The frequency distribution of species abundances [the species abundance distribution (SAD)] is considered to be a fundamental characteristic of community structure. It is almost invariably strongly right-skewed, with most species being rare. There has been much debate as to its exact properties and the processes from which it results. Here, we contend that an SAD for a study plot must be viewed as spliced from the SADs of many smaller nonoverlapping subplots covering that plot. We show that this splicing, if applied repeatedly to produce subplots of progressively larger size, leads to the observed shape of the SAD for the whole plot regardless of that of the SADs of those subplots. The widely reported shape of an SAD is thus likely to be driven by a spatial parallel of the central limit theorem, a statistically convergent process through which the SAD arises from small to large scales. Exact properties of the SAD are driven by species spatial turnover and the spatial autocorrelation of abundances, and can be predicted using this information. The theory therefore provides a direct link between SADs and the spatial correlation structure of species distributions, and thus between several fundamental descriptors of community structure. Moreover, the statistical process described may lie behind similar frequency distributions observed in many other scientific fields.log-normal distribution ͉ spatial autocorrelation ͉ spatial turnover

There have been several attempts to build a unified framework for macroecological patterns. However, these have mostly been based either on questionable assumptions or have had to be parameterized to obtain realistic predictions. Here, we propose a new model explicitly considering patterns of aggregated species distributions on multiple spatial scales, the property which lies behind all spatial macroecological patterns, using the idea we term 'generalized fractals'. Species' spatial distributions were modelled by a random hierarchical process in which the original 'habitat' patches were randomly replaced by sets of smaller patches nested within them, and the statistical properties of modelled species assemblages were compared with macroecological patterns in observed bird data. Without parameterization based on observed patterns, this simple model predicts realistic patterns of species abundance, distribution and diversity, including fractal-like spatial distributions, the frequency distribution of species occupancies/abundances and the species-area relationship. Although observed macroecological patterns may differ in some quantitative properties, our concept of random hierarchical aggregation can be considered as an appropriate null model of fundamental macroecological patterns which can potentially be modified to accommodate ecologically important variables.

The species-area relationship (SAR) is considered to be one of a few generalities in ecology, yet a universal model of its shape and slope has remained elusive. Recently, Harte et al. argued that the slope of the SAR for a given area is driven by a single parameter, the ratio between total number of individuals and number of species (i.e., the mean population size across species at a given scale). We provide a geometric interpretation of this dependence. At the same time, however, we show that this dependence cannot be universal across taxa: if it holds for a taxon composed from two subsets of species and also for one of its subsets, it cannot simultaneously hold for the other subset. Using three data sets, we show that the slope of the SAR considerably varies around the prediction. We estimate the limits of this variation by using geometric considerations, providing a theory based on species spatial turnover at different scales. We argue that the SAR cannot be strictly universal, but its slope at each particular scale varies within the constraints given by species' spatial turnover at finer spatial scales, and this variation is biologically informative.

Common species have a greater effect on observed geographical patterns of species richness than do rare ones. Here we present a theory of the relationship between individual species occurrence patterns and patterns in species richness, which allows purely geometrical and statistical causes to be distinguished from biological ones. Relationships between species occupancy and the correlation of species occurrence with overall species richness are driven by the frequency distribution of species richness among sites. Moreover, generally positive relationships are promoted by the fact that species occupancy distributions are mostly right skewed. However, biological processes can lead to deviations from the predicted pattern by changing the nestedness of a species' spatial distribution with regard to the distributions of other species in an assemblage. We have applied our theory to data for European birds at several spatial scales and have identified the species with significantly stronger or weaker correspondence with the overall richness pattern than that predicted by the null model. In sum, whereas the general macroecological pattern of a stronger influence of common species than of rare species on species richness is predicted by mathematical considerations, the theory can reveal biologically important deviations at the level of individual species.

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