A strong test of the maximum entropy theory of ecology.

The Maximum Entropy Theory of Ecology (METE), is a theoretical framework of macroecology that makes a variety of realistic ecological predictions about how species richness, abundance of species, metabolic rate distributions, and spatial aggregation of species interrelate in a given region. In the METE framework, “ecological state variables” (representing total area, total species richness, total abundance, and total metabolic energy) describe macroecological properties of an ecosystem. METE incorporates these state variables into constraints on underlying probability distributions. The method of Lagrange multipliers and maximization of information entropy (MaxEnt) lead to predicted functional forms of distributions of interest. We demonstrate how information entropy is maximized for the general case of a distribution, which has empirical information that provides constraints on the overall predictions. We then show how METE’s two core functions are derived. These functions, called the “Spatial Structure Function” and the “Ecosystem Structure Function” are the core pieces of the theory, from which all the predictions of METE follow (including the Species Area Relationship, the Species Abundance Distribution, and various metabolic distributions). Primarily, we consider the discrete distributions predicted by METE. We also explore the parameter space defined by the METE’s state variables and Lagrange multipliers. We aim to provide a comprehensive resource for ecologists who want to understand the derivations and assumptions of the basic mathematical structure of METE.

The Maximum Entropy Theory of Ecology (METE) predicts a universal species–area relationship (SAR) that can be fully characterized using only the total abundance (N) and species richness (S) at a single spatial scale. This theory has shown promise for characterizing scale dependence in the SAR. However, there are currently four different approaches to applying METE to predict the SAR and it is unclear which approach should be used due to a lack of empirical comparison. Specifically, METE can be applied recursively or non-recursively and can use either a theoretical or observed species-abundance distribution (SAD). We compared the four different combinations of approaches using empirical data from 16 datasets containing over 1000 species and 300,000 individual trees and herbs. In general, METE accurately downscaled the SAR (R2 > 0.94), but the recursive approach consistently under-predicted richness. METE’s accuracy did not depend strongly on using the observed or predicted SAD. This suggests that the best approach to scaling diversity using METE is to use a combination of non-recursive scaling and the theoretical abundance distribution, which allows predictions to be made across a broad range of spatial scales with only knowledge of the species richness and total abundance at a single scale.

Background/Question/Methods Understanding spatial patterns in ecosystems quantitatively can lead to insights regarding the relative importance of underlying physical processes, and allows us to better predict what will happen to ecosystems under natural and anthropogenic disturbances. The Maximum Entropy Theory of Ecology (METE) is a theory that can simultaneously predict many patterns and distributions, including spatial patterns. However, the predictions from METE do not seem to be as accurate in disturbed ecosystems. To compare spatial data to theory, we can bisect a plot and consider the probability that there are n individuals on one side given N total individuals. METE predicts that this probability distribution will be uniform, which is equivalent to using the Laplace rule of succession as a colonization rule. Ecologically, this leads to strong spatial aggregation. Another common theoretical approach is the random placement model, which predicts a binomial distribution. For many ecosystems METE overpredicts the amount of spatial aggregation, but there is more than predicted by random placement, which has none. We add a density dependent death rule to modify the prediction of METE, where individuals die with probability proportional to n to some exponent ɑ and are replaced with the METE colonization rule. Results/Conclusions We derive the bisection probability distribution that results from using the Laplace rule of succession as a colonization rule with a density dependent death rule. The predictions show that we can vary the amount of spatial aggregation by varying the density dependence ɑ . With ɑ = 1, the distribution is unchanged from the original predictions of METE, and with ɑ = 2, the distribution approaches the binomial distribution for large N. In the intermediate regime 1 < ɑ < 2, we show that this model predicts less spatial aggregation than METE, but more than random placement. We also show that this model fits the data well and relate it to systems where we suspect density dependence does and does not matter. In the future, we will extend this theory to finer spatial scales beyond the first bisection.

