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

TL;DR

This paper explores the dynamics of species diversity in the two-parameter Poisson-Dirichlet diffusion model, revealing its unique structure driven by state-dependent processes and providing new mathematical and biological insights.

Contribution

It introduces a finite-dimensional construction of the two-parameter model using Wright-Fisher diffusions and links species heterogeneity to a critical continuous-state branching process.

Findings

Number of species normalized by a critical branching process

Finite-dimensional construction via Wright-Fisher diffusions

Frequencies driven by state-dependent rather than constant factors

Abstract

The recently introduced two-parameter infinitely-many neutral 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. Secondly, we 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 and inhomogeneous mutation rates. 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.