Skip to main content
SpringerLink
Log in

On the extinction route of a stochastic population model under heteroclinic bifurcation

异宿分岔下随机种群模型的灭绝路径

  • Research Paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

The noise-induced transition of the augmented Lotka-Volterra system is investigated under vanishingly small noise. Populations will ultimately go extinct because of intrinsic noise, and different extinction routes may occur due to the Freidlin-Wentzell large deviation theory. The relation between the most probable extinction route (MPER) and heteroclinic bifurcation is studied in this paper. The MPERs and the quasi-potentials in different regimes of parameters are analyzed in detail. Before the bifurcation, the predator goes extinct, and the prey will survive for a long time. Then, the heteroclinic bifurcation changes the MPER wherein both species go extinct. The heteroclinic cycle plays a role in transferring the most probable extinction state. Moreover, the analyses of the weak noise limit can contribute to predicting the stochastic behavior under finite small noise. Both the heteroclinic bifurcation and the rotational deterministic vector field can reduce the action necessary for the MPER.

摘要

本文在几近于零的小噪声情况下研究了增广Lotka-Volterra系统的噪声诱导跃迁. 由于固有噪声的存在, 种群最终将灭绝, 而Freidlin-Wentzell大偏差理论可能导致不同的灭绝路径. 本文研究了最有可能灭绝路径(MPER)与异宿分岔的关系, 详细分析不同参数区间的MPER和准势. 在分岔之前, 捕食者灭绝, 猎物将存活很长时间. 然后, 异宿分岔改变了两个物种灭绝的MPER. 异宿环起着转移最有可能灭绝状态的作用. 此外, 弱噪声极限分析有助于预测有限小噪声下的随机行为. 异宿分岔和旋转确定性向量场都可以减少MPER所需的作用量.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. I. A. Khovanov, A. V. Polovinkin, D. G. Luchinsky, and P. V. E. McClintock, Noise-induced escape in an excitable system, Phys. Rev. E 87, 032116 (2013).

    Article  Google Scholar 

  2. Z. Chen, J. Zhu, and X. Liu, Crossing the quasi-threshold manifold of a noise-driven excitable system, Proc. R. Soc. A. 473, 20170058 (2017).

    Article  MathSciNet  Google Scholar 

  3. A. Kamenev, and B. Meerson, Extinction of an infectious disease: A large fluctuation in a nonequilibrium system, Phys. Rev. E 77, 061107 (2008).

    Article  Google Scholar 

  4. M. I. Dykman, I. B. Schwartz, and A. S. Landsman, Disease extinction in the presence of random vaccination, Phys. Rev. Lett. 101, 078101 (2008).

    Article  Google Scholar 

  5. O. Gottesman, and B. Meerson, Multiple extinction routes in stochastic population models, Phys. Rev. E 85, 021140 (2012).

    Article  Google Scholar 

  6. N. R. Smith, and B. Meerson, Extinction of oscillating populations, Phys. Rev. E 93, 032109 (2016).

    Article  Google Scholar 

  7. Z. Chen, and X. Liu, Noise induced transitions and topological study of a periodically driven system, Commun. Nonlinear Sci. Numer. Simul. 48, 454 (2017).

    Article  MathSciNet  Google Scholar 

  8. Y. Li, J. Wang, and X. Liu, Quasi-threshold phenomenon in noise-driven Higgins model, Commun. Nonlinear Sci. Numer. Simul. 91, 105441 (2020).

    Article  MathSciNet  Google Scholar 

  9. L. Ryashko, Noise-induced complex oscillatory dynamics in the Zeldovich-Semenov model of a continuous stirred tank reactor, Chaos 31, 013105 (2021).

    Article  MathSciNet  Google Scholar 

  10. Y. Bomze, R. Hey, H. T. Grahn, and S. W. Teitsworth, Noise-induced current switching in semiconductor superlattices: observation of nonexponential kinetics in a high-dimensional system, Phys. Rev. Lett. 109, 026801 (2012).

    Article  Google Scholar 

  11. M. I. Dykman, E. Mori, J. Ross, and P. M. Hunt, Large fluctuations and optimal paths in chemical kinetics, J. Chem. Phys. 100, 5735 (1994).

    Article  Google Scholar 

  12. B. C. Nolting, and K. C. Abbott, Balls, cups, and quasi-potentials: quantifying stability in stochastic systems, Ecology 97, 850 (2016).

    Article  Google Scholar 

  13. H. Li, Y. Xu, Y. Li, and R. Metzler, Transition path dynamics across rough inverted parabolic potential barrier, Eur. Phys. J. Plus 135, 731 (2020).

    Article  Google Scholar 

  14. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences (Springer, New York, 2004).

    Book  Google Scholar 

  15. Z. Weiqiu, and C. Guoqiang, Nonlinear stochastic dynamics: A survey of recent developments, Acta Mech. Sin. 18, 551 (2002).

    Article  MathSciNet  Google Scholar 

  16. Q. F. Lü, W. Q. Zhu, and M. L. Deng, Reliability of quasi integrable and non-resonant Hamiltonian systems under fractional Gaussian noise excitation, Acta Mech. Sin. 36, 902 (2020).

    Article  MathSciNet  Google Scholar 

  17. M. I. Freidlin, and A. D. Wentzell, Random Perturbations of Dynamical Systems (Springer, New York, 2012).

    Book  Google Scholar 

  18. M. Heymann, and E. Vanden-Eijnden, The geometric minimum action method: A least action principle on the space of curves, Comm. Pure Appl. Math. 61, 1052 (2008).

