Maximum entropy and the state-variable approach to macroecology

Species Richness Net Primary Productivity and the Water Balance Problem

Species energy theory suggests that, because of limitations on reproduction efficiency, a minimum density of plant individuals per viable species exists and that this minimum correlates the total number of plant individuals N with the number of species S. The simplest assumption is that the mean energy input per individual plant is independent of the number of individuals, making N, and thus S as well, proportional to the total energy input into the system. The primary energy input to a plant-dominated ecosystem is estimated as its Net Primary Productivity (NPP). Thus, species energy theory draws a direct correspondence from NPP to S. Although investigations have verified a strong connection between S and NPP, strong influences of other factors, such as topography, ecological processes such as competition, and historical contingencies, are also at play. The lack of a simple model of NPP expressed in terms of the principal climate variables, precipitation P, and potential evapotranspiration, PET, introduces unnecessary uncertainty to the understanding of species richness across scales. Recent research combines percolation theory with the principle of ecological optimality to derive an expression for NPP(P, PET). Consistent with assuming S is proportional to NPP, we show here that the new expression for NPP(P, PET) predicts the number of plant species S in an ecosystem as a function of P and PET. As already demonstrated elsewhere, the results are consistent with some additional variation due to non-climatic inputs. We suggest that it may be easier to infer specific deviations from species energy predictions with increased accuracy and generality of the prediction of NPP(P, PET).

Assessment of Suitable Habitat of the Demoiselle Crane (Anthropoides virgo) in the Wake of Climate Change: A Study of Its Wintering Refugees in Pakistan

The inevitable impacts of climate change have reverberated across ecosystems and caused substantial global biodiversity loss. Climate-induced habitat loss has contributed to range shifts at both species and community levels. Given the importance of identifying suitable habitats for at-risk species, it is imperative to assess potential current and future distributions, and to understand influential environmental factors. Like many species, the Demoiselle crane is not immune to climatic pressures. Khyber Pakhtunkhwa and Balochistan provinces in Pakistan are known wintering grounds for this species. Given that Pakistan is among the top five countries facing devastating effects of climate change, this study sought to conduct species distribution modeling under climate change using data collected during 4 years of field surveys. We developed a Maximum Entropy distribution model to predict the current and projected future distribution of the species across the study area. Future habitat projections for 2050 and 2070 were carried out using two representative concentration pathways (RCP 4.5 and RCP 8.5) under three global circulation models, including HADGEM2-AO, BCC-CSM1-1, and CCSM4. The most influential factors shaping Demoiselle Crane habitat suitability included the temperature seasonality, annual mean temperature, terrain ruggedness index, and human population density, all of which contributed significantly to the suitability (81.3%). The model identified 35% of the study area as moderately suitable (134,068 km2) and highly suitable (27,911 km2) habitat for the species under current climatic conditions. Under changing climate scenarios, our model predicted a major loss of the species' current suitable habitat, with shrinkage and shift towards western-central areas along the Pakistan-Afghanistan boarder. The RCP 8.5, which is the extreme climate change scenario, portrays particularly severe consequences, with habitat losses reaching 65% in 2050 and 85% in 2070. This comprehensive study provides useful insights into the Demoiselle Crane habitat's current and future dynamics in Pakistan.

What constrains food webs? A maximum entropy framework for predicting their structure with minimal biases

Food webs are complex ecological networks whose structure is both ecologically and statistically constrained, with many network properties being correlated with each other. Despite the recognition of these invariable relationships in food webs, the use of the principle of maximum entropy (MaxEnt) in network ecology is still rare. This is surprising considering that MaxEnt is a statistical tool precisely designed for understanding and predicting many types of constrained systems. This principle asserts that the least-biased probability distribution of a system's property, constrained by prior knowledge about that system, is the one with maximum information entropy. MaxEnt has been proven useful in many ecological modeling problems, but its application in food webs and other ecological networks is limited. Here we show how MaxEnt can be used to derive many food-web properties both analytically and heuristically. First, we show how the joint degree distribution (the joint probability distribution of the numbers of prey and predators for each species in the network) can be derived analytically using the number of species and the number of interactions in food webs. Second, we present a heuristic and flexible approach of finding a network's adjacency matrix (the network's representation in matrix format) based on simulated annealing and SVD entropy. We built two heuristic models using the connectance and the joint degree sequence as statistical constraints, respectively. We compared both models' predictions against corresponding null and neutral models commonly used in network ecology using open access data of terrestrial and aquatic food webs sampled globally (N = 257). We found that the heuristic model constrained by the joint degree sequence was a good predictor of many measures of food-web structure, especially the nestedness and motifs distribution. Specifically, our results suggest that the structure of terrestrial and aquatic food webs is mainly driven by their joint degree distribution.

