Absorption and fixation times for evolutionary processes on graphs

A bstract In this paper, we study the absorption and fixation times for evolutionary processes on graphs, under different updating rules. While in Moran process a single neighbour is randomly chosen to be replaced, in proliferation processes other neighbours can be replaced using Bernoulli or binomial draws depending on 0 < p ≤ 1. There is a critical value p c such that the proliferation is advantageous or disadvantageous in terms of fixation probability depending on whether p > p c or p < p c . We clarify the role of symmetries for computing the fixation time in Moran process. We show that the Maruyama-Kimura symmetry depend on the graph structure induced in each state, implying asymmetry for all graphs except cliques and cycles. There is a fitness value, not necessarily 1, beyond which the fixation time decreases monotonically. We apply Harris’ graphical method to prove that the fixation time decreases monotonically depending on p . Thus there exists another value p t for which the proliferation is advantageous or disadvantageous in terms of time. However, at the critical level p = p c , the proliferation is highly advantageous when r → +∞.