  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 $O(\log
N/(\log\log N)^2)$.
