Quantitative approximation of the discrete Moran process by a Wright-Fisher diffusion.

Area of Science:

• Population genetics
• Mathematical biology
• Evolutionary dynamics

Background:

• The Moran discrete process and Wright-Fisher model are foundational in population genetics.
• Diffusion approximations, like the Wright-Fisher diffusion, are widely used to study population genetics model dynamics.
• Understanding the accuracy of these approximations is crucial for reliable analysis.

Purpose of the Study:

• To quantitatively assess the error introduced by using diffusion approximations for population genetics models.
• To analyze the large-population limit of errors under weak selection and weak immigration in one dimension.
• To develop a robust approach applicable to Markovian selection and immigration processes.

Main Methods:

• Analysis of the large-population limit of discrete population genetics models.
• Quantitative error estimation for diffusion approximations.
• Consideration of weak selection and weak immigration dynamics.
• Extension to Markovian processes with finite state jump or diffusion limits.

Main Results:

• A quantitative bound on the error of the Wright-Fisher diffusion approximation was derived.
• The approach successfully handles weak selection and weak immigration.
• The method's robustness was demonstrated for Markovian processes.

Conclusions:

• The study provides a rigorous error analysis for diffusion approximations in population genetics.
• The findings enhance the reliability of using diffusion models for studying evolutionary dynamics.
• The developed approach offers a flexible framework for analyzing complex population processes.