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.

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.

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.

Climate warming drives local extinction: Evidence from observation and experimentation.

Despite increasing concern about elevated extinction risk as global temperatures rise, it is difficult to confirm causal links between climate change and extinction. By coupling 25 years of in situ climate manipulation with experimental seed introductions and both historical and current plant surveys, we identify causal, mechanistic links between climate change and the local extinction of a widespread mountain plant (Androsace septentrionalis). Climate warming causes precipitous declines in population size by reducing fecundity and survival across multiple life stages. Climate warming also purges belowground seed banks, limiting the potential for the future recovery of at-risk populations under ameliorated conditions. Bolstered by previous reports of plant community shifts in this experiment and in other habitats, our findings not only support the hypothesis that climate change can drive local extinction but also foreshadow potentially widespread species losses in subalpine meadows as climate warming continues.

Publisher Correction: Microbial community-level regulation explains soil carbon responses to long-term litter manipulations.

The original PDF version of this Article contained an error in Table 1. On the right-hand side of the third row, the third equation was missing a ß as an exponent on the first CB. This has now been corrected in the PDF version of the Article. The HTML version was correct from the time of publication.

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.

Microbial community-level regulation explains soil carbon responses to long-term litter manipulations.

Climatic, atmospheric, and land-use changes all have the potential to alter soil microbial activity, mediated by changes in plant inputs. Many microbial models of soil organic carbon (SOC) decomposition have been proposed recently to advance prediction of climate and carbon (C) feedbacks. Most of these models, however, exhibit unrealistic oscillatory behavior and SOC insensitivity to long-term changes in C inputs. Here we diagnose the source of these problems in four archetypal models and propose a density-dependent formulation of microbial turnover, motivated by community-level interactions, that limits population sizes and reduces oscillations. We compare model predictions to 24 long-term C-input field manipulations and identify key benchmarks. The proposed formulation reproduces soil C responses to long-term C-input changes and implies greater SOC storage associated with CO2-fertilization-driven increases in C inputs over the coming century compared to recent microbial models. This study provides a simple modification to improve microbial models for inclusion in Earth System Models.

Consumption-Based Conservation Targeting: Linking Biodiversity Loss to Upstream Demand through a Global Wildlife Footprint.

Although most conservation efforts address the direct, local causes of biodiversity loss, effective long-term conservation will require complementary efforts to reduce the upstream economic pressures, such as demands for food and forest products, which ultimately drive these downstream losses. Here, we present a wildlife footprint analysis that links global losses of wild birds to consumer purchases across 57 economic sectors in 129 regions. The United States, India, China, and Brazil have the largest regional wildlife footprints, while per-person footprints are highest in Mongolia, Australia, Botswana, and the United Arab Emirates. A US$100 purchase of bovine meat or rice products occupies approximately 0.1 km2 of wild bird ranges, displacing 1-2 individual birds, for 1 year. Globally significant importer regions, including Japan, the United Kingdom, Germany, Italy, and France, have large footprints that drive wildlife losses elsewhere in the world and represent important targets for consumption-focused conservation attention.

Temperature response of soil respiration largely unaltered with experimental warming.

The respiratory release of carbon dioxide (CO2) from soil is a major yet poorly understood flux in the global carbon cycle. Climatic warming is hypothesized to increase rates of soil respiration, potentially fueling further increases in global temperatures. However, despite considerable scientific attention in recent decades, the overall response of soil respiration to anticipated climatic warming remains unclear. We synthesize the largest global dataset to date of soil respiration, moisture, and temperature measurements, totaling >3,800 observations representing 27 temperature manipulation studies, spanning nine biomes and over 2 decades of warming. Our analysis reveals no significant differences in the temperature sensitivity of soil respiration between control and warmed plots in all biomes, with the exception of deserts and boreal forests. Thus, our data provide limited evidence of acclimation of soil respiration to experimental warming in several major biome types, contrary to the results from multiple single-site studies. Moreover, across all nondesert biomes, respiration rates with and without experimental warming follow a Gaussian response, increasing with soil temperature up to a threshold of ∼25 °C, above which respiration rates decrease with further increases in temperature. This consistent decrease in temperature sensitivity at higher temperatures demonstrates that rising global temperatures may result in regionally variable responses in soil respiration, with colder climates being considerably more responsive to increased ambient temperatures compared with warmer regions. Our analysis adds a unique cross-biome perspective on the temperature response of soil respiration, information critical to improving our mechanistic understanding of how soil carbon dynamics change with climatic warming.

