Stability, bifurcation and chaos in evolutionary dynamics of a spatiotemporal Prisoner’s Dilemma Game with payoff delay

The selection of cooperation and defection strategies among populations in social and biological systems is often influenced by their information delay and spatial location. However, classical evolutionary game studies often overlook the influence of these factors. The main objective of this study is to incorporate both time delay and spatial diffusion into the three-strategy Prisoner’s Dilemma (PD) model, which introduces a novel strategy with dual functions of punishing defection and promoting cooperation, and combines the population density dynamics with the replicator equation. Coupling the technical methods of time-delay differential equations and partial differential equations with evolutionary game dynamics theory, the temporal system undergoes transcritical and Hopf bifurcations, while the spatiotemporal system exhibits spatial stability, diffusion-induced Turing instability, and chaotic dynamics, illustrating how spatial heterogeneity disrupts the uniform steady state. Furthermore, by considering the payoff delay as a control parameter, this study analyzes the properties of the periodic solutions resulting from the Hopf bifurcation in the time system, and investigates the conditions for the Turing–Hopf bifurcation in the diffusion system. These results indicate that increasing benefits for cooperation and intensifying punishment can promote the sustainability and expansion of cooperation. Notably, an excessive time delay reverses the stable state where cooperation and defection coexist, and eventually leads to the extinction of the cooperative strategy. An interesting conclusion is that, within the current parameter regime, time delay without spatial diffusion only leads to a bifurcation, while diffusion induces chaos. This indicates that diffusion is the dominant mechanism that causes chaos in the system. This work is devoted to the theoretical study of dynamics in time-delayed partial differential game systems.