On a continuum model for random genetic drift: a dynamic boundary condition approach

Chun Liu; Jan-Eric Sulzbach; Yiwei Wang; Chun Liu; Jan-Eric Sulzbach; Yiwei Wang

doi:10.3934/cam.2025033

Communications in Analysis and Mechanics

2025, Volume 17, Issue 3: 829-848. doi: 10.3934/cam.2025033

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On a continuum model for random genetic drift: a dynamic boundary condition approach

1.
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL, USA
2.
Department of Mathematics, Technical University Munich, Munich, Germany
3.
Department of Mathematics, University of California, Riverside, Riverside, CA, USA

Received: 31 December 2024 Revised: 17 June 2025 Accepted: 06 August 2025 Published: 08 September 2025
35K65, 35K20, 92D10, 76M30

We propose a new continuum model for random genetic drift by employing a dynamic boundary condition approach. The model can be viewed as a regularized version of the Kimura equation and admits a continuous solution. We establish existence and uniqueness of a strong solution to the regularized system. Numerical experiments illustrate that, for sufficiently small regularization parameters, the model can capture key phenomena of the original Kimura equation, such as gene fixation and conservation of the first moment.
- genetic drift,
- Kimura equation,
- Wright–Fisher model,
- dynamic boundary condition,
- fixation probability,
- energetic variational approach
Citation: Chun Liu, Jan-Eric Sulzbach, Yiwei Wang. On a continuum model for random genetic drift: a dynamic boundary condition approach[J]. Communications in Analysis and Mechanics, 2025, 17(3): 829-848. doi: 10.3934/cam.2025033

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Abstract

We propose a new continuum model for random genetic drift by employing a dynamic boundary condition approach. The model can be viewed as a regularized version of the Kimura equation and admits a continuous solution. We establish existence and uniqueness of a strong solution to the regularized system. Numerical experiments illustrate that, for sufficiently small regularization parameters, the model can capture key phenomena of the original Kimura equation, such as gene fixation and conservation of the first moment.

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