An Optimal Mass Transport Method for Random Genetic Drift

TL;DR

This paper introduces a novel optimal mass transport numerical method for the Kimura equation modeling genetic drift, effectively capturing singularities and ensuring exponential convergence to equilibrium.

Contribution

It develops a new numerical scheme that accurately handles Dirac-delta singularities in genetic drift models and guarantees rapid convergence to the stationary state.

Findings

Successfully captures Dirac-delta singularities in simulations

Ensures exponential convergence to the stationary state

Preserves biologically relevant properties in numerical solutions

Abstract

We propose and analyze an optimal mass transport method for a random genetic drift problem driven by a Moran process under weak-selection. The continuum limit, formulated as a reaction-advection-diffusion equation known as the Kimura equation, inherits degenerate diffusion from the discrete stochastic process that conveys to the blow-up into Dirac-delta singularities hence brings great challenges to both the analytical and numerical studies. The proposed numerical method can quantitatively capture to the fullest possible extent the development of Dirac-delta singularities for genetic segregation on one hand, and preserves several sets of biologically relevant and computationally favored properties of the random genetic drift on the other. Moreover, the numerical scheme exponentially converges to the unique numerical stationary state in time at a rate independent of the mesh size up to a mesh error. Numerical evidence is given to illustrate and support these properties, and to demonstrate the spatio-temporal dynamics of random generic drift.