended, many of the advantages of being large were either lost or became disadvantages. For instance in a striped world, the low population densities and slow reproductive rates associated with large body size reduced the resiliency of megafaunal species to population bottlenecks. As the ice age ended, the downsides of being large in striped environments lowered the extinction thresholds of megafauna worldwide, which then increased the vulnerability of individual species to a variety of proximate threats they had previously tolerated, such as human predation, competition with other species, and habitat loss. For many megafaunal species, the plaid-to-stripes transition may have been near the base of a hierarchy of extinction causes whose relative importances varied geographically, temporally, and taxonomically.

Dynamic relationships between body size, species richness, abundance, and energy use in a shallow marine epibenthic faunal community

We study the temporal variation in the empirical relationships among body size (S), species richness (R), and abundance (A) in a shallow marine epibenthic faunal community in Coliumo Bay, Chile. We also extend previous analyses by calculating individual energy use (E) and test whether its bivariate and trivariate relationships with S and R are in agreement with expectations derived from the energetic equivalence rule. Carnivorous and scavenger species representing over 95% of sample abundance and biomass were studied. For each individual, body size (g) was measured and E was estimated following published allometric relationships. Data for each sample were tabulated into exponential body size bins, comparing species-averaged values with individual-based estimates which allow species to potentially occupy multiple size classes. For individual-based data, both the number of individuals and species across body size classes are fit by a Weibull function rather than by a power law scaling. Species richness is also a power law of the number of individuals. Energy use shows a piecewise scaling relationship with body size, with energetic equivalence holding true only for size classes above the modal abundance class. Species-based data showed either weak linear or no significant patterns, likely due to the decrease in the number of data points across body size classes. Hence, for individual-based size spectra, the SRA relationship seems to be general despite seasonal forcing and strong disturbances in Coliumo Bay. The unimodal abundance distribution results in a piecewise energy scaling relationship, with small individuals showing a positive scaling and large individuals showing energetic equivalence. Hence, strict energetic equivalence should not be expected for unimodal abundance distributions. On the other hand, while species-based data do not show unimodal SRA relationships, energy use across body size classes did not show significant trends, supporting energetic equivalence.

Fat-tailed fluctuations in the size of organizations: the role of social influence

Organizational growth processes have consistently been shown to exhibit a fatter-than-Gaussian growth-rate distribution in a variety of settings. Long periods of relatively small changes are interrupted by sudden changes in all size scales. This kind of extreme events can have important consequences for the development of biological and socio-economic systems. Existing models do not derive this aggregated pattern from agent actions at the micro level. We develop an agent-based simulation model on a social network. We take our departure in a model by a Schwarzkopf et al. on a scale-free network. We reproduce the fat-tailed pattern out of internal dynamics alone, and also find that it is robust with respect to network topology. Thus, the social network and the local interactions are a prerequisite for generating the pattern, but not the network topology itself. We further extend the model with a parameter that weights the relative fraction of an individual's neighbours belonging to a given organization, representing a contextual aspect of social influence. In the lower limit of this parameter, the fraction is irrelevant and choice of organization is random. In the upper limit of the parameter, the largest fraction quickly dominates, leading to a winner-takes-all situation. We recover the real pattern as an intermediate case between these two extremes.

Contingency and statistical laws in replicate microbial closed ecosystems

Contingency, the persistent influence of past random events, pervades biology. To what extent, then, is each course of ecological or evolutionary dynamics unique, and to what extent are these dynamics subject to a common statistical structure? Addressing this question requires replicate measurements to search for emergent statistical laws. We establish a readily replicated microbial closed ecosystem (CES), sustaining its three species for years. We precisely measure the local population density of each species in many CES replicates, started from the same initial conditions and kept under constant light and temperature. The covariation among replicates of the three species densities acquires a stable structure, which could be decomposed into discrete eigenvectors, or "ecomodes." The largest ecomode dominates population density fluctuations around the replicate-average dynamics. These fluctuations follow simple power laws consistent with a geometric random walk. Thus, variability in ecological dynamics can be studied with CES replicates and described by simple statistical laws.

