Chaos and noise in evolutionary game dynamics

Abstract: 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.