A Strong Test of the Maximum Entropy Theory of Ecology
Abstract
The maximum entropy theory of ecology (METE) is a unified theory of biodiversity that predicts a large number of macroecological patterns using information on only species richness, total abundance, and total metabolic rate of the community. We evaluated four major predictions of METE simultaneously at an unprecedented scale using data from 60 globally distributed forest communities including more than 300,000 individuals and nearly 2,000 species. METE successfully captured 96% and 89% of the variation in the rank distribution of species abundance and individual size but performed poorly when characterizing the size-density relationship and intraspecific distribution of individual size. Specifically, METE predicted a negative correlation between size and species abundance, which is weak in natural communities. By evaluating multiple predictions with large quantities of data, our study not only identifies a mismatch between abundance and body size in METE but also demonstrates the importance of conducting strong tests of ecological theories.
Introduction
The structure of ecological communities can be quantified using a variety of relationships, including many of the most well-studied patterns in ecology, such as the distribution of individuals among species (the species abundance distribution [SAD]), the increase in species richness with area (the species-area relationship [SAR]), and the distributions of energy consumption and body size (Brown 1995; Rosenzweig 1995; McGill et al. 2007; White et al. 2007). With the increasing consensus that these patterns are not fully independent, a growing number of unified theories have been proposed to identify links between the patterns and unite them under a single framework (e.g., Hanski and Gyllenberg 1997; Hubbell 2001; Harte 2011; for a review, see McGill 2010). Among these unified theories there are generally two different approaches, one based on processes and another based on constraints. With the process-based approach, characteristics of the community are captured by explicitly modeling a few key ecological processes (e.g., Hanski and Gyllenberg 1997; Hubbell 2001). While this approach has the potential to directly establish connections between patterns and processes, it has been found that the same empirical patterns can result from different processes (Cohen 1968; Pielou 1975), and process-specific parameters are often hard to obtain (Hubbell 2001; Jones and Muller-Landau 2008). Alternatively, the constraint-based approach suggests that many macroecological patterns are emergent statistical properties arising from general constraints on the system, while processes are only indirectly incorporated through their effect on the constraints (e.g., Harte 2011; Locey and White 2013). This approach attempts to provide a general explanation of the observed patterns that does not rely on specific processes, which allows predictions to be made with little detailed information about the system.
One of the newest and most parsimonious constraint-based approaches is the maximum entropy theory of ecology (METE; Harte et al. 2008, 2009; Harte 2011). METE adopts the maximum entropy principle from information theory, which identifies the most likely (least biased) state of a system given a set of constraints (Jaynes 2003). Assuming that the allocation of individuals and energy consumption within a community is constrained by three state variables (total species richness, total number of individuals, and total energy consumption), METE makes predictions for the SAD as well as multiple patterns related to energy use. Spatial patterns such as the SAR and the endemics-area relationship can also be predicted with an additional constraint on the area sampled (Harte et al. 2008, 2009; Harte 2011). METE is one of the growing number of theoretical approaches that attempt to synthesize traditionally distinct areas of macroecology dealing with the distributions of individuals and the distributions of energy and biomass (Dewar and Porté 2008; Morlon et al. 2009; O’Dwyer et al. 2009) and thus provides a very general characterization of the structure of ecological systems. With no specific assumptions about biological processes, it can potentially be applied to any community where the values of the state variables can be obtained.
Previous studies have evaluated the performance of METE with separate data sets for different patterns and have shown that METE generally provides good characterizations of these patterns across geographical locations and taxonomic groups (Harte et al. 2008, 2009; Harte 2011; White et al. 2012a; McGlinn et al. 2013). However, these tests are relatively weak as they focus on one pattern at a time (McGill 2003). As a unified theory with multiple predictions, METE allows stronger tests to be made by testing the ability of the theory to characterize multiple patterns simultaneously for the same data (McGill 2003; McGill et al. 2006). In this study, we conduct a strong test of the nonspatial predictions of METE using data from 60 globally distributed forest communities to simultaneously evaluate four predictions of the theory (fig. 1), including the SAD (the distribution of individuals among species) and energetic analogs of the individual size distribution (ISD; the distribution of body size among individuals regardless of their species identity; Enquist and Niklas 2001; Muller-Landau et al. 2006), the size-density relationship (SDR; the correlation between species abundance and average individual size within species; Cotgreave 1993), and the intraspecific ISD (iISD; the distribution of body size among individuals within a species; Gouws et al. 2011). Our analysis shows mixed support for METE across its four predictions, with METE successfully capturing the variation in some patterns while failing to do so in others. We discuss the ecological implications of our findings as well as the importance of conducting strong multipattern tests in the evaluation of ecological theories. Illustration of the four patterns with data from Barro Colorado Island. A, Species abundance distribution (presented as a rank abundance distribution); B, individual size distribution (ISD); C, size-density relationship; D, intraspecific ISD of the most abundant species, Hybanthus prunifolius. Gray circles or bars in each panel represent empirical observations, and the magenta curve represents the maximum entropy theory of ecology’s prediction. DBH = diameter at breast height.
Methods
Predicted Patterns of METE
METE assumes that allocation of individuals and energy consumption within a community is constrained by three state variables: species richness (S0), total number of individuals (N0), and total metabolic rate summed over all individuals in the community (E0; Harte et al. 2008, 2009; Harte 2011). Define 
as the joint probability that a species randomly picked from the community has abundance n and an individual randomly picked from such a species has metabolic rate between 
; two constraints are then established on the ratio between the state variables:
The forms of the four macroecological patterns that METE predicts can then be derived from 
(see Harte 2011 and app. A for a detailed derivation) and are given by the following four equations. The SAD takes the form
. Conditioned on abundance n, the species-level energy distribution (which is the energetic equivalent of the iISD) is given by
then gives the average species energy distribution (which is the energetic equivalent of the SDR), that is, the expected average metabolic rate (size) for individuals within a species with abundance n:
Data
METE predicts the iISD to be an exponential distribution (eq. [5]; also see fig. 1D), where the smallest size class is the most abundant, regardless of species identity or abundance. However, most animal species exhibit interior modes of adult body size (e.g., Koons et al. 2009; Gouws et al. 2011; but see Harte 2011) and large variation in minimum (and maximum) body size among species associated with these modal values (Gouws et al. 2011). In other words, the body sizes of conspecifics are clustered around some intermediate value, while individuals that are much larger or smaller are rare. Consequently, assembling all individuals across species in such communities often yields multimodal ISDs (Thibault et al. 2011), as opposed to the monotonically decreasing form predicted by METE (eq. [4]; also see fig. 1B). As such, animal communities are expected a priori to violate two of the predictions of METE. Therefore, to ensure that the performance of METE was not trivially rejected because of the life-history trait of determinate growth, in our analysis we focused exclusively on trees, which are known to have iISDs (Condit et al. 1998) and ISDs (Enquist and Niklas 2001; Muller-Landau et al. 2006) that are well characterized by monotonically declining distributions and which arguably have the greatest prevalence of high-quality individual-level size data among indeterminately growing taxonomic groups.
