Behavior of different numerical schemes for random genetic drift

  • Shixin Xu
  • Minxin Chen
  • Chun Liu
  • Ran Zhang
  • Xingye YueEmail author


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.


Random 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 Classification

35K65 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.


  1. 1.
    Crow, J.F., Kimura, M.: An Introduction to Population Genetics Theory. Harper & Row, New York (1970)zbMATHGoogle Scholar
  2. 2.
    Der, R., Epstein, C.L., Plotkin, J.B.: Generalized population models and the nature of genetic drift. Theor. Popul. Biol. 80(2), 80–99 (2011)CrossRefzbMATHGoogle Scholar
  3. 3.
    Duan, C., Liu, C., Wang, C., Yue, X.: Numerical complete solution for random genetic drift by energetic variational approach. arXiv:1803.09436. Mathematical modelling and numerical analysis accepted and published online (2018)
  4. 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
  5. 5.
    Fisher, R.A.: On the dominance ratio. Proc. R. Soc. Edinb. 42, 321–431 (1922)CrossRefGoogle Scholar
  6. 6.
    Hössjer, O., Tyvand, P.A., Miloh, T.: Exact Markov chain and approximate diffusion solution for hapliod genetic drift with one-way mutation. Math. Biosci. 272, 100–112 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kimura, M.: Stochastic processes and distribution of gene frequencies under natural selection. Cold Spring Harb. Symp. Quant. Biol. 20, 33–53 (1955)CrossRefGoogle Scholar
  8. 8.
    Kimura, M.: Random genetic drift in multi-allelic locus. Evolution 9(4), 419–435 (1955)CrossRefGoogle Scholar
  9. 9.
    Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47(6), 713–719 (1962)Google Scholar
  10. 10.
    Kimura, M.: Diffusion models in population genetics. J. Appl. Probab. 1, 177–232 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Kimura, M.: The Neutral Theory of Molecular Evolution: A Review of Recent Evidence. Cambridge University Press, Cambridge (1983)CrossRefGoogle Scholar
  12. 12.
    LeVeque, R.: Finite-Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  13. 13.
    McKane, A.J., Waxman, D.: Sigular solution of the diffusion equation of population genetics. J. Theor. Biol. 247, 849–858 (2007)CrossRefGoogle Scholar
  14. 14.
    Moran, P.A.P.: Random processes in genetics. Proc. Camb. Philos. Soc. 54, 60–72 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Roos, H.G., Stynes, M., Tobiska, L.: Numerical Methods for Singularly Perturbed Differential Equations. Springer, New York (1996)CrossRefzbMATHGoogle Scholar
  16. 16.
    Thomee, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25. Springer, New York (2006)zbMATHGoogle Scholar
  17. 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).
  18. 18.
    Tran, T.D., Hofrichter, J., Jost, J.: An introduction to the mathematical structure of the Wright–Fisher model of population genetics. Theory Biosci. 132, 73–82 (2013)CrossRefGoogle Scholar
  19. 19.
    Waxman, D.: Fixation at a locus with multiple alleles: structure and solution of the Wright Fisher model. J. Theor. Biol. 257, 245–251 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Wright, S.: The evolution of dominace. Am. Nat. 63(689), 556–561 (1929)CrossRefGoogle Scholar
  21. 21.
    Wright, S.: The differential equation of the distribution of gene frequencies. Proc. Natl. Acad. Sci. U.S.A. 31, 382–389 (1945)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Zhao, L., Yue, X., Waxman, D.: Complete numerical solution of the diffusion equation of random genetic drift. Genetics 194(4), 973–985 (2013)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.School of Mathematical SciencesSoochow UniversitySuzhouChina
  2. 2.Department of Applied MathematicsIllinois Institute of TechnologyChicagoUSA
  3. 3.School of MathematicsShanghai University of Finance and EconomicsShanghaiChina

Personalised recommendations