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Progressive habitat transformation causes global changes in landscape biodiversity patterns, but can be hard to quantify. Rarefaction/extrapolation approaches can quantify within-habitat biodiversity, but may not be useful for cases in which one habitat type is progressively transformed into another habitat type. To quantify biodiversity patterns in such transformed landscapes, we use Hill numbers to analyse individual-based species abundance data or replicated, sample-based incidence data. Given biodiversity data from two distinct habitat types, when a specified proportion of original habitat is transformed, our approach utilises a proportional mixture of two within-habitat rarefaction/extrapolation curves to analytically predict biodiversity changes, with bootstrap confidence intervals to assess sampling uncertainty. We also derive analytic formulas for assessing species composition (i.e. the numbers of shared and unique species) for any mixture of the two habitat types. Our analytical and numerical analyses revealed that species unique to each habitat type are the most important determinants of landscape biodiversity patterns.

Humans currently collectively use thousands of languages1,2. The number of languages in a given region (i.e. language “richness”) varies widely3–7. Understanding the processes of diversification and homogenization that produce these patterns has been a fundamental aim of linguistics and anthropology. Empirical research to date has identified various social, environmental, geographic, and demographic factors associated with language richness3. However, our understanding of causal mechanisms and variation in their effects over space has been limited by prior analyses focusing on correlation and assuming stationarity3,8. Here we use process-based, spatially-explicit stochastic models to simulate the emergence, expansion, contraction, fragmentation, and extinction of language ranges. We varied combinations of parameter settings in these computer-simulated experiments to evaluate the extent to which different processes reproduce observed patterns of pre-colonial language richness in North America. We find that the majority of spatial variation in language richness can be explained by models in which environmental and social constraints determine population density, random shocks alter population sizes more frequently at higher population densities, and population shocks are more frequently negative than positive. Language diversification occurs when populations split after reaching size limits, and when ranges fragment due to population contractions following negative shocks or due to contact with other groups that are expanding following positive shocks. These findings support diverse theoretical perspectives arguing that language richness is shaped by environmental and social conditions, constraints on group sizes, outcomes of contact among groups, and shifting demographics driven by positive innovations, such as new subsistence strategies, or negative events, such as war or disease.

In their recent critique, Qian et al. (2017) claimed that the results of structural equation modeling analysis (SEM) in Chan et al. (2016) were flawed. Here, we show that the source of the difference in their re-analysis is that Qian et al. did not follow the standard, iterative process of SEM, which allows researchers to evaluate which model offers the best account of the data in both absolute and relative senses. Here, we provide step-by-step instructions to reproduce our published results. All of Qian et al.′s concerns regarding SEM can be put to rest. Moreover, in our original paper we used three distinct statistical methods-hierarchical partitioning, SEM, and stationary bootstrap-to show that different temporal scales of environmental variability can differentially impact the elevational range size (ERS) of species. It is time to move on to probing the pressing issue of how and why climatic variability impacts ERS.

The body size of an animal is probably its most important functional trait. For arthropods, environmental drivers of body size variation are still poorly documented and understood, especially in tr ...

Macroecology has traditionally relied on descriptive characterization of large-scale ecological patterns to offer narrative explanations for the origin and maintenance of those patterns. Only recently have macroecologists begun to employ models termed ‘process-based’ and ‘mechanistic’, in contrast to other areas of ecology, where such models have a longer history. Here, we define and differentiate between process-based and mechanistic features of models, and we identify and discuss important advantages of working with models possessing such features. We describe some of the risks associated with process-based and mechanistic model-centered research programs, and we propose ways to mitigate these risks. Giving process-based and mechanistic models a more central role in research programs can reinvigorate macroecology by strengthening the link between theory and data.

Mountain regions are unusually biodiverse, with especially rich aggregations of small-30 ranged species that form centers of endemism. Mountains play an array of important roles for Earth's biodiversity, and impact neighboring lowlands through biotic interchange, changes in regional climate, and nutrient run-off. The high biodiversity of certain mountains reflects the interplay of multiple evolutionary mechanisms: enhanced speciation rates with unique opportunities for co-existence and persistence of lineages, shaped by long-term climatic changes 35 interacting with topographically dynamic landscapes. High diversity in most tropical mountains is tightly linked to bedrock geology, notably areas comprising mafic and ultramafic lithologies—rock types rich in magnesium and poor in phosphate that present special requirements for plant physiology. Mountain biodiversity bears the signature of deep-time evolutionary and ecological processes, a history worth preserving in the face of contemporary environmental changes.

Aim To identify hotspots of endemic and non-endemic avian diversity in the mountains of south-west China and delineate biodiversity corridors that connect the faunas of northern and southern Asia. To understand how biodiversity and endemism in this region has been maintained through palaeoclimate change. Location The mountains of south-west China, spanning an elevational gradient > 7000 m. Methods We used the distributional data of 752 breeding birds to investigate current patterns of diversity across elevational and geographical space. We simulated species richness under palaeoclimate models of global temperature change, assessing changes in species richness. Results Contemporary species richness of non-endemic birds peaked at 800–1800 m elevation, while endemic richness peaked at 2000–3000 m. Richness of non-endemic birds was highest in the southern Hengduan Mountains and Yungui Plateau, while endemic richness peaked further north, extending into the mountains along the western edge of the Sichuan Basin. Under global warming models, species richness remained high throughout the Hengduan Mountains region. Under global cooling models, the Sichuan Basin showed increased richness. Conclusions Endemism peaked in the mountains along the western edge of the Sichuan Basin, highlighting the importance of this region in promoting and maintaining diversity. This region has likely functioned as a biodiversity corridor, bridging the Palaearctic and Oriental biotas to the north and south. Climate simulations suggest that the mountains of south-west China can accommodate upslope range shifts in response to warming, but low elevation specialists may have experienced increased extinction probabilities during cold periods in the recent past, which may in part explain the current mid-elevation diversity peak. During glacial periods the Sichuan Basin likely served as a warm refugium for montane birds. Steep environmental heterogeneity has been a key to maintaining high diversity and endemism in the region during palaeoclimate change. These same features will likely shape the effects of future climate change on biodiversity in the region.

Mountains contribute disproportionately to the terrestrial biodiversity of Earth, especially in the tropics, where they host hotspots of extraordinary and puzzling richness. With about 25% of all land area, mountain regions are home to more than 85% of the world’s species of amphibians, birds, and mammals, many entirely restricted to mountains. Biodiversity varies markedly among these regions. Together with the extreme species richness of some tropical mountains, this variation has proven challenging to explain under traditional climatic hypotheses. However, the complex climatic characteristics of rugged mountain regions differ fundamentally from those of lowland regions, likely playing a key role in generating and maintaining diversity. With ongoing global changes in climate and land use, the role of mountains as refugia for biodiversity may well come under threat.

