Chaos and unpredictability in evolution of cooperation in continuous time

Area of Science:

• Evolutionary Game Theory
• Complex Systems Dynamics
• Computational Biology

Background:

• Cooperation is vital but costly, with its evolution depending on the cost-benefit ratio (c).
• Understanding strategy evolution in repeated games like the prisoner's dilemma is crucial.
• Memory-one strategies (e.g., Tit-for-Tat) are foundational in studying cooperation.

Purpose of the Study:

• Investigate the evolutionary dynamics of cooperation under varying cost-benefit ratios (c).
• Analyze the impact of implementation error and mutation on strategy evolution.
• Explore the potential for chaotic dynamics in evolutionary game theory.

Main Methods:

• Simulated replicator dynamics for four memory-one strategies in an infinite population.
• Incorporated implementation error and mutation rates (μ).
• Analyzed a three-dimensional continuous-time dynamical system.

Main Results:

• The system exhibits chaotic behavior via a bifurcation sequence as 'c' varies.
• Mutation rate (μ) influences the bifurcation sequence position, scaling as μ^0.1.
• Demonstrated amplification of microscopic randomness into macroscopic unpredictability.

Conclusions:

• Evolutionary dynamics of cooperation can be highly sensitive and unpredictable.
• Mutation, even at low rates, can have significant non-perturbative effects on evolutionary paths.
• Highlights the complex interplay between strategy, cost-benefit ratios, and random processes in evolution.