Incidence, or compositional, matrices are generated for a broad range of research applications in biology. Zeta diversity provides a common currency and conceptual framework that links incidence‐based metrics with multiple patterns of interest in biology, ecology, and biodiversity science. It quantifies the variation in species (or OTU) composition of multiple assemblages (or cases) in space or time, to capture the contribution of the full suite of narrow, intermediate, and wide‐ranging species to biotic heterogeneity. Here we provide a conceptual framework for the application and interpretation of patterns of continuous change in compositional diversity using zeta diversity. This includes consideration of the survey design context, and the multiple ways in which zeta diversity decline and decay can be used to examine and test turnover in the identity of elements across space and time. We introduce the zeta ratio–based retention rate curve to quantify rates of compositional change. We illustrate these applications using 11 empirical data sets from a broad range of taxa, scales, and levels of biological organization—from DNA molecules and microbes to communities and interaction networks—including one of the original data sets used to express compositional change and distance decay in ecology. We show (1) how different sample selection schemes used during the calculation of compositional change are appropriate for different data types and questions, (2) how higher orders of zeta may in some cases better detect shifts and transitions, and (3) the relative roles of rare vs. common species in driving patterns of compositional change. By exploring the application of zeta diversity decline and decay, including the retention rate, across this broad range of contexts, we demonstrate its application for understanding continuous turnover in biological systems.

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The PD measure of phylogenetic diversity interprets branch lengths cladistically to make inferences about feature diversity. PD calculations extend conventional species-level ecological indices to the features level. The “phylogenetic beta diversity” framework developed by microbial ecologists calculates PD-dissimilarities between community localities. Interpretation of these PD-dissimilarities at the feature level explains the framework’s success in producing ordinations revealing environmental gradients. An example gradients space using PD-dissimilarities illustrates how evolutionary features form unimodal response patterns to gradients. This features model supports new application of existing species-level methods that are robust to unimodal responses, plus novel applications relating to climate change, commercial products discovery, and community assembly.

Summary1. Traditional species diversity measures do not make distinctions among species. Faith's phylogenetic diversity (PD), which is defined as the sum of the branch lengths of a phylogenetic tree connecting all species, takes into account phylogenetic differences among species and has found many applications in various research fields. In this paper, we extend Faith's PD to represent the total length of a phylogenetic tree from any fixed point on its main trunk. 2. Like species richness, Faith's PD tends to be an increasing function of sampling effort and thus tends to increase with sample completeness. We develop in this paper the 'PD accumulation curve' (an extension of the species accumulation curve) to depict how PD increases with sampling size and sample completeness. 3. To make fair comparisons of Faith's PD among several assemblages based on sampling data from each assemblage, we derive both theoretical formulae and analytic estimators for seamless rarefaction (interpolation) and extrapolation (prediction). We develop a lower bound of the undetected PD for an incomplete sample to guide the extrapolation; the PD estimator for an extrapolated sample is generally reliable up to twice the size of the empirical sample. 4. We propose an integrated curve that smoothly links rarefaction and extrapolation to standardize samples on the basis of sample size or sample completeness. A bootstrap method is used to obtain the unconditional variances of PD estimators and to construct the confidence interval of the expected PD for a fixed sample size or fixed degree of sample completeness. This facilitates comparison of multiple assemblages of both rarefied and extrapolated samples. 5. We illustrate our formulae and estimators using empirical data sets from Australian birds in two sites. We discuss the extension of our approach to the case of multiple incidence data and to incorporate species abundances.

