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


Yu, Xiaoquan; Li, Xiang-Yi (2019). Applications of WKB and Fokker–Planck Methods in Analyzing Population Extinction Driven by Weak Demographic Fluctuations. Bulletin of Mathematical Biology, 81(11):4840-4855.

Abstract

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.

Abstract

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.

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Additional indexing

Item Type:Journal Article, refereed, original work
Communities & Collections:07 Faculty of Science > Institute of Evolutionary Biology and Environmental Studies
Dewey Decimal Classification:570 Life sciences; biology
590 Animals (Zoology)
Scopus Subject Areas:Life Sciences > General Neuroscience
Life Sciences > Immunology
Physical Sciences > General Mathematics
Life Sciences > General Biochemistry, Genetics and Molecular Biology
Physical Sciences > General Environmental Science
Life Sciences > Pharmacology
Life Sciences > General Agricultural and Biological Sciences
Physical Sciences > Computational Theory and Mathematics
Uncontrolled Keywords:Immunology, General Biochemistry, Genetics and Molecular Biology, Computational Theory and Mathematics, General Neuroscience, Pharmacology, General Agricultural and Biological Sciences, General Mathematics, General Environmental Science
Language:English
Date:1 November 2019
Deposited On:26 Feb 2019 17:15
Last Modified:15 Apr 2020 23:06
Publisher:Springer
ISSN:0092-8240
OA Status:Closed
Publisher DOI:https://doi.org/10.1007/s11538-018-0483-6

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