The Convergence to Equilibrium of Neutral Genetic Models

• Published in: Stochastic analysis and applications Vol. 28; no. 1; pp. 123 - 143
• Main Authors: Moral, P. Del, Miclo, L., Patras, F., Rubenthaler, S.
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Taylor & Francis Group 01-01-2010; Taylor & Francis; Taylor & Francis: STM, Behavioural Science and Public Health Titles
• Subjects: Coalescent trees; Convergence; Decay; Design engineering; Exact sciences and technology; Experimental design; Genetics; Lyapunov exponent; Markov processes; Mathematical analysis; Mathematics; Mutations; Neutral genetic models; Norms; Probability; Probability and statistics; Probability theory and stochastic processes; Representations; Sciences and techniques of general use; Stationary distribution; Statistics; Stochastic analysis; Stochastic processes; Wright-Fisher model; Stochastic analysis; Experimental design; Lyapunov exponent; Distribution function; Population size; Finite population; Application; Population genetics; Equilibrium model; Numerical stability; Convergence; neutral genetic models; stationary distribution; Wright-Fisher model; coalescent trees
• ISSN: 0736-2994; 1532-9356
• DOI: 10.1080/07362990903415833

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 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.