AimThe maximum entropy theory of ecology (METE) is a unified theory of biodiversity that attempts to simultaneously predict patterns of species abundance, size and spatial structure. The spatial predictions of this theory have repeatedly performed well at predicting diversity patterns across scales. However, the theoretical development and evaluation of METE has focused on predicting patterns that ignore intersite spatial correlations. As a result the theory has not been evaluated using one of the core components of spatial structure. We develop and test a semi‐recursive version of the spatially explicit predictions of METE for the distance–decay relationship of community similarity and compare its performance with the classic random placement model of completely random species distributions. This provides a better understanding and stronger test of the spatial community predictions of METE.LocationNew World tropical and temperate plant communities.MethodsWe analytically derived and simulated the spatially explicit expectations of METE for the Sørensen index of community similarity. We then compared the distance–decay of community similarity of 16 mapped plant communities with METE and the random placement model.ResultsThe version of METE we examined was successful at capturing the general functional form of the observed distance–decay relationships, a negative power function relationship between community similarity and distance. However, the semi‐recursive approach consistently predicted lower intercepts and higher slopes than observed in the empirical distance–decay relationships and yielded worse predictions than the random placement model.Main conclusionsOur results suggest that while the current spatial models of METE accurately predict the spatial scaling of species occupancy, and therefore core ecological patterns like the species–area relationship, its semi‐recursive form does not accurately characterize spatially explicit patterns of correlation. More generally, this suggests that tests of spatial theories based only on the species–area relationship may appear to support the underlying theory despite significant deviations in important aspects of spatial structure.

Summary Macroecological patterns appear to follow consistent forms across a range of natural systems; however, the origin of their regularity remains obscured. The maximum entropy theory of ecology (METE) predicts macroecological patterns of abundance, metabolic rates and their distribution within communities and across space using an information theoretic approach. METE's success in predicting empirical patterns demands that we further press the theory's predictions to determine how (or whether) predictability depends on attributes of the system and the temporal, spatial and biological scales at which we study it. Maximum entropy theory of ecology predicts multiple macroecological metrics using statistical idealizations from information theory; thus, confronting METE with data represents a strong test of the underlying biological mechanisms that could drive real communities away from statistical idealizations. METE has remained somewhat inaccessible due to its highly mathematical nature and a lack of software for model construction/evaluation. To remedy this, we have developed an r package implementation of METE. Our open‐source (GNU General Public License v2) r package, meteR (version 1.2; https://cran.r-project.org/package=meteR), (i) directly calculates all of METE's predictions from a variety of data formats; (ii) automatically handles approximations and other technical details; and (iii) provides high‐level plotting and model comparison functions to explore and interrogate models. With these tools in hand, ecologists can more readily test the predictions of METE for their data sets. By facilitating tests of METE, we expect that a better understanding of its strengths and limitations will emerge. A better understanding of the strengths and limitations of METE will offer insight into how biological mechanisms and statistical constraints combine to drive macroecological patterns.

Metabolic scaling provides valuable information about the physiological and ecological functions of organisms, although few studies have quantified the metabolic scaling exponent (b) of communities under natural conditions. Maximum entropy theory of ecology (METE) is a constraint-based unified theory with the potential to empirically assess the spatial variation of the metabolic scaling. Our main goal is to develop a novel method of estimating b within a community by integrating metabolic scaling and METE. We also aim to study the relationships between the estimated b and environmental variables across communities. We developed a new METE framework to estimate b in 118 stream fish communities in the north-eastern Iberian Peninsula. We first extended the original maximum entropy model by parameterizing b in the model prediction of the community-level individual size distributions and compared our results with empirical and theoretical predictions. We then tested the effects of abiotic conditions, species composition and human disturbance on the spatial variation of community-level b. We found that community-level b of the best maximum entropy models showed great spatial variability, ranging from 0.25 to 2.38. The mean exponent (b = 0.93) resembled the community-aggregated mean values from three previous metabolic scaling meta-analyses, all of which were greater than the theoretical predictions of 0.67 and 0.75. Furthermore, the generalized additive model showed that b reached maximum at the intermediate mean annual precipitation level and declined significantly as human disturbance intensified. The parameterized METE is proposed here as a novel framework for estimating the metabolic pace of life of stream fish communities. The large spatial variation of b may reflect the combined effects of environmental constraints and species interactions, which likely have important feedback on the structure and function of natural communities. Our newly developed framework can also be applied to study the impact of global environmental pressures on metabolic scaling and energy use in other ecosystems.

A major goal of ecology is to predict patterns and changes in the abundance, distribution, and energetics of individuals and species in ecosystems. The maximum entropy theory of ecology (METE) predicts the functional forms and parameter values describing the central metrics of macroecology, including the distribution of abundances over all the species, metabolic rates over all individuals, spatial aggregation of individuals within species, and the dependence of species diversity on areas of habitat. In METE, the maximum entropy inference procedure is implemented using the constraints imposed by a few macroscopic state variables, including the number of species, total abundance, and total metabolic rate in an ecological community. Although the theory adequately predicts pervasive empirical patterns in relatively static ecosystems, there is mounting evidence that in ecosystems in which the state variables are changing rapidly, many of the predictions of METE systematically fail. Here we discuss the underlying logic and predictions of the static theory and then describe progress toward achieving a dynamic theory (DynaMETE) of macroecology capable of describing ecosystems undergoing rapid change as a result of disturbance. An emphasis throughout is on the tension between, and reconciliation of, two legitimate perspectives on ecology: that of the natural historian who studies the uniqueness of every ecosystem and the theorist seeking unification and generality.