Universality of Form: The Case of Retinal Cone Photoreceptor Mosaics

Cone photoreceptor cells are wavelength-sensitive neurons in the retinas of vertebrate eyes and are responsible for color vision. The spatial distribution of these nerve cells is commonly referred to as the cone photoreceptor mosaic. By applying the principle of maximum entropy, we demonstrate the universality of retinal cone mosaics in vertebrate eyes by examining various species, namely, rodent, dog, monkey, human, fish, and bird. We introduce a parameter called retinal temperature, which is conserved across the retinas of vertebrates. The virial equation of state for two-dimensional cellular networks, known as Lemaître's law, is also obtained as a special case of our formalism. We investigate the behavior of several artificially generated networks and the natural one of the retina concerning this universal, topological law.

Diversity and metabolic energy in bacteria

Why are some groups of bacteria more diverse than others? We hypothesize that the metabolic energy available to a bacterial functional group (a biogeochemical group or 'guild') has a role in such a group's taxonomic diversity. We tested this hypothesis by looking at the metacommunity diversity of functional groups in multiple biomes. We observed a positive correlation between estimates of a functional group's diversity and their metabolic energy yield. Moreover, the slope of that relationship was similar in all biomes. These findings could imply the existence of a universal mechanism controlling the diversity of all functional groups in all biomes in the same way. We consider a variety of possible explanations from the classical (environmental variation) to the 'non-Darwinian' (a drift barrier effect). Unfortunately, these explanations are not mutually exclusive, and a deeper understanding of the ultimate cause(s) of bacterial diversity will require us to determine if and how the key parameters in population genetics (effective population size, mutation rate, and selective gradients) vary between functional groups and with environmental conditions: this is a difficult task.

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.

Combining metabolic and maximum entropy theories of ecology, the authors derive an equation of state capable of capturing the relationships between multiple ecological variables across varied spatial scales and habitats.

Information theory: A foundation for complexity science

Modeling and inference are central to most areas of science and especially to evolving and complex systems. Critically, the information we have is often uncertain and insufficient, resulting in an underdetermined inference problem; multiple inferences, models, and theories are consistent with available information. Information theory (in particular, the maximum information entropy formalism) provides a way to deal with such complexity. It has been applied to numerous problems, within and across many disciplines, over the last few decades. In this perspective, we review the historical development of this procedure, provide an overview of the many applications of maximum entropy and its extensions to complex systems, and discuss in more detail some recent advances in constructing comprehensive theory based on this inference procedure. We also discuss efforts at the frontier of information-theoretic inference: application to complex dynamic systems with time-varying constraints, such as highly disturbed ecosystems or rapidly changing economies.

Inferring spatial source of disease outbreaks using maximum entropy

Mathematical modeling of disease outbreaks can infer the future trajectory of an epidemic, allowing for making more informed policy decisions. Another task is inferring the origin of a disease, which is relatively difficult with current mathematical models. Such frameworks, across varying levels of complexity, are typically sensitive to input data on epidemic parameters, case counts, and mortality rates, which are generally noisy and incomplete. To alleviate these limitations, we propose a maximum entropy framework that fits epidemiological models, provides calibrated infection origin probabilities, and is robust to noise due to a prior belief model. Maximum entropy is agnostic to the parameters or model structure used and allows for flexible use when faced with sparse data conditions and incomplete knowledge in the dynamical phase of disease-spread, providing for more reliable modeling at early stages of outbreaks. We evaluate the performance of our model by predicting future disease trajectories based on simulated epidemiological data in synthetic graph networks and the real mobility network of New York State. In addition, unlike existing approaches, we demonstrate that the method can be used to infer the origin of the outbreak with accurate confidence. Indeed, despite the prevalent belief on the feasibility of contact-tracing being limited to the initial stages of an outbreak, we report the possibility of reconstructing early disease dynamics, including the epidemic seed, at advanced stages.

