A critical branching process model for biodiversity

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, ∞); thereafter, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on there being the prescribed number n of species at the present time. We study various mathematical properties of this model as n → ∞: namely the time of origin and of the most recent common ancestor, the pattern of divergence times within lineage trees, the time series of the number of species, the total number of extinct species, the total number of species ancestral to the extant ones, and the 'local' structure of the tree itself. We emphasize several mathematical techniques: the association of walks with trees; a point process representation of lineage trees; and Brownian limits. © Applied Probability Trust 2005