AimBoth macroecology and disturbance ecology have long been used to characterize population‐ and community‐level patterns across scales, but the integration of both approaches in characterizing disturbed ecosystems is rare. Here, we use the maximum entropy theory of ecology (METE) to model the individual size distribution (ISD) of trees in pre‐ and post‐disturbance tree populations and estimate the corresponding metabolic scaling exponents.LocationNew Zealand.Time Period1987–1999.Major Taxa StudiedMountain beech (Fuscospora cliffortioidesNothofagaceae).MethodsMETE uses information entropy and empirical macro‐state variables to constrain predictions of ecological distributions related to biodiversity. METE has successfully predicted a range of biodiversity metrics in static or relatively undisturbed conditions. However, METE can fail to accurately model ecological patterns in disturbed ecosystems. We extend existing theoretical predictions to a highly disturbed ecosystem by treating the metabolic scaling exponent and Lagrange multipliers as free parameters in METE.ResultsWe showed that the fully parameterized METE (FP‐METE) model reasonably predicted the ISD of mountain beech populations in a monodominant forest after a strong earthquake, which restructured the forest. Furthermore, the FP‐METE model revealed that decreasing metabolic scaling exponent drove the substantial decline of total metabolic rate energy and the redistribution of energy towards smaller trees after the earthquake. Increased number of small trees was not sufficient to capture the full impact of disturbance on forest energy use.Main ConclusionsOur FP‐METE model applies an informatics approach to estimate the metabolic scaling relationship. We find that instead of maintaining a fixed value, the metabolic scaling exponent is variable among populations, and declines significantly after an earthquake disturbance. This leads to major shifts in the total population metabolic energy and energy distribution. With this approach, we now have the opportunity to advance beyond categorizing forms of mathematical distributions that describe biodiversity patterns and move into a predictive framework where the true constraints on ecosystems and their dynamics emerge.

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.

Macroecological studies have established widespread patterns of species diversity and abundance in ecosystems but have generally restricted their scope to relatively steady‐state systems. As a result, how macroecological metrics are expected to scale in ecosystems that experience natural disturbance regimes is unknown. We examine macroecological patterns in a fire‐dependent forest of Bishop pine (Pinus muricata). We target two different‐aged stands in a stand‐replacing fire regime: a mature stand with a diverse understory and with no history of major disturbance for at least 40 yr, and one disturbed by a stand‐replacing fire 17 yr prior to measurement. We compare properties of these stands with macroecological predictions from the Maximum Entropy Theory of Ecology (METE), an information entropy‐based theory that has proven highly successful in predicting macroecological metrics in multiple ecosystems and taxa. Ecological patterns in the mature stand more closely matchMETEpredictions than do data from the more recently disturbed, mid‐seral stage stand. This suggestsMETE's predictions are more robust in late‐successional, slowly changing, or steady‐state systems than those in rapid flux with respect to species composition, abundances, and organisms’ sizes. Our findings highlight the need for a macroecological theory that incorporates natural disturbance, perturbations, and ecological dynamics into its predictive capabilities, because most natural systems are not in a steady state.

Ecological patterns arise from the interplay of many different processes, and yet the emergence of consistent phenomena across a diverse range of ecological systems suggests that many patterns may in part be determined by statistical or numerical constraints. Differentiating the extent to which patterns in a given system are determined statistically, and where it requires explicit ecological processes, has been difficult. We tackled this challenge by directly comparing models from a constraint-based theory, the Maximum Entropy Theory of Ecology (METE) and models from a process-based theory, the size-structured neutral theory (SSNT). Models from both theories were capable of characterizing the distribution of individuals among species and the distribution of body size among individuals across 76 forest communities. However, the SSNT models consistently yielded higher overall likelihood, as well as more realistic characterizations of the relationship between species abundance and average body size of conspecific individuals. This suggests that the details of the biological processes contain additional information for understanding community structure that are not fully captured by the METE constraints in these systems. Our approach provides a first step towards differentiating between process- and constraint-based models of ecological systems and a general methodology for comparing ecological models that make predictions for multiple patterns.

