Ethan P. White

On estimating the exponent of power-law frequency distributions

Power-law frequency distributions characterize a wide array of natural phenomena. In ecology, biology, and many physical and social sciences, the exponents of these power laws are estimated to draw inference about the processes underlying the phenomenon, to test theoretical models, and to scale up from local observations to global patterns. Therefore, it is essential that these exponents be estimated accurately. Unfortunately, the binning-based methods traditionally used in ecology and other disciplines perform quite poorly. Here we discuss more sophisticated methods for fitting these exponents based on cumulative distribution functions and maximum likelihood estimation. We illustrate their superior performance at estimating known exponents and provide details on how and when ecologists should use them. Our results confirm that maximum likelihood estimation outperforms other methods in both accuracy and precision. Because of the use of biased statistical methods for estimating the exponent, the conclusions of several recently published papers should be revisited.

On the use of log‐transformation vs. nonlinear regression for analyzing biological power laws

Power-law relationships are among the most well-studied functional relationships in biology. Recently the common practice of fitting power laws using linear regression (LR) on log-transformed data has been criticized, calling into question the conclusions of hundreds of studies. It has been suggested that nonlinear regression (NLR) is preferable, but no rigorous comparison of these two methods has been conducted. Using Monte Carlo simulations, we demonstrate that the error distribution determines which method performs better, with NLR better characterizing data with additive, homoscedastic, normal error and LR better characterizing data with multiplicative, heteroscedastic, lognormal error. Analysis of 471 biological power laws shows that both forms of error occur in nature. While previous analyses based on log-transformation appear to be generally valid, future analyses should choose methods based on a combination of biological plausibility and analysis of the error distribution. We provide detailed guidelines and associated computer code for doing so, including a model averaging approach for cases where the error structure is uncertain.

Evidence for a general species-time-area relationship

The species-area relationship (SAR) plays a central role in biodiversity re- search, and recent work has increased awareness of its temporal analogue, the species- time relationship (STR). Here we provide evidence for a general species-time-area rela- tionship (STAR), in which species number is a function of the area and time span of sampling, as well as their interaction. For eight assemblages, ranging from lake zooplankton to desert rodents, this model outperformed a sampling-based model and two simpler models in which area and time had independent effects. In every case, the interaction term was negative, meaning that rates of species accumulation in space decreased with the time span of sampling, while species accumulation rates in time decreased with area sampled. Al- though questions remain about its precise functional form, the STAR provides a tool for scaling species richness across time and space, for comparing the relative rates of species turnover in space and time at different scales of sampling, and for rigorous testing of mechanisms proposed to drive community dynamics. Our results show that the SAR and STR are not separate relationships but two dimensions of one unified pattern.

Zero-sum, the niche,and metacommunities: long-term dynamics of community assembly

Recent models of community assembly, structure, and dynamics have incorporated, to varying degrees, three mechanistic processes: resource limitation and interspecific competition, niche requirements of species, and exchanges between a local community and a regional species pool. Synthesizing 30 years of data from an intensively studied desert rodent community, we show that all of these processes, separately and in combination, have influenced the structural organization of this community and affected its dynamical response to both natural environmental changes and experimental perturbations. In addition, our analyses suggest that zero‐sum constraints, niche differences, and metacommunity processes are inextricably linked in the ways that they affect the structure and dynamics of this system. Explicit consideration of the interaction of these processes should yield a deeper understanding of the assembly and dynamics of other ecological communities. This synthesis highlights the role that long‐term data, especially when coupled with experimental manipulations, can play in assessing the fundamental processes that govern the structure and function of ecological communities.

Integrating spatial and temporal approaches to understanding species richness

Understanding species richness patterns represents one of the most fundamental problems in ecology. Most research in this area has focused on spatial gradients of species richness, with a smaller area of emphasis dedicated to understanding the temporal dynamics of richness. However, few attempts have been made to understand the linkages between the spatial and temporal patterns related to richness. Here, we argue that spatial and temporal richness patterns and the processes that drive them are inherently linked, and that our understanding of richness will be substantially improved by considering them simultaneously. The species–time–area relationship provides a case in point: successful description of the empirical spatio-temporal pattern led to a rapid development and testing of new theories. Other areas of research on species richness could also benefit from an explicitly spatio-temporal approach, and we suggest future directions for understanding the processes common to these two traditionally isolated fields of research.

