Spatial evolutionary games provide a valuable framework for elucidating the emergence and maintenance of cooperative behavior. However, most previous studies assume that individuals are profiteers and neglect to consider the effects of memory. To bridge this gap, in this paper, we propose a memory-based spatial evolutionary game with dynamic interaction between learners and profiteers. Specifically, there are two different categories of individuals in the network, including profiteers and learners with different strategy updating rules. Notably, there is a dynamic interaction between profiteers and learners, i.e., each individual has the transition probability between profiteers and learners, which is portrayed by a Markov process. Besides, the payoff of each individual is not only determined by a single round of the game but also depends on the memory mechanism of the individual. Extensive numerical simulations validate the theoretical analysis and uncover that dynamic interactions between profiteers and learners foster cooperation, memory mechanisms facilitate the emergence of cooperative behaviors among profiteers, and increasing the learning rate of learners promotes a rise in the number of cooperators. In addition, the robustness of the model is verified through simulations across various network sizes. Overall, this work contributes to a deeper understanding of the mechanisms driving the formation and evolution of cooperation.

翻译：空间演化博弈为阐明合作行为的涌现与维持提供了有价值的理论框架。然而，先前研究大多假设个体为利己者，且未考虑记忆效应的影响。为弥补这一不足，本文提出了一种基于记忆的、利己者与学习者之间存在动态交互的空间演化博弈模型。具体而言，网络中个体分为两类：利己者和学习者，二者采用不同的策略更新规则。值得注意的是，利己者与学习者之间存在动态交互，即每个个体均具有在利己者与学习者状态间转换的概率，该过程由马尔可夫链描述。此外，个体的收益不仅由单次博弈决定，还依赖于其记忆机制。大量数值模拟验证了理论分析，并揭示：利己者与学习者间的动态交互促进了合作；记忆机制有助于利己者群体中合作行为的涌现；提高学习者的学习率可增加合作者数量。此外，通过在不同网络规模下的仿真验证了模型的鲁棒性。总体而言，本研究有助于深化对合作形成与演化驱动机制的理解。