The statistical framework of rarefaction curves and asymptotic estimators allows for an effective standardization of biodiversity measures. However, most statistical analyses still consist of point comparisons of diversity estimators for a particular sampling level. We introduce new randomization methods that incorporate sampling variability encompassing the entire length of the rarefaction curve and allow for statistical comparison of i ≥2 individual-based, sample-based, or coverage-based rarefaction curves. These methods distinguish between two distinct null hypotheses: the ecological null hypothesis (H0eco) and the biogeographical null hypothesis (H0biog). H0eco states that the i samples were drawn from a single assemblage, and any differences among them in species richness, composition, or relative abundance reflect only sampling effects. H0biog states that the i samples were drawn from assemblages that differ in their species composition but share similar species richness and species abundance distri...

Ecologists and biogeographers usually rely on a single phylogenetic tree to study evolutionary processes that affect macroecological patterns. This approach ignores the fact that each phylogenetic tree is a hypothesis about the evolutionary history of a clade, and cannot be directly observed in nature. Also, trees often leave out many extant species, or include missing species as polytomies because of a lack of information on the relationship among taxa. Still, researchers usually do not quantify the effects of phylogenetic uncertainty in ecological analyses. We propose here a novel analytical strategy to maximize the use of incomplete phylogenetic information, while simultaneously accounting for several sources of phylogenetic uncertainty that may distort statistical inferences about evolutionary processes. We illustrate the approach using a clade-wide analysis of the hummingbirds, evaluating how different sources of uncertainty affect several phylogenetic comparative analyses of trait evolution and biogeographic patterns. Although no statistical approximation can fully substitute for a complete and robust phylogeny, the method we describe and illustrate enables researchers to broaden the number of clades for which studies informed by evolutionary relationships are possible, while allowing the estimation and control of statistical error that arises from phylogenetic uncertainty. Software tools to carry out the necessary computations are offered.

Variability for a day or a seasonSpecies that experience larger seasonal climatic fluctuations are likely to be more physiologically flexible and thus likely to occur across a wider elevational range. Daily changes in temperature are also common but have rarely been considered. Chanet al.used a global data set of vertebrates to look at how these two different sets of variation affect a species' elevational distribution (see the Perspective by Perezet al.). Unexpectedly, larger daily fluctuations were associated with smaller elevational distributions. Thus, specialists are favored where daily fluctuations are dominant, whereas generalists are favored where seasonal fluctuations are the main climate influence.Science, this issue p.1437; see also p.1392

© 2017 Cambridge Philosophical Society Climate change is driving a pervasive global redistribution of the planet's species. Species redistribution poses new questions for the study of ecosystems, conservation science and human societies that require a coordinated and integrated approach. Here we review recent progress, key gaps and strategic directions in this nascent research area, emphasising emerging themes in species redistribution biology, the importance of understanding underlying drivers and the need to anticipate novel outcomes of changes in species ranges. We highlight that species redistribution has manifest implications across multiple temporal and spatial scales and from genes to ecosystems. Understanding range shifts from ecological, physiological, genetic and biogeographical perspectives is essential for informing changing paradigms in conservation science and for designing conservation strategies that incorporate changing population connectivity and advance adaptation to climate change. Species redistributions present challenges for human well-being, environmental management and sustainable development. By synthesising recent approaches, theories and tools, our review establishes an interdisciplinary foundation for the development of future research on species redistribution. Specifically, we demonstrate how ecological, conservation and social research on species redistribution can best be achieved by working across disciplinary boundaries to develop and implement solutions to climate change challenges. Future studies should therefore integrate existing and complementary scientific frameworks while incorporating social science and human-centred approaches. Finally, we emphasise that the best science will not be useful unless more scientists engage with managers, policy makers and the public to develop responsible and socially acceptable options for the global challenges arising from species redistributions.

Simulating South American biodiversityThe emergence, distribution, and extinction of species are driven by interacting factors—spatial, temporal, physical, and biotic. Rangelet al.simulated the past 800,000 years of evolution in South America, incorporating these factors into a spatially explicit dynamic model to explore the geographical generation of diversity. Their simulations, based on a paleoclimate model on a 5° latitude-longitude scale, result in shifting maps of speciation, persistence, and extinction (or cradles, museums, and graves). The simulations culminate in a striking resemblance to contemporary distribution patterns across the continent for birds, mammals, and plants—despite having no target patterns and no empirical data parameterizing them.Science, this issue p.eaar5452

Measures of niche breadth and overlap that depend on the distribution of individual among resource states (ecological categories) should be independent of the relative abundance of the species and of the number of resource states considered. Such measures should also take into account the degree of distinctness of the resource states from the point of view of the organisms concerned. An ecoassay of the distinctness of resource states may well be easier and more meaningful than measurements of physical and chemical factors. We propose that the species composition of communities utilizing different resource states may be used to develop weighting factors with which each state may be weighted in proportion to its degree of distinctness. The weighting factors are used in the development of indices of niche breadth and overlap that correct for variation in the range and distinctness of resource states and that suffer less from human subjectivity than do the measures used to date. The use of such indices and the relationship of niche overlap to competition are discussed.

AIMS: We aim to document elevational richness patterns of geometrid moths in a globally replicated, multi-gradient setting, and to test general hypotheses on environmental and spatial effects (i.e. productivity, temperature, precipitation, area, mid-domain effect and human habitat disturbance) on these richness patterns. LOCATION: Twenty-six elevational gradients world-wide (latitudes 28° S to 51° N). METHODS: We compiled field datasets on elevational gradients for geometrid moths, a lepidopteran family, and documented richness patterns across each gradient while accounting for local undersampling of richness. Environmental and spatial predictor variables as well as habitat disturbance were used to test various hypotheses. Our analyses comprised two pathways: univariate correlations within gradients, and multivariate modelling on pooled data after correcting for overall variation in richness among different gradients. RESULTS: The majority of gradients showed midpeak patterns of richness, irrespective of climate and geographical location. The exclusion of human-affected sampling plots did not change these patterns. Support for univariate main drivers of richness was generally low, although there was idiosyncratic support for particular predictors on single gradients. Multivariate models, in agreement with univariate results, provided the strongest support for an effect of area-integrated productivity, or alternatively for an elevational area effect. Temperature and the mid-domain effect received support as weaker, modulating covariates, while precipitation-related variables had no explanatory potential. MAIN CONCLUSIONS: Despite the predicted decreasing diversity–temperature relationship in ectotherms, geometrid moths are similar to ants and salamanders as well as small mammals and ferns in having predominantly their highest diversity at mid-elevations. As in those comparative analyses, single or clear sets of drivers are elusive, but both productivity and area appear to be influential. More comparative elevational studies for various insect taxa are necessary for a more comprehensive understanding of elevational diversity and productivity.

