Maximum Entropy Theory Of Ecology Research Articles

An Ecoinformatic Model Using Rényi's Entropy Predicts Soil Chemistry Effect on Tree Species Abundance Distributions

ABSTRACTAimAn ecological community consists of species of various abundances that reflect their responses to the environmental conditions. A classic macroecological pattern, the species abundance distribution (SAD), has been studied for diverse taxa and communities and integrated into numerous modelling tools. Despite its widespread use, a mathematical model that can capture variations in the empirical SAD and describe its response to environmental changes is still lacking. By integrating the Maximum Entropy Theory of Ecology (METE) with a generalised entropy called Rényi's entropy, we aim to develop a new ecoinformatic model that can predict the variation of empirical SAD along multiple environmental gradients.LocationPanama.TaxonAngiosperms.MethodsWe extend the METE using the Rényi's entropy as an uncertainty measure. We apply this extended METE, called Rényi model, to the tree abundance data from 49 plots in Panama and predict the SAD within each plot. We estimate Rényi's parameter q by fitting the predicted SAD to the empirical SAD in each plot. We further compile climate and soil data from the Panama plots and analyse their relationships with the estimated q using multiple regressions.ResultsRényi model provides adequate description of the empirical SADs and outperforms lognormal or log‐series models in 40 of the 49 tree plots, according to the Akaike information criterion. Variations in Renyi's q estimates (from 1/2 to 1) reflect shifts in the empirical SADs. Multiple regressions reveal that P, Al and NH4, three soil chemicals that are important for tree growth and species distribution, significantly affect Renyi's q across plots.Main ConclusionsThese findings suggest that the Rényi model and Rényi's q can characterise the SAD of communities under environmental changes. They also indicate the potential of using generalised entropies to predict macroecological patterns in stressed ecosystems.

Generalized Core Functions of Maximum Entropy Theory of Ecology

Core distributions of Maximum Entropy Theory of Ecology (METE) are the Spatial Structure Function (SSF) and the Ecosystem Structure Function (ESF). SSF is a by-species prediction of the clustering of individuals over space. ESF is a kind of container function that describes the probability space of how abundances are assigned to species and how metabolic energy is parti-tioned over individuals in a community. In this study, these core functions of METE are generalized by deriving the corresponding functions in the Tsallis q-entropy. Derivation used the method of Lagrange multipliers. The generalized SSF and ESF are expressed in terms of the q-exponential function. Numerical examples are provided to illustrate the generalized SSF.

Earthquake disturbance shifts metabolic energy use and partitioning in a monodominant forest

AbstractAimBoth 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 cliffortioides Nothofagaceae).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.

Maximum entropy models reveal spatial variation of metabolic scaling in stream fish communities.

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.

Enhancing the predictability of ecology in a changing world: A call for an organism-based approach

Ecology is usually very good in making descriptive explanations of what is observed, but is often unable to make predictions of the response of ecosystems to change. This has implications in a human-dominated world where a suite of anthropogenic stresses are threatening the resilience and functioning of ecosystems that sustain mankind through a range of critical regulating and supporting services. In ecosystems, cause-and-effect relationships are difficult to elucidate because of complex networks of negative and positive feedbacks. Therefore, being able to effectively predict when and where ecosystems could pass into different (and potentially unstable) new states is vitally important under rapid global change. Here, we argue that such better predictions may be reached if we focus on organisms instead of species, because organisms are the principal biotic agents in ecosystems that react directly on changes in their environment. Several studies show that changes in ecosystems may be accurately described as the result of changes in organisms and their interactions. Organism-based theories are available that are simple and derived from first principles, but allow many predictions. Of these we discuss Trait-based Ecology, Agent Based Models, and Maximum Entropy Theory of Ecology and show that together they form a logical sequence of approaches that allow organism-based studies of ecological communities. Combining and extending them makes it possible to predict the spatiotemporal distribution of groups of organisms in terms of how metabolic energy is distributed over areas, time, and resources. We expect that this “Organism-based Ecology” (OE) ultimately will improve our ability to predict ecosystem dynamics.

