Chaos and noise in evolutionary game dynamics

TL;DR

This work examines how demographic noise from finite population size interacts with chaotic population dynamics in evolutionary game theory using the ACT-Skyrms payoff matrix. By comparing deterministic PCP trajectories with stochastic, finite-N realizations, it shows that large populations retain signatures of chaos, with the deterministic strange attractor qualitatively persisting in the stochastic model. A time-rescaling framework and a comprehensive set of nonlinear-dynamics metrics reveal a critical transition at β* ≈ 7 from chaotic to periodic dynamics, while noise dominates in small populations and diminishes as N grows. The findings reveal a robust chaos–noise interplay, demonstrating that chaotic structure can endure under demographic stochasticity and offering insights into how diversity and non-equilibrium dynamics are maintained in finite populations.

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.

Figures (9)

• Figure 1: The replicator equation for the ACT-Skyrms payoff matrix exhibits aperiodic dynamics characterised by a Shilnikov strange attractor. A. ACT-Skyrms payoff matrix and the corresponding relations between different strategies. B. The replicator equation is the standard update mechanism to model the evolution of an infinitely large, well-mixed population. It states that the change in abundance of a given strategy $x_i$ is proportional to the difference between the individual fitness $f_i$ and the average fitness of the population $\langle f \rangle$. C. The replicator equation for the ACT-Skyrms payoff matrix exhibits aperiodic dynamics characterised by a Shilnikov strange attractor. The trajectories display 10,000 time steps starting from the initial condition $\boldsymbol{x}=(0.25,0.25,0.25,0.25)$.
• Figure 2: Deterministic chaos arises under low selection intensity. A. Ternary plots of the deterministic PCP dynamics for different selection intensity values ($\beta$). For low selection strength, the trajectories display a strange attractor, characterised by its fractal structure. For higher selection intensities, the system exhibits periodic dynamics, where the attractor gradually contracts in the state space towards the equilibrium point $x^*$. The trajectories display $1\times 10^4$ effective time steps starting from the initial condition $\boldsymbol{x(0)} = (0.25,0.25,0.25,0.25)$. B. Quantification of the chaotic dynamics using various numerical measures. A critical value, $\beta^* \approx 7$, is identified, which distinguishes chaotic from non-chaotic behavior. For $\beta < \beta^*$, the system exhibits chaotic behavior characterised by a positive maximum Lyapunov exponent, $\lambda_1$, indicating sensitive dependence on initial conditions. Additionally, the system displays a higher Lempel-Ziv complexity, $C_{LZ}$, suggesting a lack of periodicity. In this chaotic regime, the system possesses a strange attractor with a non-integer fractal dimension, $\Delta^C$, that spans a larger region of the state space, as indicated by a higher standard deviation, $\sigma$.
• Figure 3: The underlying chaotic structure is reflected in the stochastic dynamics for large populations. The stochastic trajectories show that for small populations demographic noise largely dominates the dynamics. Specifically, demographic noise overshadows selection effects. In contrast, for large population sizes the dynamics approximate the deterministic limit because the step size becomes increasingly small, $\Delta \boldsymbol{x} \rightarrow 0$. However, the transition between strange attractor and limit cycle in the presence of demographic noise is less distinct than in the deterministic case. Finally, regardless of the population size the system's dynamics are primarily governed by the underlying attractor. All the trajectories start from the same initial condition $\boldsymbol{x(0)} = (0.25,0.25,0.25,0.25)$ and display an equivalent number of time steps according to the scaling described in the SI Appendix.
• Figure 4: Quantification of the stochastic trajectories. A. Dynamics for smaller population sizes result in faster fixation. This occurs because the step size is larger for smaller populations, delaying the time for the trajectories to reach the boundaries signifying the extinction of one of the population's types. B. Demographic noise overshadows selection effects, therefore there is no attractor's contraction for smaller populations, as shown by the attractor's standard deviation. C. The fractal dimension increases as the selection intensity becomes larger. The perturbations generated by demographic noise prevent the trajectories from forming a well-defined limit cycle under high selection. The plot shows the average quantifier values for 100 stochastic runs.
• Figure S1: Shilnikov attractor in 3D phase space. The trajectories are drawn toward the saddle-focus equilibrium along its two-dimensional stable manifold, $W^s(O)$, but are repelled by its one-dimensional unstable manifold, $W^u(O)$. The Shilnikov attractor is created when the trajectories return to the equilibrium point through a homoclinic orbit, $\gamma$.