The recently introduced two-parameter infinitely-many-neutral-alleles model extends the celebrated one-parameter version (which is related to Kingman's distribution) to diffusive two-parameter Poisson-Dirichlet frequencies. In this paper 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 frequency dynamics are driven by state-dependent rather than constant quantities.

A new coalescent is introduced to study the genealogy of a sample from the infinite-alleles model of population genetics. This coalescent also records the age ordering of alleles in the sample. The distribution of this process is found explicitly for the Moran model, and is shown to be robust for a wide class of reproductive schemes.

Properties of the ages themselves and the relationship between ages and class sizes then follow readily.

A study is made of the joint distribution of the number of alleles and segregating sites in a random sample of genes from the infinite-alleles model of population genetics. Tables of the joint distribution are given for representative values of the mutation parameter. A conclusion is that each segregating site has a high probability of delineating an allele type.