Predicting extinction debt from community patterns.

A significant challenge in both measuring and predicting species extinction rates at global and local scales is the possibility of extinction debt, time-delayed extinctions that occur gradually following an initial impact. Here we examine how relative abundance distributions and spatial aggregation combine to influence the likely magnitude of future extinction debt following habitat loss or climate-driven range contraction. Our analysis is based on several fundamental premises regarding abundance distributions, most importantly that species abundances immediately following habitat loss are a sample from an initial relative abundance distribution and that the long-term, steady-state form of the species abundance distribution is a property of the biology of a community and not of area. Under these two hypotheses, the results show that communities following canonical lognormal and broken-stick abundance distributions are prone to exhibit extinction debt, especially when species exhibit low spatial aggregation. Conversely, communities following a logseries distribution with a constant Fisher's α parameter never demonstrate extinction debt and often show an "immigration credit," in which species richness rises in the long term following an initial decrease. An illustration of these findings in 25 biodiversity hotspots suggests a negligible immediate extinction rate for bird communities and eventual extinction debts of 30-50% of initial species richness, whereas plant communities are predicted to immediately lose 5-15% of species without subsequent extinction debt. These results shed light on the basic determinants of extinction debt and provide initial indications of the magnitude of likely debts in landscapes where few empirical data are available.

Integrating macroecological metrics and community taxonomic structure.

We extend macroecological theory based on the maximum entropy principle from species level to higher taxonomic categories, thereby predicting distributions of species richness across genera or families and the dependence of abundance and metabolic rate distributions on taxonomic tree structure. Predictions agree with qualitative trends reported in studies on hyper-dominance in tropical tree species, mammalian body size distributions and patterns of rarity in worldwide plant communities. Predicted distributions of species richness over genera or families for birds, arthropods, plants and microorganisms are in excellent agreement with data. Data from an intertidal invertebrate community, but not from a dispersal-limited forest, are in excellent agreement with a predicted new relationship between body size and abundance. Successful predictions of the original species level theory are unmodified in the extended theory. By integrating macroecology and taxonomic tree structure, maximum entropy may point the way towards a unified framework for understanding phylogenetic community structure.

Population dynamics and competitive outcome derive from resource allocation statistics: The governing influence of the distinguishability of individuals.

Model predictions for species competition outcomes highly depend on the assumed form of the population growth function. In this paper we apply an alternative inferential method based on statistical mechanics, maximizing Boltzmann entropy, to predict resource-constrained population dynamics and coexistence. Within this framework, population dynamics and competition outcome can be determined without assuming any particular form of the population growth function. The dynamics of each species is determined by two parameters: the mean resource requirement θ (related to the mean metabolic rate) and individual distinguishability Dr (related to intra- compared to interspecific functional variation). Our theory clarifies the condition for the energetic equivalence rule (EER) to hold, and provide a statistical explanation for the importance of species functional variation in determining population dynamics and coexistence patterns.

Inferring regional-scale species diversity from small-plot censuses.

Estimation of the number of species at spatial scales too large to census directly is a longstanding ecological challenge. A recent comprehensive census of tropical arthropods and trees in Panama provides a unique opportunity to apply an inference procedure for up-scaling species richness and thereby make progress toward that goal. Confidence in the underlying theory is first established by showing that the method accurately predicts the species abundance distribution for trees and arthropods, and in particular accurately captures the rare tail of the observed distributions. The rare tail is emphasized because the shape of the species-area relationship is especially influenced by the numbers of rare species. The inference procedure is then applied to estimate the total number of arthropod and tree species at spatial scales ranging from a 6000 ha forest reserve to all of Panama, with input data only from censuses in 0.04 ha plots. The analysis suggests that at the scale of the reserve there are roughly twice as many arthropod species as previously estimated. For the entirety of Panama, inferred tree species richness agrees with an accepted empirical estimate, while inferred arthropod species richness is significantly below a previous published estimate that has been criticized as too high. An extension of the procedure to estimate species richness at continental scale is proposed.

