Upper bound on the rate of adaptation in an asexual population

We consider a model of asexually reproducing individuals. The birth and death rates of the individuals are affected by a fitness parameter. The rate of mutations that cause the fitnesses to change is proportional to the population size, N. The mutations may be either beneficial or deleterious. In a paper by Yu, Etheridge and Cuthbertson [Ann. Appl. Probab. 20 (2010) 978-1004] it was shown that the average rate at which the mean fitness increases in this model is bounded below by ${\log }^{1-\delta }N$ for any $\delta >0$. We achieve an upper bound on the average rate at which the mean fitness increases of $O(\log N/(\log\log N)^2)$.