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Species Dynamics in the Two-Parameter Poisson-Dirichlet Diffusion Model

Part of: Markov processes Stochastic processes

Published online by Cambridge University Press:  30 January 2018

Matteo Ruggiero
Matteo Ruggiero*
Affiliation:
University of Torino and Collegio Carlo Alberto
*
∗ Postal address: Department of Economics and Statistics, University of Torino, Corso Unione Sovietica 218/bis, 10134 Torino, Italy, Email address: [email protected]
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Abstract

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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.

Keywords

Alpha diversityinfinite-alleles modelinfinite-dimensional diffusionmutation ratePoisson-Dirichlet distributionweak convergence

MSC classification

Primary: 60J60: Diffusion processes
Secondary: 60G57: Random measures 92D25: Population dynamics (general)
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 1 , March 2014 , pp. 174 - 190
Copyright
© Applied Probability Trust 

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