We compiled forest plot data from previous publications, publicly available databases, and personal communications (Table 1). All plots have been fully surveyed, with size measurements for all individuals above plot-specific minimum thresholds. For those plots where surveys have been conducted multiple times, we adopted data from the most recent survey unless otherwise specified (see Table 1). We excluded records of ferns, palms, and herbs if they existed. Individuals that were dead, not identified to species/morphospecies, and/or missing size measurements were excluded. Individuals with size measurements below or equal to the designated minimum thresholds were excluded as well, because it is unclear whether these size classes were thoroughly surveyed. Overall, our analysis encompassed 60 plots that were at least 1 ha in size and had a richness of at least 14 (Table 1), with 1,943 species/morphospecies and 379,022 individuals in total.
| Data set | Description | Area of individual plots (ha) | No. plots | Survey year | References |
|---|---|---|---|---|---|
| Serimbu | Tropical rain forest | 1 | 2 | 1995a | Kohyama et al. 2001, 2003; Lopez-Gonzalez 2009, 2011 |
| La Selva | Tropical wet forest | 2.24 | 5 | 2009 | Baribault et al. 2011a, 2011b |
| ACA Amazon Forest Inventories | Tropical moist forest | 1 | 1 | 2000–2001 | Pitman et al. 2005 |
| Barro Colorado Island | Tropical moist forest | 50 | 1 | 2010 | Condit 1998b; Hubbell et al. 1999, 2005 |
| DeWalt Bolivia forest plots | Tropical moist forest | 1 | 2 | NA | DeWalt et al. 1999 |
| Lahei | Tropical moist forest | 1 | 3 | 1998 | Nishimura and Suzuki 2001; Nishimura et al. 2006; Lopez-Gonzalez 2009, 2011 |
| Luquillo | Tropical moist forest | 16 | 1 | 1994–1996b | Zimmerman et al. 1994; Thompson et al. 2002 |
| Sherman | Tropical moist forest | 5.96 | 1 | 1999 | Condit 1998a; Pyke et al. 2001; Condit et al. 2004 |
| Cocoli | Tropical moist forest | 4 | 1 | 1998 | Condit 1998a; Pyke et al. 2001; Condit et al. 2004 |
| Western Ghats | Wet evergreen/moist/dry deciduous forests | 1 | 34 | 1996–1997 | Ramesh et al. 2010 |
| UCSC FERP | Mediterranean mixed evergreen forest | 6 | 1 | 2007 | Gilbert et al. 2010 |
| Shirakami | Beech forest | 1 | 2 | 2006 | Nakashizuka et al. 2003; Lopez-Gonzalez 2009, 2011 |
| Oosting | Hardwood forest | 6.55 | 1 | 1989 | Reed et al. 1993; Palmer et al. 2007 |
| North Carolina forest plots | Mixed hardwoods/pine forest | 1.3–5.65 | 5 | 1990–1993c | Peet and Christensen 1987; McDonald et al. 2002; Xi et al. 2008 |
Analyses
The scaling relationship between diameter and metabolic rate can be described with good approximation by metabolic theory as 
, where B is metabolic rate, D is diameter, T is temperature, E is activation energy, and k is Boltzmann’s constant (West et al. 1999; Gillooly et al. 2001). Assuming that E is constant across species and that T is constant within a community, the temperature-dependent term 
is constant within a community and can be dropped when the metabolic rates of individuals are rescaled. We thus used 
as the surrogate for individual metabolic rate, where Dmin is the diameter of the smallest individual in the community, which sets the minimal individual metabolic rate to be 1 following METE’s assumption (see eq. [2]). Applying alternative models that more accurately capture nonlinearities between diameter, mass and metabolic rate did not have any qualitative effect on our results (app. B). For individuals with multiple stems, we adopted the pipe model to combine the records, that is, 
, where di’s were diameters of individual stems (Ernest et al. 2009). Since metabolic rate scales as D2, the pipe model preserves the total area as well as the total metabolic rate for all stems combined.
We obtained the Lagrange multipliers λ1 and λ2 in each community with inputs S0, N0, and E0 (i.e., the sum over the rescaled individual metabolic rates; see app. A). Predictions for the four ecological patterns were obtained from equations (3)–(6) and further transformed to facilitate comparison with observations. For the SAD and the ISD, we converted the predicted probability distributions (eqq. [3], [4]) to rank distributions of abundance (i.e., abundance at each rank from the most abundant species to the least abundant species) and size (i.e., scaled metabolic rate at each rank from the largest individual to the smallest individual across all species; Harte et al. 2008, 2011; White et al. 2012a), which were compared with the empirical rank distributions of abundance and size. For the SDR, predicted average metabolic rate was obtained from equation (6) for species with abundance n, which was compared with the observed average metabolic rate for that species. For the iISD, we converted the predicted exponential distribution (eq. [5]) into a rank distribution of individual size for each species and compared the scaled metabolic rate predicted at each rank to the observed value. Alternative analyses for the two continuous distributions, the ISD and the iISD, did not change our results (app. C).
The explanatory power of METE for each pattern was quantified using the coefficient of determination R2, which was calculated as
While R2 between predicted and observed values provides an intuitive measure of the predictive power of the theory, it ignores the variation that can arise from random sampling even when the predicted distribution is valid. To address this issue, we conducted a bootstrap analysis, where we drew 500 random samples from the predicted distribution for each pattern (eq. [3] for the SAD, eq. [4] for the ISD, and eq. [5] for the SDR and the iISD) and examined the fit of the theory to the bootstrap samples using both R2 and the Kolmogorov-Smirnov statistic (see app. E for details). If METE fits the empirical data as well as it fits the bootstrap samples, then the theory matches the data, and the residual variation is consistent with random sampling. If instead METE fits the bootstrap samples better than the empirical data, it indicates that there are meaningful deviations of empirical data from the theory’s predictions. By comparing the fits to empirical data to those for data simulated from the theory, this analysis provides additional insights into patterns like the SAD that are expected to be fit well by many theories (Connolly et al. 2009; Locey and White 2013) and into patterns like the iISD, where large amounts of variation about the predicted values may be expected due to sampling.
