  In large asexual populations, beneficial mutations have to compete with each
other for fixation. Here, I derive explicit analytic expressions for the rate
of substitution and the mean beneficial effect of fixed mutations, under the
assumptions that the population size N is large, that the mean effect of new
beneficial mutations is smaller than the mean effect of new deleterious
mutations, and that new beneficial mutations are exponentially distributed. As
N increases, the rate of substitution approaches a constant, which is equal to
the mean effect of new beneficial mutations. The mean effect of fixed mutations
continues to grow logarithmically with N. The speed of adaptation, measured as
the change of log fitness over time, also grows logarithmically with N for
moderately large N, and it grows double-logarithmically for extremely large N.
Moreover, I derive a simple formula that determines whether at given N
beneficial mutations are expected to compete with each other or go to fixation
independently. Finally, I verify all results with numerical simulations.
