The Theory of Island Biogeography (TIB) has been successful in predicting variation in species richness among islands, and has recently been advanced through the incorporation of island ontogenies, speciation, trophic interactions and environmental gradients into the classical McArthur-Wilson framework. Differences in community composition (beta diversity) have long been recognized as an important element of the TIB, and is crucial for understanding community assembly rules and setting up conservation priorities, but have to date not been mathematically incorporated into the theory. We derived the first analytical solution for the expected pairwise community dissimilarities (for both Jaccard and Sørensen families) as well as their partitioned beta diversity components under a neutral scenario where all species have equal colonization and extinction rates.
We obtain a set of non-trivial testable predictions, which directly link extinction rate e and colonization rate c to beta diversity patterns: 1) We find that the expected pairwise Jaccard dissimilarity is a monotonic increasing function of Θ (Θ = e/c) which is independent of the size of the mainland species pool M, while the partitioned components of pairwise Jaccard dissimilarity are not independent of M. 2) Turnover component increases monotonically with Θ while nestedness component has a unimodal relationship with Θ (increases first and then decreases). 3) Turnover component dominates when the size of mainland species pool increases, but the ratio of expected turnover and expected nestedness increases linearly with Θ as Θ approaches infinity. 4) Asymmetries of isolation distances and areas of two islands increase expected pairwise Jaccard dissimilarity and the corresponding turnover components. Dissimilarity measures of the Sørensen family show similar quantative behaviors. We suggest that these new predictions will allow stronger tests of the island biogeography theory and facilitate the incorporation of beta diversity patterns into future developments of a unified biodiversity theory.