In the classical evolutionary game theory, mutation is usually considered as a constant, however strategy mutation is affected by strategies in the real game process. Therefore, the main purpose of this paper is to study the effects of mutation feedback and time delays on strategy dynamics, where mutation is a linear feedback related to strategy. First, we construct a co-evolutionary game model with two time delays induced by mutation feedback and analyze the existence and stability of the equilibria of the non-time delay system. The conditions for the coexistence of the two strategies and the mutation rate are obtained. Second, Sotomayor’s theorem is used to explore the transcritical bifurcation of the system. Then, the existence of Hopf bifurcation is investigated by using feedback delay and payoff delay as bifurcation parameters in the time-delay system. Furthermore, we discuss the direction of the Hopf bifurcation, stability and periodic change of the periodic solution in detail. Finally, a series of numerical simulations are used to describe the theoretical analysis. The main results are as follows: (i) Mutation causes negative feedback to cooperative strategy. (ii) As the time delay increases, the stable equilibrium point becomes unstable, and which branches a stable limit cycle. When the time delay continues to increase sufficiently, the stable limit cycle becomes unstable and produces irregular oscillation and chaos. (iii) When the two time delays are large enough, the coexistence of the two strategies becomes that defective strategy is dominant, and the mutation rate also reaches the maximum.

中文翻译：

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时滞和变异反馈引起的演化博弈策略的稳定性、周期和混沌

在经典的进化博弈论中，变异通常被认为是一个常数，而策略变异在实际博弈过程中会受到策略的影响。因此，本文的主要目的是研究突变反馈和时间延迟对策略动态的影响，其中突变是与策略相关的线性反馈。首先，我们构建了由突变反馈引起的两次时滞的协同进化博弈模型，并分析了非时滞系统平衡点的存在性和稳定性。得到了两种策略共存的条件和突变率。其次，利用索托马约尔定理来探讨系统的跨临界分岔。然后，以反馈时滞和支付时滞作为时滞系统的分岔参数，研究了Hopf分岔的存在性。此外，我们还详细讨论了周期解的Hopf分岔的方向、稳定性和周期变化。最后，通过一系列数值模拟来描述理论分析。主要结果如下：（i）变异对合作策略造成负反馈。(ii) 随着时间延迟的增加，稳定平衡点变得不稳定，从而分支出一个稳定的极限环。当时滞继续充分增加时，稳定的极限环变得不稳定，并产生不规则的振荡和混沌。(iii) 当两个时间延迟足够大时，两种策略共存时，缺陷策略占主导地位，变异率也达到最大值。