Browsing by Subject "FIXATION"

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    Behavior of different numerical schemes for random genetic drift
    (BIT Numerical Mathematics, 2019-09-01) Xu, S; Chen, M; Liu, C; Zhang, R; Yue, X
    In the problem of random genetic drift, the probability density of one gene is governed by a degenerated convection-dominated diffusion equation. Dirac singularities will always be developed at boundary points as time evolves, which is known as the fixation phenomenon in genetic evolution. Three finite volume methods: FVM1-3, one central difference method: FDM1 and three finite element methods: FEM1-3 are considered. These methods lead to different equilibrium states after a long time. It is shown that only schemes FVM3 and FEM3, which are the same, preserve probability, expectation and positiveness and predict the correct probability of fixation. FVM1-2 wrongly predict the probability of fixation due to their intrinsic viscosity, even though they are unconditionally stable. Contrarily, FDM1 and FEM1-2 introduce different anti-diffusion terms, which make them unstable and fail to preserve positiveness.
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    Numerical method for multi-alleles genetic drift problem
    (SIAM Journal on Numerical Analysis, 2019-01-01) Xu, S; Chen, X; Liu, C; Yue, X
    Genetic drift describes random fluctuations in the number of genes variants in a population. One of the most popular models is the Wright-Fisher model. The diffusion limit of this model is a degenerate diffusion-convection equation. Due to the degeneration and convection, Dirac singularities will always develop at the boundaries as time evolves, i.e., the fixation phenomenon occurs. Theoretical analysis has proven that the weak solution of this equation, regarded as measure, conserves total probability and expectations. In the current work, we propose a scheme for 3-alleles model with absolute stability and generalize it to N-alleles case (N > 3). Our method can conserve not only total probability and expectations, but also positivity. We also prove that the discrete solution converges to a measure as the mesh size tends to zero, which is the exact measure solution of the original problem. The simulations illustrate that the probability density decays to zero first on the inner nodes, then also on the edge nodes except at the three vertex nodes, on which the density finally concentrates. The results correctly predict the fixation probability and are consistent with theoretical ones and with direct Monte Carlo simulations.