On a Modified Random Genetic Drift Model: Derivation and a Structure-Preserving Operator-Splitting Discretization

TL;DR

This paper introduces a modified Kimura equation for genetic drift that admits classical solutions by domain modification and boundary conditions, and develops a hybrid numerical scheme to accurately simulate its dynamics while preserving key properties.

Contribution

It proposes a domain modification and boundary condition approach for the Kimura equation, along with a hybrid operator-splitting scheme that preserves mass, positivity, and structure.

Findings

The modified model admits classical solutions with biologically meaningful boundary conditions.

The hybrid scheme accurately captures fixation and stationary distributions.

Numerical tests confirm the scheme's efficiency and structure-preserving qualities.

Abstract

One of the fundamental mathematical models for studying random genetic drift is the Kimura equation, derived as the large-population limit of the discrete Wright-Fisher model. However, due to the degeneracy of the diffusion coefficient, it is impossible to impose a suitable boundary condition that ensures the Kimura equation admits a classical solution while preserving biological significance. In this work, we propose a modified model for random genetic drift that admits classical solutions by modifying the domain of the Kimura equation from $(0, 1)$ to $(\delta, 1 - \delta)$ with $\delta$ being a small parameter, which allows us to impose a Robin-type boundary condition. By introducing two additional variables for the probabilities in the boundary region, we effectively capture the conservation of mass and the fixation dynamics in the original model. To numerically investigate the modified model, we develop a hybrid Eulerian-Lagrangian operator splitting scheme. The scheme first solves the flow map equation in the bulk region using a Lagrangian approach with a no-flux boundary condition, followed by handling the boundary dynamics in Eulerian coordinates. This hybrid scheme ensures mass conservation, maintains positivity, and preserves the first moment. Various numerical tests demonstrate the efficiency, accuracy, and structure-preserving properties of the proposed scheme. Numerical results demonstrate the key qualitative features of the original Kimura equation, including the fixation behavior and the correct stationary distribution in the small-$\delta$ limit.