  We consider a model of asexually reproducing individuals with random
mutations and selection. The rate of mutations 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 conjectured that the
average rate at which the mean fitness increases in this model is $O(\log
N/(\log\log N)^2)$. In this paper we show that for any time $t > 0$ there exist
values $\epsilon_N \rightarrow 0$ and a fixed $c > 0$ such that the maximum
fitness of the population is greater than $cs\log N/(\log\log N)^2$ for all
times $s \in [\epsilon_N,t]$ with probability tending to 1 as $N$ tends to
infinity.

Probability;Populations and Evolution