Understanding the causes of spatial variation in species richness is a major research focus of biogeography and macroecology. Gridded environmental data and species richness maps have been used in increasingly sophisticated curve-fitting analyses, but these methods have not brought us much closer to a mechanistic understanding of the patterns. During the past two decades, macroecologists have successfully addressed technical problems posed by spatial autocorrelation, intercorrelation of predictor variables and non-linearity. However, curve-fitting approaches are problematic because most theoretical models in macroecology do not make quantitative predictions, and they do not incorporate interactions among multiple forces. As an alternative, we propose a mechanistic modelling approach. We describe computer simulation models of the stochastic origin, spread, and extinction of speciesÕ geographical ranges in an environmentally heterogeneous, gridded domain and describe progress to date regarding their implementation. The output from such a general simulation model (GSM) would, at a minimum, consist of the simulated distribution of species ranges on a map, yielding the predicted number of species in each grid cell of the domain. In contrast to curve-

The diversity of sites and the distribution of species are fundamental pieces in the analysis of biogeographic and macroecological questions. A link between these two variables is the correlation between the species diversity of sites and the mean range size of species occurring there. Alternatively, one could correlate the range sizes of species and the mean species diversity within those ranges. Here we show that both approaches are mirror images of the same patterns, reflecting fundamental mathematical and biological relationships. We develop a theory and analyze data for North American mammals to interpret range-diversity plots in which the species diversity of sites and the geographic range of species can be depicted simultaneously. We show that such plots contain much more information than traditional correlative approaches do, and we demonstrate that the positions of points in the plots are determined to a large extent by the average, minimum, and maximum values of range and diversity but that the dispersion of points depends on the association among species and the similitude among sites. These generalizations can be applied to biogeographic studies of diversity and distribution and in the identification of hotspots of diversity and endemism.

Local density and size of distributional range have been used to characterize rarity, but conclusions are weakened by their possible lack of independence. The usefulness and validity of using these two variables were tested with data on distribution, local density, body size, and feeding habits for a set of 100 Neotropical forest mammals. In a bivariate plot of distributional range against local density, species clustered according to their trophic or taxonomic groups. This indicates that diet and phylogenetic history have an influence on rarity. A negative correlation was found between distribution and abundance. However this correlation was weaker within trophic or taxonomic groups, and vanished when body size was held constant. These results show that both distribution and abundance are valid and independent estimators of rarity when comparing species with similar sizes and ecological traits. Regression analysis showed that larger animals tend to have lower densities and wider distributional ranges. Rarity is clearly associated with body size. A dichotomous classification of rarity based on area of distribution and local density is suitable for Neotropical forest mammals. Species in each of four categories created by such a scheme require different conservation and management policies that are determined by the ecological characteristics of the species. Final conservation strategies must also be shaped by political and economic constraints.

The study of the relative roles of local and regional processes in determining the scaling of species diversity is a very active field in current ecology. The importance of species turnover and the species‐range‐size frequency distributions in determining how local and regional species diversity are linked has been recognised by recent approaches. Here we present a model, based on a system of fully nested sampling quadrats, to analyse species diversity at several scales. Using a recursive procedure that incorporates increasingly smaller scales and a multiplicative formula for relating local and regional diversity, the model allows the simultaneous depiction of alpha, beta and gamma diversity in a single “species‐scale plot”. Species diversity is defined as the number of ranges that are intersected by sampling quadrats of various sizes. The size, shape and location of individual species ranges determine diversity at any scale, but the average point diversity, measured at hypothetical zero‐area localities, is determined solely by the size of individual ranges, regardless of their shape and location. The model predicts that if the species‐area relationship is a power function, then beta diversity must be scale invariant if measured at constant scale increments. Applying the model to the mammal fauna of four Mexican regions with contrasting environmental conditions, we found that: 1) the species‐range‐size frequency distribution at the scale of the Mexican regions differs from the log‐normal pattern reported for the national and continental scales. 2) Beta diversity is not scale‐invariant within each region, implying that the species‐area relationship (SAR) does not follow a power function. 3) There is geographic variation in beta diversity. 4) The scaling of diversity is directly linked to patterns of species turnover rate, and ultimately determined by patterns in the geographic distribution of species. The model shows that regional species diversity and the average distribution range of species are the two basic data necessary to predict patterns in the scaling of species diversity.