## Introduction

The term ‘beta diversity’ is applied in a broad sense to any measure of variation in species composition (Anderson *et al.* 2011). In the narrowest sense, it is the simple ratio between gamma and alpha diversities (Jost 2007; Tuomisto 2010; Jurasinski & Koch 2011), which only differs from 1 when local sites differ in species composition. A wide range of broader measures exist (see Anderson *et al.* 2011), including measures of differentiation and proportional diversity (Jurasinski, Retzer & Beierkuhnlein 2009; Jurasinski & Koch 2011), but all broadly aim at providing a measure of the difference between the assemblages present at each site, taking into account the identities of all species. This last characteristic makes beta diversity studies complementary of analyses of the variation in species richness, which ignores species identity. Therefore, compared to species richness, the analysis of beta diversity allows testing of different hypotheses about the processes driving species distributions and biodiversity.

The concept of ‘change in species composition’ or the question ‘how different are two species assemblages’ may apparently seem straightforward, but, as argued elsewhere (Baselga 2007, 2010, 2012; Baselga, Jiménez-Valverde & Niccolini 2007), there are two potential ways in which two species assemblages can be ‘different’. One is species replacement (i.e. turnover), which consists in the substitution of species in one site by different species in the other site. The second way is species loss (or gain), which implies the elimination (or addition) of species in only one of the sites, and leads to the poorest assemblage being a strict subset of the richest one (a pattern called nestedness). Therefore, the selection of the dissimilarity measure used to quantify the differences between assemblages can be crucial, because different dissimilarity indices account for the two phenomena in different ways. For example, the strict sense definition of beta diversity (the ratio of gamma and alpha diversity: Whittaker 1960; Tuomisto 2010) yields a measure that accounts for turnover and nestedness as being equivalent, as both turnover and nested patterns make alpha diversity lower than gamma diversity. The same applies for the widely used Jaccard and Sørensen indices, which are monotonic transformations of gamma/alpha (Jost 2007; Chao, Chiu & Hsieh 2012). In contrast, the Simpson index of dissimilarity (Simpson 1943; Lennon *et al.* 2001) accounts only for turnover (species replacement), and building on this, Baselga (2010, 2012) proposed a method for partitioning total dissimilarity (i.e. Sørensen and Jaccard indices, both monotonic transformations of beta diversity) into two separate components accounting for the dissimilarity derived solely from turnover and the dissimilarity derived from nestedness. The two decompositions for a single pair of cells are shown below for the Sørensen (eqn 1) and Jaccard (eqn 2) indices, where *a* is the number of shared species between two cells, *b* the number of species unique to the poorest site and *c* the number of species unique to the richest site.

where β_{sor} is Sørensen dissimilarity, β_{sim} is Simpson dissimilarity (= turnover component of Sørensen dissimilarity), β_{sne} is the nestedness component of Sørensen dissimilarity, β_{jac} is Jaccard dissimilarity, β_{jtu} is the turnover component of Jaccard dissimilarity, and β_{jne} is the nestedness component of Jaccard dissimilarity. Pairwise dissimilarity between all pairs of sites can be used to investigate spatial patterning of turnover and nestedness-resultant dissimilarity. In addition, multiple-site measures of compositional dissimilarity across a set of sites can be calculated by substituting the multiple-site analogues of the values *a*, *b* and *c* into the two equations (see Baselga 2010, 2012). Note that these are the sum across all pairs of sites for *b* and *c* analogues, but not for the shared species *a*, whose multiple-site analogue is , where *S*_{i} is the number of species in site *i*, and *S*_{T} is the number of species in the total pool of sites. This component makes the indices real multiple-site measures and not averaged pairwise dissimilarities. We use capital letters to differentiate these multiple-site measures from pairwise measures (Baselga 2012): β_{SOR} = β_{SIM} + β_{SNE} and β_{JAC} = β_{JTU} + β_{JNE.}