Hopf Bifurcations in Replicator Dynamics with Distributed Delays

In this paper, we study the existence and the property of the Hopf bifurcation in the two-strategy replicator dynamics with distributed delays. In evolutionary games, we assume that a strategy would take an uncertain time delay to have a consequence on the fitness (or utility) of the players. As the mean delay increases, a change in the stability of the equilibrium (Hopf bifurcation) may occur at which a periodic oscillation appears. We consider Dirac, uniform, Gamma, and discrete delay distributions, and we use the Poincar\'e- Lindstedt's perturbation method to analyze the Hopf bifurcation. Our theoretical results are corroborated with numerical simulations

Global Convergence for Replicator Dynamics of Repeated Snowdrift Games

To understand the emergence and sustainment of cooperative behavior in interacting collectives, we perform global convergence analysis for replicator dynamics of a large, well-mixed population of individuals playing a repeated snowdrift game with four typical strategies, which are always cooperate (ALLC), tit-for-tat (TFT), suspicious tit-for-tat (STFT) and always defect (ALLD). The dynamical model is a three-dimensional ODE system that is parameterized by the payoffs of the base game. Instead of routine searches for evolutionarily stable strategies and sets, we expand our analysis to determining the asymptotic behavior of solution trajectories starting from any initial state, and in particular show that for the full range of payoffs, every trajectory of the system converges to an equilibrium point. The convergence results highlight three findings that are of particular importance for understanding the cooperation mechanisms among self-interested agents playing repeated snowdrift games. First, the inclusion of TFT- and STFT-players, the two types of conditional strategy players in the game, increases the share of cooperators of the overall population compared to the situation when the population consists of only ALLC- and ALLD-players. This confirms findings in biology and sociology that reciprocity may promote cooperation in social collective actions, such as reducing traffic jams and division of labor, where each individual may gain more to play the opposite of what her opponent chooses. Second, surprisingly enough, regardless of the payoffs, there always exists a set of initial conditions under which ALLC players do not vanish in the long run, which does not hold for all the other three types of players. So an ALLC-player, although perceived as the one that can be easily taken advantage of in snowdrift games, has certain endurance in the long run