Lower Bound on the Rate of Adaptation in an Asexual Population

TL;DR

This paper establishes a lower bound on the maximum fitness increase in an asexual population model, showing it grows at least as fast as a specific logarithmic rate with high probability as population size increases.

Contribution

It proves a lower bound on the adaptation rate in an asexual population, confirming the conjectured order of fitness increase over time.

Findings

Maximum fitness exceeds c s log N / (log log N)^2 with high probability

The lower bound holds uniformly over a time interval [ε_N, t]

The result supports the conjectured rate of adaptation in the model.

Abstract

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.