标题： 演化博弈动力学中的混沌与噪声 显示英文标题

标题： Chaos and noise in evolutionary game dynamics

摘要： 演化博弈论传统上采用确定性模型来描述种群动力学。 这些模型由于其固有的非线性，可以表现出确定性混沌，其中种群波动遵循复杂的非周期模式。 最近，研究重点转向了随机模型，量化固定概率并分析具有守恒量的系统。 然而，在演化博弈论中，对于具有混沌动力学的系统中随机效应的作用仍基本未被探索。 本研究探讨了源于概率性出生和死亡事件的人口噪声如何影响有限种群中的混沌动力学。 我们表明，尽管存在随机性，大型种群仍然保留了混沌动力学的特征，这一点可以通过比较混沌确定性系统与其随机对应物来证明。 更具体地说，确定性模型中观察到的奇怪吸引子在随机模型中以定性方式恢复，此时“确定性混沌”这一术语失去了意义。 我们利用非线性动力学工具来量化种群规模如何影响动力学。 我们观察到，对于小种群，随机性占主导地位，掩盖了确定性选择效应。 然而，随着种群规模的增加，动力学越来越反映出潜在的混沌结构。 这种对人口噪声的抗性对于维持种群多样性至关重要，即使在非平衡动力学中也是如此。 总体而言，我们的结果拓宽了我们对种群动力学的理解，并重新审视了混沌与噪声之间的界限，展示了当考虑在确定性极限下呈现混沌的有限种群时，它们如何保持结构。 显示英文摘要

摘要： Evolutionary game theory has traditionally employed deterministic models to describe population dynamics. These models, due to their inherent nonlinearities, can exhibit deterministic chaos, where population fluctuations follow complex, aperiodic patterns. Recently, the focus has shifted towards stochastic models, quantifying fixation probabilities and analysing systems with constants of motion. Yet, the role of stochastic effects in systems with chaotic dynamics remains largely unexplored within evolutionary game theory. This study addresses how demographic noise -- arising from probabilistic birth and death events -- impacts chaotic dynamics in finite populations. We show that despite stochasticity, large populations retain a signature of chaotic dynamics, as evidenced by comparing a chaotic deterministic system with its stochastic counterpart. More concretely, the strange attractor observed in the deterministic model is qualitatively recovered in the stochastic model, where the term deterministic chaos loses its meaning. We employ tools from nonlinear dynamics to quantify how the population size influences the dynamics. We observe that for small populations, stochasticity dominates, overshadowing deterministic selection effects. However, as population size increases, the dynamics increasingly reflect the underlying chaotic structure. This resilience to demographic noise can be essential for maintaining diversity in populations, even in non-equilibrium dynamics. Overall, our results broaden our understanding of population dynamics, and revisit the boundaries between chaos and noise, showing how they maintain structure when considering finite populations in systems that are chaotic in the deterministic limit.