An equation of state unifies diversity, productivity, abundance and biomass

To advance understanding of biodiversity and ecosystem function, ecologists seek widely applicable relationships among species diversity and other ecosystem characteristics such as species productivity, biomass, and abundance. These metrics vary widely across ecosystems and no relationship among any combination of them that is valid across habitats, taxa, and spatial scales, has heretofore been found. Here we derive such a relationship, an equation of state, among species richness, energy flow, biomass, and abundance by combining results from the Maximum Entropy Theory of Ecology and the Metabolic Theory of Ecology. It accurately captures the relationship among these state variables in 42 data sets, including vegetation and arthropod communities, that span a wide variety of spatial scales and habitats. The success of our ecological equation of state opens opportunities for estimating difficult-to-measure state variables from measurements of others, adds support for two current theories in ecology, and is a step toward unification in ecology.

Land use change through the lens of macroecology: insights from Azorean arthropods and the maximum entropy theory of ecology

Human activity and land management practices, in particular land use change, have resulted in the global loss of biodiversity. These types of disturbance affect the shape of macroecological patterns, and therefore analyzing these patterns can provide insights into how ecosystems are affected by land use change. We here use arthropod census data from 96 sites at Terceira Island in the Azores archipelago across four different land uses of increasing management intensity: native forest, exotic forest, semi‐natural pasture and intensive pasture, to examine the effects of land use type on three macroecological patterns: the species abundance distribution, the metabolic rate distribution of individuals and the species–area relationship. The maximum entropy theory of ecology (METE) has successfully predicted these patterns across habitats and taxa in undisturbed ecosystems, and thus provides a null expectation for their shapes. Across these patterns, we find that the forest habitats are the best fit by METE predictions, while the semi‐natural pasture is consistently the worst fit, and the intensive pasture is intermediately well fit. We show that the direction of failure of the METE predictions at the pasture sites is likely due to the hyper‐dominance of introduced spider species present there. We hypothesize that the particularly poor fit for the semi‐natural pasture is due to the mix of arthropod communities out of equilibrium, leading to greater heterogeneity in composition and complex dynamics that violate METE's assumption of static state variables. The comparative better fit for the intensive pasture plausibly results from more homogeneous arthropod communities that are well adapted to intensive management, and thus whose state variables are less in flux. Analyzing deviations from theoretical predictions across land use type provides useful information about how land use and disturbance affect ecosystems, and such comparisons could be useful across other habitats and taxa.

Relating the Strength of Density Dependence and the Spatial Distribution of Individuals

Spatial patterns in ecology contain useful information about underlying mechanisms and processes. Although there are many summary statistics used to quantify these spatial patterns, there are far fewer models that directly link explicit ecological mechanisms to observed patterns easily derived from available data. We present a model of intraspecific spatial aggregation that quantitatively relates static spatial patterning to negative density dependence. Individuals are placed according to the colonization rule consistent with the Maximum Entropy Theory of Ecology (METE), and die with probability proportional to their abundance raised to a power α, a parameter indicating the degree of density dependence. This model can therefore be interpreted as a hybridization of MaxEnt and mechanism. Our model shows quantitatively and generally that increasing density dependence randomizes spatial patterning. α = 1 recovers the strongly aggregated METE distribution that is consistent with many ecosystems empirically, and as α → 2 our prediction approaches the binomial distribution consistent with random placement. For 1 < α < 2, our model predicts more aggregation than random placement but less than METE. We additionally relate our mechanistic parameter α to the statistical aggregation parameter k in the negative binomial distribution, giving it an ecological interpretation in the context of density dependence. We use our model to analyze two contrasting datasets, a 50 ha tropical forest and a 64 m2 serpentine grassland plot. For each dataset, we infer α for individual species as well as a community α parameter. We find that α is generally larger in the tightly packed forest than the sparse grassland, and the degree of density dependence increases at smaller scales. These results are consistent with current understanding in both ecosystems, and we infer this underlying density dependence using only empirical spatial patterns. Our model can easily be applied to other datasets where spatially explicit data are available.

