Mutation in populations governed by a Galton-Watson branching process

Population genetics models are usually based on the Wright-Fisher model, in which each individual in a population randomly chooses their parent from the previous generation.  One could argue that a more realistic description of population dynamics is a Galton-Watson branching process, in which each individual in a population produces a random number of offspring to create the next generation.  

In this talk I will describe a multi-type branching process, in which the population is divided into a finite number of allele types.  At each generation individuals are able to mutate to a different type.  By considering the the diffusion limit forward Kolmogorov equation for the case of neutral mutations we find that the population asymptotically partitions into subpopulations whose relative sizes are determined by mutation rates. An approximate time-dependent solution is obtained in the limit of low mutation rates. This solution has the property that the system undergoes a rapid transition from a perturbation of the model with zero mutation rates to a phase in which the distribution collapses onto the asymptotic stationary distribution. The changeover point of the transition is determined by the per-generation growth factor and mutation rate.