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Feedback loops between strategies and the environment are commonly observed in socio-ecological, evolution-ecological, and psychology-economic systems. However, the impact of mutations in these feedback processes is often overlooked. This study proposes a novel model that integrates the public goods game with environmental feedback, considering the presence of mutations. In our model, the enhancement factor of the public goods game combines positive and negative incentives from the environment. By employing replicator–mutator (RM) equations, we provide an objective understanding of the system's evolutionary state, focusing on identifying conditions that foster cooperation and prevent the tragedy of the commons. Specifically, mutations play a crucial role in the RM dynamics, leading to the emergence of an oscillatory tragedy of the commons. By verifying the Hopf bifurcation condition, we establish the existence of a stable limit cycle, providing valuable insights into sustained oscillation strategies. Moreover, the feedback mechanism inherent in the public goods game model offers a fresh perspective on effectively addressing the classic dilemma of the tragedy of the commons. [ABSTRACT FROM AUTHOR]

Complex networks are widespread in real-world environments across diverse domains. Real-world networks tend to form spontaneously through interactions between individual agents. Inspired by this, we design an evolutionary game model in which agents participate in a prisoner's dilemma game (PDG) with their neighboring agents. Agents can autonomously modify their connections with neighbors using reinforcement learning to avoid unfavorable environments. Interestingly, our findings reveal some remarkable results. Exploiting reinforcement learning-based adaptive networks improves cooperation when juxtaposed with existing PDGs performed on homogeneous networks. At the same time, the network's topology evolves from homogeneous to heterogeneous states. This change occurs as players gain experience from past games and become more astute in deciding whether to join PDGs with their current neighbors or disconnect from the least profitable neighbors. Instead, they seek out more favorable environments by establishing connections with second-order neighbors with higher rewards. By calculating the degree distribution and modularity of the adaptive network in a steady state, we confirm that the adaptive network follows a power law and has a clear community structure, indicating that the adaptive network is similar to networks in the real world. Our study reports a new phenomenon in evolutionary game theory on networks. It proposes a new perspective to generate scale-free networks, which is generating scale-free networks by the evolution of homogeneous networks rather than typical ways of network growth and preferential connection. Our results provide new aspects to understanding the network structure, the emergence of cooperation, and the behavior of actors in nature and society. [ABSTRACT FROM AUTHOR]

We investigate evolving dynamics of cyclically competing species on spatially extended systems with considering a specific region, which is called the "wildlife refuge," one of the institutional ways to preserve species biodiversity. Through Monte-Carlo simulations, we found that the refuge can play not groundbreaking but an important role in species survival. Species coexistence is maintained at a moderate mobility regime, which traditionally leads to the collapse of coexistence, and eventually, the extinction is postponed depending on the competition rate rather than the portion of the refuge. Incorporating the extinction probability and Fourier transform supported our results in both stochastic and analogous ways. Our findings may provide valuable evidence to assist fields of ecological/biological sciences in understanding the presence and construction of refuges for wildlife with associated effects on species biodiversity. [ABSTRACT FROM AUTHOR]

While prosocial exclusion has been proposed as a mechanism to maintain cooperation in one-shot social dilemma games, the evolution of prosocial peer exclusion in response to the threat of antisocial peer exclusion, particularly in structured populations, remains insufficiently understood. In this study, we employ an extended spatial public goods game to investigate the evolution of prosocial peer exclusion and its impact on cooperation in the presence of both prosocial and antisocial peer exclusion. Our model encompasses four primary strategies: traditional cooperation and defection, prosocial peer exclusion targeting defectors, and antisocial peer exclusion targeting cooperators. Our findings illuminate that the presence of antisocial peer exclusion significantly disrupts network reciprocity and suppresses cooperation. However, when coexisting with prosocial peer exclusion, it does not undermine the latter's efficacy in upholding cooperation, except in scenarios with low exclusion costs Unlike the cooperation-sustaining cyclic dominance pattern observed in the exclusive presence of prosocial peer exclusion, the co-presence of prosocial and antisocial peer exclusion gives rise to more intricate pathways for maintaining cooperation. These pathways include cyclic dominance involving traditional cooperation, prosocial peer exclusion, and antisocial peer exclusion, or a similar pattern involving traditional defection and the two exclusion strategies, or even cyclic dominance among all four strategies. In essence, our study enhances the theoretical framework concerning the effectiveness of the prosocial exclusion strategy, contributing to a more comprehensive understanding of its dynamics. [ABSTRACT FROM AUTHOR]