Migration is a common behavior in both biology and human societies for seeking more suitable environments or driven by greater benefits. Under the theoretical framework of evolutionary games, migration has been proved to be an effective mechanism for promoting cooperation. In this work, we introduce the migration based on historical payoffs into an evolutionary public goods game in two-dimensional space. Specifically, we introduce a parameter α to account for the ambition of individuals. During the evolution, individuals compare their current payoffs with the threshold values determined by the historical payoffs obtained in last round and the parameter α, and then decide whether to migrate. In particular, if the current payoff is greater than the threshold value, the individual will remain in his position, otherwise he will move in a randomly chosen direction with the migration speed v. The results show that cooperation could be promoted by the historical-payoff–dependent migration when compared with the situation of random migration, and there exists an optimal α which is most favorable to cooperation. Moreover, we provide some phenomenological explanations based on the characteristic snapshots of strategy pattern during the evolution. We also study the effects of the synergy factor r and the migration speed v on cooperation, and find that the moderate values of v could facilitate cooperation best. Besides, the robustness of the results has been tested by investigating the model in a sparse population and in a larger range of α, respectively.

迁徙是生物学和人类社会为寻求更合适的环境或受更大利益驱动的常见行为。在演化博弈的理论框架下，迁移被证明是促进合作的有效机制。在这项工作中，我们将基于历史收益的迁移引入到二维空间中的进化公共产品博弈中。具体来说，我们引入了一个参数α来说明个人的野心。在进化过程中，个体将自己当前的收益与上一轮获得的历史收益和参数α确定的阈值进行比较，然后决定是否迁移。特别地，如果当前收益大于阈值，则个体将保持在他的位置，否则他将以迁移速度v向随机选择的方向移动。结果表明，与随机迁移的情况相比，历史收益依赖迁移可以促进合作，并且存在一个最有利于合作的最优α。此外，我们根据演化过程中策略模式的特征快照提供了一些现象学解释。我们还研究了协同因子r和迁移速度v对合作的影响，发现v的适中值可以最好地促进合作。此外，通过分别在稀疏种群和较大范围的α 中研究模型，测试了结果的稳健性。