Imitation dynamics with time delay

Based on the classic imitation dynamics (Hofbauer and Sigmund, 1998, Evolutionary Games and Population Dynamics, Cambridge University Press), the imitation dynamics with time delay is investigated, where the probability that an individual will imitate its opponent's own strategy is assumed to depend on the comparison between the past expected payoff of this individual's own strategy and the past expected payoff of its opponent's own strategy, i.e. there is a time delay effect. For the two-phenotype model, we show that if the system has an interior equilibrium and this interior equilibrium is stable when there is no time delay, then there must be a critical value of time delay such that the system tends to a stable periodic solution when the time delay is larger than the critical value. On the other hand, for three-phenotype (rock-scissors-paper) model, the numerical analysis shows that for the stable periodic solution induced by the time delay, the amplitude and the period will increase with the increase of the time delay. These results should help to understand the evolution of behavior based on the imitation dynamics with time delay.