Aim To understand the causes of historical and current elevational richness patterns of Leiothrichinae babblers, a diverse and mostly endemic group of birds. Location A 5000-m elevational gradient in the Hengduan Mountains, China. Methods By means of a dated phylogenetic tree and reconstructed ancestral states, we estimated elevation-specific diversification rate, applied a new method to estimate colonization frequency and age and, for the first time, modelled historical species richness patterns that take account of temporal patterns of palaeotemperature. As explanations for current richness patterns, we assessed area, geometric constraints, temperature, precipitation, seasonality and productivity. Results The current elevational pattern of species richness is a hump-shaped curve with a peak at about 1000‐2500 m. The reconstructed palaeopatterns of species richness suggest that babblers, as a clade, first occupied the Hengduan Mountains at low to mid-elevations, although the method of ancestral state reconstruction cannot conclusively reject origins outside the current elevational distribution of the group.Diversification rates varied little along the elevational gradient, and thus cannot explain the richness pattern, but historical colonization frequency and colonization age were highly correlated with present-day species richness. Seasonality and productivity had greater power than area and geometric constraints in explaining the present-day richness pattern of babblers along the elevational gradient. Conclusions Historical and modern factors have both played important roles in shaping species richness patterns. Reconstructed historical richness patterns suggest that babblers first diversified in the Hengduan Mountains at low to mid elevations,but richness patterns almost certainly shifted substantially under changing climates of the past 10 Myr. The current richness patterns of babblers are associated with seasonality and productivity, but they are also a product of historical evolutionary and ecological dynamics. The methods we introduce for assessing historical colonization rates and past patterns of richness offer promise for understanding other elevational richness gradients.

Climate change is transforming the structure of biological communities through the geographic extension and contraction of species’ ranges. Range edges are naturally dynamic, and shifts in the location of range edges occur at different rates and are driven by different mechanisms. This leads to challenges when seeking to generalize responses among taxa and across systems. We focus on warming-related range shifts in marine systems to describe extensions and contractions as stages. Range extensions occur as a sequence of (1) arrival, (2) population increase, and (3) persistence. By contrast, range contractions occur progressively as (1) performance decline, (2) population decrease and (3) local extinction. This stage-based framework can be broadly applied to geographic shifts in any species, life-history stage, or population subset. Ideally the probability of transitioning through progressive range shift stages could be estimated from empirical understanding of the various factors influencing range shift rates. Nevertheless, abundance and occupancy data at the spatial resolution required to quantify range shifts are often unavailable and we suggest the pragmatic solution of considering observations of range shifts within a confidence framework incorporating the type, amount and quality of data. We use case studies to illustrate how diverse evidence sources can be used to stage range extensions and contractions and assign confidence that an observed range shift stage has been reached. We then evaluate the utility of trait-based risk (invasion) and vulnerability (extinction) frameworks for application in a range shift context and find inadequacies, indicating an important area for development. We further consider factors that influence rates of extension and contraction of range edges in marine habitats. Finally, we suggest approaches required to increase our capacity to observe and predict geographic range shifts under climate change.

EstimateS offers statistical tools for analyzing and comparing the diversity and composition of species assemblages, based on sampling data. The latest version computes a wide range of biodiversity statistics for both sample-based and individual-based data, including analytical rarefaction and non-parametric extrapolation, estimators of asymptotic species richness, diversity indices, Hill numbers, and (for sample-based data) measures of compositional similarity among assemblages. In the first 20 yr of its existence, EstimateS has been downloaded more than 70 000 times by users in 140 countries, who have cited it in 5000 publications in studies of taxa from microbes to mammals in every biome.

Physiological thermal-tolerance limits of terrestrial ectotherms often exceed local air temperatures, implying a high degree of thermal safety (an excess of warm or cold thermal tolerance). However, air temperatures can be very different from the equilibrium body temperature of an individual ectotherm. Here, we compile thermal-tolerance limits of ectotherms across a wide range of latitudes and elevations and compare these thermal limits both to air and to operative body temperatures (theoretically equilibrated body temperatures) of small ectothermic animals during the warmest and coldest times of the year. We show that extreme operative body temperatures in exposed habitats match or exceed the physiological thermal limits of most ectotherms. Therefore, contrary to previous findings using air temperatures, most ectotherms do not have a physiological thermal-safety margin. They must therefore rely on behavior to avoid overheating during the warmest times, especially in the lowland tropics. Likewise, species living at temperate latitudes and in alpine habitats must retreat to avoid lethal cold exposure. Behavioral plasticity of habitat use and the energetic consequences of thermal retreats are therefore critical aspects of species’ vulnerability to climate warming and extreme events.

Quantifying and assessing changes in biological diversity are central aspects of many ecological studies, yet accurate methods of estimating biological diversity from sampling data have been elusive. Hill numbers, or the effective number of species, are increasingly used to characterize the taxonomic, phylogenetic, or functional diversity of an assemblage. However, empirical estimates of Hill numbers, including species richness, tend to be an increasing function of sampling effort and, thus, tend to increase with sample completeness. Integrated curves based on sampling theory that smoothly link rarefaction (interpolation) and prediction (extrapolation) standardize samples on the basis of sample size or sample completeness and facilitate the comparison of biodiversity data. Here we extended previous rarefaction and extrapolation models for species richness (Hill number q D, where q ¼ 0) to measures of taxon diversity incorporating relative abundance (i.e., for any Hill number q D, q . 0) and present a unified approach for both individual-based (abundance) data and sample- based (incidence) data. Using this unified sampling framework, we derive both theoretical formulas and analytic estimators for seamless rarefaction and extrapolation based on Hill numbers. Detailed examples are provided for the first three Hill numbers: q ¼ 0 (species richness), q ¼ 1 (the exponential of Shannon's entropy index), and q ¼ 2 (the inverse of Simpson's concentration index). We developed a bootstrap method for constructing confidence intervals around Hill numbers, facilitating the comparison of multiple assemblages of both rarefied and extrapolated samples. The proposed estimators are accurate for both rarefaction and short-range extrapolation. For long-range extrapolation, the performance of the estimators depends on both the value of q and on the extrapolation range. We tested our methods on simulated data generated from species abundance models and on data from large species inventories. We also illustrate the formulas and estimators using empirical data sets from biodiversity surveys of temperate forest spiders and tropical ants.