Python code to replicate our analyses together with a processed subset of data sets are deposited in the Dryad Digital Repository:
Results
The results for all forest plots combined are summarized in figure 2, with observations plotted against predictions for each macroecological pattern. METE provides excellent predictions for the SAD ( Maximum entropy theory of ecology’s predictions plotted against empirical observations across 60 communities for the species abundance distribution (A; each data point is the abundance of a species at a single rank in one community), the individual size distribution (ISD; B; each data point is the metabolic rate of an individual at a single rank in one community), the size-density relationship (SDR; C; each data point is the average metabolic rate within one species in one community), and the intraspecific ISD (iISD; D; each data point is the metabolic rate of an individual at a single rank belonging to a specific species in one community). The diagonal black line in each panel is the 1∶1 line. The points are color coded to reflect the density of neighboring points, with warm (red) colors representing higher densities and cold (blue) colors representing lower densities. Each inset reflects the distribution of R2 among 60 communities from below 0 (left) to 1 (right). DBH = diameter at breast height.
) and the ISD (
), although the largest size classes deviate slightly but consistently in the ISD. However, the SDR (
) and the iISD (
) are not well characterized by the theory.
Further examination of the four macroecological patterns within each community (see the supplementary figures; also see insets in fig. 2) confirms METE’s ability to consistently characterize the SAD (all 
, 59 of 60 
) and the ISD (all 
, 49 of 60 
) as well as its inadequacy in characterizing the SDR (all 
) and the iISD (maximal 
, 49 of 60 
). The high R2 for the SAD and the ISD is not an artifact of ranking the data (figs. C1, D1). Results from bootstrap analysis (app. E) are also largely consistent with the direct interpretation of the goodness-of-fit statistics. METE provides comparable characterization for the empirical and the bootstrap SADs in most communities, while its fit to the empirical SDRs and the iISDs is consistently worse than that to the bootstrap samples (fig. E2). For the ISD, however, the analysis reveals that METE characterizes bootstrap samples consistently better than its fit to empirical data (fig. E2), which implies that the empirical ISD significantly deviates from METE’s prediction despite the theory’s ability to capture the general shape of the pattern (fig. 2B; supplementary figures). This is consistent with model comparison in appendix F, where we show that alternative models provide a better fit to the distribution (Table F1).
Discussion
Macroecological theories increasingly attempt to make predictions across numerous ecological patterns (McGill 2010) by either directly modeling ecological processes or imposing constraints on the system. Among the constraint-based theories, METE is unique in that it makes simultaneous predictions for two distinct sets of ecological patterns, synthesizing traditionally separate areas of macroecology dealing with distributions of individuals and distributions related to body size and energy use (see also Dewar and Porté 2008; Morlon et al. 2009; O’Dwyer et al. 2009). Using only information on species richness, total abundance, and total energy use as inputs, METE attempts to characterize various aspects of community structure without additional tunable parameters or assumptions, making it one of the most parsimonious of the current unified theories.
Our analysis shows that METE accurately captures the general shape of the SAD (allocation of individuals among species) and the ISD (allocation of energy/biomass among individuals) within and among 60 forest communities (fig. 2A, 2B; supplementary figures). The SAD and the ISD are among the most well-studied patterns in ecology, and numerous models exist for both patterns. For instance, with metabolic theory and demographic equilibrium models, Muller-Landau et al. (2006) identified four possible predictions for the ISD under different assumptions of growth and mortality rates. For the SAD, more than 20 models have been proposed (Marquet et al. 2003; McGill et al. 2007), ranging from purely statistical to mechanistic.
Our study demonstrates METE’s high predictive power for these two patterns, but it does not imply that it is the best model when each pattern is considered independently. Indeed, our results reveal a consistent departure of individuals in the largest size class from the ISD predicted by METE (fig. 2B, supplementary figures), which may result from mortality unrelated to energy use (Muller-Landau et al. 2006). Bootstrap analysis (app. E) further confirms that such deviation is more severe than expected from the effect of random sampling alone. The discrepancy between the high R2 of the ISD both within and across communities and the seemingly poor fit of the pattern revealed by bootstrapping results from the two different ways that goodness of fit is evaluated by the two analyses. While METE is able to predict the rank size of individuals (fig. 2B; supplementary figures) as well as the relative frequency of size bins (fig. C1) with high accuracy (illustrated by the high R2 between predicted and observed values), the empirical ISDs are still significantly different from the predicted distribution (illustrated by higher deviation of empirical data from the predicted form when compared with bootstrap samples). Indeed, while METE has been shown to frequently outperform the most common model of the SAD (the lognormal) for a variety of taxonomic groups, including plants (White et al. 2012a), model comparisons for the ISD using Akaike’s information criterion suggest that the maximum likelihood Weibull distribution (one of the distributions for tree diameter in Muller-Landau et al. 2006) almost always outperforms METE (although METE’s performance is comparable to that of the other two distributions, the exponential and the Pareto; see app. F).
Quantitatively comparing theories that make multiple predictions is challenging, and there is no general approach for properly comparing models that make different numbers of predictions. When comparing general theories to single prediction models with multiple tunable parameters, it is not surprising that theories such as METE fail to provide the best quantitative fit (White et al. 2012b). However, as a constraint-based unified theory, METE’s strength lies in its ability to link together ecological phenomena that were previously considered distinct and to make predictions based on first principles with minimal inputs. The general agreement between METE’s predictions and the observed SAD and ISD (as measured by the R2 for the rank distributions) supports the notion that the majority of variation in these macroecological patterns can be characterized by variation in the state variables S0, N0, and E0 alone (Harte 2011; Supp et al. 2012; White et al. 2012a).