Setting the absolute tempo of biodiversity dynamics

Neutral biodiversity theory has the potential to contribute to our understanding of how macroevolutionary dynamics influence contemporary biodiversity, but there are issues regarding its dynamical predictions that must first be resolved. Here we address these issues by extending the theory in two ways using a novel analytical approach: (1) we set the absolute tempo of biodiversity dynamics by explicitly incorporating population-level stochasticity in abundance; (2) we allow new species to arise with more than one individual. Setting the absolute tempo yields quantitative predictions on biodiversity dynamics that can be tested using contemporary and fossil data. Allowing incipient-species abundances greater than one individual yields predictions on how these dynamics, and the form of the species-abundance distribution, are affected by multiple speciation modes. We apply this new model to contemporary and fossil data that encompass 30 Myr of macroevolution for planktonic foraminifera. By synthesizing the model with these empirical data, we present evidence that dynamical issues with neutral biodiversity theory may be resolved by incorporating the effects of environmental stochasticity and incipient-species abundance on biodiversity dynamics.

Scaling metabolic rate fluctuations

Complex ecological and economic systems show fluctuations in macroscopic quantities such as exchange rates, size of companies or populations that follow non-Gaussian tent-shaped probability distributions of growth rates with power-law decay, which suggests that fluctuations in complex systems may be governed by universal mechanisms, independent of particular details and idiosyncrasies. We propose here that metabolic rate within individual organisms may be considered as an example of an emergent property of a complex system and test the hypothesis that the probability distribution of fluctuations in the metabolic rate of individuals has a "universal" form regardless of body size or taxonomic affiliation. We examined data from 71 individuals belonging to 25 vertebrate species (birds, mammals, and lizards). We report three main results. First, for all these individuals and species, the distribution of metabolic rate fluctuations follows a tent-shaped distribution with power-law decay. Second, the standard deviation of metabolic rate fluctuations decays as a power-law function of both average metabolic rate and body mass, with exponents -0.352 and -1/4 respectively. Finally, we find that the distributions of metabolic rate fluctuations for different organisms can all be rescaled to a single parent distribution, supporting the existence of general principles underlying the structure and functioning of individual organisms.

QUANTIFYING POPULATION SUBSTRUCTURE: EXTENDING THE GRAPH-THEORETIC APPROACH

Among the few universal themes in ecology is that resources, energy, and organisms themselves, are patchily distributed. This patchy distribution imposes a need for some level of dispersal or connectivity among spatially separate patches in order to allow organisms to acquire sufficient resources for survival. To date, general patterns of connectivity have not emerged. This is, in part, because different species respond to different scales of patchiness. I propose an extension of the graph-theoretic approach to control for such differences and reveal potential generalities about how natural populations are organized. Using statistical methods and simple applications of graph theory, continuum percolation, and metapopulation models, I demonstrate a pattern of hierarchical clustering among populations in both a plant-pathogen system at an extent of 1000 m and gene flow in a salamander species across a subcontinental range. Results suggest that some patches or populations have a disproportionately high importance to the maintenance of overall connectivity in the system within and across scales.

Power-law scaling in dimension-to-biomass relationship of fish schools

Motivated by the finding that there is some biological universality in the relationship between school geometry and school biomass of various pelagic fishes in various conditions, I here establish a scaling law for school dimensions: the school diameter increases as a power-law function of school biomass. The power-law exponent is extracted through the data collapse, and is close to 35. This value of the exponent implies that the mean packing density decreases as the school biomass increases, and the packing structure displays a mass-fractal dimension of 53. By exploiting an analogy between school geometry and polymer chain statistics, I examine the behavioral algorithm governing the swollen conformation of large-sized schools of pelagics, and I explain the value of the exponent.

Scaling and power-laws in ecological systems

Scaling relationships (where body size features as the independent variable) and power-law distributions are commonly reported in ecological systems. In this review we analyze scaling relationships related to energy acquisition and transformation and power-laws related to fluctuations in numbers. Our aim is to show how individual level attributes can help to explain and predict patterns at the level of populations that can propagate at upper levels of organization. We review similar relationships also appearing in the analysis of aquatic ecosystems (i.e. the biomass spectra) in the context of ecological invariant relationships (i.e. independent of size) such as the `energetic equivalence rule' and the `linear biomass hypothesis'. We also discuss some power-law distributions emerging in the analysis of numbers and fluctuations in ecological attributes as they point to regularities that are yet to be integrated with traditional scaling relationships and which we foresee as an exciting area of future research.