Speed of evolution on graphs

Area of Science:

• Evolutionary Game Theory
• Population Genetics
• Mathematical Biology

Background:

• Fixation probability is a key concept in evolutionary game theory, but fixation time is crucial for understanding evolutionary speed.
• The impact of population structure on fixation times remains poorly understood, especially compared to well-mixed populations.

Purpose of the Study:

• To theoretically investigate how population structure, specifically regular graphs, affects fixation times in evolutionary dynamics.
• To analyze fixation times under neutral and weak selection scenarios.

Main Methods:

• Employed pair approximation and diffusion approximation methods.
• Analyzed fixation times on regular graphs for various selection strengths and game parameters.

Main Results:

• Under neutral selection, increased neighborhood size shortens both unconditional and conditional fixation times.
• Under weak selection in a prisoner's dilemma, fixation time can slow down if the benefit-to-cost ratio exceeds graph degree.
• Limited neighborhood size mitigates stochastic slowdowns observed in well-mixed populations.

Conclusions:

• All findings can be unified by interpreting results within a well-mixed population model using a transformed payoff matrix.
• This provides a bridge between fixation time dynamics in structured and well-mixed populations, enabling the use of existing well-mixed population theory for structured cases.