Ecological patterns arise from the interplay of many different processes, and yet the emergence of consistent phenomena across a diverse range of ecological systems suggests that many patterns may in part be determined by statistical or numerical constraints. Differentiating the extent to which patterns in a given system are determined statistically, and where it requires explicit ecological processes, has been difficult. We tackled this challenge by directly comparing models from a constraint-based theory, the Maximum Entropy Theory of Ecology (METE) and models from a process-based theory, the size-structured neutral theory (SSNT). Models from both theories were capable of characterizing the distribution of individuals among species and the distribution of body size among individuals across 76 forest communities. However, the SSNT models consistently yielded higher overall likelihood, as well as more realistic characterizations of the relationship between species abundance and average body size of conspecific individuals. This suggests that the details of the biological processes contain additional information for understanding community structure that are not fully captured by the METE constraints in these systems. Our approach provides a first step towards differentiating between process- and constraint-based models of ecological systems and a general methodology for comparing ecological models that make predictions for multiple patterns.

A goal of every science is comprehensive theory that is predictive, realistic, and parsimonious. Is such theory possible in ecology? The sheer complexity, historical contingency, and scale-dependence of organisms and their interactions with their surroundings suggest to many a negative answer. This book answers yes. Rather than building and combining mechanistic models of ecosystems, the approach here is grounded in information theory and the logic of inference. Paralleling the derivation of thermodynamics from the maximum entropy principle, the state variable theory of ecology developed in the book predicts realistic forms for all metrics of ecology that describe patterns in the distribution, abundance, and energetics of species across multiple spatial scales. Part I is foundational, discussing the nature of theory, the relationship of ecology to other sciences, and the concept of the logic of inference. Parts II and III, respectively, present the fundamentals of macroecology and of maximum information entropy from the ground up. Part IV integrates the fundamentals, leading to the derivation and testing of the predictions of the maximum entropy theory of ecology (METE). Part V widens the perspective by showing how METE can help clarify several major issues in conservation biology, placing METE in context with other theories, and pointing readers along avenues for future research.

Species abundance distributions (SADs) describe community structure and are a key component of biodiversity theory and research. Although different distributions have been proposed to represent SADs at different scales, a systematic empirical assessment of how SAD shape varies across wide scale gradients is lacking. Here, we examined 11 empirical large-scale datasets for a wide range of taxa and used maximum likelihood methods to compare the fit of the logseries, lognormal, and multimodal (i.e., with multiple modes of abundance) models to SADs across a scale gradient spanning several orders of magnitude. Overall, there was a higher prevalence of multimodality for larger spatial extents, whereas the logseries was exclusively selected as best fit for smaller areas. For many communities the shape of the SAD at the largest spatial extent (either lognormal or multimodal) was conserved across the scale gradient, despite steep declines in area and taxonomic diversity sampled. Additionally, SAD shape was affected by species richness, but we did not detect a systematic effect of the total number of individuals. Our results reveal clear departures from the predictions of two major macroecological theories of biodiversity for SAD shape. Specifically, neither the Neutral Theory of Biodiversity (NTB) nor the Maximum Entropy Theory of Ecology (METE) are able to accommodate the variability in SAD shape we encountered. This is highlighted by the inadequacy of the logseries distribution at larger scales, contrary to predictions of the NTB, and by departures from METE expectation across scales. Importantly, neither theory accounts for multiple modes in SADs. We suggest our results are underpinned by both inter- and intraspecific spatial aggregation patterns, highlighting the importance of spatial distributions as determinants of biodiversity patterns. Critical developments for macroecological biodiversity theories remain in incorporating the effect of spatial scale, ecological heterogeneity and spatial aggregation patterns in determining SAD shape.

In the maximum entropy theory of ecology (METE), the form of a function describing the distribution of abundances over species and metabolic rates over individuals in an ecosystem is inferred using the maximum entropy inference procedure. Favretti shows that an alternative maximum entropy model exists that assumes the same prior knowledge and makes predictions that differ from METE’s. He shows that both cannot be correct and asserts that his is the correct one because it can be derived from a classic microstate-counting calculation. I clarify here exactly what the core entities and definitions are for METE, and discuss the relevance of two critical issues raised by Favretti: the existence of a counting procedure for microstates and the choices of definition of the core elements of a theory. I emphasize that a theorist controls how the core entities of his or her theory are defined, and that nature is the final arbiter of the validity of a theory.