Multistability and bifurcation analysis for a three-strategy game system with public goods feedback and discrete delays

The interaction between evolution and public goods exists in social and biological systems. Exploring mathematical models that combine evolution and public goods facilitates insights and discoveries about the evolution of cooperation. We propose a third strategy called the encouragement strategy, which can reward and punish, and bear the related costs, in addition to the classical cooperation and defection strategies. We incorporate strategy and public goods dynamics to formulate a high-dimensional and high-order three-strategy evolutionary game system with public goods feedback. For this non-delayed system, we evaluate the existence of equilibria and the conditions for the occurrence of transcritical and Hopf bifurcations. Interestingly, the non-delayed system exhibits multiple multistability due to different payoffs and initial conditions. The results suggest that the environment with rich public goods is suitable for cooperators to survive. Surprisingly, raising reward does not always promote the survival of cooperators. Cooperators prevail in environments with moderate reward and severe punishment. Further, we develop a delayed system by considering information delay and production delay. For the delayed system, we investigate the properties of Hopf bifurcation, including existence, direction and stability. We find that small delays do not affect the evolutionary outcome of the strategies. However, delays larger than the critical value lead to periodic oscillations of the strategies, in which case the cooperators still have a dominant advantage. Unfortunately, larger delays can cause the disappearance of cooperators. Theoretical and numerical results show that public goods feedback, reward, punishment, payoffs, initial conditions, and delays are all significant factors in determining the outcome of strategic evolution.