Bulletin of Mathematical Biology

, Volume 81, Issue 11, pp 4840–4855 | Cite as

Applications of WKB and Fokker–Planck Methods in Analyzing Population Extinction Driven by Weak Demographic Fluctuations

  • Xiaoquan Yu
  • Xiang-Yi LiEmail author
Special Issue: Modelling Biological Evolution: Developing Novel Approaches


In large but finite populations, weak demographic stochasticity due to random birth and death events can lead to population extinction. The process is analogous to the escaping problem of trapped particles under random forces. Methods widely used in studying such physical systems, for instance, Wentzel–Kramers–Brillouin (WKB) and Fokker–Planck methods, can be applied to solve similar biological problems. In this article, we comparatively analyse applications of WKB and Fokker–Planck methods to some typical stochastic population dynamical models, including the logistic growth, endemic SIR, predator-prey, and competitive Lotka–Volterra models. The mean extinction time strongly depends on the nature of the corresponding deterministic fixed point(s). For different types of fixed points, the extinction can be driven either by rare events or typical Gaussian fluctuations. In the former case, the large deviation function that governs the distribution of rare events can be well-approximated by the WKB method in the weak noise limit. In the later case, the simpler Fokker–Planck approximation approach is also appropriate.


Demographic stochasticity Fokker–Planck equation Mean extinction time WKB 



We thank Ping Ao, George Constable, Andrew Morozov, Ira B. Schwartz, Xiaomei Zhu, and an anonymous reviewer for useful discussions and/or comments. Both authors are grateful to the Institute of Theoretical Physics of the Chinese Academy of Sciences, the collaboration was made possible through its Young Scientists’ Forum on Theoretical Physics and Interdisciplinary Studies.


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© Society for Mathematical Biology 2018

Authors and Affiliations

  1. 1.Department of Physics, Centre for Quantum Science, and Dodd-Walls Centre for Photonic and Quantum TechnologiesUniversity of OtagoDunedinNew Zealand
  2. 2.Department of Evolutionary Biology and Environmental StudiesUniversity of ZurichZurichSwitzerland

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