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 qD, where q = 0) to measures of taxon diversity incorporating relative abundance (i.e., for any Hill number qD, 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 = 0 (species richness), q = 1 (the exponential of Shannon's entropy index), and q = 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.

Summary1. Hill numbers (or the effective number of species) have been increasingly used to quantify the species/taxonomic diversity of an assemblage. The sample-size-and coverage-based integrations of rarefaction (interpolation) and extrapolation (prediction) of Hill numbers represent a unified standardization method for quantifying and comparing species diversity across multiple assemblages. 2. We briefly review the conceptual background of Hill numbers along with two approaches to standardization. We present an R package iNEXT (iNterpolation/EXTrapolation) which provides simple functions to compute and plot the seamless rarefaction and extrapolation sampling curves for the three most widely used members of the Hill number family (species richness, Shannon diversity and Simpson diversity). Two types of biodiversity data are allowed: individual-based abundance data and sampling-unit-based incidence data. 3. Several applications of the iNEXT packages are reviewed: (i) Non-asymptotic analysis: comparison of diversity estimates for equally large or equally complete samples. (ii) Asymptotic analysis: comparison of estimated asymptotic or true diversities. (iii) Assessment of sample completeness (sample coverage) across multiple samples. (iv) Comparison of estimated point diversities for a specified sample size or a specified level of sample coverage. 4. Two examples are demonstrated, using the data (one for abundance data and the other for incidence data) included in the package, to illustrate all R functions and graphical displays.

Shannon entropy H and related measures are increasingly used in molecular ecology and population genetics because (1) unlike measures based on heterozygosity or allele number, these measures weigh alleles in proportion to their population fraction, thus capturing a previously-ignored aspect of allele frequency distributions that may be important in many applications; (2) these measures connect directly to the rich predictive mathematics of information theory; (3) Shannon entropy is completely additive and has an explicitly hierarchical nature; and (4) Shannon entropy-based differentiation measures obey strong monotonicity properties that heterozygosity-based measures lack. We derive simple new expressions for the expected values of the Shannon entropy of the equilibrium allele distribution at a neutral locus in a single isolated population under two models of mutation: the infinite allele model and the stepwise mutation model. Surprisingly, this complex stochastic system for each model has an entropy expressable as a simple combination of well-known mathematical functions. Moreover, entropy- and heterozygosity-based measures for each model are linked by simple relationships that are shown by simulations to be approximately valid even far from equilibrium. We also identify a bridge between the two models of mutation. We apply our approach to subdivided populations which follow the finite island model, obtaining the Shannon entropy of the equilibrium allele distributions of the subpopulations and of the total population. We also derive the expected mutual information and normalized mutual information (“Shannon differentiation”) between subpopulations at equilibrium, and identify the model parameters that determine them. We apply our measures to data from the common starling (Sturnus vulgaris) in Australia. Our measures provide a test for neutrality that is robust to violations of equilibrium assumptions, as verified on real world data from starlings.

The repertoire of IgG antibody responses to infection and vaccination varies depending on the characteristics of the immunogen and the ability of the host to mount a protective immune response. Chronic hepatitis B virus (HBV) infections are marked by persistent infection and immune tolerance to vaccination. This disease offers a unique opportunity to discover key repertoire signatures during infection and in response to vaccination. Complementarity determining region 3 of an antibody heavy chain (CDR-H3) has a major impact on the antigenic specificity of an antibody. We used next-generation sequencing to characterize the CDR-H3 sequences in paired siblings of 4 families in which only one member of each pair had chronic HBV infection. Blood samples were obtained before and 2 weeks after HBV vaccination. The analysis revealed a huge network of sequence-related CDR-H3 clones found almost exclusively among carriers. In contrast, vaccination induced significant increases of CDR-H3 cluster diversities among siblings without hepatitis B. Several vaccination-associated clone clusters were identified. Similar findings of vaccination-associated clone networks were observed in healthy adults receiving HBV boosters. These strategies can be used to identify signatures of other infectious diseases and accelerate discoveries of antibody sequences with important biomedical implications.

Whether successional forests converge towards an equilibrium in species composition remains an elusive question, hampered by high idiosyncrasy in successional dynamics. Based on long-term tree monitoring in second-growth (SG) and old-growth (OG) forests in Costa Rica, we show that patterns of convergence between pairs of forest stands depend upon the relative abundance of species exhibiting distinct responses to the successional gradient. For instance, forest generalists contributed to convergence between SG and OG forests, whereas rare species and old-growth specialists were a source of divergence. Overall, opposing trends in taxonomic similarity among different subsets of species nullified each other, producing a net outcome of stasis over time. Our results offer an explanation for the limited convergence observed between pairwise communities and suggest that rare species and old-growth specialists may be prone to dispersal limitation, while the dynamics of generalists and second-growth specialists are more predictable, enhancing resilience in tropical secondary forests.