The species–area relationship (SAR) is often expressed as a power law, which indicates scale invariance. It has been claimed that the scale invariance – or self‐similarity at the community level – is not compatible with the self‐similarity at the level of spatial distribution of individual species, because the power law would only emerge if distributions for all species had identical fractal dimensions (FD). Here we show that even if species differ in their FD, the resulting SAR is approximately linear on a log–log scale because observed spatial distributions are inevitably spatially restricted – a phenomenon we term the ‘finite‐area effect’. Using distribution atlases, we demonstrate that the apparent power law of SARs for central European birds is attributable to this finite‐area effect affecting species that indeed reveal self‐similar distributions. We discuss implications of this mechanism producing the SAR.

Summary 1.The objective of this study was to test the theoretical prediction that the thermal tolerance range for development in insects should be about 20 ° C. 2. The data on the thermal requirements for development of 66 species from eight orders of insects was obtained from the literature. The temperatures at which the developmental rates are at their minimum and maximum was obtained for each population by defining the relationship between developmental rate (1/ D ) and temperature, using either Lactin et al. 's (1995) or Briére et al. 's (1999) model. 3. Thermal windows, i.e. the range in temperature between the minimum and maximum rate of development for individual species, and the relationship between the minimum and maximum temperatures, were examined. 4. The mean thermal window, 19·8 ° C with 95% confidence interval 19·1-20·5 and range 13·3 − 28·6, was influenced by species phylogeny, with the windows narrower for species having a true pupal stage, but not by ecological traits thought to affect species thermal requirements. The relationship between the minimum and maximum temperatures was highly significant and independent of species phylogeny. 5. Theory and this analysis of empirical data indicate that each species of insect can only develop over a limited range of temperatures independent of species traits. In addition, the relationship between the minimum and maximum developmental rates co-vary independent of species phylogeny. This may help identify the precise nature of the physiological mechanism underlying the seasonal development and distribution of insects, and possibly other ectotherms.

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.