A critical branching process model for biodiversity

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Motivated as a null model for comparison with data, we study the following model for a phylogenetic tree on n extant species. The origin of the clade is a random time in the past, whose (improper) distribution is uniform on (0,∞). After that origin, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on having the prescribed number n of species at the present time. We study various mathematical properties of this model as n → ∞ limits: time of origin and of most recent common ancestor; pattern of divergence times within lineage trees; time series of numbers of species; number of extinct species in total, or ancestral to extant species; and "local" structure of the tree itself. We emphasize several mathematical techniques: associating walks with trees, a point process representation of lineage trees, and Brownian limits.