Fractional coalescent

Significance The fractional coalescent is a generalization of Kingman’s n -coalescent. It facilitates the development of the theory of population genetic processes that deviate from Poisson-distributed waiting times. It also marks the use of methods developed in fractional calculus in population genetics. The fractional coalescent is an extension of Canning’s model, where the variance of the number of offspring per parent is a random variable. The distribution of the number of offspring depends on a parameter α , which is a potential measure of the environmental heterogeneity that is commonly ignored in current inferences. An approach to the coalescent, the fractional coalescent ( f -coalescent), is introduced. The derivation is based on the discrete-time Cannings population model in which the variance of the number of offspring depends on the parameter α . This additional parameter α affects the variability of the patterns of the waiting times; values of α < 1 lead to an increase of short time intervals, but occasionally allow for very long time intervals. When α = 1 , the f -coalescent and the Kingman’s n -coalescent are equivalent. The distribution of the time to the most recent common ancestor and the probability that n genes descend from m ancestral genes in a time interval of length T for the f -coalescent are derived. The f -coalescent has been implemented in the population genetic model inference software M igrate . Simulation studies suggest that it is possible to accurately estimate α values from data that were generated with known α values and that the f -coalescent can detect potential environmental heterogeneity within a population. Bayes factor comparisons of simulated data with α < 1 and real data (H1N1 influenza and malaria parasites) showed an improved model fit of the f -coalescent over the n -coalescent. The development of the f -coalescent and its inclusion into the inference program M igrate facilitates testing for deviations from the n -coalescent.