Asymptotic behavior of mean fixation times in the Moran process with frequency-independent fitnesses

• Published in: Journal of mathematical biology 2024-01, Vol.88 (1), p.12-12, Article 12
• Main Authors: Pires, Rosângela A., Neves, Armando G. M.
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Asymptotic properties; Fixation; Gene Frequency; Humans; Hypotheses; Infinity; Mathematical and Computational Biology; Mathematics; Mathematics and Statistics; Population Density; Population number; Probability
• ISSN: 0303-6812; 1432-1416
• DOI: 10.1007/s00285-023-02032-1

We derive asymptotic formulae in the limit when population size N tends to infinity for mean fixation times (conditional and unconditional) in a population with two types of individuals, A and B, governed by the Moran process. We consider only the case in which the fitness of the two types do not depend on the population frequencies. Our results start with the important cases in which the initial condition is a single individual of any type, but we also consider the initial condition of a fraction x ∈ ( 0 , 1 ) of A individuals, where x is kept fixed and the total population size tends to infinity. In the cases covered by Antal and Scheuring (Bull Math Biol 68(8):1923–1944, 2006), i.e. conditional fixation times for a single individual of any type, it will turn out that our formulae are much more accurate than the ones they found. As quoted, our results include other situations not treated by them. An interesting and counterintuitive consequence of our results on mean conditional fixation times is the following. Suppose that a population consists initially of fitter individuals at fraction x and less fit individuals at a fraction 1 - x . If population size N is large enough, then in the average the fixation of the less fit individuals is faster (provided it occurs) than fixation of the fitter individuals, even if x is close to 1, i.e. fitter individuals are the majority.