An explicit transition density expansion for a multi-allelic Wright-Fisher diffusion with general diploid selection

Area of Science:

• Population genetics
• Mathematical biology
• Evolutionary dynamics

Background:

• Understanding allele frequency changes over time is crucial in population genetics.
• The Wright-Fisher diffusion model describes these dynamics using a transition density function.
• Explicit formulas for multi-allelic models with diploid selection have been challenging to derive.

Purpose of the Study:

• To extend a technique for deriving explicit transition densities to multi-allelic models.
• To provide a method for analyzing population genetics under general diploid selection and mutation.
• To enable accurate computation of key population genetic parameters.

Main Methods:

• Extension of a previously developed technique for diallelic models.
• Analysis of the generator for multi-allelic diffusion.
• Spectral representation using eigenvalues and eigenfunctions.

Main Results:

• An explicit formula for the transition density function in multi-allelic diffusion models.
• Accurate spectral representation of the transition density.
• Efficient computation of stationary distribution normalizing constants and convergence rates.

Conclusions:

• The developed method provides a significant advancement in analyzing complex population genetic models.
• This approach facilitates a deeper understanding of evolutionary dynamics under selection and mutation.
• The findings offer practical tools for computing essential population genetic parameters.