Species dynamics in the two-parameter Poisson-Dirichlet diffusion model

The recently introduced two-parameter infinite alleles model extends the celebrated one-parameter version, related to Kingman’s distribution, to diffusive two-parameter Poisson-Dirichlet frequencies. Here we investigate the dynamics driving the species heterogeneity underlying the two-parameter model. First we show that a suitable normalization of the number of species is driven by a critical continuous state branching process with immigration, or a squared Bessel process with dimension given by the immigration rate. Secondly, we identify an instance of the intensity rates driving the finite-dimensional mutation process that gives rise to the two-parameter model, which turn out to be inhomogeneous and unbounded. These, together with additional restrictions, allow to provide a finite-dimensional construction of the two-parameter model, obtained by means of a sequence of Feller diffusions of Wright-Fisher flavor which feature finitely-many types. Both results provide insight into the mathematical properties and biological interpretation of the two-parameter model, showing that it is structurally different from the one-parameter case in that the frequencies dynamics are driven by state-dependent rather than constant quantities.