  The infinitely-many-neutral-alleles model has recently been extended to a
class of diffusion processes associated with Gibbs partitions of two-parameter
Poisson-Dirichlet type. This paper introduces a family of infinite-dimensional
diffusions associated with a different subclass of Gibbs partitions, induced by
normalized inverse-Gaussian random probability measures. Such diffusions
describe the evolution of the frequencies of infinitely-many types together
with the dynamics of the time-varying mutation rate, which is driven by an
alpha-diversity diffusion. Constructed as a dynamic version, relative to this
framework, of the corresponding notion for Gibbs partitions, the latter is
explicitly derived from an underlying population model and shown to coincide,
in a special case, with the diffusion approximation of a critical Galton-Watson
branching process. The class of infinite-dimensional processes is characterized
in terms of its infinitesimal generator on an appropriate domain, and shown to
be the limit in distribution of a certain sequence of Feller diffusions with
finitely-many types. Moreover, a discrete representation is provided by means
of appropriately transformed Moran-type particle processes, where the particles
are samples from a normalized inverse-Gaussian random probability measure. The
relationship between the limit diffusion and the two-parameter model is also
discussed.