Aim To document the species richness pattern of birds in the Hengduan Mountains and to understand its causes. Location Hengduan Mountains, China. Methods Species richness of 738 breeding bird species was calculated for each 100-m elevational band along a gradient from 100 to 6000 m a.s.l. Climate data were compiled based on monthly records from 182 meteorological stations in the Hengduan Mountains from 1959 to 2004. We calculated the planimetric area, predicted richness under geometric constraints, three-year average NDVI (normalized difference vegetation index) and EVI (enhanced vegetation index) in each elevational band. Simple and multiple regression models were used to test the explanatory power of variables associated with different factors proposed to account for elevational species richness gradients. Results The elevational pattern in species richness, for all breeding birds, was hump-shaped, with the peak occurring at 800–1800 m elevation. Endemic and non-endemic species, as well as four elevational range size categories of birds, also showed the general hump-shaped patterns of species richness, but with peaks at different elevations. In most data sets, species richness correlated well with climatic and energy factors along the elevational gradients; seasonality and productivity had a strong statistical relationship with species richness of montane birds in this study, with geometric constraints contributing to richness patterns for larger-ranged species endemic to the gradient. Main conclusions We found that climatic and energy factors correlate well with the richness pattern of birds, and that on the surveyed subtropical mountain, the elevational bands with highest seasonality harbour fewer species than areas with less seasonal variation in temperature. The results, however, vary somewhat among taxonomic groups. The most diverse species groups and species with the broadest ranges have a disproportionate influence on our perception of the overall diversity pattern and its underlying explanatory factors.

Our species displays remarkable linguistic diversity. Although the uneven distribution of this diversity demands explanation, the drivers of these patterns have not been conclusively determined. We address this issue in two steps: First, we review previous empirical studies whose authors have suggested environmental, geographical, and sociocultural drivers of linguistic diversification. However, contradictory results and methodological variation make it difficult to draw general conclusions. Second, we outline a program for future research. We suggest that future analyses should account for interactions among causal factors, the lack of spatial and phylogenetic independence of the data, and transitory patterns. Recent analytical advances in biogeography and evolutionary biology, such as simulation modeling of diversity patterns, hold promise for testing four key mechanisms of language diversification proposed here: neutral change, population movement, contact, and selection. Future modeling approaches should also evaluate how the outcomes of these processes are influenced by demography, environmental heterogeneity, and time.

AimTwo fundamental questions about human language demand answers: why are so many languages spoken today and why is their geographical distribution so uneven? Although hypotheses have been proposed for centuries, the processes that determine patterns of linguistic and cultural diversity remain poorly understood. Previous studies, which relied on correlative, curve-fitting approaches, have produced contradictory results. Here we present the first application of process-based simulation modelling, derived from macroecology, to examine the distribution of human groups and their languages. LocationThe Australian continent is used as a case study to demonstrate the power of simulation modelling for identifying processes shaping the diversity and distribution of human languages. MethodsProcess-based simulation models allow investigators to hold certain factors constant in order to isolate and assess the impact of modelled processes. We tested the extent to which a minimal set of processes determines the number and spatial distribution of languages on the Australian continent. Our model made three basic assumptions based on previously proposed, but untested, hypotheses: groups fill unoccupied spaces, rainfall limits population density and groups divide after reaching a maximum population. ResultsRemarkably, this simple model accurately predicted the total number of languages (average estimate 406, observed 407), and explained 56% of spatial variation in language richness on the Australian continent. Main conclusionsOur results present strong evidence that current climatic conditions and limits to group size are important processes shaping language diversity patterns in Australia. Our study also demonstrates how simulation models from macroecology can be used to understand the processes that have shaped human cultural diversity across the globe.

Estimating the species, phylogenetic, and functional diversity of a community is challenging because rare species are often undetected, even with intensive sampling. The Good-Turing frequency formula, originally developed for cryptography, estimates in an ecological context the true frequencies of rare species in a single assemblage based on an incomplete sample of individuals. Until now, this formula has never been used to estimate undetected species, phylogenetic, and functional diversity. Here, we first generalize the Good-Turing formula to incomplete sampling of two assemblages. The original formula and its two-assemblage generalization provide a novel and unified approach to notation, terminology, and estimation of undetected biological diversity. For species richness, the Good-Turing framework offers an intuitive way to derive the non-parametric estimators of the undetected species richness in a single assemblage, and of the undetected species shared between two assemblages. For phylogenetic diversity, the unified approach leads to an estimator of the undetected Faith's phylogenetic diversity (PD, the total length of undetected branches of a phylogenetic tree connecting all species), as well as a new estimator of undetected PD shared between two phylogenetic trees. For functional diversity based on species traits, the unified approach yields a new estimator of undetected Walker et al.'s functional attribute diversity (FAD, the total species-pairwise functional distance) in a single assemblage, as well as a new estimator of undetected FAD shared between two assemblages. Although some of the resulting estimators have been previously published (but derived with traditional mathematical inequalities), all taxonomic, phylogenetic, and functional diversity estimators are now derived under the same framework. All the derived estimators are theoretically lower bounds of the corresponding undetected diversities; our approach reveals the sufficient conditions under which the estimators are nearly unbiased, thus offering new insights. Simulation results are reported to numerically verify the performance of the derived estimators. We illustrate all estimators and assess their sampling uncertainty with an empirical dataset for Brazilian rain forest trees. These estimators should be widely applicable to many current problems in ecology, such as the effects of climate change on spatial and temporal beta diversity and the contribution of trait diversity to ecosystem multi-functionality. This article is protected by copyright. All rights reserved.

The extinction of a single species is rarely an isolated event. Instead, dependent parasites, commensals, and mutualist partners (affiliates) face the risk of coextinction as their hosts or partners decline and fail. Species interactions in ecological networks can transmit the effects of primary extinctions within and between trophic levels, causing secondary extinctions and extinction cascades. Documenting coextinctions is complicated by ignorance of host specificity, limitations of historical collections, incomplete systematics of affiliate taxa, and lack of experimental studies. Host shifts may reduce the rate of coextinctions, but they are poorly understood. In the absence of better empirical records of coextinctions, statistical models estimate the rates of past and future coextinctions, and based on primary extinctions and interactions among species, network models explore extinction cascades. Models predict and historical evidence reveals that the threat of coextinction is influenced by both host and affiliate traits and is exacerbated by other threats, including habitat loss, climate change, and invasive species.

