Modelling the species-area relationship using extreme value theory.

The nested species-area relationship, obtained by counting species in increasingly larger areas in a nested fashion, exhibits robust and recurring qualitative and quantitative patterns. When plotted in double logarithmic scales it shows three phases: rapid species increase at small areas, slower growth at intermediate scales, and faster rise at large scales. Despite its significance, the theoretical foundations of this pattern remain incompletely understood. Here, we develop a theory for the species-area relationship using extreme value theory, and show that the species-area relationship is a mixture of the distributions of minimum distances to a starting sampling focal point for each individual species. A key insight of our study is that each phase is determined by the geographical distributions of the species, i.e., their ranges, relative to the focal point, enabling us to develop a formula for estimating the number of species at phase transitions. We test our approach by comparing empirical species-area relationships for different continents and taxa with our predictions using Global Biodiversity Information Facility data. Although a SAR reflects the underlying biological attributes of the constituent species, our interpretations and use of the extreme value theory are general and can be widely applicable to systems with similar spatial features.