Reciprocity-induced proliferation: Survival resonance and equilibrium beyond the tragedy of the commons.

We investigate the mechanisms sustaining population survival beyond the tragedy of the commons (TOC) in finite spatial populations with dynamic environmental feedback. Environmental collapse and population contraction arise when the social dilemma intensifies or environmental carrying capacity declines, driving the system toward the TOC state. Near the boundary separating the collapse regime from a survival-favorable equilibrium, we uncover a survival resonance response to mutation noise. Both the equilibrium state and the resonance, driven by reciprocity of tightly knit cooperator agglomerates, enhance population persistence and environmental quality. Furthermore, our pair-approximation dynamics successfully explains the dynamic transition from the survival-favorable equilibrium state to the TOC state and offers insights into resonance mechanisms. Generally, these results reveal how collective dynamical responses enable cooperative populations to persist under common-resource constraints.