Convergent ecosystem responses to 23-year ambient and manipulated warming link advancing snowmelt and shrub encroachment to transient and long-term climate-soil carbon feedback.

Ecosystem responses to climate change can exert positive or negative feedbacks on climate, mediated in part by slow-moving factors such as shifts in vegetation community composition. Long-term experimental manipulations can be used to examine such ecosystem responses, but they also present another opportunity: inferring the extent to which contemporary climate change is responsible for slow changes in ecosystems under ambient conditions. Here, using 23 years of data, we document a shift from nonwoody to woody vegetation and a loss of soil carbon in ambient plots and show that these changes track previously shown similar but faster changes under experimental warming. This allows us to infer that climate change is the cause of the observed shifts in ambient vegetation and soil carbon and that the vegetation responses mediate the observed changes in soil carbon. Our findings demonstrate the realism of an experimental manipulation, allow attribution of a climate cause to observed ambient ecosystem changes, and demonstrate how a combination of long-term study of ambient and experimental responses to warming can identify mechanistic drivers needed for realistic predictions of the conditions under which ecosystems are likely to become carbon sources or sinks over varying timescales.

Responses of high-altitude graminoids and soil fungi to 20 years of experimental warming.

High-elevation ecosystems are expected to be particularly sensitive to climate warming because cold temperatures constrain biological processes. Deeper understanding of the consequences of climate change will come from studies that consider not only the direct effects of temperature on individual species, but also the indirect effects of altered species interactions. Here we show that 20 years of experimental warming has changed the species composition of graminoid (grass and sedge) assemblages in a subalpine meadow of the Rocky Mountains, USA, by increasing the frequency of sedges and reducing the frequency of grasses. Because sedges typically have weak interactions with mycorrhizal fungi relative to grasses, lowered abundances of arbuscular mycorrhizal (AM) fungi or other root-inhabiting fungi could underlie warming-induced shifts in plant species composition. However, warming increased root colonization by AM fungi for two grass species, possibly because AM fungi can enhance plant water uptake when soils are dried by experimental warming. Warming had no effect on AM fungal colonization of three other graminoids. Increased AM fungal colonization of the dominant shrub Artemisia tridentata provided further grounds for rejecting the hypothesis that reduced AM fungi caused the shift from grasses to sedges. Non-AM fungi (including dark septate endophytes) also showed general increases with warming. Our results demonstrate that lumping grasses and sedges when characterizing plant community responses can mask significant shifts in the responses of primary producers, and their symbiotic fungi, to climate change.

Maximum information entropy: a foundation for ecological theory.

The maximum information entropy (MaxEnt) principle is a successful method of statistical inference that has recently been applied to ecology. Here, we show how MaxEnt can accurately predict patterns such as species-area relationships (SARs) and abundance distributions in macroecology and be a foundation for ecological theory. We discuss the conceptual foundation of the principle, why it often produces accurate predictions of probability distributions in science despite not incorporating explicit mechanisms, and how mismatches between predictions and data can shed light on driving mechanisms in ecology. We also review possible future extensions of the maximum entropy theory of ecology (METE), a potentially important foundation for future developments in ecological theory.

Taxon categories and the universal species-area relationship (a comment on Sizling et al., "between geometry and biology:the problem of universality of the species-area relationship").

A theory of macroecology based on the maximum information entropy (MaxEnt) inference procedure predicts that the log-log slope of the species-area relationship (SAR) at any spatial scale is a specified function of the ratio of abundance, N(A), to species richness, S(A), at that scale. The theory thus predicts, in generally good agreement with observation, that all SARs collapse onto a specified universal curve when local slope, z(A), is plotted against N(A)/S(A). A recent publication, however, argues that if it is assumed that patterns in macroecology are independent of the taxonomic choices that define assemblages of species, then this principle of "taxon invariance" precludes the MaxEnt-predicted universality of the SAR. By distinguishing two dimensions of the notion of taxon invariance, we show that while the MaxEnt-based theory predicts universality regardless of the taxonomic choices that define an assemblage of species, the biological characteristics of assemblages should under MaxEnt, and do in reality, influence the realism of the predictions.