\indent 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 (2009) it was shown that the average rate at which the mean fitness increases in this model is bounded below by $\text{log}^{1-\delta}N$ for any $\delta > 0$. We achieve an upper bound on the average rate at which the mean fitness increases of $O(\text{log}\hspace{1pt}N/\text{log log} \hspace{1pt}N)$.