Summary 1.Little consensus has been reached as to general features of spatial variation in beta diversity, a fundamental component of species diversity. This could reflect a genuine lack of simple gradients in beta diversity, or a lack of agreement as to just what constitutes beta diversity. Unfortunately, a large number of approaches have been applied to the investigation of variation in beta diversity, which potentially makes comparisons of the findings difficult. 2. We review 24 measures of beta diversity for presence/absence data (the most frequent form of data to which such measures are applied) that have been employed in the literature, express many of them for the first time in common terms, and compare some of their basic properties. 3. Four groups of measures are distinguished, with a fundamental distinction arising between 'broad sense' measures incorporating differences in composition attributable to species richness gradients, and 'narrow sense' measures that focus on compositional differences independent of such gradients. On a number of occasions on which the former have been employed in the literature the latter may have been more appropriate, and there are many situations in which consideration of both kinds of measures would be valuable. 4. We particularly recommend (i) considering beta diversity measures in terms of matching/mismatching components (usually denoted a , b and c ) and thereby identifying the contribution of different sources of variation in species composition, and (ii) the use of ternary plots to express the relationship between the values of these measures and of the components, and as a way of understanding patterns in beta diversity.

Summary1. Using data on the spatial distribution of the British avifauna, we address three basic questions about the spatial structure of assemblages: (i) Is there a relationship between species richness (alpha diversity) and spatial turnover of species (beta diversity)? (ii) Do high richness locations have fewer species in common with neighbouring areas than low richness locations?, and (iii) Are any such relationships contingent on spatial scale (resolution or quadrat area), and do they reflect the operation of a particular kind of species-area relationship (SAR)? 2. For all measures of spatial turnover, we found a negative relationship with species richness. This held across all scales, with the exception of turnover measured as β sim . 3. Higher richness areas were found to have more species in common with neighbouring areas. 4. The logarithmic SAR fitted better than the power SAR overall, and fitted significantly better in areas with low richness and high turnover. 5. Spatial patterns of both turnover and richness vary with scale. The finest scale richness pattern (10 km) and the coarse scale richness pattern (90 km) are statistically unrelated. The same is true of the turnover patterns. 6. With coarsening scale, locations of the most species-rich quadrats move north. This observed sensitivity of richness 'hotspot' location to spatial scale has implications for conservation biology, e.g. the location of a reserve selected on the basis of maximum richness may change considerably with reserve size or scale of analysis. 7. Average turnover measured using indices declined with coarsening scale, but the average number of species gained or lost between neighbouring quadrats was essentially scale invariant at 10-13 species, despite mean richness rising from 80 to 146 species (across an 81-fold area increase). We show that this kind of scale invariance is consistent with the logarithmic SAR.

There is little understanding in ecology as to how biodiversity patterns emerge from the distribution patterns of individual species. Here we consider the question of the contributions of rare (restricted range) and common (widespread) species to richness patterns. Considering a species richness pattern, is most of the spatial structure, in terms of where the peaks and troughs of diversity lie, caused by the common species or the rare species (or neither)? Using southern African and British bird richness patterns, we show here that commoner species are most responsible for richness patterns. While rare and common species show markedly different species richness patterns, most spatial patterning in richness is caused by relatively few, more common, species. The level of redundancy we found suggests that a broad understanding of what determines the majority of spatial variation in biodiversity may be had by considering only a minority of species.

Mid‐domain models have been argued to provide a default explanation for the best known spatial pattern in biodiversity, namely the latitudinal gradient in species richness. These models assume no environmental gradients, but merely a random latitudinal association between the size and placement of the geographic ranges of species. A mid‐domain peak in richness is generated because when the latitudinal extents of species in a given taxonomic group are bounded to north and south, perhaps by a physical constraint such as a continental edge or perhaps by a climatic constraint such as a critical temperature or precipitation threshold, then the number of ways in which ranges can be distributed changes systematically between the bounds. In addition, such models make predictions about latitudinal variation in the latitudinal extents of the distributions of species, and in beta diversity (the spatial turnover in species identities). Here we test how well five mid‐domain models predict observed latitudinal patterns of species richness, latitudinal extent and beta diversity in two groups of birds, parrots and woodpeckers, across the New World. Whilst both groups exhibit clear gradients in richness and beta diversity and the general trend in species richness is acceptably predicted (but not accurately, unless substantial empirical information is assumed), the fit of these models is uniformly poor for beta diversity and latitudinal range extent. This suggests either that, at least for these data, as presently formulated mid‐domain models are too simplistic, or that in practice the mid‐domain effect is not significant in determining geographical variation in diversity.