Intrinsic noise in structured replicator dynamics modelling time delays

Abstract

We construct and analyze structured replicator dynamics of the Snowdrift game. In our model, the offspring is put in juvenile compartments and then mature and join adult compartments with strategy-dependent rates. This is augmented by death rates and hence the population size is bounded. In the corresponding birth-death Markov jump process, rates of leaving juvenile compartments may be interpreted as inverses of averages of exponentially distributed time delays. We observe a novel behavior: for equal average time delays of both strategies, the frequency of cooperators in the quasi-stationary state of a stochastic dynamics is bigger than that in the corresponding stationary state of the deterministic structured replicator dynamics which is actually equal to the critical point of the original replicator equation for the Snowdrift game. In short, time delays favor cooperation in the presence of intrinsic stochastic fluctuations.