Replicator dynamics of division of labor games with delayed payoffs in infinite populations

Abstract

In recent years, the division of labor game has received widespread attention, however, the time delay in payoffs has been ignored. Hence, in this paper, we consider that there is a time delay for the player to gain the payoffs in the game, and the delay in payoffs leads to a Hopf bifurcation. When the time delay is greater than the critical time delay, the replicator dynamics oscillates near the equilibrium point, the critical time delay is determined by the cost difference between the two tasks and the cooperation benefits between the two players. We obtain the expression of the critical time delay by applying the characteristic equation analysis method, and clearly analyze various cases through observation and numerical simulation. With the increase of time delays, the dynamic system shows a transition from asymptotic stability to oscillation near equilibrium.