A geometric approach to scaling individual distributions to macroecological patterns

Understanding macroecological patterns across scales is a central goal of ecology and a key need for conservation biology. Much research has focused on quantifying and understanding macroecological patterns such as the species-area relationship (SAR), the endemic-area relationship (EAR) and relative species abundance curve (RSA). Understanding how these aggregate patterns emerge from underlying spatial pattern at individual level, and how they relate to each other, has both basic and applied relevance. To address this challenge, we develop a novel spatially explicit geometric framework to understand multiple macroecological patterns, including the SAR, EAR, RSA, and their relationships, using theory of point processes. The geometric approach provides a theoretical framework to derive SAR, EAR, and RSA from species range distributions and the pattern of individual distribution patterns therein. From this model, various well-documented macroecological patterns are recovered, including the tri-phasic SAR on a log-log plot with its asymptotic slope, and various RSAs (e.g., Fisher {boxtimes} s logseries and the Poisson lognormal distribution). Moreover, this approach can provide new insights such as a single equation describing the RSA at an arbitrary spatial scale, and explicit forms of the EAR with its asymptotic slope. The theory, which links spatial distributions of individuals and species with macroecological patterns, is ambiguous with regards to the mechanism(s) responsible for the statistical properties of individual distributions and species range sizes. However, our approach can be connected to mechanistic models that make such predictions about lower-level patterns and be used to scale them up to aggregate patterns, and therefore is applicable to many ecological questions. We demonstrate an application of the geometric model to scaling issue of beta diversity.