Red Junglefowl Resource Management Guide: Bioresource Reintroduction for Sustainable Food Security in Thailand

The domestication of wild animals represents a major milestone for human civilization. Chicken is the largest domesticated livestock species and used for both eggs and meat. Chicken originate from the red junglefowl (Gallus gallus). Its adaptability to diverse environments and ease of selective breeding provides a unique genetic resource to address the challenges of food security in a world impacted by climatic change and human population growth. Habitat loss has caused population declines of red junglefowl in Thailand. However, genetic diversity is likely to remain in captive stocks. We determine the genetic diversity using microsatellite genotyping and the mitochondrial D-loop sequencing of wild red junglefowl. We identified potential distribution areas in Thailand using maximum entropy models. Protected areas in the central and upper southern regions of Thailand are highly suitable habitats. The Bayesian clustering analysis of the microsatellite markers revealed high genetic diversity in red junglefowl populations in Thailand. Our model predicted that forest ranges are a highly suitable habitat that has enabled the persistence of a large gene pool with a nationwide natural distribution. Understanding the red junglefowl allows us to implement improved resource management, species reintroduction, and sustainable development to support food security objectives for local people.

How do African elephants utilize the landscape during wet season? A habitat connectivity analysis for Sioma Ngwezi landscape in Zambia

The influence of environmental factors on the distribution and persistence of African elephants (Loxodonta africana) is pertinent to policy makers and managers to formulate balanced plans for different land-use types.The study focuses on movement of elephants and how they utilize foraging areas in Sioma Ngwezi landscape in Zambia by answering the following questions: (1) Which environmental variables and land-cover class predict the movement of elephants during the wet season in Sioma Ngwezi landscape? (2) What is the wet season suitable habitat for elephants in Sioma Ngwezi landscape? (3) What are the major wet season movement corridors for elephants in Sioma Ngwezi landscape?We used GPS telemetry data from the collared elephants to assess habitat connectivity. Maximum entropy (MaxEnt) and linkage mapper were the tools used to predict habitat suitability, movement corridors, and barriers in the landscape during the wet season.The study identified elevation, land cover, and NDVI as the most important environmental predictors that modify the dispersal of elephants in the landscape during the wet season. Additionally, a total of 36 potential wet season corridors were identified connecting 15 core areas mainly used for foraging and protection from poachers in the landscape. Of these, 24 corridors were highly utilized and are suggested as priority corridors for elephant movement in the landscape.The identified wet season habitats and functional corridors may help to combat elephant poaching by patrolling areas with high relative probability of elephant presence. The findings may also help abate human-elephant conflict such as crop-raiding by managing identified corridors that run into agriculture zones in the game management area.

A unified framework for species spatial patterns: Linking the occupancy area curve, Taylor's Law, the neighborhood density function and two-plot species turnover

The description of spatial patterns in species distributions is central to research throughout ecology. In this manuscript, we demonstrate that five of the most widely used species‐level spatial patterns are not only related, but can in fact be quantitatively derived from each other under minimal assumptions: the occupancy area curve, Taylor's Law, the neighborhood density function, a two‐plot variant of Taylor's Law and two‐plot single‐species turnover. We present an overarching mathematical framework and derivations for several theoretical example cases, along with a simulation study and empirical analysis that applies the framework to data from the Barro Colorado Island tropical forest plot. We discuss how knowledge of this mathematical relationship can support the testing of ecological theory, suggest efficient field sampling schemes, highlight the relative importance of plot area and abundance in driving turnover patterns and lay the groundwork for future unified theories of community‐level spatial metrics and multi‐patch spatial patterns.

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.

Empirical abundance distributions are more uneven than expected given their statistical baseline

Exploring and accounting for the emergent properties of ecosystems as complex systems is a promising horizon in the search for general processes to explain common ecological patterns. For example the ubiquitous hollow-curve form of the species abundance distribution is frequently assumed to reflect ecological processes structuring communities, but can also emerge as a statistical phenomenon from the mathematical definition of an abundance distribution. Although the hollow curve may be a statistical artefact, ecological processes may induce subtle deviations between empirical species abundance distributions and their statistically most probable forms. These deviations may reflect biological processes operating on top of mathematical constraints and provide new avenues for advancing ecological theory. Examining ~22,000 communities, we found that empirical SADs are highly uneven and dominated by rare species compared to their statistical baselines. Efforts to detect deviations may be less informative in small communities-those with few species or individuals-because these communities have poorly resolved statistical baselines. The uneven nature of many empirical SADs demonstrates a path forward for leveraging complexity to understand ecological processes governing the distribution of abundance, while the issues posed by small communities illustrate the limitations of using this approach to study ecological patterns in small samples.