    Article  MathSciNet  Google Scholar 

  19. M. Cameron, Finding the quasipotential for nongradient SDEs, Phys. D 241, 1532 (2012).

    Article  MathSciNet  Google Scholar 

  20. D. Dahiya, and M. Cameron, Ordered line integral methods for computing the quasi-potential, J. Sci. Comput. 75, 1351 (2018).

    Article  MathSciNet  Google Scholar 

  21. D. Dahiya, and M. Cameron, An Ordered Line Integral Method for Computing the Quasi-potential in the case of Variable Anisotropic Diffusion, Phys. D-Nonlinear Phenomena 382–383, 33 (2018).

    Article  MathSciNet  Google Scholar 

  22. S. Yang, S. F. Potter, and M. K. Cameron, Computing the quasipotential for nongradient SDEs in 3D, J. Comput. Phys. 379, 325 (2018).

    Article  MathSciNet  Google Scholar 

  23. S. Beri, R. Mannella, D. G. Luchinsky, A. N. Silchenko, and P. V. E. McClintock, Solution of the boundary value problem for optimal escape in continuous stochastic systems and maps, Phys. Rev. E 72, 036131 (2005).

    Article  MathSciNet  Google Scholar 

  24. B. S. Lindley, and I. B. Schwartz, An iterative action minimizing method for computing optimal paths in stochastic dynamical systems, Phys. D-Nonlinear Phenomena 255, 22 (2013).

    Article  MathSciNet  Google Scholar 

  25. D. Ludwig, Persistence of dynamical systems under random perturbations, Siam Rev. 17, 605 (1975).

    Article  MathSciNet  Google Scholar 

  26. T. Grafke, and E. Vanden-Eijnden, Numerical computation of rare events via large deviation theory, Chaos 29, 063118 (2019).

    Article  MathSciNet  Google Scholar 

  27. M. Assaf, and B. Meerson, Extinction of metastable stochastic populations, Phys. Rev. E 81, 021116 (2010).

    Article  Google Scholar 

  28. D. G. Schaeffer, and J. W. Cain, Ordinary Differential Equations: Basics and Beyond (Springer, New York, 2016).

    Book  Google Scholar 

  29. W. E, and X. Zhou, Study of noise-induced transitions in the lorenz system using the minimum action method, Commun. Math. Sci. 8, 341 (2010).

    Article  MathSciNet  Google Scholar 

  30. E. Forgoston, and R. O. Moore, A primer on noise-induced transitions in applied dynamical systems, SIAM Rev. 60, 969 (2018).

    Article  MathSciNet  Google Scholar 

  31. R. S. Maier, and D. L. Stein, A scaling theory of bifurcations in the symmetric weak-noise escape problem, J Stat Phys 83, 291 (1996).

    Article  MathSciNet  Google Scholar 

  32. R.V. Roy, Asymptotic analysis of first-passage problems, Int. J. Nonlin. Mech. 32, 173 (1997).

    Article  MathSciNet  Google Scholar 

  33. N. Berglund, and B. Gentz, On the noise-induced passage through an unstable periodic orbit II: general case, SIAM J. Math. Anal. 46, 310 (2014).

    Article  MathSciNet  Google Scholar 

  34. R. de la Cruz, R. Perez-Carrasco, P. Guerrero, T. Alarcon, and K. M. Page, Minimum action path theory reveals the details of stochastic transitions out of oscillatory states, Phys. Rev. Lett. 120, 128102 (2018).

    Article  Google Scholar 

  35. S. J. B. Einchcomb, and A. J. McKane, Use of Hamiltonian mechanics in systems driven by colored noise, Phys. Rev. E 51, 2974 (1995).

    Article  Google Scholar 

  36. M. I. Dykman, Large fluctuations and fluctuational transitions in systems driven by colored Gaussian noise: A high-frequency noise, Phys. Rev. A 42, 2020 (1990).

    Article  MathSciNet  Google Scholar 

  37. H. Li, Y. Xu, R. Metzler, and J. Kurths, Transition path properties for one-dimensional systems driven by Poisson white noise, Chaos Solitons Fractals 141, 110293 (2020).

    Article  MathSciNet  Google Scholar 

  38. W. Zan, Y. Xu, R. Metzler, and J. Kurths, First-passage problem for stochastic differential equations with combined parametric Gaussian and Lévy white noises via path integral method, J. Comput. Phys. 435, 110264 (2021).

    Article  Google Scholar 

  39. S. S. Pan, and W. Q. Zhu, Dynamics of a prey-predator system under Poisson white noise excitation, Acta Mech. Sin. 30, 739 (2014).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Qing Yu  (虞晴) & Xianbin Liu  (刘先斌)

  2. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yang Li  (李扬)

Authors
  1. Qing Yu  (虞晴)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yang Li  (李扬)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xianbin Liu  (刘先斌)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xianbin Liu  (刘先斌).

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 11772149, and 12172167), A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD) and The Research Fund of State Key Laboratory of Mechanics and Control of Mechanical Structures (Grant No. MCMS-I-19G01).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yu, Q., Li, Y. & Liu, X. On the extinction route of a stochastic population model under heteroclinic bifurcation. Acta Mech. Sin. 38, 221333 (2022). https://doi.org/10.1007/s10409-021-09062-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10409-021-09062-x

Keywords

Search

Navigation