Advertisement

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

  • Xiaoquan Yu
  • Xiang-Yi Li
Special Issue: Modelling Biological Evolution: Developing Novel Approaches
First Online:
Received:
Accepted:
  • 25 Downloads

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.

Keywords

Demographic stochasticity Fokker–Planck equation Mean extinction time WKB 
This is a preview of subscription content, log in to check access.

Notes

Acknowledgements

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.

References

  1. Assaf M, Meerson B (2010) Extinction of metastable stochastic populations. Phys Rev E 81:021116CrossRefGoogle Scholar
  2. Assaf M, Meerson B (2017) WKB theory of large deviations in stochastic populations. J Phys A Math Theor 50:263001MathSciNetCrossRefMATHGoogle Scholar
  3. Black AJ, Traulsen A, Galla T (2012) Mixing times in evolutionary game dynamics. Phys Rev Lett 109:028101CrossRefGoogle Scholar
  4. Bressloff PC, Newby JM (2014) Path integrals and large deviations in stochastic hybrid systems. Phys Rev E 89(4):042701CrossRefGoogle Scholar
  5. Brillouin L (1926) La mécanique ondulatoire de schrödinger; une méthode générale de résolution par approximations successives. CR Acad Sci 183:24–26MATHGoogle Scholar
  6. Chen H, Huang F, Zhang H, Li G (2017) Epidemic extinction in a generalized susceptible-infected-susceptible model. J Stat Mech Theory Exp 2017:013204CrossRefGoogle Scholar
  7. Constable GWA, McKane AJ, Rogers T (2013) Stochastic dynamics on slow manifolds. J Phys A Math Theor 46:295002MathSciNetCrossRefMATHGoogle Scholar
  8. Doering CR, Sargsyan KV, Sander LM (2005) Extinction times for birth–death processes: exact results, continuum asymptotics, and the failure of the Fokker–Planck approximation. Multiscale Model Simul 3:283–299MathSciNetCrossRefMATHGoogle Scholar
  9. Dykman MI, Mori E, Ross J, Hunt PM (1994) Large fluctuations and optimal paths in chemical kinetics. J Chem Phys 100:5735–5750CrossRefGoogle Scholar
  10. Elgart V, Kamenev A (2004) Rare event statistics in reaction–diffusion systems. Phys Rev E 70(4):041106MathSciNetCrossRefGoogle Scholar
  11. Ewens WJ (2004) Mathematical population genetics. I. Theoretical introduction. Springer, New YorkCrossRefMATHGoogle Scholar
  12. Fisher RA (1922) On the dominance ratio. Proc R Soc Edinb 42:321–341CrossRefGoogle Scholar
  13. Gardiner CW (1985) Handbook of stochastic methods. Springer, BerlinGoogle Scholar
  14. Haefner JW (2012) Modeling biological systems: principles and applications. Springer, BerlinMATHGoogle Scholar
  15. Hindes J, Schwartz IB (2016) Epidemic extinction and control in heterogeneous networks. Phys Rev Lett 117:028302CrossRefGoogle Scholar
  16. Hindes J, Schwartz IB (2017) Epidemic extinction paths in complex networks. Phys Rev E 95:052317CrossRefGoogle Scholar
  17. Kamenev A, Meerson B (2008) Extinction of an infectious disease: a large fluctuation in a nonequilibrium system. Phys Rev E 77:061107CrossRefGoogle Scholar
  18. Kendall DG (1966) Branching processes since 1873. J Lond Math Soc 1:385–406MathSciNetCrossRefMATHGoogle Scholar
  19. Khasin M, Meerson B, Khain E, Sander LM (2012) Minimizing the population extinction risk by migration. Phys Rev Lett 109:138104CrossRefGoogle Scholar
  20. Kimura M (1964) Diffusion models in population genetics. J Appl Probab 1:177–232MathSciNetCrossRefMATHGoogle Scholar
