The mechanics of stochastic slowdown in evolutionary games

Abstract

We study the stochastic dynamics of evolutionary games, and focus on the so-called 'stochastic slowdown ' effect, previously observed in (Altrock et al., 2010) for simple evolutionary dynamics. Slowdown here refers to the fact that a beneficial mutation may take longer to fixate than a neutral one. More precisely, the fixation time conditioned on the mutant taking over can show a maximum at intermediate selection strength. We show that this phenomenon is present in the Prisoner's Dilemma, and also discuss counterintuitive slowdown and speedup in coexistence games. In order to establish the microscopic origins of these phenomena, we calculate the average sojourn times. This allows us to identify the transient states which contribute most to the slowdown effect, and enables us to provide an understanding of slowdown in the takeover of a small group of cooperators by defectors in the Prisoner's Dilemma: Defection spreads fast initially, but the final steps to takeover can be delayed substantially. The analysis of coexistence games reveals even more intricate non-monotonic behavior. In small populations, the conditional average fixation time can show multiple extrema as a function of the selection strength, e.g., slowdown, speedup, and slowdown again. We classify generic 2 × 2 games with respect to the possibility to observe non-monotonic behavior of the conditional average fixation time as a function of selection strength.

Selective Strolls: Fixation and Extinction in Diploids Are Slower for Weakly Selected Mutations Than for Neutral Ones

“…We model a birth-death Moran process as in Altrock et al 2012, for which fixation and extinction times can be calculated analytically. We consider a biallelic haploid well-mixed population of size 2N: We denote the number of A alleles as i and B alleles as 2N 2 i: To mimic the fitness structure of a diploid population we consider the birth and death probability of birth of type A respectively equal to…”

“…Altrock et al (2012) pointed out that in haploids the sojourn times, the amount of time spent at the different population frequencies, increase at higher frequencies. Here we provide a simple explanation for the stochastic slowdown in diploids, by looking at the conditional transition probabilities.…”

“…Kimura and Ohta (1969a,b) and Ewens (2004) applied diffusion theory to model finite populations, obtaining keystone approximations for the neutral case, in the absence of selection, or when selection is fairly strong. Recently it has been pointed out that in haploid models, in the presence of frequency-dependent fitness, the time to fixation of a positively selected allele can increase with the strength of selection (Altrock et al 2010(Altrock et al , 2012. In diploids, a newly arising mutation is usually expected to have a time to fixation longer than that of a neutral one in the case of overdominance, when the heterozygote has an higher fitness than the two homozygotes-a case usually referred to as heterozygote advantage or more generally balancing selection (Key et al 2014).…”

The effect of population structure on the rate of evolution

“…before it reaches a boundary or the moments of the distribution of fixation times [2,27,33,34]. We also consider the probability that there are ever K mutants in the system, f K (1).…”

“…As this is straightforward to compute but somewhat technical, we refer to the literature for the concrete equations [27,33,34].…”