When the mean is not enough: Calculating fixation time distributions in birth-death processes

Area of Science:

• Mathematical Biology
• Evolutionary Dynamics
• Stochastic Processes

Background:

• Traditional fixation dynamics studies in Markov processes primarily use mean time to absorption.
• This mean-based approach can be insufficient for broad and skewed fixation time distributions.

Purpose of the Study:

• To compute and analyze the distribution of fixation times in one-step birth-death processes with two absorbing states.
• To explore efficient methods for sampling these fixation time distributions.
• To apply these methods to evolutionary game dynamics.

Main Methods:

• Calculating fixation time distributions using the spectrum of the birth-death process.
• Representing processes in eigenspace for efficient forward-only sampling.
• Analyzing evolutionary game dynamics with invading mutants.

Main Results:

• Fixation time distributions are derived and expressed via the process spectrum.
• Efficient sampling methods for these distributions are provided.
• The median fixation time is identified as a relevant analog for mixing times in specific evolutionary scenarios.

Conclusions:

• The distribution of fixation times provides a more comprehensive understanding than the mean alone.
• Median fixation time offers a valuable metric for evolutionary game dynamics, particularly with low mutation rates.
• The developed methods enable efficient analysis of fixation dynamics in stochastic systems.