While METE performs well in characterizing the SAD and the ISD, it performs poorly when predicting the distribution of energy at the species level (fig. 2C, 2D; supplementary figures). This is not that surprising given that the iISD and the SDR (which is the expectation of the iISD) provide a more detailed perspective on the community structure by examining the intercorrelation of abundance and size. The deviations of the empirical patterns from the predictions reveal a mismatch between the predicted metabolic rate of individuals and their species’ abundances. METE predicts a monotonically decreasing relationship between species abundance and average intraspecific metabolic rate, that is, species with higher abundance are also smaller in size on average and are more likely to contain smaller individuals (eqq. [5, 6]; fig. 1C). Evaluating the total (instead of average) intraspecific metabolic rate, this relationship translates roughly into Damuth’s energetic equivalence rule (Damuth 1981), where the total energy consumption within a species does not depend on species identity or abundance (Harte et al. 2008; Harte 2011). While Damuth’s rule has been argued to apply at global scales (Damuth 1981; White et al. 2007), our results indicate that it does not hold locally, in concordance with a number of previous studies (Brown and Maurer 1987; Blackburn and Gaston 1997; White et al. 2007).
The consistency of our results across 60 forest communities (as well as confirmative evidence from a concurrent study of a single herbaceous plant community; Newman et al. 2014) provides strong evidence for METE’s mixed performance among the four macroecological patterns. However, several limitations of the study are worth noting. First, we analyzed only a single taxonomic group (trees). This was in part because individual-level size data collected in standardized ways is available for a large number of tree communities and in part based on a prior knowledge that the form of the ISD and the iISD (Condit et al. 1998; Enquist and Niklas 2001; Muller-Landau et al. 2006) had a reasonable chance of being well characterized by the theory (see “Methods”). While we know that the SAD predictions of the theory perform well in general (White et al. 2012a), further tests are necessary to determine whether the simultaneous good fit of the ISD predictions is supported in other taxonomic groups. There is some evidence that this result holds in invertebrate communities (Harte 2011). Second, we estimated the metabolic rate of individuals based on predictions of metabolic theory rather than direct measurement. While our results were not sensitive to the use of other equations used for estimating metabolic rate (app. B), it is possible that directly measured metabolic rates could result in different fits to the theory (but see Newman et al. 2014, which adopts a different method to obtain metabolic rate yet reaches similar conclusions).
Models and theories can be evaluated at multiple levels that yield different strengths of inference (McGill 2003; McGill et al. 2006), progressing from matching theory to empirical observations on a single pattern, testing against a null hypothesis, evaluating multiple a priori predictions, and eventually comparing between multiple competing models. With quantitative predictions on various ecological patterns, METE and other unified theories allow for simultaneous examination of multiple predictions, which provides a much stronger test than curve fitting for a single pattern and can often reveal important insight into theories that are otherwise overlooked by single-pattern tests (e.g., Adler 2004). As a comprehensive analysis of the performance of METE in predicting abundance and energy distributions in the same data sets, our study demonstrates the importance of moving toward stronger tests in ecology, especially when multiple intercorrelated predictions are available; while previous studies have shown that METE does an impressive job characterizing a single pattern (White et al. 2012a; McGlinn et al. 2013), concurrently evaluating all predictions of the theory identifies a slight yet consistent discrepancy between the observed and the predicted size distribution as well as a mismatch between species’ abundance and individual size.
That METE fails to provide good characterization of all four patterns of community structure and performs more poorly than alternative models in some cases can be interpreted in two ways. First, the aspects of community structure that are poorly characterized by the theory may be more adequately characterized by explicitly modeling ecological processes. For example, O’Dwyer et al. (2009) has developed a model that incorporates individual demographic rates of birth, death, and growth, which likewise yields predictions of abundance and body size distributions. It is worth noting, however, that the process-based approach and the constraint-based approach do not have to be mutually exclusive. While O’Dwyer et al. (2009) suggested that size-related patterns may reflect ecological processes, the agreement between their model and METE in the predicted SAD (both log series) as well as METE’s performance for the ISD support the idea that information in the underlying processes can be summarized in constraints alone for some macroecological patterns. Alternatively, the constraint-based approach may be sufficient in characterizing patterns of abundance and body size, but the current form of METE may be incorrect. Specifically, the limitations revealed in our analyses may be remedied either by relaxing the current constraints to remove the implicit negative correlation between species-level average body size and abundance (fig. 1C) from the theory or by adding additional constraints to the system so that energetic equivalence among species no longer holds (Harte and Newman 2014). While the success of METE in characterizing the general shape of the SAD and the ISD adds to the growing support for the constraint-based approach for studying macroecological patterns, further work is clearly needed to develop unified theories for community structure whether they are based on specific biological processes or emergent statistical properties.
Acknowledgments
We thank J. Harte, E. Newman, and the rest of the Harte laboratory as well as members of the Weecology laboratory for extensive feedback on this research, for general insights into MaxEnt, and for being incredibly supportive of our efforts to evaluate the maximum entropy theory of ecology. We thank S. R. Connolly and two anonymous reviewers for their extremely helpful comments, which have helped to make our conclusions more robust. N. G. Swenson provided data for wood density in the Luquillo forest plot and gave insightful comments. R. K. Peet provided data for the North Carolina forest plots. The Serimbu (provided by T. Kohyama), Lahei (provided by T. B. Nishimura), and Shirakami (provided by T. Nakashizuka) data sets were obtained from the PlotNet Forest Database. The ACA Amazon Forest Inventories (provided by N. Pitman) and DeWalt Bolivia (provided by S. DeWalt) data sets where obtained from SALVIAS (Synthesis and Analysis of Local Vegetation Inventories across Scales). The Barro Colorado Island Forest Dynamics Research Project was made possible by US National Science Foundation (NSF) grants to S. P. Hubbell (DEB-0640386, DEB-0425651, DEB-0346488, DEB-0129874, DEB-00753102, DEB-9909347, DEB-9615226, DEB-9615226, DEB-9405933, DEB-9221033, DEB-9100058, DEB-8906869, DEB-8605042, DEB-8206992, and DEB-7922197); by support from the Center for Tropical Forest Science, the Smithsonian Tropical Research Institute, the John D. and Catherine T. MacArthur Foundation, the Mellon Foundation, the Small World Institute Fund, and numerous private individuals; and through the hard work of more than 100 people from 10 countries over the last 2 decades. The University of California, Santa Cruz (UCSC), Forest Ecology Research Plot was made possible by NSF grants to G. S. Gilbert (DEB-0515520 and DEB-084259), the Pepper-Giberson Chair Fund, the University of California, and the hard work of dozens of UCSC students. These two projects are part of the Center for Tropical Forest Science, a global network of large-scale demographic tree plots. The Luquillo Experimental Forest Long-Term Ecological Research Program was supported by grants BSR-8811902, DEB-9411973, DEB-0080538, DEB-0218039, DEB-0620910, and DEB-0963447 from the NSF to the Institute for Tropical Ecosystem Studies, University of Puerto Rico, and to the International Institute of Tropical Forestry, USDA Forest Service, as part of the Luquillo Long-Term Ecological Research Program. Funds were contributed for the 2000 census by the Andrew Mellon Foundation and by the Center for Tropical Forest Science. The US Forest Service (Department of Agriculture) and the University of Puerto Rico gave additional support. We also thank the many technicians, volunteers, and interns who have contributed to data collection in the field. This research was supported by a CAREER award from the NSF to E.P.W. (DEB-0953694).
