# Modelling the species-area relationship using extreme value theory

## TL;DR

This paper explains how the number of species increases with area using extreme value theory, revealing patterns based on species' spatial distributions.

## Contribution

A novel theoretical framework using extreme value theory to model species-area relationships and predict phase transitions.

## Key findings

- The species-area relationship phases are determined by species' geographical ranges relative to a focal point.
- A formula was developed to estimate species numbers at phase transitions using extreme value theory.
- Empirical data from different continents and taxa align with the predicted species-area patterns.

## Abstract

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.

The species-area relationship describes how the number of species increases with the size of the area sampled, showing a consistent triphasic pattern. This study shows that these phases arise from species’ spatial distributions, providing a unified framework to predict phase transitions across ecosystems.