A critical branching process model for biodiversity

Abstract

Motivated as a null model for comparison with data, 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, ∞). After that origin, the process of extinctions and speciations is a continuous-time critical branching process of constant rate, conditioned on having the prescribed number n of species at the present time. We study various mathematical properties of this model as n → ∞ limits: time of origin and of most recent common ancestor; pattern of divergence times within lineage trees; time series of numbers of species; number of extinct species in total, or ancestral to extant species; and "local" structure of the tree itself. We emphasize several mathematical techniques: associating walks with trees, a point process representation of lineage trees, and Brownian limits. * MSC 2000 subject classification. Primary: 60J85; Secondary: 60J65,92D15 Standard modelsOurs is, roughly speaking, the third simplest model one might devise, so let us first recall the two simpler models.The Yule model. Yule [24] proposed the basic model for speciations without extinctions. Initially there is one species. Thereafter, independently for each existing species, new species originate as "daughter" species at constant rate λ (i.e. at the times of a Poisson (rate λ) process). So for given n one can get a model for an n-species tree by taking the present as a random time at which the number of species equals n. (The associated continuous-time Markov chain counting number of species is often called the Yule process, though its origin as a model for species is often forgotten.)

Sampling-through-time in birth–death trees

“…Using a uniform (improper) prior on ð0,1Þ for the time of origin, we show how to condition on other characteristics of the tree. This prior is explicitly used when considering the birth-death process conditioned on n extant sampled species without sampling in the past (Aldous and Popovic, 2005;Stadler, 2009). These studies show that the uniform prior actually yields a posterior density on n-species trees.…”

“…We need to assume some distribution for the time of origin when conditioning on a fixed number of sampled extant individuals. We assume a uniform (improper) prior on ð0,1Þ as has been done in Popovic (2004), Aldous and Popovic (2005), and Gernhard (2008). The time of origin of the tree corresponds to the origin of the epidemic or the origin of the first species in the considered clade.…”

Recovering speciation and extinction dynamics based on phylogenies

“…Lambert (2010) introduced a mathematical framework calculating the likelihood for a reconstructed phylogenetic tree, assuming any species lifetime distribution and assuming a constant rate of asymmetric speciation. It turns out that the reconstructed trees induced by this model can be interpreted as a so-called point process (Popovic, 2004;Aldous & Popovic, 2005;Gernhard, 2008), resulting in particular in the uniform distribution on ranked tree shapes with n tips.…”

Coalescences in continuous-state branching processes

“…A third approach is to represent the backwards genealogy through point processes. Aldous and Popovic [AP05] and Popovic [Pop04] have shown how to encode the genealogy of a critical Feller diffusion with a Poisson point process on R + ×R + called Coalescent Point Process. Atoms of the coalescent point process represents times of coalescences between two "consecutive" individuals in the boundary of the Brownian tree.…”