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Questions: How can a resemblance (similarity or dissimilarity) measure be formulated to include information on both the evolutionary relationships and abundances of organisms, and how does it compare to measures lacking such information? Methods: We extend the family of Phylogenetic Diversity (PD) measures to include a generalized method for calculating pair-wise resemblance of ecological assemblages. Building on previous work, we calculate the matching/mismatching components of the 2Â2 contingency table so as to incorporate information on both phylogeny and abundance. We refer to the class of measures so defined as ''PD resemblance'' and use the term ''SD resemblance'' for the traditional class of measures based on species diversity alone. As an illustration, we employ data on the diversity and stem density of shrubs of Toohey Forest, Australia, to compare PD resemblance to its SD resemblance equivalent for both incidence and abundance data. Results: While highly correlated, PD resemblance consistently measures assemblages as more similar than does SD resemblance, and tends to ''smooth out'' the otherwise skewed and truncated distribution of pair-wise resemblance indices of our highturnover data set, resulting in nMDS ordinations with lower stress. Randomization of species distributions across assemblages indicates that phylogeny has made a significant contribution to the ordination pattern. Conclusions: PD resemblance measures, in addition to providing an evolutionary perspective, have great potential to improve distance-based analyses of community patterns, particularly if species responses to ecological gradients are unimodal and phylogenetically conserved.

Climate change is expected to have substantial impacts on the composition of freshwater communities, and many species are threatened by the loss of climatically suitable habitat. In this study we identify Australian Odonata (dragonflies and damselflies) vulnerable to the effects of climate change on the basis of exposure, sensitivity and pressure to disperse in the future. We used an ensemble of species distribution models to predict the distribution of 270 (85%) species of Australian Odonata, continent-wide at the subcatchment scale, and for both current and future climates using two emissions scenarios each for 2055 and 2085. Exposure was scored according to the departure of temperature, precipitation and hydrology from current conditions. Sensitivity accounted for change in the area and suitability of projected climatic habitat, and pressure to disperse combined measurements of average habitat shifts and the loss experienced with lower dispersal rates. Streams and rivers important to future conservation efforts were identified based on the sensitivity-weighted sum of habitat suitability for the most vulnerable species. The overall extent of suitable habitat declined for 56–69% of the species modelled by 2085 depending on emissions scenario. The proportion of species at risk across all components (exposure, sensitivity, pressure to disperse) varied between 7 and 17% from 2055 to 2085 and a further 3–17% of species were also projected to be at high risk due to declines that did not require range shifts. If dispersal to Tasmania was limited, many south-eastern species are at significantly increased risk. Conservation efforts will need to focus on creating and preserving freshwater refugia as part of a broader conservation strategy that improves connectivity and promotes adaptive range shifts. The significant predicted shifts in suitable habitat could potentially exceed the dispersal capacity of Odonata and highlights the challenge faced by other freshwater species.

Summary Phylogenetic diversity (PD) depends on sampling depth, which complicates the comparison of PD between samples of different depth. One approach to dealing with differing sample depth for a given diversity statistic is to rarefy, which means to take a random subset of a given size of the original sample. Exact analytical formulae for the mean and variance of species richness under rarefaction have existed for some time but no such solution exists for PD.We have derived exact formulae for the mean and variance of PD under rarefaction. We confirm that these formulae are correct by comparing exact solution mean and variance to that calculated by repeated random (Monte Carlo) subsampling of a dataset of stem counts of woody shrubs of Toohey Forest, Queensland, Australia. We also demonstrate the application of the method using two examples: identifying hotspots of mammalian diversity in Australasian ecoregions, and characterising the human vaginal microbiome.There is a very high degree of correspondence between the analytical and random subsampling methods for calculating mean and variance of PD under rarefaction, although the Monte Carlo method requires a large number of random draws to converge on the exact solution for the variance.Rarefaction of mammalian PD of ecoregions in Australasia to a common standard of 25 species reveals very different rank orderings of ecoregions, indicating quite different hotspots of diversity than those obtained for unrarefied PD. The application of these methods to the vaginal microbiome shows that a classical score used to quantify bacterial vaginosis is correlated with the shape of the rarefaction curve.The analytical formulae for the mean and variance of PD under rarefaction are both exact and more efficient than repeated subsampling. Rarefaction of PD allows for many applications where comparisons of samples of different depth is required.