The convergence to equilibrium of neutral genetic models

2007/01/10 by Pierre Del Moral, Del Moral, Pierre, Laurent Miclo +5
Biochemistry, Genetics and Molecular Biology · Mathematics · #60F17 #60J80 #65C35 #92D10 #92D15. #Evolution and Genetic Dynamics #FOS: Mathematics #Gene Regulatory Network Analysis #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.math/0701284

openalex publication_date 2007/01/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This article is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter.

Citations

• The Neutral Theory of Molecular Evolution

Related

• Birth and death processes with neutral mutations
• Splitting trees with neutral mutations at birth
• A limit theorem for trees of alleles in branching processes with rare neutral mutations
• Splitting trees with neutral Poissonian mutations I: Small families
• Exchangeable coalescents beyond the Cannings class