Abstract
Aim The variation in species composition among sites, or beta diversity, can be decomposed into replacement and richness difference. A debate is ongoing in the literature concerning the best ways of computing and interpreting these indices. This paper first reviews the historical development of the formulae for decomposing dissimilarities into replacement, richness difference and nestedness indices. These formulae are presented for species presence–absence and abundance using a unified algebraic framework. The indices decomposing beta play different roles in ecological analysis than do beta‐diversity indices. Innovation Replacement and richness difference indices can be interpreted and related to ecosystem processes. The pairwise index values can be summed across all pairs of sites; these sums form a valid decomposition of total beta diversity into total replacement and total richness difference components. Different communities and study areas can be compared: some may be dominated by replacement, others by richness/abundance difference processes. Within a region, differences among sites measured by these indices can then be analysed and interpreted using explanatory variables or experimental factors. The paper also shows that local contributions of replacement and richness difference to total beta diversity can be computed and mapped. A case study is presented involving fish communities along a river. Main conclusions The different forms of indices are based upon the same functional numerators. These indices are complementary; they can help researchers understand different aspects of ecosystem functioning. The methods of analysis used in this paper apply to any of the indices recently proposed. Further work, based on ecological theory and numerical simulations, is required to clarify the precise meaning and domain of application of the different forms. The forms available for presence–absence and quantitative data are both useful because these different data types allow researchers to answer different types of ecological or biogeographic questions.