[Paper Review] Sensitivity to Initial Conditions and Nonextensivity in Biological Evolution

Francisco A. Tamarit, Sergio A. Cannas+1 more|ArXiv.org|Dec 20, 1996

Complex Systems and Time Series Analysis2 references3 citations

TL;DR

This paper investigates sensitivity to initial conditions in the Bak-Sneppen model of biological evolution, demonstrating power-law dependence at the self-organized critical state. It links this behavior to nonextensive statistical mechanics via Tsallis statistics, finding a critical exponent consistent with nonextensivity and drawing parallels to chaotic systems near onset of chaos.

ABSTRACT

We consider biological evolution as described within the Bak and Sneppen 1993 model. We exhibit, at the self-organized critical state, a power-law sensitivity to the initial conditions, calculate the associated exponent, and relate it to the recently introduced nonextensive thermostatistics. The scenario which here emerges without tuning strongly reminds that of the tuned onset of chaos in say logistic-like onedimensional maps. We also calculate the dynamical exponent z.

Motivation & Objective

• To analyze the sensitivity of the Bak-Sneppen model of biological evolution to initial conditions at the self-organized critical state.
• To investigate whether this sensitivity follows a power-law behavior, characteristic of systems near criticality.
• To relate the observed power-law exponent to the framework of nonextensive thermostatistics introduced by Tsallis.
• To compare the dynamical behavior of the model to that of tuned chaotic systems, such as logistic maps at the onset of chaos.
• To calculate the dynamical exponent z in the context of the model’s critical dynamics.

Proposed method

• Simulates the Bak-Sneppen model of biological evolution, which evolves a system of species with fitness values subject to selective removal and random replacement.
• Analyzes the time evolution of the distance between trajectories starting from slightly different initial conditions to quantify sensitivity.
• Fits the divergence of initial condition trajectories to a power law to extract the sensitivity exponent.
• Applies the formalism of nonextensive statistical mechanics (Tsallis entropy) to interpret the observed power-law behavior.
• Calculates the dynamical exponent z by analyzing the scaling of correlation length and time scales near the critical state.
• Compares the results with known behavior in one-dimensional maps at the onset of chaos, particularly the power-law divergence of trajectories.

Experimental results

Research questions

• RQ1 Does the Bak-Sneppen model exhibit power-law sensitivity to initial conditions at the self-organized critical state?
• RQ2 What is the value of the sensitivity exponent, and how does it relate to nonextensive statistical mechanics?
• RQ3 Can the dynamics of the model be analogized to the onset of chaos in logistic-like maps?
• RQ4 What is the value of the dynamical exponent z in this critical evolutionary model?
• RQ5 How does the nonextensive framework of Tsallis statistics describe the observed scaling behavior in biological evolution?

Key findings

• The model exhibits power-law sensitivity to initial conditions at the self-organized critical state, with a divergence exponent consistent with nonextensive statistics.
• The sensitivity exponent obtained matches the predictions of nonextensive thermostatistics, suggesting a deep connection between evolutionary dynamics and nonextensive thermodynamics.
• The dynamical exponent z is calculated, providing a quantitative measure of the system's critical scaling behavior.
• The observed behavior strongly resembles the tuned onset of chaos in one-dimensional maps, despite the absence of tuning in the model.
• The results support the idea that biological evolution, as modeled by Bak and Sneppen, may be described by nonextensive statistical mechanics.
• The power-law dependence of trajectory divergence is robust and independent of initial conditions, indicating universal critical behavior.