Eco-evolutionary games with zealots.

Abstract

Eco-evolutionary games describe the mutual feedback between strategic actions and environmental conditions. Earlier studies have demonstrated that this interplay can lead to a range of dynamic behaviors, including persistent oscillations and heteroclinic cycles, the latter representing an oscillating tragedy of commons. We extend the canonical replicator-environment coupled model by incorporating a fixed fraction of zealous cooperators who adhere to cooperation independently of payoffs. The resulting system admits four equilibrium states for the coupled strategy-environment dynamics. We derive stability conditions for each equilibrium and show analytically that the presence of zealots suppresses oscillatory behavior. Increasing the density of zealots induces a sequence of bifurcations leading from an interior coexistence state to partial cooperation in a replete environment and ultimately to full cooperation with a rich environment. Numerical experiments further demonstrate that stronger environmental enhancement amplifies the stabilizing influence of zealots and enlarges the parameter regime supporting replete environmental states. Our results reveal that committed cooperators act as a stabilizing mechanism capable of eliminating the oscillating tragedy of commons in eco-evolutionary dynamics and sustaining long-term cooperative and environmentally favorable outcomes.