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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., (© 2024 Author(s). Published under an exclusive license by AIP Publishing.)

The existence of neutral species carries profound ecological implications that warrant further investigation. In this paper, we study the impact of neutral species on biodiversity in a spatial tritrophic system of cyclic competition, in which the neutral species are identified as the fourth species that may affect the competition process of the other three species under the rock-paper-scissors (RPS) rule. Extensive simulations showed that neutral species can promote coexistence in a high mobility regime within the system. When coexistence occurs, we found that the state can be maintained by two mechanisms: Species can either (i) adhere to traditional RPS rule or (ii) form patches to resist invasion. Our findings might aid in understanding the impact of neutral species on biodiversity in ecosystems.