Appendix A: Derivation for the Equations
The equations we adopted in our analysis (see “Predicted Patterns of METE” in the main text) are largely identical to those in Harte (2011), except for a few minor modifications. Below we briefly summarize the derivations and derive those that are slightly different. See Harte (2011) for the step-by-step procedure.
The distribution of central significance on which all other predictions are based is 
, the joint probability that a species randomly picked from the community has abundance n and an individual randomly picked from such a species has a metabolic rate between 
. By maximizing information entropy
can be obtained as
Derivation for Equations Not Found in Harte (2011)
Species-Abundance Distribution (SAD; Eq. [3] in This Study)
From equation (7.23) in Harte (2011):
.Given that E0 is large, the second term in the numerator, 
, is much smaller than the first term, 
. Dropping the second term,
The Energetic Analog of the Individual Size Distribution (Eq. [4] in This Study)
From equation (7.6) in Harte (2011):
. Note that equation (A4) is not identical to equation (7.24) in Harte (2011), which contains a minor error (J. Harte, personal communication). However, the trivial difference is unlikely to invalidate or significantly change any published results.
While we converted diameter (D) to metabolic rate (B) with 
in our analyses, alternative relationships between diameter and metabolic rate have been proposed. Specifically, it has been suggested that the aboveground biomass of tropical trees is a function of diameter, wood density, and forest type (Chave et al. 2005), while the relationship between aboveground biomass and metabolic rate is a biphasic mixed-power function (Mori et al. 2010). Here, we demonstrate that adopting this alternative scaling relationship does not quantitatively change our results.
We compiled species-specific wood densities (wood-specific gravity [WSG]) from previous publications (Reyes et al. 1992; Chave et al. 2009; Zanne et al. 2009; Wright et al. 2010; Swenson et al. 2012). Since WSG information is not available for every species, we included only communities of tropical forest where no less than 70% of individuals belonged to species with known WSG to ensure the accuracy of our analysis. This criterion was met by five communities (Barro Colorado Island, Cocoli, plots 4 and 5 in La Selva, and Luquillo) of all 60 that we examined. Individuals in these communities for which WSG information was not available were assigned average WSG values across all species in the WSG compilation.
We obtained the metabolic rate of each individual using the alternative scaling relationships specified in Chave et al. (2005) and Mori et al. (2010). The maximum entropy theory of ecology (METE) was then applied to each community following the steps described in “Methods,” and its predictions were compared with the observed values for the individual size distribution (ISD), the size-density relationship (SDR), and the intraspecific ISD (iISD; figs. B1–B5). Although the patterns differ slightly in shape with metabolic rates obtained from the alternative method, the explanatory power of METE for each pattern does not change qualitatively, that is, METE characterizes the ISD with high accuracy but is unable to explain much variation in the SDR or the iISD regardless of the method used to calculate metabolic rate (compare figs. B1–B5 with corresponding communities in the supplementary figures).
Maximum entropy theory of ecology’s (METE’s) predictions plotted against observed values for the species abundance distribution, which remains unchanged (A); the individual size distribution (ISD; B); the size-density relationship (SDR; C); and the intraspecific ISD (iISD; D) for Barro Colorado Island. Here, the metabolic rate was obtained with the alternative scaling method, which slightly changes the shape of the ISD, the SDR, and the iISD without significantly impacting the explanatory power of METE. DBH = diameter at breast height.
Maximum entropy theory of ecology’s predictions plotted against observed values for the species abundance distribution (A), the individual size distribution (ISD; B), the size-density relationship (C), and the intraspecific ISD (iISD; D) for Cocoli, with the alternative scaling method used for metabolic rate. DBH = diameter at breast height.
Maximum entropy theory of ecology’s predictions are plotted against observed values for the species abundance distribution (A), the individual size distribution (ISD; B), the size-density relationship (C), and the intraspecific ISD (iISD; D) for plot 4 in La Selva, with the alternative scaling method used for metabolic rate. DBH = diameter at breast height.
Maximum entropy theory of ecology’s predictions plotted against observed values for the species abundance distribution (A), the individual size distribution (ISD; B), the size-density relationship (C), and the intraspecific ISD (iISD; D) for plot 5 in La Selva, with the alternative scaling method used for metabolic rate. DBH = diameter at breast height.
Maximum entropy theory of ecology’s predictions plotted against observed values for the species abundance distribution (A), the individual size distribution (ISD; B), the size-density relationship (C), and the intraspecific ISD (iISD; D) for Luquillo, with the alternative scaling method used for metabolic rate. DBH = diameter at breast height.
Appendix C: Alternative Analyses for the Individual Size Distribution (ISD) and the Intraspecific ISD (iISD)
In our analyses in the main text, we converted all three probability distributions (species abundance distribution [SAD], ISD, and iISD) into distributions of rank and compared the predicted values at each rank against the observed values. While this approach has been widely adopted (Harte et al. 2008; Harte 2011; White et al. 2012a), it may not be entirely adequate for continuous distributions such as the ISD and the iISD, where empirical measurements are usually rounded off to decimals and thus may not be directly comparable to the truly continuous values obtained from the predicted distributions of rank. Here, we conduct additional analyses for the ISD and the iISD with alternative approaches applied directly on the probability distributions without converting them to distributions of rank to demonstrate the robustness of our results.
For the ISD, we grouped the scaled individual metabolic rates into log(1.7) bins (1–1.7, 1.7–2.89, 2.89–4.913, etc.), which resulted in 10–21 bins for each forest community. The predicted frequency for each bin was then calculated from the cumulative distribution of Ψ(ε) (eq. [4]) and compared with the observed frequency. The predictive power of the maximum entropy theory of ecology (METE) for the ISD does not change qualitatively when the ISD is analyzed as frequencies (
; fig. C1) instead of as ranked metabolic rates (
; fig. 2B).
