2004/10/18 by David Aldous, David J. Aldous, Aldous, David J. +2 · 1 citation
Biochemistry, Genetics and Molecular Biology · Environmental Science · Mathematics · Social Sciences · #60J85 #92D15 #Ecosystem dynamics and resilience #Evolution and Genetic Dynamics #Evolutionary Game Theory and Cooperation #FOS: Biological sciences #FOS: Mathematics #Populations and Evolution (q-bio.PE) #Probability (math.PR) #math.PR #msc:60J85 #msc:92D15 #q-bio.PE
paper · pdf · doi:10.48550/arxiv.math/0410402
31 pages, 7 figures
arxiv created 2004/10/18 · openalex publication_date 2004/10/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.