Advances in Applied Probability

A critical branching process model for biodiversity

David Aldous and Lea Popovic

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Abstract

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.

Article information

Source
Adv. in Appl. Probab. Volume 37, Number 4 (2005), 1094-1115.

Dates
First available in Project Euclid: 14 December 2005

Permanent link to this document
http://projecteuclid.org/euclid.aap/1134587755

Digital Object Identifier
doi:10.1239/aap/1134587755

Mathematical Reviews number (MathSciNet)
MR2193998

Zentralblatt MATH identifier
05033681

Subjects
Primary: 60J85: Applications of branching processes [See also 92Dxx]
Secondary: 60J65: Brownian motion [See also 58J65] 92D15: Problems related to evolution

Keywords
Biodiversity Brownian excursion contour process critical branching process genealogy local weak convergence phylogenetic tree point process

Citation

Aldous, David; Popovic, Lea. A critical branching process model for biodiversity. Adv. in Appl. Probab. 37 (2005), no. 4, 1094--1115. doi:10.1239/aap/1134587755. http://projecteuclid.org/euclid.aap/1134587755.


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