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Behavior of different numerical schemes for random genetic drift

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

Abstract

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.

Keywords

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 

Notes

Acknowledgements

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.

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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

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