Intrinsic noise in structured replicator dynamics modelling time delays

TL;DR

This paper models structured replicator dynamics with time delays in the Snowdrift game, revealing that intrinsic stochastic fluctuations and delays can promote cooperation.

Contribution

It introduces a novel structured replicator model incorporating strategy-dependent time delays and analyzes their impact on cooperation.

Findings

Time delays increase cooperation levels in stochastic dynamics.

Stochastic fluctuations favor cooperation over deterministic predictions.

Delayed dynamics differ significantly from delay-free models in population behavior.

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.