Aims In ecology and conservation biology, the number of species counted in a biodiversity study is a key metric but is usually a biased underestimate of total species richness because many rare species are not detected. Moreover, comparing species richness among sites or samples is a statistical challenge because the observed number of species is sensitive to the number of individuals counted or the area sampled. For individual-based data, we treat a single, empirical sample of species abundances from an investigator-defined species assemblage or community as a reference point for two estimation objectives under two sampling models: estimating the expected number of species (and its unconditional variance) in a random sample of (i) a smaller number of individuals (multinomial model) or a smaller area sampled (Poisson model) and (ii) a larger number of individuals or a larger area sampled. For sample-based incidence (presence–absence) data, under a Bernoulli product model, we treat a single set of species incidence frequencies as the reference point to estimate richness for smaller and larger numbers of sampling units. Methods The first objective is a problem in interpolation that we address with classical rarefaction (multinomial model) and Coleman rarefaction (Poisson model) for individual-based data and with sample-based rarefaction (Bernoulli product model) for incidence frequencies. The second is a problem in extrapolation that we address with sampling-theoretic predictors for the number of species in a larger sample (multinomial model), a larger area (Poisson model) or a larger number of sampling units (Bernoulli product model), based on an estimate of asymptotic species richness. Although published methods exist for many of these objectives, we bring them together here with some new estimators under a unified statistical and notational framework. This novel integration of mathematically distinct approaches allowed us to link interpolated (rarefaction) curves and extrapolated curves to plot a unified species accumulation curve for empirical examples. We provide new, unconditional variance estimators for classical, individual-based rarefaction and for Coleman rarefaction, long missing from the toolkit of biodiversity measurement. We illustrate these methods with datasets for tropical beetles, tropical trees and tropical ants.

Mountains are centres of global biodiversity, endemism and threatened species. Elevational gradients present opportunities for species currently living near their upper thermal limits to track cooler temperatures upslope in warming climates, but only if changes in precipitation are sufficiently in step with temperature. We model local population extirpation risk for a range of temperature and precipitation scenarios over the next 100 years for 16 848 vertebrate species populations distributed along 156 elevational gradients. Average population extirpation risks due to warming alone were < 5%, but increased 10-fold, on average, when changes in precipitation were also considered. Under the driest scenarios (minimum predicted precipitation), local extirpation risks increased sharply (50‐60%) and were especially worrisome for hydrophilic amphibians and montane Latin America (c. 80%). Realistic assessment of risks urgently requires improved monitoring of precipitation, better regional precipitation models and more research on the effects of changes in precipitation on montane distributions.

Assessing species survival status is an essential component of conservation programs. We devised a new statistical method for estimating the probability of species persistence from the temporal sequence of collection dates of museum specimens. To complement this approach, we developed quantitative stopping rules for terminating the search for missing or allegedly extinct species. These stopping rules are based on survey data for counts of co-occurring species that are encountered in the search for a target species. We illustrate both these methods with a case study of the Ivory-billed Woodpecker (Campephilus principalis), long assumed to have become extinct in the United States in the 1950s, but reportedly rediscovered in 2004. We analyzed the temporal pattern of the collection dates of 239 geo-referenced museum specimens collected throughout the southeastern United States from 1853 to 1932 and estimated the probability of persistence in 2011 as

We provide a cross-taxon and historical analysis of what makes tropical forest species vulnerable to extinction. Several traits have been important for species survival in the recent and distant geological past, including seed dormancy and vegetative growth in plants, small body size in mammals, and vagility in insects. For major past catastrophes, such as the five mass extinction events, large range size and vagility or dispersal were key to species survival. Traits that make some species more vulnerable to extinction are consistent across time scales. Terrestrial organisms, particularly animals, are more extinction prone than marine organisms. Plants that persist through dramatic changes often reproduce vegetatively and possess mechanisms of die back. Synergistic interactions between current anthropogenic threats, such as logging, fire, hunting, pests and diseases, and climate change are frequent. Rising temperatures threaten all organisms, perhaps particularly tropical organisms adapted to small temperature ranges and isolated by distance from suitable future climates. Mutualist species and trophic specialists may also be more threatened because of such range-shift gaps. Phylogenetically specialized groups may be collectively more prone to extinction than generalists. Characterization of tropical forest species' vulnerability to anthropogenic change is constrained by complex interactions among threats and by both taxonomic and ecological impediments, including gross undersampling of biotas and poor understanding of the spatial patterns of taxa at all scales.

Biodiversity sampling is labor intensive, and a substantial fraction of a biota is often represented by species of very low abundance, which typically remain undetected by biodiversity surveys. Statistical methods are widely used to estimate the asymptotic number of species present, including species not yet detected. Additional sampling is required to detect and identify these species, but richness estimators do not indicate how much sampling effort (additional individuals or samples) would be necessary to reach the asymptote of the species accumulation curve. Here we develop the first statistically rigorous nonparametric method for estimating the minimum number of additional individuals, samples, or sampling area required to detect any arbitrary proportion (including 100%) of the estimated asymptotic species richness. The method uses the Chao1 and Chao2 nonparametric estimators of asymptotic richness, which are based on the frequencies of rare species in the original sampling data. To evaluate the performance of the proposed method, we randomly subsampled individuals or quadrats from two large biodiversity inventories (light trap captures of Lepidoptera in Great Britain and censuses of woody plants on Barro Colorado Island [BCI], Panama). The simulation results suggest that the method performs well but is slightly conservative for small sample sizes. Analyses of the BCI results suggest that the method is robust to nonindependence arising from small-scale spatial aggregation of species occurrences. When the method was applied to seven published biodiversity data sets, the additional sampling effort necessary to capture all the estimated species ranged from 1.05 to 10.67 times the original sample (median approximately equal to 2.23). Substantially less effort is needed to detect 90% of the species (0.33-1.10 times the original effort; median approximately equal to 0.80). An Excel spreadsheet tool is provided for calculating necessary sampling effort for either abundance data or replicated incidence data.

Many studies suggest that global warming is driving species ranges poleward and toward higher elevations at temperate latitudes, but evidence for range shifts is scarce for the tropics, where the shallow latitudinal temperature gradient makes upslope shifts more likely than poleward shifts. Based on new data for plants and insects on an elevational transect in Costa Rica, we assess the potential for lowland biotic attrition, range-shift gaps, and mountaintop extinctions under projected warming. We conclude that tropical lowland biotas may face a level of net lowland biotic attrition without parallel at higher latitudes (where range shifts may be compensated for by species from lower latitudes) and that a high proportion of tropical species soon faces gaps between current and projected elevational ranges.

