Asexual Evolution Waves: Fluctuations and Universality
  In large asexual populations, multiple beneficial mutations arise in the population, compete, interfere with each other, and accumulate on the same genome, before any of them fix. The resulting dynamics, although studied by many authors, is still not fully understood, fundamentally because the effects of fluctuations due to the small numbers of the fittest individuals are large even in enormous populations. In this paper, branching processes and various asymptotic methods for analyzing the stochastic dynamics are further developed and used to obtain information on fluctuations, time dependence, and the distributions of sizes of subpopulations, jumps in the mean fitness, and other properties. The focus is on the behavior of a broad class of models: those with a distribution of selective advantages of available beneficial mutations that falls off more rapidly than exponentially. For such distributions, many aspects of the dynamics are universal - quantitatively so for extremely large populations. On the most important time scale that controls coalescent properties and fluctuations of the speed, the dynamics is reduced to a simple stochastic model that couples the peak and the high-fitness "nose" of the fitness distribution. Extensions to other models and distributions of available mutations are discussed briefly. 
Daniel S. Fisher
q-bio.PE