Diversity without borders: partitioning continuous spaces using probabilistic equivalent numbers
Equivalent numbers represent biodiversity as the effective number of equally distinct units, typically species, and can be partitioned across scales. In practice, they summarize each unit of biodiversity by a single value and compare units pairwise, misrepresenting units that are better described as distributions and the relationships between several units that share the same space. We introduce an equivalent-number index for assemblages of units represented as probability density functions (PDFs) over a continuous space, estimated as the integral of the pointwise maximum across abundance-weighted PDFs. Resulting equivalent PDF numbers fulfil elementary properties of classical equivalent numbers, and support additive partitioning across any number of nested scales. We illustrate the framework with case studies across three domains: (1) measuring trait diversity considering intraspecific variability in grasslands, (2) partitioning realized bioclimatic niches among clades of Carnivora, and (3) understanding seasonal changes in the partitioning of fish home ranges in geographic space.