Robert K. Colwell

INTERPOLATING, EXTRAPOLATING, AND COMPARING INCIDENCE‐BASED SPECIES ACCUMULATION CURVES

A general binomial mixture model is proposed for the species accumulation function based on presence-absence (incidence) of species in a sample of quadrats or other sampling units. The model covers interpolation between zero and the observed number of samples, as well as extrapolation beyond the observed sample set. For interpolation (sample- based rarefaction), easily calculated, closed-form expressions for both expected richness and its confidence limits are developed (using the method of moments) that completely eliminate the need for resampling methods and permit direct statistical comparison of richness between sample sets. An incidence-based form of the Coleman (random-placement) model is developed and compared with the moment-based interpolation method. For ex- trapolation beyond the empirical sample set (and simultaneously, as an alternative method of interpolation), a likelihood-based estimator with a bootstrap confidence interval is de- scribed that relies on a sequential, AIC-guided algorithm to fit the mixture model parameters. Both the moment-based and likelihood-based estimators are illustrated with data sets for temperate birds and tropical seeds, ants, and trees. The moment-based estimator is confi- dently recommended for interpolation (sample-based rarefaction). For extrapolation, the likelihood-based estimator performs well for doubling or tripling the number of empirical samples, but it is not reliable for estimating the richness asymptote. The sensitivity of individual-based and sample-based rarefaction to spatial (or temporal) patchiness is dis- cussed.

On the Measurement of Niche Breadth and Overlap

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.

The mid-domain effect: geometric constraints on the geography of species richness

Geographic patterns of species richness are influenced by many factors, but the role of shared physiographical and physiological boundaries in relation to range-size distributions has been surprisingly neglected, in spite of the fact that such geometric constraints lead to mid-domain richness peaks even without environmental gradients (the mid-domain effect). Relying on null models, several recent studies have begun to quantify this problem using simulated and empirical data. This approach promises to transform how we perceive geographic variation in diversity, including the long unresolved latitudinal gradient in species richness. The question is not whether geometry affects such patterns, but by how much.

Global Warming, Elevational Range Shifts, and Lowland Biotic Attrition in the Wet Tropics

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.

Rarefaction and extrapolation with Hill numbers: a framework for sampling and estimation in species diversity studies

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 Shannons entropy index), and q ¼ 2 (the inverse of Simpsons 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.

Predictability, Constancy, and Contingency of Periodic Phenomena

Temporal patterns in fluctuating physical and biological phenomena are of great interest in several fields of biology, primarily because of their importance as evolutionary constraints. To clarify and simplify the wide variety of terms used to describe aspects of temporal pattern, simple measures of predictability, constancy, and contingency are proposed. These are sufficient to describe the general characteristics of periodic phenomena. The measures are based on the mathematics of information theory. Methods of testing the statistical significance of these measures are given. Predictability, constancy, and contingency can be determined for either qualitative (categorical) or quantitative (discrete or continuous) variables measured over a period of time. Alternative patterns of seasonal flowering and fruiting of tropical trees are given as an example of a qualitative variable; precipitation data are analyzed as an example of a quantitative variable.

Species Loss and Aboveground Carbon Storage in a Tropical Forest

Tropical forest biodiversity is declining, but the resulting effects on key ecosystem services, such as carbon storage and sequestration, remain unknown. We assessed the influence of the loss of tropical tree species on carbon storage by simulating 18 possible extinction scenarios within a well-studied 50-hectare tropical forest plot in Panama, which contains 227 tree species. Among extinction scenarios, aboveground carbon stocks varied by more than 600%, and biological insurance varied by more than 400%. These results indicate that future carbon storage in tropical forests will be influenced strongly by future species composition.

Thermal-safety margins and the necessity of thermoregulatory behavior across latitude and elevation

Significance We find that most terrestrial ectotherms are insufficiently tolerant of high temperatures to survive the warmest potential body temperatures in exposed habitats and must therefore thermoregulate by using shade, burrows, or evaporative cooling. Our results reveal that exposure to extreme heat can occur even at high elevations and latitudes and show why heat-tolerance limits are relatively invariant in comparison with cold limits. To survive climate warming, ectotherms in most areas may need to rely on behaviors—and have access to habitats—that provide a reprieve from extreme operative temperatures. 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.

Hutchinson's duality: The once and future niche

The duality between “niche” and “biotope” proposed by G. Evelyn Hutchinson provides a powerful way to conceptualize and analyze biogeographical distributions in relation to spatial environmental patterns. Both Joseph Grinnell and Charles Elton had attributed niches to environments. Attributing niches, instead, to species, allowed Hutchinsons key innovation: the formal severing of physical place from environment that is expressed by the duality. In biogeography, the physical world (a spatial extension of what Hutchinson called the biotope) is conceived as a map, each point (or cell) of which is characterized by its geographical coordinates and the local values of n environmental attributes at a given time. Exactly the same n environmental attributes define the corresponding niche space, as niche axes, allowing reciprocal projections between the geographic distribution of a species, actual or potential, past or future, and its niche. In biogeographical terms, the realized niche has come to express not only the effects of species interactions (as Hutchinson intended), but also constraints of dispersal limitation and the lack of contemporary environments corresponding to parts of the fundamental niche. Hutchinsons duality has been used to classify and map environments; model potential species distributions under past, present, and future climates; study the distributions of invasive species; discover new species; and simulate increasingly more realistic worlds, leading to spatially explicit, stochastic models that encompass speciation, extinction, range expansion, and evolutionary adaptation to changing environments.

Sufficient sampling for asymptotic minimum species richness estimators

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.