Journal of Applied Probability

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

Matteo Ruggiero

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Abstract

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.

Article information

Source
J. Appl. Probab., Volume 51, Number 1 (2014), 174-190.

Dates
First available in Project Euclid: 25 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1395771422

Digital Object Identifier
doi:10.1239/jap/1395771422

Mathematical Reviews number (MathSciNet)
MR3189450

Zentralblatt MATH identifier
1291.60162

Subjects
Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60G57: Random measures 92D25: Population dynamics (general)

Keywords
Alpha diversity infinite-alleles model infinite-dimensional diffusion mutation rate Poisson-Dirichlet distribution weak convergence

Citation

Ruggiero, Matteo. Species dynamics in the two-parameter Poisson-Dirichlet diffusion model. J. Appl. Probab. 51 (2014), no. 1, 174--190. doi:10.1239/jap/1395771422. https://projecteuclid.org/euclid.jap/1395771422


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