  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.
