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