Replicator dynamics of evolutionary games with different delays on costs and benefits

The replicator dynamics of two-player evolutionary games with delays are examined. In contrast to situations with a common delay, we examine the resulting stability for the situation where the delays on the cost contributions to the payoff may be different from the delays on the benefit contributions to the payoff. In such scenarios we show that increasing one delay, while holding the other constant, can lead to conditions where the increasing delay causes instability, yet further increases in the delay can bring back stability, while still further increases in the delay can lead to instability again, and so on. This is in contrast to the case where costs and benefits share a common delay, in which case we show that only a simple destabilization can occur with an increasing delay. We develop a Hopf bifurcation analysis of this phenomenon for a general version of evolutionary games that have a stable mixed strategy equilibrium in the nominal (no delay) case. From the Hopf bifurcation analysis we show conditions under which the stabilizing and destabilizing phenomena occur. We then illustrate the results by applying them to specific examples of both a snowdrift game and a division of labor game. Numerical examples are presented to validate the analysis claims.