Branching with Selection and Mutation I: Mutant Fitness of Fréchet Type

Abstract

We investigate two stochastic models of a growing population with discrete and non-overlapping generations, subject to selection and mutation. In our models each individual carries a fitness which determines its mean offspring number. Many of these offspring inherit their parent’s fitness, but some are mutants and obtain a fitness randomly sampled, as in Kingman’s house-of-cards model, from a distribution in the domain of attraction of the Fréchet distribution. We give a rigorous proof for the precise rate of superexponential growth of these stochastic processes and support the argument by a heuristic and numerical study of the mechanism underlying this growth. This study yields in particular that the empirical fitness distribution of one model in the long time limit displays periodic behaviour.