Gamma-distributed delays in the prisoner’s dilemma with defection penalty: Stability hierarchy of memory kernels

A penalty on mutual defection creates an interior coexistence equilibrium in the prisoner's dilemma, but payoff information received with delay can destabilise that equilibrium. We examine how this outcome depends on the temporal distribution of memory. Replacing the discrete delay of Wang and Meng [1] with gamma-distributed payoff feedback, we use the linear chain trick to obtain finite-dimensional ODE representations. Exponentially fading (weak) memory makes the coexistence equilibrium locally asymptotically stable for every positive mean delay. By contrast, a peaked (strong) gamma kernel undergoes a supercritical Hopf bifurcation, established by an explicit first-Lyapunov-coefficient calculation, at a critical mean delay 8/π times larger than for the discrete-delay benchmark. Thus, memory dispersion can delay, and in the weak-kernel case eliminate, delay-induced oscillations. A distributed-delay replicator–mutator extension, spectral-abscissa calculations, two-parameter stability maps, phase portraits, and bifurcation diagrams support the resulting stability hierarchy.