Population size effects in evolutionary dynamics on neutral networks and toy landscapes

We study the dynamics of a population subject to selective pressures, evolving either on RNA neutral networks or in toy fitness landscapes. We discuss the spread and the neutrality of the population in the steady state. Different limits arise depending on whether selection or random drift are dominant. In the presence of strong drift we show that observables depend mainly on $M \mu$, $M$ being the population size and $\mu$ the mutation rate, while corrections to this scaling go as 1/M: such corrections can be quite large in the presence of selection if there are barriers in the fitness landscape. Also we find that the convergence to the large $M \mu$ limit is linear in $1/M \mu$. Finally we introduce a protocol that minimizes drift; then observables scale like 1/M rather than $1/(M\mu)$, allowing one to determine the large $M$ limit faster when $\mu$ is small; furthermore the genotypic diversity increases from $O(\ln M)$ to $O(M)$.