AN OPTIMAL MASS TRANSPORT METHOD FOR RANDOM GENETIC DRIFT*

摘要

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 blowup into Dirac-delta singularities and hence brings great challenges to both 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 the one hand and preserve 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 spatiotemporal dynamics of random genetic drift.