We analysed the distribution patterns of megabenthic species of marine molluscs from the Uruguayan shelf in relation to 3 interacting gradients, namely depth, salinity and temperature of sea bottom waters. Our results were generated from a data set comprising the complete list of megabenthic gastropods (adult shell length >50 mm; 22 species) from a study area that encompassed latitudes from 34 to 39° south, between 4 and 800 m depth. Multiple regressions were used to evalu- ate the contribution of 3 classes of explanatory variables to gastropod richness at sampling points along the gradients: (1) geometric constraints on the distribution of species' tolerance ranges for envi- ronmental variables (the mid-domain effect, MDE), (2) the species-area effect driven by the spatial extent of scaled environmental variables, and (3) the environmental variables themselves. In addi- tion, we explored multivariate explanations for the bathymetric pattern of gastropod species richness, incorporating depth-based area, temperature, salinity and geometric constraints as predictor vari- ables. Examination of Moran's I for regression residuals confirmed that results were not biased by spatial autocorrelation. Overall, the environmental variables per se were the most important contrib- utors to the observed patterns. However, a significant signature of the MDE was revealed, by the entry of geometric constraints into nearly all models. In contrast, the species-area effect was less important. These results suggest that: (1) MDE may operate in any kind of bounded gradient, (2) both stochastic and deterministic processes are responsible for the pattern observed, and (3) the relative contribution of the MDE to the patterns is dependent on the gradient considered and the way in which species richness is measured.

Two hummingbird species inhabit the Juan Fernandez Islands, nearly 700 km off the Chilean coast in the Pacific Ocean—the endangered endemic Juan Fernandez Firecrown Sephanoides fernandensis and the continental Green-backed Firecrown S. sephaniodes. In terms of body size, the endemic species is the most sexually dimorphic species of hummingbird known; it also displays an extraordinary degree of sexual dichromatism. Both sexes hold feeding territories, within which courtship probably occurs. It is suggested that sexual selection, and selection for gender recognition and the absence of indigenous predators may explain the evolution of sexual dimorphism and dichromatism in the Juan Fernandez Firecrown. In spite of a more than twofold difference in body size, the bill lengths of both sexes in both species are nearly identical and closely match the flower tube length of the several species of endemic plants they pollinate. The endemic Juan Fernandez Firecrown has become extinct on one of the two main islands (Isla Alejandro Selkirk) and its population on the other main island (Isla Robinson Crusoe) has greatly declined in recent decades. In contrast, the population of the Green-backed Firecrown has probably increased on Robinson Crusoe and the species has recently become established on Alejandro Selkirk. Because historical records show that Green-backed Firecrown survived centuries of potential competition from Juan Fernandez Firecrown, massive habitat destruction, plagues of rats, feral cats and dogs and the effects of feral livestock before beginning its recent decline, it is suggested that the introduction earlier in this century of the bramble Rubus ulmifalius, the coati Nasua nasua and possibly the rabbit Oryctolagus cuniculus may have differentially favoured the Green-backed Firecrown at the expense of the Juan Fernandez Firecrown.

RangeModel is a computer application that offers animated demonstrations of the mechanism behind the mid-domain effect. The program also provides analytical tools for the assessment of geometric constraints in empirical datasets for one-dimensional domains (transects). The mid-domain effect (MDE) is the increasing overlap of species ranges towards the center of a shared, bounded domain due to geometric boundary constraints in relation to the distribution of range sizes, producing a peak or plateau of species richness towards the center of the domain. Domains may be spatial, temporal, or functional. RangeModel is a stand-alone, graphical-interface, freeware application for PC and Mac OS platforms.

We sampled 14,603 geometrid moths along a forested elevational gradient from 1020–3021 m in the southern Ecuadorian Andes, and then employed DNA barcoding to refine decisions on species boundaries initially made by morphology. We compared the results with those from an earlier study on the same but slightly shorter gradient that relied solely on morphological criteria to discriminate species. The present analysis revealed 1857 putative species, an 80% increase in species richness from the earlier study that detected only 1010 species. Measures of species richness and diversity that are less dependent on sample size were more than twice as high as in the earlier study, even when analysis was restricted to an identical elevational range. The estimated total number of geometrid species (new dataset) in the sampled area is 2350. Species richness at single sites was 32–43% higher, and the beta diversity component rose by 43–51%. These impacts of DNA barcoding on measures of richness reflect its capacity to reveal cryptic species that were overlooked in the first study. The overall results confirmed unique diversity patterns reported in the first investigation. Species diversity was uniformly high along the gradient, declining only slightly above 2800 m. Species turnover also showed little variation along the gradient, reinforcing the lack of evidence for discrete faunal zones. By confirming these major biodiversity patterns, the present study establishes that incomplete species delineation does not necessarily conceal trends of biodiversity along ecological gradients, but it impedes determination of the true magnitude of diversity and species turnover.

© 2016 John Wiley & Sons Ltd/CNRS. We introduce a novel framework for conceptualising, quantifying and unifying discordant patterns of species richness along geographical gradients. While not itself explicitly mechanistic, this approach offers a path towards understanding mechanisms. In this study, we focused on the diverse patterns of species richness on mountainsides. We conjectured that elevational range midpoints of species may be drawn towards a single midpoint attractor - a unimodal gradient of environmental favourability. The midpoint attractor interacts with geometric constraints imposed by sea level and the mountaintop to produce taxon-specific patterns of species richness. We developed a Bayesian simulation model to estimate the location and strength of the midpoint attractor from species occurrence data sampled along mountainsides. We also constructed midpoint predictor models to test whether environmental variables could directly account for the observed patterns of species range midpoints. We challenged these models with 16 elevational data sets, comprising 4500 species of insects, vertebrates and plants. The midpoint predictor models generally failed to predict the pattern of species midpoints. In contrast, the midpoint attractor model closely reproduced empirical spatial patterns of species richness and range midpoints. Gradients of environmental favourability, subject to geometric constraints, may parsimoniously account for elevational and other patterns of species richness.

Evolutionary processes underlying spatial patterns in species richness remain largely unexplored, and correlative studies lack the theoretical basis to explain these patterns in evolutionary terms. In this study, we develop a spatially explicit simulation model to evaluate, under a pattern-oriented modeling approach, whether evolutionary niche dynamics (the balance between niche conservatism and niche evolution processes) can provide a parsimonious explanation for patterns in species richness. We model the size, shape, and location of species' geographical ranges in a multivariate heterogeneous environmental landscape by simulating an evolutionary process in which environmental fluctuations create geographic range fragmentation, which, in turn, regulates speciation and extinction. We applied the model to the South American domain, adjusting parameters to maximize the correspondence between observed and predicted patterns in richness of about 3,000 bird species. Predicted spatial patterns, which closely resemble observed ones (r2=0.795), proved sensitive to niche dynamics processes. Our simulations allow evaluation of the roles of both evolutionary and ecological processes in explaining spatial patterns in species richness, revealing the enormous potential of the link between ecology and historical biogeography under integrated theoretical and methodological frameworks.