Emergent Rarity Properties in Carabid Communities From Chinese Steppes With Different Climatic Conditions

Species abundance distributions (SADs) are increasingly used to investigate how species community structure changes in response to environmental variations. SAD models depict the relative abundance of species recorded in a community and express fundamental aspects of the community structure, namely patterns of commonness and rarity. However, the influence of differences in environmental conditions on SAD characteristics is still poorly understood. In this study we used SAD models of carabid beetles (Coleoptera: Carabidae) in three grassland ecosystems (desert, typical, and meadow steppes) in China. These ecosystems are characterized by different aridity conditions, thus offering an opportunity to investigate how SADs are influenced by differences in environmental conditions (mainly aridity and vegetation cover, and hence productivity). We used various SAD models, including the meta-community zero sum multinomial (mZSM), the lognormal (PLN) and Fisher’s logseries (LS), and uni- and multimodal gambin models. Analyses were done at the level of steppe type (coarse scale) and for different sectors within the same steppe (fine scale). We found that the mZSM model provided, in general, the best fit at both analysis scales. Model parameters were influenced by the scale of analysis. Moreover, the LS was the best fit in desert steppe SAD. If abundances are rarefied to the smallest sample, results are similar to those without rarefaction, but differences in models estimates become more evident. Gambin unimodal provided the best fit with the lowest α-value observed in desert steppe and higher values in typical and meadow steppes, with results which were strongly affected by the scale of analysis and the use of rarefaction. Our results indicate that all investigated communities are adequately modeled by two similar distributions, the mZSM and the LS, at both scales of analyses. This indicates (1) that all communities are characterized by a relatively small number of species, most of which are rare, and (2) that the meta-communities at the large scale maintain the basic SAD shape of the local communities. The gambin multimodal models produced exaggerated α-values, which indicates that they overfit simple communities. Overall, Fisher’s α, mZSM θ, and gambin α-values were substantially lower in the desert steppe and higher in the typical and meadow steppes, which implies a decreasing influence of environmental harshness (aridity) from the desert steppe to the typical and meadow steppes.

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.

Maximum entropy method for estimating the reproduction number: An investigation for COVID-19 in China and the United States

The key parameter that characterizes the transmissibility of a disease is the reproduction number R. If it exceeds 1, the number of incident cases will inevitably grow over time, and a large epidemic is possible. To prevent the expansion of an epidemic, R must be reduced to a level below 1. To estimate the reproduction number, the probability distribution function of the generation interval of an infectious disease is required to be available; however, this distribution is often unknown. In this paper, given the incomplete information for the generation interval, we propose a maximum entropy method to estimate the reproduction number. Based on this method, given the mean value and variance of the generation interval, we first determine its probability distribution function and in turn estimate the real-time values of the reproduction number of COVID-19 in China and the United States. By applying these estimated reproduction numbers into the susceptible-infectious-removed epidemic model, we simulate the evolutionary tracks of the epidemics in China and the United States, both of which are in accordance with that of the real incident cases.

Parameterized maximum entropy models predict variability of metabolic scaling across tree communities and populations

The maximum entropy theory of ecology (METE) applies the concept of "entropy" from information theory to predict macroecological patterns. The energetic predictions of the METE rely on predetermined metabolic scaling from external theories, and this reliance diminishes the testability of the theory. In this work, I build parameterized METE models by treating the metabolic scaling exponent as a free parameter, and I use the maximum-likelihood method to obtain empirically plausible estimates of the exponent. I test the models using the individual tree data from an oak-dominated deciduous forest in the northeastern United States and from a tropical forest in central Panama. My analysis shows that the metabolic scaling exponents predicted from the parameterized METE models deviate from that of the metabolic theory of ecology and exhibit large variation, at both community and population levels. Assemblage and population abundance may act as ecological constraints that regulate the individual-level metabolic scaling behavior. This study provides a novel example of the use of the parameterized METE models to reveal the biological processes of individual organisms. The implication and possible extensions of the parameterized METE models are discussed.

