Adaptive punishment in public goods games

The Public Goods Game (PGG) serves as the foundational paradigm for modeling collective action, yet existing evolutionary models overwhelmingly treat punishment as a static parameter or rely on discrete algorithmic updates. This structural rigidity severs the continuous macroscopic feedback loops inherent to real-world adaptive systems. In this work, we elevate adaptive punishment in PGGs to a rigorous nonlinear dynamical framework by formulating the collective punitive intensity (β) as a continuously coevolving macroscopic state variable within a coupled replicator system. Driven by the environmental synergy factor (R) and modulated by redistributive coupling (η) and self-regulating elasticity (θ), this adaptive mechanism continuously reshapes the effective payoff landscape. Our exact global bifurcation analysis reveals that highly efficient adaptive punishment acts as a non-conservative restoring force. Crucially, this mechanism successfully sustains a stable interior focus of cooperation—even in severely resource-deprived environments (synergy factor R<1) where static PGG models inevitably collapse to a global boundary attractor of absolute defection. Furthermore, we mathematically demonstrate that the elasticity of adaptive punishment strictly governs the system’s topological phase transitions, giving rise to non-dissipative neutral centers, macroscopic heteroclinic cycles, and a Bogdanov–Takens codimension-2 bifurcation hub. Ultimately, these findings provide a purely analytical perspective on how continuous adaptive punishment stabilizes metastable order and drives complex oscillatory dynamics in coevolutionary PGGs, advancing the theoretical taxonomy beyond discrete empirical approximations.