By reanalysing inaccurately presented data of Kerr et al. (2006), we refute their claims that area-corrected species richness of endemic Madagascan birds and mammals increases toward the Equator and is best explained by environmental factors, and that the rainforest mid-domain effect (MDE) Lees et al. (1999) demonstrated is artefactual.

We examine co-occurrence patterns of microorganisms to evaluate community assembly ''rules.'' We use methods previously applied to macroorganisms, both to evaluate their applicability to microorganisms and to allow comparison of co-occurrence patterns observed in microorganisms to those found in macroorganisms. We use a null model analysis of 124 incidence matrices from microbial communities, including bacteria, archaea, fungi, and algae, and we compare these results to previously published findings from a meta-analysis of almost 100 macroorganism data sets. We show that assemblages of microorganisms demon- strate nonrandom patterns of co-occurrence that are broadly similar to those found in assemblages of macroorganisms. These results suggest that some taxon co-occurrence patterns may be general characteristics of communities of organisms from all domains of life. We also find that co-occurrence in microbial communities does not vary among taxonomic groups or habitat types. However, we find that the degree of co-occurrence does vary among studies that use different methods to survey microbial communities. Finally, we discuss the potential effects of the undersampling of microbial communities on our results, as well as processes that may contribute to nonrandom patterns of co-occurrence in both macrobial and microbial communities such as competition, habitat filtering, historical effects, and neutral processes.

Aim The biodiversity of geometrid moths (Lepidoptera) along a complete tropical elevational gradient was studied for the first time. The patterns are described, and the role of geometric constraints and environmental factors is explored. Location The study was carried out along the Barva Transect (10° N, 84° W), a complete elevational gradient ranging from 40 to 2730 m a.s.l. in Braulio Carrillo National Park, Costa Rica, and adjacent areas. Methods Moths were sampled manually in 2003 and 2004 at 12 rain forest sites using light ‘towers’, each with two 15 W ultraviolet fluorescent tubes. We used abundance-based rarefaction, statistical estimation of true richness (Chao 1), geographically interpolated observed richness and Fisher's alpha as measures of local diversity. Results A total of 13,765 specimens representing 739 species were analysed. All four measures showed a hump-shaped pattern with maxima between 500 and 2100 m elevation. The two subfamilies showed richness and diversity maxima at either lower (Ennominae) or higher (Larentiinae) elevation than Geometridae as a whole. Among the four environmental factors tested, relative humidity yielded the highest correlation over the transect with the rarefaction-based richness estimates as well as with estimated true species richness of Geometridae as a whole and of Larentiinae, while rainfall explained the greatest variation of Ennominae richness. The elevational pattern of moth richness was discordant with both temperature and with tree species richness. A combination of all environmental factors in a stepwise multiple regression produced high values of r2 in Geometridae. The potential effects of geometric constraints (mid-domain effect, MDE) were investigated by comparing them with observed, interpolated richness. Overall, models fitted very well for Geometridae as a whole and for Ennominae, but less well for Larentiinae. Small-ranged species showed stronger deviations from model predictions than large-ranged species, and differed strikingly between the two subfamilies, suggesting that environmental factors play a more pronounced role for small-ranged species. We hypothesize that small-ranged species (at least of the Ennominae) may tend to be host specialists, whereas large-ranged species tend to be polyphagous. Based on interpolated ranges, mean elevational range for these moths was larger with increasing elevation, in accordance with Rapoport's elevational rule, although sampling effects may have exaggerated this pattern. The underlying mechanism remains unknown because Rapoport's ‘rescue’ hypothesis could not explain the observed pattern. Conclusions The results clearly show that moth diversity shows a hump-shaped pattern. However, remarkable variation exists with regard to taxon and range size. Both environmental and geometric factors are likely to contribute to the observed patterns.

Based on a sample of individuals, we focus on inferring the vector of species relative abundance of an entire assemblage and propose a novel estimator of the complete species-rank abundance distribution (RAD). Nearly all previous estimators of the RAD use the conventional "plug-in" estimator Pi (sample relative abundance) of the true relative abundance pi of species i. Because most biodiversity samples are incomplete, the plug-in estimators are applied only to the subset of species that are detected in the sample. Using the concept of sample coverage and its generalization, we propose a new statistical framework to estimate the complete RAD by separately adjusting the sample relative abundances for the set of species detected in the sample and estimating the relative abundances for the set of species undetected in the sample but inferred to be present in the assemblage. We first show that P, is a positively biased estimator of pi for species detected in the sample, and that the degree of bias increases with increasing relative rarity of each species. We next derive a method to adjust the sample relative abundance to reduce the positive bias inherent in j. The adjustment method provides a nonparametric resolution to the longstanding challenge of characterizing the relationship between the true relative abundance in the entire assemblage and the observed relative abundance in a sample. Finally, we propose a method to estimate the true relative abundances of the undetected species based on a lower bound of the number of undetected species. We then combine the adjusted RAD for the detected species and the estimated RAD for the undetected species to obtain the complete RAD estimator. Simulation results show that the proposed RAD curve can unveil the true RAD and is more accurate than the empirical RAD. We also extend our method to incidence data. Our formulas and estimators are illustrated using empirical data sets from surveys of forest spiders (for abundance data) and soil ciliates (for incidence data). The proposed RAD estimator is also applicable to estimating various diversity measures and should be widely useful to analyses of biodiversity and community structure.

The causes of global variation in species richness have been debated for nearly two centuries with no clear resolution in sight. Competing hypotheses have typically been evaluated with correlative models that do not explicitly incorporate the mechanisms responsible for biotic diversity gradients. Here, we employ a fundamentally different approach that uses spatially explicit Monte Carlo models of the placement of cohesive geographical ranges in an environmentally heterogeneous landscape. These models predict species richness of endemic South American birds (2248 species) measured at a continental scale. We demonstrate that the principal single-factor and composite (species-energy, water-energy and temperature-kinetics) models proposed thus far fail to predict ( r 2 ⩽0.05) the richness of species with small to moderately large geographical ranges (first three range-size quartiles). These species constitute the bulk of the avifauna and are primary targets for conservation. Climate-driven models performed reasonably well only for species with the largest geographical ranges (fourth quartile) when range cohesion was enforced. Our analyses suggest that present models inadequately explain the extraordinary diversity of avian species in the montane tropics, the most species-rich region on Earth. Our findings imply that correlative climatic models substantially underestimate the importance of historical factors and small-scale niche-driven assembly processes in shaping contemporary species-richness patterns.

