Mutation is an unavoidable and indispensable phenomenon in both biological and social systems undergoing evolution through replication-selection processes. Here we show that mutation in a generation-wise nonoverlapping population with two-player-two-strategy symmetric game gives rise to coexisting stable population states, one of which can even be chaotic; the chaotic state prevents the cooperators in the population from going extinct. Specifically, we use replicator maps with additive and multiplicative mutations, and rigorously find all possible two dimensional payoff matrices for which physically allowed solutions can be achieved in the equations. Subsequently, we discover the various possibilities of bistable outcomes—e.g., coexistences of fixed point and periodic orbit, periodic orbit and chaos, and chaos and fixed point—in the resulting replicator-mutator maps.

在通过复制选择过程进行进化的生物和社会系统中，突变是不可避免且不可或缺的现象。在这里，我们展示了在具有两个玩家、两个策略的对称博弈的逐代非重叠种群中的突变会产生共存的稳定种群状态，其中一个甚至可能是混乱的；混乱状态阻止了群体中的合作者灭绝。具体来说，我们使用具有加法和乘法突变的复制器映射，并严格找到所有可能的二维支付矩阵，这些矩阵可以在方程中实现物理上允许的解。随后，我们发现了双稳态结果的各种可能性——例如，不动点和周期轨道共存，周期轨道和混沌，