Two-time-scale population evolution on a singular landscape

Song Xu, Shuyun Jiao, Pengyao Jiang, and Ping Ao

Phys. Rev. E 89, 012724 – Published 31 January 2014

Abstract

Under the effect of strong genetic drift, it is highly probable to observe gene fixation or gene loss in a population, shown by singular peaks on a potential landscape. The genetic drift-induced noise gives rise to two-time-scale diffusion dynamics on the bipeaked landscape. We find that the logarithmically divergent (singular) peaks do not necessarily imply infinite escape times or biological fixations by iterating the Wright-Fisher model and approximating the average escape time. Our analytical results under weak mutation and weak selection extend Kramers's escape time formula to models with B (Beta) function-like equilibrium distributions and overcome constraints in previous methods. The constructed landscape provides a coherent description for the bistable system, supports the quantitative analysis of bipeaked dynamics, and generates mathematical insights for understanding the boundary behaviors of the diffusion model.

Received 21 August 2013
Revised 23 November 2013

DOI:https://doi.org/10.1103/PhysRevE.89.012724

Authors & Affiliations

Song Xu^1,*, Shuyun Jiao^2,3, Pengyao Jiang⁴, and Ping Ao^2,5,†

¹Department of Biomathematics, University of California at Los Angeles, Los Angeles, California 90095-1766, USA
²Shanghai Center for Systems Biomedicine, Key Laboratory of Systems Biomedicine of Ministry of Education, Shanghai Jiao Tong University, Shanghai 200240, China
³Department of Mathematics, Xinyang Normal University, Xinyang 464000, Henan, China
⁴Department of Ecology and Evolution, University of Chicago, 1101 E. 57th Street, Chicago, Illinois 60637, USA
⁵Department of Physics, Shanghai Jiao Tong University, Shanghai 200240, China

^*sxu11@ucla.edu
^†Corresponding author: aoping@sjtu.edu.cn

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Vol. 89, Iss. 1 — January 2014

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Physical Review E

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