The iISDs for most species contain too few individuals for the above-described analysis with binned frequencies. Instead, we directly looked at the shape of the distribution. METE predicts that the iISD for each species within a community follows an exponential distribution left truncated at 1, with the parameter of the distribution proportional to the abundance of the species (see eq. [5]). Deviation from METE’s prediction can occur in one or both of two ways: (1) the observed iISDs are not well characterized by exponential distributions; and (2) assuming that the iISDs can be characterized by exponential distributions (which may or may not be true), the parameter of the distributions that best capture the observed iISDs differ from those predicted by METE (eq. [5]). Here, we show that METE’s prediction for iISD fails in both aspects, which is consistent with our results in the main text (fig. 2D).
Characterizing iISDs with Exponential Distributions
In each community, we fit an exponential distribution left truncated at 1 (the minimal rescaled metabolic rate within each community) to rescaled individual metabolic rates for each species with at least 5 individuals and obtained the maximum likelihood (MLE) parameter of the distribution. For each species, 5,000 independent samples were drawn from a left-truncated exponential distribution with the MLE parameter, where the sample size was equal to the abundance of the species. The two-sample Kolmogorov-Smirnov test was then applied to evaluate whether the empirical iISD differs significantly from each sample drawn from the left-truncated exponential distribution. If the proportion of tests (among all 5,000) where the empirical iISD and the randomly generated sample differ in distribution is higher than the significance level (α) of the tests, the empirical iISD for the focal species does not conform to a left-truncated exponential distribution.
Figure C2 shows a histogram of proportions of Kolmogorov-Smirnov tests that are significant at 
among species (with abundance ≥5) across all 60 communities. Overall, the iISDs for more than half the species are deemed to be significantly different from the left-truncated exponential distribution, which implies that the form of iISD predicted by METE does not hold.
Comparing the MLE Parameter with METE’s Predicted Parameter
We further compared the MLE parameter of the left-truncated exponential distribution for each species to the parameter predicted by METE (λ2n; see eq. [5]; fig. C3). Note that this analysis is biased in favor of METE, as we have already shown that left-truncated exponential distribution does not provide a good characterization of empirical iISD for most species (fig. C2). That the R2 value for the iISD is below 0 even when METE is evaluated with this biased analysis further strengthens our conclusion that METE is unable to meaningfully capture any variation in the iISD.
Plot of maximum entropy theory of ecology’s predictions against empirical observations across 60 communities for the individual size distribution, which is analyzed as binned frequencies. The diagonal black line is the 1∶1 line. The points are color coded to reflect the density of neighboring points, with warm (red) colors representing higher densities and cold (blue) colors representing lower densities. The inset in the lower right corner shows the distribution of R2 among individual communities from below 0 (left) to 1 (right).
Histogram of the proportion of Kolmogorov-Smirnov tests that are significant for each species. The dashed vertical line represents the significance level of the tests (
). Species for which the proportion of tests (among 5,000) with significant results is higher than 0.05 have intraspecific individual size distributions (iISDs) that differ significantly from the left-truncated exponential distribution.
Intraspecific individual size distribution (iISD) parameter predicted by the maximum entropy theory of ecology plotted against maximum likelihood parameter for the empirical distribution for each species (with no fewer than five individuals) in each of the 60 communities. The diagonal black line is the 1∶1 line. The points are color coded to reflect the density of neighboring points, with warm (red) colors representing higher densities and cold (blue) colors representing lower densities. The inset reflects the distribution of R2 among 60 communities from negative (left) to 1 (right).
Appendix D: Validation of R2 with Simulations
In the main text, we adopted the coefficient of determination R2 between the empirical data and the values predicted by the maximum entropy theory of ecology (METE) to evaluate the performance of the theory. For the three patterns that are probability distributions (species abundance distribution [SAD], individual size distribution [ISD], and intraspecific ISD [iISD]), we ranked both the observed and the predicted values for comparison, resulting in two monotonically nondecreasing sequences (see “Analyses”). Such ranking could, in concept, lead to spuriously high coefficients of determination between the predictions and the observations, resulting in models appearing to perform well at prediction when in fact they did not.
We explored this possibility using simulations. For each community with state variables S0, N0, and E0, we first constructed a simulated SAD by sampling S0 abundance values from a discrete uniform distribution between 1 and 
, where the upper bound was chosen so that the expected total abundance of the simulated community equaled N0. Then, for such a simulated community with S0 species and 
individuals (with 
centered around but in most cases not equal to N0), we further constructed a simulated ISD by sampling 
size values from a continuous uniform distribution between 1 and 
. Again, the upper bound was chosen so that the expected total metabolic rate of the simulated community equaled E0. This simulation procedure thus largely preserved the values of the state variables for each community, while the shape of the SAD and the ISD differed markedly from METE’s predictions.
We conducted 100 simulations for each empirical community and applied METE to each simulated community with the values of the state variables S0, 
, and 
that resulted from the simulation. The performance of METE on the SAD and the ISD was evaluated in the same way as in the main text, with R2 between the “observed” (simulated) and predicted rank values. If the high R2 values that we obtained for the SAD and the ISD in empirical communities are an artifact of ranking, we would expect equally high R2 values for the simulated communities. In contrast, we found that R2 values for the SAD and the ISD in the vast majority of the simulated communities were below 0, and the R2 for the two patterns in real communities were higher than those in any of the 100 simulated communities (fig. D1). This shows that METE’s predictive power for the SAD and the ISD is not an artifact of ranking, consistent with previous studies of the SAD (White et al. 2012a) as well as our alternative analysis for the ISD where the distribution is not converted to ranks (app. C, fig. C1).
R2 for the species abundance distribution (SAD) and the individual size distribution (ISD) in the empirical communities (black circles) versus the full range of R2 for the two patterns in 100 simulated communities (gray area), where both the SAD and the ISD were generated from uniform distributions.
