Absorption and fixation times for evolutionary processes on graphs

In this paper, we study the absorption and fixation times for evolutionary processes on graphs, with relative fitness $r>0$ , under different updating rules. While in the 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 $0<p\le 1$ . 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 the Moran process. We show that the Maruyama-Kimura symmetry depends 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's 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 $r$ tends to infinity.