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Aim: To derive null models for the expected number of species shared among multiple samples or habitat patches, allowing exploration of the geometric effects of subdivision on species diversity. Location: Global. Major taxa studied: Predominantly sessile organisms. Methods: The occurrence probability of a species in a subdivided area depends on its abundance and spatial pattern over a known habitat extent. The joint probability that two subareas share a species is the product of the probability of species occurrence in each subarea provided that the latter probability is independent. The sum of this probability over all species is the number of species the two subareas share, or zeta diversity of order 2. Generalizing from 2 to m subareas yields a null model for zeta diversity of order m. From zeta diversity, many metrics (e.g., beta and gamma diversity) for the m habitat patches can be calculated, revealing the effects of increasing habitat fragmentation. Results: The null models show the geometric effects of subdivision depend on patterns of spatial distribution of species within a landscape and evenness of species abundance distribution. For aggregated assemblages, increasing subdivision decreases shared species, increases beta diversity and results in higher total species richness in subdivided habitat than an equal contiguous area. Main conclusions: To correctly interpret diversity patterns in fragmented habitat the geometric effects of subdivision must be considered. Our models explain why fragmented habitat could have higher diversity than continuous habitat of equal area but predict a threshold patch‐size above which this will not occur (here c. 100 ha). Apparently positive diversity effects of subdivision, including more species in groups of small patches, are probable outcomes of spatial aggregation of assemblages. The shared species null models offer an analytical tool for exploring the geometric effects of subdivision on diversity while controlling for total habitat area. [ABSTRACT FROM AUTHOR]

The classic Jaccard and Sørensen indices of compositional similarity (and other indices that depend upon the same variables) are notoriously sensitive to sample size, especially for assemblages with numerous rare species. Further, because these indices are based solely on presence–absence data, accurate estimators for them are unattainable. We provide a probabilistic derivation for the classic, incidence-based forms of these indices and extend this approach to formulate new Jaccard-type or Sørensen-type indices based on species abundance data. We then propose estimators for these indices that include the effect of unseen shared species, based on either (replicated) incidence- or abundance-based sample data. In sampling simulations, these new estimators prove to be considerably less biased than classic indices when a substantial proportion of species are missing from samples. Based on species-rich empirical datasets, we show how incorporating the effect of unseen shared species not only increases accuracy but also can change the interpretation of results. [ABSTRACT FROM AUTHOR]