Geometric causes of species rarity

Understanding the limits of species distributions is a central objective of biogeography and macroecology and has become increasingly important as climate change drives rapid shifts in geographic ranges. Species range sizes follow a highly skewed frequency distribution, with most species occupying ranges orders of magnitude smaller than those of the most widespread species. Range sizes also exhibit pronounced geographic patterns, with small-ranged species concentrated near continental margins and other geographic boundaries. No universally accepted explanation has been proposed for these patterns. Here we present a simple geometric model showing that species range size patterns emerge from the random placement of dispersal barriers within continental domains. The model predicts both the observed frequency distribution and the spatial distribution of range sizes across amphibians, birds, and mammals. It therefore provides a first-order explanation for global patterns of species rarity and can be refined by incorporating elevational barriers and spatial variation in species richness. Our findings suggest that species range size is constrained by the geometry of dispersal barriers and the geographic domain, with proximity to domain boundaries acting as a primary determinant of species rarity. These results have important implications for understanding species' evolutionary potential and vulnerability to extinction.