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.

翻译：演化博弈论传统上采用确定性模型描述种群动态。由于固有的非线性特性，这些模型可能呈现确定性混沌现象，即种群涨落遵循复杂且非周期性的模式。近期研究焦点逐渐转向随机模型，量化固定概率并分析具有运动常数的系统。然而，在演化博弈论框架内，随机效应对混沌动力学系统的作用仍亟待探索。本研究探讨了由概率性出生死亡事件产生的种群规模噪声如何影响有限种群中的混沌动力学。通过对比混沌确定性系统与其随机对应模型，我们发现：尽管存在随机性，大规模种群仍保留混沌动力学的特征印记。具体而言，确定性模型中观察到的奇异吸引子在随机模型中得以定性复现，此时"确定性混沌"这一术语已失去其本义。我们运用非线性动力学工具量化了种群规模对动力学行为的影响机制。研究显示：在小规模种群中，随机性占据主导地位，掩盖了确定性选择效应；但随着种群规模扩大，动力学行为逐渐显现出底层混沌结构。这种对种群规模噪声的鲁棒性对于维持种群多样性至关重要，即使在非平衡动力学中亦是如此。总体而言，我们的研究拓展了对种群动力学的理解，重新审视了混沌与噪声的界限，揭示了在确定性极限下呈混沌特性的系统中，有限种群如何维持其动力学结构。