## Advances in Applied Probability

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

### A critical branching process model for biodiversity

#### 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.