  Game theory deals with strategic interactions among players and evolutionary
game dynamics tracks the fate of the players' populations under selection. In
this paper, we consider the replicator equation for two-player-two-strategy
games involving cooperators and defectors. We modify the equation to include
the effect of mutation and also delay that corresponds either to the delayed
information about the population state or in realizing the effect of
interaction among players. By focusing on the four exhaustive classes of
symmetrical games -- the Stag Hunt game, the Snowdrift game, the Prisoners'
Dilemma game, and the Harmony game -- we analytically and numerically analyze
delayed replicator-mutator equation to find the explicit condition for the Hopf
bifurcation bringing forth stable limit cycle. The existence of the
asymptotically stable limit cycle imply the coexistence of the cooperators and
the defectors; and in the games, where defection is a stable Nash strategy, a
stable limit cycle does provide a mechanism for evolution of cooperation. We
find that while mutation alone can never lead to oscillatory cooperation state
in two-player-two-strategy games, the delay can change the scenario. On the
other hand, there are situations when delay alone cannot lead to the Hopf
bifurcation in the absence of mutation in the selection dynamics.