Integrating metabolic scaling variation into the maximum entropy theory of ecology explains Taylor's law for individual metabolic rate in tropical forests

Individual trait variation has important ecological implications for species populations and communities. In particular, individual variation of metabolic rate links directly with the energy use estimation and scaling patterns in community ecology. Here, we examine the mean-variance relationship of individual metabolic rate by testing Taylor's law (a power-function relationship between mean and variance) for individual rescaled metabolic rate across tree communities in tropical forests. We use a constraint-based model called maximum entropy theory of ecology (METE) to estimate and predict the parameters of Taylor's law. Our results show that, when assuming a universal metabolic scaling between metabolic rate and aboveground biomass, the METE generates the form of Taylor's law but fails to predict its slope. When setting the metabolic scaling exponent as a community-level parameter in the METE model, the estimated and predicted slopes of Taylor's law agree with each other. Our parameterized METE model reveals the positive effect of number of individuals on the metabolic scaling exponent. These results suggest that fluctuation scaling of individual metabolic rate can be explained solely by the macroecological constraints of communities, without relying on the physiological or genetic characters of individual organisms.

Shifting macroecological patterns and static theory failure in a stressed alpine plant community

AbstractAccumulating evidence suggests that ecological communities undergoing change in response to either anthropogenic or natural disturbances exhibit macroecological patterns that differ from those observed in similar types of communities in relatively undisturbed sites. In contrast to such cross‐site comparisons, however, there are few empirical studies of shifts over time in the shapes of macroecological patterns. Here, we provide a dramatic example of a plant community in which the species–area relationship and the species‐abundance distribution change markedly over a period of six years. These patterns increasingly deviate from the predictions of the maximum entropy theory of ecology (METE), which successfully predicts macroecological patterns in relatively static systems. The error in the species–area relationship prediction additionally correlates over time with increased stress measured as mortality minus recruitment, providing a link between demography and the failure of macroecological theory. Information on the dynamic state of an ecosystem inferred from snapshot measurements of macroecological community structure can potentially assist in identifying causes and consequences of disturbance and extending the domain of current theories and models to disturbed ecosystems.

The Shape of Species Abundance Distributions Across Spatial Scales

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.

DynaMETE: a hybrid MaxEnt-plus-mechanism theory of dynamic macroecology.

The Maximum Entropy Theory of Ecology (METE) predicts the shapes of macroecological metrics in relatively static ecosystems, across spatial scales, taxonomic categories and habitats, using constraints imposed by static state variables. In disturbed ecosystems, however, with time‐varying state variables, its predictions often fail. We extend macroecological theory from static to dynamic by combining the MaxEnt inference procedure with explicit mechanisms governing disturbance. In the static limit, the resulting theory, DynaMETE, reduces to METE but also predicts a new scaling relationship among static state variables. Under disturbances, expressed as shifts in demographic, ontogenic growth or migration rates, DynaMETE predicts the time trajectories of the state variables as well as the time‐varying shapes of macroecological metrics such as the species abundance distribution and the distribution of metabolic rates over individuals. An iterative procedure for solving the dynamic theory is presented. Characteristic signatures of the deviation from static predictions of macroecological patterns are shown to result from different kinds of disturbance. By combining MaxEnt inference with explicit dynamical mechanisms of disturbance, DynaMETE is a candidate theory of macroecology for ecosystems responding to anthropogenic or natural disturbances.

Disturbance macroecology: a comparative study of community structure metrics in a high‐severity disturbance regime

AbstractMacroecological 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 match METE predictions than do data from the more recently disturbed, mid‐seral stage stand. This suggests METE'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.