Invariance in ecological pattern

Background: The abundance of different species in a community often follows the log series distribution. Other ecological patterns also have simple forms. Why does the complexity and variability of ecological systems reduce to such simplicity? Common answers include maximum entropy, neutrality, and convergent outcome from different underlying biological processes.  Methods: This article proposes a more general answer based on the concept of invariance, the property by which a pattern remains the same after transformation. Invariance has a long tradition in physics. For example, general relativity emphasizes the need for the equations describing the laws of physics to have the same form in all frames of reference.  Results: By bringing this unifying invariance approach into ecology, we show that the log series pattern dominates when the consequences of processes acting on abundance are invariant to the addition or multiplication of abundance by a constant. The lognormal pattern dominates when the processes acting on net species growth rate obey rotational invariance (symmetry) with respect to the summing up of the individual component processes. Conclusions: Recognizing how these invariances connect pattern to process leads to a synthesis of previous approaches. First, invariance provides a simpler and more fundamental maximum entropy derivation of the log series distribution. Second, invariance provides a simple derivation of the key result from neutral theory: the log series at the metacommunity scale and a clearer form of the skewed lognormal at the local community scale. The invariance expressions are easy to understand because they uniquely describe the basic underlying components that shape pattern.

Habitat suitability model with maximum entropy approach for European roe deer (Capreolus capreolus) in the Black Sea Region

Evaluating the relationships between wildlife species and their habitats helps to predict effects of habitat change for present and future management of wild animal populations. Building ecological models are good ways to understand and manage wildlife populations and to predict various environmental scenarios. Recently, management of ungulates is becoming more important in Europe due to a high demand of hunting and their role in biodiversity. European roe deer (Capreolus capreolus) is the smallest species of cervids and has a widespread distribution in Turkey. In this study, two habitat suitability models of roe deers, living in the Black Sea Region in Turkey, were created by using a maximum entropy (MaxEnt) approach. Two wildlife development areas, which have widely different habitat types, were selected as study sites. As a result of this study, area under the curve (AUC) values were found to be above 0.80. According to the modeling results, in two different habitat types, ecological variables are quite similar in general. This study is the first study on modeling European roe deers in Turkey.

A Comparison of the Maximum Entropy Principle Across Biological Spatial Scales

Despite their differences, biological systems at different spatial scales tend to exhibit common organizational patterns. Unfortunately, these commonalities are often hard to grasp due to the highly specialized nature of modern science and the parcelled terminology employed by various scientific sub-disciplines. To explore these common organizational features, this paper provides a comparative study of diverse applications of the maximum entropy principle, which has found many uses at different biological spatial scales ranging from amino acids up to societies. By presenting these studies under a common approach and language, this paper aims to establish a unified view over these seemingly highly heterogeneous scenarios.

Thermodynamics Beyond Molecules: Statistical Thermodynamics of Probability Distributions

Statistical thermodynamics has a universal appeal that extends beyond molecular systems, and yet, as its tools are being transplanted to fields outside physics, the fundamental question, what is thermodynamics, has remained unanswered. We answer this question here. Generalized statistical thermodynamics is a variational calculus of probability distributions. It is independent of physical hypotheses but provides the means to incorporate our knowledge, assumptions and physical models about a stochastic processes that gives rise to the probability in question. We derive the familiar calculus of thermodynamics via a probabilistic argument that makes no reference to physics. At the heart of the theory is a space of distributions and a special functional that assigns probabilities to this space. The maximization of this functional generates the mathematical network of thermodynamic relationship. We obtain statistical mechanics as a special case and make contact with Information Theory and Bayesian inference.

Habitat preference and potential distribution of Magnolia officinalis subsp. officinalis and M. o. subsp. biloba in China