Taking species abundance distributions beyond individuals

The species abundance distribution (SAD) is one of the few universal patterns in ecology. Research on this fundamental distribution has primarily focused on the study of numerical counts, irrespective of the traits of individuals. Here we show that considering a set of Generalized Species Abundance Distributions (GSADs) encompassing several abundance measures, such as numerical abundance, biomass and resource use, can provide novel insights into the structure of ecological communities and the forces that organize them. We use a taxonomically diverse combination of macroecological data sets to investigate the similarities and differences between GSADs. We then use probability theory to explore, under parsimonious assumptions, theoretical linkages among them. Our study suggests that examining different GSADs simultaneously in natural systems may help with assessing determinants of community structure. Broadening SADs to encompass multiple abundance measures opens novel perspectives in biodiversity research and warrants future empirical and theoretical developments.

TEMPORAL DYNAMICS IN THE STRUCTURE AND COMPOSITION OF A DESERT RODENT COMMUNITY

The rank-abundance distribution (RAD) represents the manner in which spe- cies divide resources. Community-specific division rules that determine resource allocation among species, and thereby the shape of the RAD, have been hypothesized to account for observed stability of local species richness over time. While the shape of the RAD has been well studied, the temporal dynamics of this distribution have received much less attention. Here we assess changes in the shape of the RAD through time in a desert rodent community in Arizona (USA). Because energy use may be more appropriate for studying resource division than abundance, we also evaluate an energetic equivalent of the RAD. Significant, directional trends in the shapes of both distributions are present in this com- munity. These changes are driven by trends in the relative abundances (or energy use) of Ranks 2, 3, and 4, and are significantly correlated with variation in total energy use by the community and with compositional change. Our results suggest that (1) rank-abundance and rank-energy distributions are not static and can change directionally through time, (2) species richness and rank distributions are not necessarily as intimately connected as early studies assumed, and (3) rank-abundance and rank-energy distributions are influenced by both the amount of energy available to the community and species-specific interactions.

The Combined Influence of the Local Environment and Regional Enrichment on Bird Species Richness

It is generally accepted that local species richness at a site reflects the combined influence of local and regional processes. However, most empirical studies evaluate the influence of either local environmental variables or regional enrichment but not both simultaneously. Here we demonstrate the importance of combining these processes to understand continental‐scale richness patterns in breeding birds. We show that neither regional enrichment nor the local environment in isolation is sufficient to characterize observed patterns of species richness. Combining both sets of variables into a single model results in improved model fit and the removal of residual spatial autocorrelation. At short timescales, local processes are most important for determining local richness, but as the timescale of analysis increases, regional enrichment becomes increasingly important. These results emphasize the need for increased integration of multiple scales of processes into models of species richness.

Characterizing species abundance distributions across taxa and ecosystems using a simple maximum entropy model.

The species abundance distribution (SAD) is one of themost studied patterns in ecology due to its potential insights into commonness and rarity, community assembly, and patterns of biodiversity. It is well established that communities are composed of a few common and many rare species, and numerous theoretical models have been proposed to explain this pattern. However, no attempt has been made to determine how well these theoretical characterizations capture observed taxonomic and global-scale spatial variation in the general form of the distribution. Here, using data of a scope unprecedented in community ecology, we show that a simple maximum entropy model produces a truncated log-series distribution that can predict between 83% and 93% of the observed variation in the rank abundance of species across 15 848 globally distributed communities including birds, mammals, plants, and butterflies. This model requires knowledge of only the species richness and total abundance of the community to predict the full abundance distribution, which suggests that these factors are sufficient to understand the distribution for most purposes. Since geographic patterns in richness and abundance can often be successfully modeled, this approach should allow the distribution of commonness and rarity to be characterized, even in locations where empirical data are unavailable.

Evaluating scaling models in biology using hierarchical Bayesian approaches

Theoretical models for allometric relationships between organismal form and function are typically tested by comparing a single predicted relationship with empirical data. Several prominent models, however, predict more than one allometric relationship, and comparisons among alternative models have not taken this into account. Here we evaluate several different scaling models of plant morphology within a hierarchical Bayesian framework that simultaneously fits multiple scaling relationships to three large allometric datasets. The scaling models include: inflexible universal models derived from biophysical assumptions (e.g. elastic similarity or fractal networks), a flexible variation of a fractal network model, and a highly flexible model constrained only by basic algebraic relationships. We demonstrate that variation in intraspecific allometric scaling exponents is inconsistent with the universal models, and that more flexible approaches that allow for biological variability at the species level outperform universal models, even when accounting for relative increases in model complexity.