Behavior of different numerical schemes for random genetic drift
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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.
KeywordsRandom genetic drift Degenerate equation Conservations of probability and expectation Finite volume method Finite difference method Finite element method Numerical viscosity and numerical anti-diffusion
Mathematics Subject Classification35K65 65M06 92D10
The authors benefitted a great deal from discussions with Prof. David Waxman, Prof. Xinfu Chen and Prof. Xiaobing Feng. The authors thank the anonymous referees for their most valuable comments which improve the paper.
- 4.Eymard, R., Gallouët, T., Herbin, R.: The finite volume method. In: Ciarlet, P., Lions, J.L. (eds.) Handbook for Numerical Analysis, pp. 715–1022. North Holland, Amsterdam (2000)Google Scholar
- 9.Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47(6), 713–719 (1962)Google Scholar
- 17.Tran, T.D., Hofrichter, J., Jost, J.: A general solution of the Wright-Fisher model of random genetic drift. Differ. Equ. Dyn. Syst. (2012). https://doi.org/10.1007/s12591-016-0289-7