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Extinction scenarios in evolutionary processes: a multinomial Wright-Fisher approach.

Alexander Roitershtein¹, Reza Rastegar², Robert S Chapkin³

¹Department of Statistics, Texas A &M University, College Station, TX, 77843, USA.

Journal of Mathematical Biology

|September 26, 2023

Summary

This study examines a multi-type Wright-Fisher population process, revealing how mean-field dynamics influence extinction. The research shows the least fit type disappears first due to system metastability near equilibrium.

Keywords:

Evolutionary dynamics Fitness landscape Lyapunov functions Metastability Quasi-equilibria Wright–Fisher process

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Area of Science:

Population Genetics
Mathematical Biology
Stochastic Processes

Background:

The Wright-Fisher model is a cornerstone of population genetics, describing random genetic drift.
Understanding multi-type population dynamics is crucial for evolutionary studies.
Mean-field approximations simplify complex population interactions.

Purpose of the Study:

To analyze a discrete-time multi-type Wright-Fisher process with general replicator dynamics.
To investigate the asymptotic behavior and extinction patterns of the model.
To explore the implications of a proposed maximization principle for deterministic replicator dynamics.

Main Methods:

Mathematical modeling of a discrete-time multi-type Wright-Fisher process.
Analysis of mean-field dynamics using a general replicator difference equation.
Development of limit theorems to describe asymptotic behavior and extinction.
Investigation of system metastability and its effect on population dynamics.

Main Results:

A limit theorem is established for almost certain extinction, identifying the least fit type at mean-field equilibrium as the first to disappear.
The study demonstrates that system metastability explains the prolonged time spent near equilibrium before extinction.
A maximization principle for deterministic replicator dynamics is proposed and its impact on the stochastic model is analyzed.

Conclusions:

Mean-field dynamics significantly impact the extinction pathways in multi-type Wright-Fisher populations.
Metastability plays a key role in the pre-extinction dynamics of the stochastic system.
The proposed maximization principle offers new insights into the behavior of deterministic replicator dynamics and their stochastic counterparts.