Appendix E: Bootstrap Analysis
We conducted a bootstrap analysis to examine whether the deviation of the empirical data from the distributions predicted by the maximum entropy theory of ecology (METE) was comparable to that of random samples drawn from the distributions themselves. In each community, we obtained the Lagrange multipliers λ1 and λ2 with empirically observed state variables S0, N0, and E0, which determined the form of the predicted patterns (eqq. [3]–[5]). We drew 500 bootstrap samples from the predicted distribution for each pattern. For the species abundance distribution (SAD), samples of size S0 were drawn from the upper-truncated log-series distribution defined by equation (3). For the individual size distribution (ISD), samples of size N0 were drawn from the distribution defined by equation (4). For the size-density relationship (SDR) and the intraspecific ISD (iISD), samples for each species given its abundance n were drawn from the exponential distribution defined by equation (5). The SDR of each sample community was then obtained by taking the average body size within a bootstrap sample for each species. Note that this sampling scheme assumes independence among values within each bootstrap sample. As a result, the values of the original state variables are unlikely to be preserved in the bootstrap samples. However, given that the discrepancy is not systematic and that the results of the bootstrap analysis are highly consistent both across samples and across communities (see fig. E2), we conclude that the assumption of independence should not qualitatively affect our results.
The deviation between empirical data or bootstrap samples and METE’s predictions were quantified using R2 and the Kolmogorov-Smirnov (K-S) statistic. The K-S statistic is defined as
We converted the test statistics within each community into quantiles so that results can be pooled across communities. The quantile for each pattern in a community was calculated for each of the two statistics as the proportion of bootstrap samples that had larger deviations from METE’s prediction (i.e., lower value of R2 or higher value of K-S statistic) than the empirical data. For example, figure E1A shows that 77% of the bootstrap SADs have a lower R2 than the empirical SAD in the community UCSC FERP. For the iISD, where bootstrap samples were independently generated for each species, the quantile of the K-S statistic for a given community was calculated as the average quantile across all species having 10 or more individuals, weighted by their abundances.
Comparisons between the empirical data and the bootstrap samples for the four ecological patterns across all 60 communities are summarized in figure E2. Results from the two statistics are qualitatively consistent (although note again that the K-S statistic cannot be applied to the SDR, which is not a probability distribution). While the bootstrap analysis confirms that METE provides a satisfactory characterization for the empirical SAD but not for the empirical SDR or iISD, it shows that the empirical ISD cannot be fully accounted for by METE’s prediction, despite the relatively high R2 within and across communities for this pattern (see fig. 2B and the supplementary figures).
Illustration of the comparison between empirical data and bootstrap samples using data from the University of California, Santa Cruz, Forest Ecology Research Plot as an example. A and B show the results for the species abundance distribution (SAD) when evaluated with R2 (A) and the Kolmogorov-Smirnov (K-S) statistic (B), while C and D show the results for the intraspecific individual size distribution (iISD). In each panel, the histogram represents the frequency distribution of the test statistic among the 500 bootstrap samples, while the red vertical line represents the test statistic of the empirical data. Note that for the iISD the K-S statistic was individually obtained for each species, and the illustration in D includes the results for only one species, Pseudotsuga menziesii.
Results of bootstrap analysis across all 60 communities evaluated with R2 and the Kolmogorov-Smirnov (K-S) statistic for the four patterns (except for the size-density relationship [SDR], where only R2 is available). The histogram in each panel is the frequency distribution of the quantile values across the 60 communities for one pattern using one statistic (R2 or K-S statistic), where each quantile value represents the quantile of the empirical statistic among that of the 500 bootstrap communities. The number of communities where the quantile equals 0 (i.e., where the empirical data have a larger deviation from the predicted pattern than any of the bootstrap samples) is also given. Note that for the intraspecific individual size distribution (iISD), the quantile of the K-S statistic is a pooled value across all species with abundance >10 in a community and thus can equal 0 only when the quantiles for all species are 0. ISD = individual size distribution, SAD = species abundance distribution.
Muller-Landau et al. (2006) proposed four possible distributions (exponential, Pareto, Weibull, and quasi-Weibull) for diameter in old-growth forests under different assumptions of growth and mortality. Here, we compare the fit of three of the four distributions (exponential, Pareto, and Weibull) to the fit of the ISD predicted by the maximum entropy theory of ecology (METE; eq. [4]) using data from the 60 forest communities. The quasi-Weibull distribution, which has been shown to provide the best fit for the majority of communities (Muller-Landau et al. 2006), is not evaluated due to the difficulty in obtaining its maximum likelihood parameters when it is left truncated.
All distributions are left truncated to account for the fact that individuals below the minimal threshold in each community were excluded from the data sets. With the minimal size rescaled as 1 across communities (see “Methods”), the left-truncated exponential distribution takes the form
for all three distributions.Parameters in equations (F1), (F2), and (F3) were obtained with the maximum likelihood (MLE) method for each community. While analytical solutions exist for parameters in equation (F1) and equation (F2), MLE solutions for parameters in equation (F3) can only be obtained numerically. The three distributions of D were then transformed into distributions of D2 (surrogate for metabolic rate; see “Methods”) to be consistent with METE’s prediction (eq. [4]) as
The fit of the ISD predicted by METE and the other three distributions was evaluated with Akaike’s information criterion (AIC; Burnham and Anderson 2002). Corrected AIC (AICc), a second-order variant of AIC that corrects for finite sample size, was computed for each distribution as
Our results show that overall the Weibull distribution provides the best fit for the ISD, which outperforms the other three distributions (i.e., has the smallest AICc value) in 50 of 60 communities. While METE is exceeded by the Weibull distribution in all except three communities, its performance is comparable to that of the other two distributions, with METE outperforming the exponential distribution in 24 communities and the Pareto distribution in 33 (Table F1).