Magnolia officinalis subsp. officinalis and M. officinalis subsp. biloba are important medicinal plants in China. The bark of these two subspecies is commonly used in the production of a widely-used Chinese traditional medicine named ‘Houpu’. In recent years, M. o. subsp. officinalis and M. o. subsp. biloba have become increasingly threatened owing to the over-harvesting of their bark and the fragmentation of their habitats. In this study, we aimed to support the conservation and cultivation of these two subspecies in China by: (1) assessing the relationship between numerous environmental variables and the geographical distributions of the subspecies; (2) analysing the environmental characteristics of suitable habitats for both subspecies and predicting the spatial distribution of these habitats in China; and (3) identifying conservation areas of both subspecies in China via overlay analysis. We also assessed the degree of human disturbance within suitable habitats. We found that temperature was a major determinant for the distribution of M. o. subsp. officinalis. Conversely, the distribution of M. o. subsp. biloba was primarily dependent on precipitation rather than temperature. Distinct habitat preferences were observed between M. o. subsp. officinalis and M. o. subsp. biloba. Suitable habitats of M. o. subsp. officinalis were primarily distributed in the northern subtropical areas of China, with greater fluctuations in ambient temperature, lower extreme temperatures, less precipitation and greater fluctuations in precipitation. Habitats suitable for M. o. subsp. biloba were highly fragmented and were distributed in the central subtropical areas of China. We found that a large proportion of suitable habitats were not in the protected areas and that they were significantly disturbed by human activity. This analysis could provide useful information for the conservation of both M. o. subsp. officinalis and M. o. subsp. biloba and could aid in the selection of cultivation sites.

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.

Species with moderate intraspecific trait variability are locally abundant within an environmentally heterogeneous subtropical forest

Species with large intraspecific trait variability (ITV) have larger niche breadth than species with low ITV and thus are expected to be more abundant at the local scale. However, whether the positive ITV-abundance relationship holds in heterogeneous local environments remains uncertain. Using an individual-based trait dataset encompassing three leaf traits (leaf area, specific leaf area, and leaf dry mass content) of 20,248 individuals across 80 species in an environmentally heterogeneous subtropical forest in eastern China, ITV for each trait of each species was estimated by rarefaction. Resource-based niche breadth and marginality (the absolute distance between the mean resource states used by a species and the mean plot-wise resource states) were estimated simultaneously by the K-S method and the outlying mean index, respectively. Species with moderate ITV were often locally abundant, while species with large or small ITV were locally rare. This unimodal relationship between ITV and species abundance persisted when traits were analyzed separately and for all tree size classes. There was also a hump-backed relationship between niche breadth and marginality, and ITV was positively associated with niche breadth. The combined results suggest either a trade-off between the benefit from expanding niche breadth to adapt to multiple habitats and the disadvantage of reducing competitive ability, or a scarcity of favorable resources. Our results do not support the traditional thought that ITV positively correlates with species abundance in heterogeneous local environments. Instead, our study suggests that moderate-rather than large-intraspecific trait variability increases species abundance at local scales.

Habitat differentiation and conservation gap of Magnolia biondii, M. denudata, and M. sprengeri in China

The flower buds of Magnolia biondii, M. denudata, and M. sprengeri are the materials of Xinyi, a traditional Chinese medicine. The harvest of flower buds and habitat fragmentation caused by human disturbance heavily threatens the natural regeneration and survival of these three Magnolia species. With the aim to support the conservation and improve the effectiveness of conservation, we performed an assessment on habitat suitability, influences of environmental variables on habitat suitability, and the conservation gap of these three Magnolia species, based on the Maxent modeling method. The results indicated that: (1) altitude, annual mean temperature, extreme temperature, temperature fluctuation, annual precipitation, and extreme precipitation are the most influential environmental variables for the distribution of M. sprengeri, M. biondii, and M. denudata; (2) obvious habitat differentiations were observed among M. biondii, M. denudata, and M. sprengeri. M. sprengeri tends to be located in further northern areas with higher altitudes, lower temperatures, and lower precipitation compared to M. biondii and M. denudata; and (3) a large proportion of suitable habitats have been left without protection. Woodland and forest shared the largest area out of the suitable habitats. However, grassland, agricultural land, residential land, and mining and industry areas also occupied large areas of suitable habitats.

Estimating species richness in hyper-diverse large tree communities

Species richness estimation is one of the most widely used analyses carried out by ecologists, and nonparametric estimators are probably the most used techniques to carry out such estimations. We tested the assumptions and results of nonparametric estimators and those of a logseries approach to species richness estimation for simulated tropical forests and five data sets from the field. We conclude that nonparametric estimators are not suitable to estimate species richness in tropical forests, where sampling intensity is usually low and richness is high, because the assumptions of the methods do not meet the sampling strategy used in most studies. The logseries, while also requiring substantial sampling, is much more effective in estimating species richness than commonly used nonparametric estimators, and its assumptions better match the way field data is being collected.