259.Biologists have long noted higher levels of species diversity in the longitudinal middle-courses of river systems and have proposed many explanations. As a new explanationfor this widespread pattern, we suggest that many middle-course peaks in richness maybe, at least in part, a consequence of geometric constraints on the location of species’ranges along river courses, considering river headwaters and mouths as boundaries forthe taxa considered. We demonstrate this extension of the mid-domain effect (MDE) toriver systems for riparian plants along two rivers in Sweden, where a previous studyfound a middle-course peak in richness of natural (non-ruderal) species. We comparepatterns of empirical richness of these species to null model predictions of speciesrichness along the two river systems and to spatial patterns for six environmentalvariables (channel width, substrate fineness, substrate heterogeneity, ice scour, bankheight, and bank area). In addition, we examine the independent prediction of mid-domain effects models that species with large ranges, because the location of theirranges is more constrained, are more likely to produce a mid-domain peak in richnessthan are species with small ranges. Species richness patterns of riparian plants were bestpredicted by models including both null model predictions and environmentalvariables. When species were divided into large-ranged and small-ranged groups, themid-domain effect was more prominent and the null model predictions were a better fitto the empirical richness patterns of large-ranged species than those of small-rangedspecies. Our results suggest that the peak in riparian plant species richness in the middlecourses of the rivers studied can be explained by an underlying mid-domain effect(driven by geometric constraints on large-ranged species), together with environmentaleffects on richness patterns (particularly on small-ranged species). We suggest that themid-domain effect may help to explain similar middle-course richness peaks alongother rivers.R. R. Dunn (Rob_Dunn@ncsu.edu), Dept of Zoology, North Carolina State Univ.,Raleigh, NC 27695-7617, USA.

We review the biogeography of microorganisms in light of the biogeography of macroorganisms. A large body of research supports the idea that free-living microbial taxa exhibit biogeographic patterns. Current evidence confirms that, as proposed by the Baas-Becking hypothesis, 'the environment selects' and is, in part, responsible for spatial variation in microbial diversity. However, recent studies also dispute the idea that 'everything is everywhere'. We also consider how the processes that generate and maintain biogeographic patterns in macroorganisms could operate in the microbial world.

Summary 1 We examined in situ diversity and distribution of vascular epiphytes, as well as site environmental variables at six sites along a continuous elevational gradient (30‐2600 m a.s.l.) of old-growth forest in Costa Rica. 2 A total of 555 species of vascular epiphytes from 130 genera of 53 families were identified to species or morphospecies. The ferns were the most diverse group, with 138 species, followed by orchids (112 species). Cloud forest at 1000 m was the richest site, representing the maximum of a pronounced mid-elevation peak in epiphyte species richness. 3 Spatial randomizations of recorded elevational ranges suggest that the overall elevational richness pattern of most epiphyte groups on this transect is substantially influenced by the mid-domain effect (MDE, the mid-elevation overlap of large-ranged species). Among the environmental factors considered (rainfall, temperature and canopy light environment), only rainfall was significantly correlated with richness. 4 Different patterns of richness for vascular epiphytes and for trees indicate that mechanisms differ between life forms. 5 We collected 26% of the estimated epiphyte species of Costa Rica along a single mountain transect. This, together with the finding that different groups and life forms varied in the elevation at which species richness peaked, highlights the need to conserve the few remaining intact elevational gradients in Latin America.

To better understand changes in the distribution and diversity within plant functional types along an elevational gradient and the potential mechanisms driving such changes, we studied species richness of ferns at six elevations along a forested elevational gradient in Costa Rica, from La Selva Biological Station at 30 m a.s.l. up the slopes of Volcan Barva to 2960 m a.s.l. Among the samples from all the sites combined, we found 264 species from 60 genera. Sixty-nine species were terrestrial, 113 were canopy epiphytes, and 121 were low-trunk epiphytes. Only one species occupied both canopy and terrestrial habitats at any of the study sites. Overlap of canopy and low-trunk species composition was relatively low (18%), and lower still was the overlap of terrestrial and low-trunk species (12%). Total species richness peaked at the 1000-m site for canopy and low-trunk epiphytic species. In contrast, the richness of terrestrial species rose to a mid-elevation maximum and remained relatively constant at higher elevations. In an effort to explain elevational patterns of species richness, we examined mean annual rainfall and temperature, light intensities in the canopy and at ground level, and the mid-domain effect. Of the explanatory factors examined, the mid-domain effect accounted for most of the elevational pattern. We found little evidence that environmental gradients drive patterns of fern species richness on this spatial scale.

We examine the role of rare species in the problem of estimating within- habitat species richness based on sampling data. Richness estimation can be modeled re- alistically for abundance-based and incidence-based data using Poisson or binomial mix- tures, respectively. The problem can be reduced to estimation of the odds of the probability of a species remaining undetected in the sample or sample set. Within this rigorous statistical framework, we explore existing methods of richness estimation and assess their limitations. We do this by modeling the addition of increasing numbers of rare, undetected species to a reference assemblage, assessing the power of different methods to distinguish the modified species assemblages from the reference assemblage. (We use empirical example data sets for birds, seeds, and beetles as reference assemblages.) By considering the contributions of rare species and the role of undetected species for a fixed sampling effort, we show why the problem of richness estimation is so difficult, and we discuss what statistical models can provide.

Although the mid-domain effect (MDE) model for species richness in bound- ed geographical domains has proved controversial, several studies have revealed its ex- planatory potential for patterns of species richness. This paper investigates unexplored aspects of one-dimensional MDE, based on a data set of latitudinal distributions of New World birds (3706 species) on a 10 scale. Two previously published data sets for other taxa are also considered. We adjusted band sums (number of species per latitudinal band) for longitudinal area by constructing species-area curves for each band. Area-corrected richness patterns differed substantially from raw band sums, although both confirmed a strong, mid- tropical peak in richness. An MDE model accounted for 47% of the adjusted pattern, whereas area alone explained 13% of variation. Area-adjusted band sum data proved preferable to coastal transect data from the same data set. MDE was relatively more important in smaller latitudinal domains and/or for taxa with relatively large ranges. On fundamental grounds, we concluded that MDE randomizations based on empirical ranges are more appropriate than those based on theoretical range size distributions. Models that, like MDE, produce quantitative richness predictions should be evaluated statistically against the null hypothesis of equality, not simply correlation, of empirical vs. predicted richness values.