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.

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.

Approaching a state shift in Earth's biosphere.

Localized ecological systems are known to shift abruptly and irreversibly from one state to another when they are forced across critical thresholds. Here we review evidence that the global ecosystem as a whole can react in the same way and is approaching a planetary-scale critical transition as a result of human influence. The plausibility of a planetary-scale 'tipping point' highlights the need to improve biological forecasting by detecting early warning signs of critical transitions on global as well as local scales, and by detecting feedbacks that promote such transitions. It is also necessary to address root causes of how humans are forcing biological changes.

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.

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.

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.

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.

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.

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.

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.

Global assessment of experimental climate warming on tundra vegetation: heterogeneity over space and time.

Understanding the sensitivity of tundra vegetation to climate warming is critical to forecasting future biodiversity and vegetation feedbacks to climate. In situ warming experiments accelerate climate change on a small scale to forecast responses of local plant communities. Limitations of this approach include the apparent site-specificity of results and uncertainty about the power of short-term studies to anticipate longer term change. We address these issues with a synthesis of 61 experimental warming studies, of up to 20 years duration, in tundra sites worldwide. The response of plant groups to warming often differed with ambient summer temperature, soil moisture and experimental duration. Shrubs increased with warming only where ambient temperature was high, whereas graminoids increased primarily in the coldest study sites. Linear increases in effect size over time were frequently observed. There was little indication of saturating or accelerating effects, as would be predicted if negative or positive vegetation feedbacks were common. These results indicate that tundra vegetation exhibits strong regional variation in response to warming, and that in vulnerable regions, cumulative effects of long-term warming on tundra vegetation - and associated ecosystem consequences - have the potential to be much greater than we have observed to date.

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.

Incipient criticality in ecological communities.

In ecology, there have been attempts to establish links between the relative species abundance (RSA), the fraction of species in a community with a given abundance, and a power-law form of the species area relationship (SAR), the dependence of species richness on sampling area. However the SAR and other patterns in ecology often do not exhibit power-law behavior over an appreciable range of scales. This raises the question whether a scaling framework can be applied when the system under analysis does not exhibit power-law behavior. Here, we derive a general finite-size scaling framework applicable to such systems that can be used to identify incipient critical behavior and links the scale dependence of the RSA and the SAR. We confirm the generality of our theory by using data from a serpentine grassland plot, which exhibits a power-law SAR, and the Barro Colorado Island plot in Panama, whose SAR shows deviations from power-law behavior at every scale. Our results demonstrate that scaling provides a model-independent framework for analyzing and unifying ecological data and that, despite the absence of power laws, ecosystems are poised in the vicinity of a critical point.

The debt of nations and the distribution of ecological impacts from human activities.

As human impacts to the environment accelerate, disparities in the distribution of damages between rich and poor nations mount. Globally, environmental change is dramatically affecting the flow of ecosystem services, but the distribution of ecological damages and their driving forces has not been estimated. Here, we conservatively estimate the environmental costs of human activities over 1961-2000 in six major categories (climate change, stratospheric ozone depletion, agricultural intensification and expansion, deforestation, overfishing, and mangrove conversion), quantitatively connecting costs borne by poor, middle-income, and rich nations to specific activities by each of these groups. Adjusting impact valuations for different standards of living across the groups as commonly practiced, we find striking imbalances. Climate change and ozone depletion impacts predicted for low-income nations have been overwhelmingly driven by emissions from the other two groups, a pattern also observed for overfishing damages indirectly driven by the consumption of fishery products. Indeed, through disproportionate emissions of greenhouse gases alone, the rich group may have imposed climate damages on the poor group greater than the latter's current foreign debt. Our analysis provides prima facie evidence for an uneven distribution pattern of damages across income groups. Moreover, our estimates of each group's share in various damaging activities are independent from controversies in environmental valuation methods. In a world increasingly connected ecologically and economically, our analysis is thus an early step toward reframing issues of environmental responsibility, development, and globalization in accordance with ecological costs.

Biodiversity conservation: climate change and extinction risk.

Thomas et al. have carried out a useful analysis of the extinction risk from climate warming. Their overall conclusion, that a large fraction of extant species could be driven to extinction by expected climate trends over the next 50 years, is compelling: it adds to the many other reasons why new energy policies are needed to reduce the pace of warming.

Convergence across biomes to a common rain-use efficiency.

Water availability limits plant growth and production in almost all terrestrial ecosystems. However, biomes differ substantially in sensitivity of aboveground net primary production (ANPP) to between-year variation in precipitation. Average rain-use efficiency (RUE; ANPP/precipitation) also varies between biomes, supposedly because of differences in vegetation structure and/or biogeochemical constraints. Here we show that RUE decreases across biomes as mean annual precipitation increases. However, during the driest years at each site, there is convergence to a common maximum RUE (RUE(max)) that is typical of arid ecosystems. RUE(max) was also identified by experimentally altering the degree of limitation by water and other resources. Thus, in years when water is most limiting, deserts, grasslands and forests all exhibit the same rate of biomass production per unit rainfall, despite differences in physiognomy and site-level RUE. Global climate models predict increased between-year variability in precipitation, more frequent extreme drought events, and changes in temperature. Forecasts of future ecosystem behaviour should take into account this convergent feature of terrestrial biomes.