Rate of adaptation of large populations

We consider the accumulation of beneficial and deleterious mutations in asexual populations. The rate of adaptation is affected by the total mutation rate, proportion of beneficial mutations, and population size N. We describe two models: a strong selection model for large population size and a weak selection model for moderate population size. For the strong selection model, we give an argument to show that regardless of mutation rates, as long as the proportion of beneficial mutations is strictly positive, the adaptation rate is at least O(log1−δ N), if the population size is sufficiently large. For the weak selection model, we use a Girsanov transform to perform some preliminary calculations that shed new light on the adaptation rate.