The effects of density dependence on spatial aggregation in the maximum entropy theory of ecology

The effects of density dependence on spatial aggregation in the maximum entropy theory of ecology

Derivations of the Core Functions 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.

Comparison of two maximum entropy models highlights the metabolic structure of metacommunities as a key determinant of local community assembly

The principle of Maximum Entropy (MaxEnt) promises a novel approach for understanding community assembly. Despite reproducing a variety of observed species abundance patterns, MaxEnt models in ecology have been hampered by disparate model assumptions and interpretations. A recurring challenge is that MaxEnt predictions are highly sensitive to the level of detail used to describe the community being modeled, and there seems to be no reason to prefer one level of detail over another. Here we present of formal unification of two previously developed MaxEnt models which differ in their level of detail, but which are otherwise mathematically similar. The less detailed model, “Maximum Entropy Theory of Ecology” (METE), does not resolve species identity or explicitly represent species-specific traits. The more detailed model, “Very Entropic Growth” (VEG), defines each separate species by its per capita metabolic rate ε and assumes a “density of species” function ρ(ε) representing the distribution of ε in the metacommunity. A formal comparison of METE and VEG then highlights ρ(ε) as a key determinant of local community assembly. In particular, appropriate choice of ρ(ε) in VEG can produce more realistic predictions for the metabolic-rank distribution of local communities than METE, which does not explicitly account for metacommunity structure. This opens new avenues of inquiry about what determines metacommunity structure in nature and suggests possible ways to improve METE.

Building up biogeography: Pattern to process

AbstractLinking pattern to process across spatial and temporal scales has been a key goal of the field of biogeography. In January 2017, the 8th biennial conference of the International Biogeography Society sponsored a symposium on Building up biogeography—process to pattern that aimed to review progress towards this goal. Here we present a summary of the symposium, in which we identified promising areas of current research and suggested future research directions. We focus on (1) emerging types of data such as behavioural observations and ancient DNA, (2) how to better incorporate historical data (such as fossils) to move beyond what we term “footprint measures” of past dynamics and (3) the role that novel modelling approaches (e.g. maximum entropy theory of ecology and approximate Bayesian computation) and conceptual frameworks can play in the unification of disciplines. We suggest that the gaps separating pattern and process are shrinking, and that we can better bridge these aspects by considering the dimensions of space and time simultaneously.

Maximum Entropy Theory of Ecology: A Reply to Harte.

In a paper published in this journal, I addressed the following problem: under which conditions will two scientists, observing the same system and sharing the same initial information, reach the same probabilistic description upon the application of the Maximum Entropy inference principle (MaxEnt) independent of the probability distribution chosen to set up the MaxEnt procedure. This is a minimal objectivity requirement which is generally asked for scientific investigation. In the same paper, I applied the findings to a critical examination of the application of MaxEnt made in Harte’s Maximum Entropy Theory of Ecology (METE). Prof. Harte published a comment to my paper and this is my reply. For the sake of the reader who may be unaware of the content of the papers, I have tried to make this reply self-contained and to skip technical details. However, I invite the interested reader to consult the previously published papers.

Maximum Entropy and Theory Construction: A Reply to Favretti.

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.

Remarks on the Maximum Entropy Principle with Application to the Maximum Entropy Theory of Ecology

In the first part of the paper we work out the consequences of the fact that Jaynes’ Maximum Entropy Principle, when translated in mathematical terms, is a constrained extremum problem for an entropy function expressing the uncertainty associated with the probability distribution p. Consequently, if two observers use different independent variables p or , the associated entropy functions have to be defined accordingly and they are different in the general case. In the second part we apply our findings to an analysis of the foundations of the Maximum Entropy Theory of Ecology (M.E.T.E.) a purely statistical model of an ecological community. Since the theory has received considerable attention by the scientific community, we hope to give a useful contribution to the same community by showing that the procedure of application of MEP, in the light of the theory developed in the first part, suffers from some incongruences. We exhibit an alternative formulation which is free from these limitations and that gives different results.