Evaluating Sustainability of Regional Water Resources Based on Improved Generalized Entropy Method

The sustainability of regional water resources has important supporting data needed for establishing policies on the sustainable development of the social economy. The purpose of this paper is to propose an assessment method to accurately reflect the sustainability of regional water resources in various areas. The method is based on the relative entropy of the information entropy theory. The steps are as follows. Firstly, the pretreatment of the evaluation sample data is required, before the relative entropy of each standard evaluation sample and evaluation grade (SEG) is calculated to obtain the entropy weight of each evaluation index. After this, the entropy weighted comprehensive index (WCI) of the standard evaluation grade sample is obtained. The function relation between WCI and SEG can be fitted by the cubic polynomial to construct the evaluation function. Using the above steps, a generalized entropy method (GEM) for the sustainable assessment of regional water resources is established and it is used to evaluate the sustainability of water resources in the Pingba and Huai River areas in China. The results show that the proposed GEM model can accurately reflect the sustainable water resources in the two regions. Compared with the other evaluation models, such as the Shepherd method, Artificial Neural Network and Fuzzy comprehensive evaluation, the GEM model has larger differences in its evaluation results, which are more reasonable. Thus, the proposed GEM model can provide scientific data support for coordinating the relationship between the sustainable development and utilization of regional water resources in order to improve the development of regional population, society and economy.

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.

The more-individuals hypothesis revisited: the role of community abundance in species richness regulation and the productivity-diversity relationship

Species richness increases with energy availability, yet there is little consensus as to the exact processes driving this species-energy relationship. The most straightforward explanation is the more-individuals hypothesis (MIH). It states that higher energy availability promotes a higher total number of individuals in a community, which consequently increases species richness by allowing for a greater number of species with viable populations. Empirical support for the MIH is mixed, partially due to the lack of proper formalisation of the MIH and consequent confusion as to its exact predictions. Here, we review the evidence of the MIH and evaluate the reliability of various predictions that have been tested. There is only limited evidence that spatial variation in species richness is driven by variation in the total number of individuals. There are also problems with measures of energy availability, with scale-dependence, and with the direction of causality, as the total number of individuals may sometimes itself be driven by the number of species. However, even in such a case the total number of individuals may be involved in diversity regulation. We propose a formal theory that encompasses these processes, clarifying how the different factors affecting diversity dynamics can be disentangled.

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.

An introduction to the maximum entropy approach and its application to inference problems in biology

A cornerstone of statistical inference, the maximum entropy framework is being increasingly applied to construct descriptive and predictive models of biological systems, especially complex biological networks, from large experimental data sets. Both its broad applicability and the success it obtained in different contexts hinge upon its conceptual simplicity and mathematical soundness. Here we try to concisely review the basic elements of the maximum entropy principle, starting from the notion of 'entropy', and describe its usefulness for the analysis of biological systems. As examples, we focus specifically on the problem of reconstructing gene interaction networks from expression data and on recent work attempting to expand our system-level understanding of bacterial metabolism. Finally, we highlight some extensions and potential limitations of the maximum entropy approach, and point to more recent developments that are likely to play a key role in the upcoming challenges of extracting structures and information from increasingly rich, high-throughput biological data.

Modeling the spatial and temporal dynamics of riparian vegetation induced by river flow fluctuation

River flow fluctuation has an important influence on riparian vegetation dynamics. A temporally segmented stochastic model focusing on a same‐aged population is developed for the purpose of describing both spatial and temporal dynamics of riparian vegetation. In the model, the growth rate of population, rather than carrying capacity, is modeled as the random variable. This model has explicit physical meaning. The model deduces a process‐based solution. From the solution process, the probability density of spatial distribution can be derived; therefore, the spatial distribution of population abundance can be described. The lifespan of a same‐aged population and the age structure of the species‐specific population can also be studied with the aid of this temporally segmented model. The influence of correlation time of river flow fluctuation is also quantified according to the model. The calibration of model parameters and model application are discussed. The model provides a computer‐aided method to simulate and predict vegetation dynamics during river flow disturbances. Meanwhile, the model is open and allows for more accurate and concrete modeling of growth rate. Because of the Markov property involved in the process‐based solution, the model also has the ability to deal with cases of nonstationary disturbances.

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 H(p) expressing the uncertainty associated with the probability distribution p. Consequently, if two observers use different independent variables p or g(p), 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.