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A critical branching process model for biodiversity

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

David Aldous  and
Lea Popovic
David Aldous*
Affiliation:
University of California, Berkeley
Lea Popovic*
Affiliation:
University of Minnesota
*
∗ Postal address: Department of Statistics, University of California, 367 Evans Hall #3860, Berkeley, CA 94720-3860, USA. Email address: [email protected]
∗∗ Current address: Department of Mathematics, 310 Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA.
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Abstract

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We study the following model for a phylogenetic tree on n extant species: the origin of the clade is a random time in the past whose (improper) distribution is uniform on (0,∞); thereafter, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on there being the prescribed number n of species at the present time. We study various mathematical properties of this model as n→∞: namely the time of origin and of the most recent common ancestor, the pattern of divergence times within lineage trees, the time series of the number of species, the total number of extinct species, the total number of species ancestral to the extant ones, and the ‘local’ structure of the tree itself. We emphasize several mathematical techniques: the association of walks with trees; a point process representation of lineage trees; and Brownian limits.

Keywords

BiodiversityBrownian excursioncontour processcritical branching processgenealogylocal weak convergencephylogenetic treepoint process

MSC classification

Primary: 60J85: Applications of branching processes
Secondary: 60J65: Brownian motion 92D15: Problems related to evolution
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 37 , Issue 4 , December 2005 , pp. 1094 - 1115
Copyright
Copyright © Applied Probability Trust 2005 

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