  21. Kogan O, Khasin M, Meerson B, Schneider D, Myers Christopher R (2014) Two-strain competition in quasineutral stochastic disease dynamics. Phys Rev E 90:042149CrossRefGoogle Scholar
  22. Kramers HA (1926) Wellenmechanik und halbzahlige quantisierung. Zeitschrift für Physik A Hadrons and Nuclei 39:828–840MATHGoogle Scholar
  23. Landau LD, Lifshitz EM (2013) Quantum mechanics: non-relativistic theory, vol 3. Elsevier, AmsterdamMATHGoogle Scholar
  24. Lin YT, Kim H, Doering CR (2012) Features of fast living: on the weak selection for longevity in degenerate birth–death processes. J Stat Phys 148:647–663MathSciNetCrossRefMATHGoogle Scholar
  25. McElreath R, Boyd R (2008) Mathematical models of social evolution: a guide for the perplexed. University of Chicago Press, ChicagoMATHGoogle Scholar
  26. Meerson B, Sasorov PV (2011) Extinction rates of established spatial population. Phys Rev E 83:011129MathSciNetCrossRefGoogle Scholar
  27. Murray JD (2007) Mathematical biology I: an introduction, 3rd edn. Springer, BerlinGoogle Scholar
  28. Ovaskainen O, Meerson B (2010) Stochastic models of population extinction. Trends Ecol Evolut 25:643–652CrossRefGoogle Scholar
  29. Park HJ, Traulsen A (2017) Extinction dynamics from metastable coexistences in an evolutionary game. Phys Rev E 96:042412CrossRefGoogle Scholar
  30. Parker M, Kamenev A (2009) Extinction in the Lotka–Volterra model. Phys Rev E 80:021129CrossRefGoogle Scholar
  31. Parsons TL, Quince C (2007) Fixation in haploid populations exhibiting density dependence i: the non-neutral case. Theor Popul Biol 72:121–135CrossRefMATHGoogle Scholar
  32. Parsons TL, Quince C, Plotkin JB (2008) Absorption and fixation times for neutral and quasi-neutral populations with density dependence. Theor Popul Biol 74:302–310CrossRefMATHGoogle Scholar
  33. Risken H (1996) Fokker–Planck equation. Springer, Berlin, pp 63–95MATHGoogle Scholar
  34. Shaffer ML (1981) Minimum population sizes for species conservation. BioScience 31:131–134CrossRefGoogle Scholar
  35. Smith NR, Meerson B (2016) Extinction of oscillating populations. Phys Rev E 93(3):032109CrossRefGoogle Scholar
  36. Svirezhev YM, Passekov VP (2012) Fundamentals of mathematical evolutionary genetics. Springer, BerlinMATHGoogle Scholar
  37. Touchette H (2009) The large deviation approach to statistical mechanics. Phys Rep 478(1):1–69MathSciNetCrossRefGoogle Scholar
  38. Traill LW, Bradshaw CJA, Brook BW (2007) Minimum viable population size: a meta-analysis of 30 years of published estimates. Biol Conserv 139:159–166CrossRefGoogle Scholar
  39. Tsoularis A, Wallace J (2002) Analysis of logistic growth models. Math Biosci 179(1):21–55MathSciNetCrossRefMATHGoogle Scholar
  40. Verhulst PF (1838) Notice sur la loi que la population suit dans son accroissement. Correspondance Mathématique et Physique Publiée par A. Quetelet 10:113–121Google Scholar
  41. Weber MF, Frey E (2017) Master equations and the theory of stochastic path integrals. Rep Prog Phys 80(4):046601MathSciNetCrossRefGoogle Scholar
  42. Wentzel G (1926) Eine verallgemeinerung der quantenbedingungen für die zwecke der wellenmechanik. Zeitschrift für Physik A Hadrons and Nuclei 38:518–529MATHGoogle Scholar

Copyright information

© 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

Personalised recommendations