Chaos and noise in evolutionary game dynamics

2025-03-28

AI Summary

This work investigates the impact of demographic noise—stochasticity in birth and death events—on chaotic dynamics in evolutionary games within finite populations. Addressing the lack of understanding regarding the robustness of chaotic structures under stochasticity, the authors integrate nonlinear dynamical analysis (Lyapunov exponents, phase-space reconstruction), stochastic process modeling, and large-scale numerical simulations, complemented by analytical fixation probability calculations and attractor quantification. Their approach reveals three key contributions: first, rigorous demonstration that, in the large-population limit, the stochastic system qualitatively reproduces the deterministic strange attractor; second, identification of a critical population-size–dependent phase transition governing the dominance between chaos and noise; and third, discovery of a novel noise-mediated mechanism that nondestructively sustains nonequilibrium biodiversity. Collectively, these findings refine the theoretical boundary between chaos and stochasticity in evolutionary dynamics.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Machine Learning: Evolutionary Learning
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Social Networks and Social Media: Applications and emergent phenomena

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.

Problem

• Investigates how demographic noise affects chaotic dynamics in finite populations
• Compares deterministic chaos with stochastic models in evolutionary game theory
• Explores resilience of chaotic structures to noise in population dynamics

Innovation

• Studying demographic noise in chaotic game dynamics
• Comparing deterministic and stochastic chaotic systems
• Analyzing population size impact on chaotic dynamics