Evolutionary dynamics in spatial public goods games with environmental feedbacks

Collective actions aimed at achieving goals such as resource sustainability and environmental protection often face conflicting interests between individuals and groups. These social dilemmas can be modeled using public goods games and collective risk dilemmas. However, in reality, multiple generations share a common pool of resources, leading to high costs for today's overexploitation that impacts future generations' welfare. This delayed effect creates a multigenerational conflict. To address this, we develop a coupled social-ecological coevolutionary model by establishing a relationship between individual payoffs and regional environmental quality. Our goal is to study how cooperative behaviors spread in a public goods game with environmental feedback and to identify the factors influencing this spread. We achieve this by examining the mechanisms behind certain phases and phase transitions, monitoring the spatial distribution of strategies, and assessing the environmental quality of all regions. Our findings reveal some counterintuitive results. For instance, despite cooperators' ability to enhance the environment, the overall level of cooperation in the system sometimes decreases. This is linked to cooperative clusters being invaded by defectors within the clusters' cracks. Additionally, the destructive power of defection and the cost of cooperation have more complex effects on the system.