| AICc | |||||
|---|---|---|---|---|---|
| Data set | Site | Exponential | Pareto | Weibull | METE |
| UCSC FERP | FERP | 85,971.15 | 82,823.11 | 81,893.76 | 88,390.74 |
| ACA Amazon Forest Inventories | eno-2 | 3,047.892 | 3,123.951 | 3,037.737 | 3,048.544 |
| Western Ghats | BSP104 | 8,447.378 | 8,232.82 | 8,147.375 | 8,597.933 |
| Western Ghats | BSP11 | 9,670.786 | 9,737.739 | 9,565.319 | 9,756.008 |
| Western Ghats | BSP12 | 8,072.348 | 7,580.985 | 7,580.105 | 8,005.097 |
| Western Ghats | BSP16 | 6,505.854 | 6,465.984 | 6,371.536 | 6,473.227 |
| Western Ghats | BSP27 | 4,158.854 | 4,352.934 | 4,154.657 | 4,168.587 |
| Western Ghats | BSP29 | 5,200.085 | 5,601.832 | 5,186.167 | 5,246.872 |
| Western Ghats | BSP30 | 5,228.032 | 5,550.478 | 5,229.22 | 5,272.148 |
| Western Ghats | BSP36 | 5,363.257 | 4,997.568 | 4,994.507 | 5,613.485 |
| Western Ghats | BSP37 | 6,648.723 | 5,882.951 | 5,940.894 | 6,702.201 |
| Western Ghats | BSP42 | 4,862.353 | 4,579.541 | 4,572.774 | 4,912.597 |
| Western Ghats | BSP5 | 6,316.684 | 5,868.932 | 5,879.056 | 6,344.512 |
| Western Ghats | BSP6 | 8,362.132 | 8,224.467 | 8,144.515 | 8,368.706 |
| Western Ghats | BSP65 | 10,730.14 | 10,597.32 | 10,418.12 | 10,323.55 |
| Western Ghats | BSP66 | 6,127.039 | 6,078.716 | 5,969.159 | 6,118.758 |
| Western Ghats | BSP67 | 5,733.979 | 6,116.641 | 5,713.447 | 5,970.901 |
| Western Ghats | BSP69 | 9,639.039 | 9,839.743 | 9,566.506 | 9,677.272 |
| Western Ghats | BSP70 | 7,568.366 | 7,643.62 | 7,475.877 | 7,471.337 |
| Western Ghats | BSP73 | 13,866.8 | 14,638.34 | 13,867.97 | 14,056.6 |
| Western Ghats | BSP74 | 10,384.88 | 10,164.99 | 10,043.66 | 10,178.07 |
| Western Ghats | BSP75 | 3,828.718 | 4,032.776 | 3,830.225 | 3,844.366 |
| Western Ghats | BSP79 | 10,012.15 | 10,192.38 | 9,943.069 | 10,014.63 |
| Western Ghats | BSP80 | 10,351.04 | 10,721.97 | 10,333.53 | 10,392.1 |
| Western Ghats | BSP82 | 7,775.241 | 8,109.038 | 7,766.727 | 7,779.842 |
| Western Ghats | BSP83 | 10,080.84 | 10,603.67 | 10,082.84 | 10,184.62 |
| Western Ghats | BSP84 | 9,941.77 | 10,676.22 | 9,906.56 | 10,087.81 |
| Western Ghats | BSP85 | 4,090.759 | 4,051.023 | 3,986.417 | 4,092.965 |
| Western Ghats | BSP88 | 9,539.878 | 10,007.25 | 9,532.9 | 9,468.538 |
| Western Ghats | BSP89 | 7,758.469 | 8,040.773 | 7,746.257 | 7,749.632 |
| Western Ghats | BSP90 | 7,802.77 | 8,287.765 | 7,800.707 | 7,891.673 |
| Western Ghats | BSP91 | 8,443.673 | 9,081.623 | 8,392.871 | 8,709.277 |
| Western Ghats | BSP92 | 5,010.321 | 5,156.128 | 4,980.47 | 5,037.136 |
| Western Ghats | BSP94 | 4,995.435 | 5,113.566 | 4,949.09 | 4,997.738 |
| Western Ghats | BSP98 | 6,338.305 | 6,535.699 | 6,312.535 | 6,336.033 |
| Western Ghats | BSP99 | 8,329.191 | 8,461.831 | 8,238.427 | 8,268.363 |
| BCI | bci | 1,663,761 | 1,595,835 | 1,580,094 | 1,616,953 |
| BVSF | BVPlot | 2,801.075 | 2,851.043 | 2,790.895 | 2,792.688 |
| BVSF | SFPlot | 2,452.828 | 2,427.723 | 2,409.388 | 2,413.466 |
| Cocoli | cocoli | 7,3752.32 | 68,152.93 | 67,835.59 | 75,938.32 |
| Lahei | heath1 | 9,947.228 | 9,966.227 | 9,841.178 | 9,888.052 |
| Lahei | heath2 | 9,795.598 | 9,650.197 | 9,595.179 | 9,618.001 |
| Lahei | peat | 9,183.332 | 9,040.189 | 8,961.699 | 9,030.188 |
| La Selva | 1 | 5,518.14 | 5,434.672 | 5,376.494 | 5,555.8 |
| La Selva | 2 | 5,504.011 | 5,548.332 | 5,444.005 | 5,489.366 |
| La Selva | 3 | 6,337.174 | 6,328.63 | 6,237.519 | 6,294.73 |
| La Selva | 4 | 5,445.745 | 5,527.303 | 5,402.815 | 5,409.85 |
| La Selva | 5 | 4,410.166 | 4,318.777 | 4,281.463 | 4,440.427 |
| Luquillo | lfdp | 534,427.2 | 515,126.9 | 509,926.5 | 525,725.7 |
| NC | 12 | 45,716.48 | 44,860.83 | 44,212.08 | 45,592.31 |
| NC | 13 | 36,251.18 | 34,948.55 | 34,539.55 | 36,220.19 |
| NC | 14 | 56,695.06 | 52,506.98 | 52,273.61 | 55,964.15 |
| NC | 4 | 36,203.17 | 36,553.64 | 35,587.05 | 36,447.78 |
| NC | 93 | 34,667.37 | 33,277.48 | 32,934.38 | 34,730.18 |
| Oosting | oosting | 74,293.18 | 69,837.5 | 69,718.9 | 74,739.21 |
| Serimbu | S-1 | 7,887.232 | 7,471.463 | 7,463.06 | 7,981.97 |
| Serimbu | S-2 | 8,507.118 | 8,123.406 | 8,102.843 | 8,614.922 |
| Shirakami | Akaishizawa | 3,105.173 | 3,104.759 | 3,057.59 | 3,188.967 |
| Shirakami | Kumagera | 3,473.692 | 3,680.852 | 3,473.805 | 3,597.692 |
| Sherman | sherman | 191,735.8 | 188,206 | 185,424 | 190,339.9 |
Notes
1 Code that appears in The American Naturalist is provided as a convenience to the readers. It has not necessarily been tested as part of the peer review.
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Supplements
Appendix A: Derivation for the Equations
Appendix B: Alternative Scaling between Diameter and Metabolic Rate
Appendix C: Alternative Analyses for the Individual Size Distribution (ISD) and the Intraspecific ISD (iISD)
Appendix D: Validation of R2 with Simulations
Appendix E: Bootstrap Analysis
Appendix F: Model Comparison for the Individual Size Distribution (ISD)
