  We study the $(1,\lambda)$-EA with mutation rate $c/n$ for $c\le 1$, where
the population size is adaptively controlled with the $(1:s+1)$-success rule.
Recently, Hevia Fajardo and Sudholt have shown that this setup with $c=1$ is
efficient on \onemax for $s<1$, but inefficient if $s \ge 18$. Surprisingly,
the hardest part is not close to the optimum, but rather at linear distance. We
show that this behavior is not specific to \onemax. If $s$ is small, then the
algorithm is efficient on all monotone functions, and if $s$ is large, then it
needs superpolynomial time on all monotone functions. In the former case, for
$c<1$ we show a $O(n)$ upper bound for the number of generations and $O(n\log
n)$ for the number of function evaluations, and for $c=1$ we show $O(n\log n)$
generations and $O(n^2\log\log n)$ evaluations. We also show formally that
optimization is always fast, regardless of $s$, if the algorithm starts in
proximity of the optimum. All results also hold in a dynamic environment where
the fitness function changes in each generation.
