Noise-induced extinction in Bazykin-Berezovskaya population model

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Abstract

A nonlinear Bazykin-Berezovskaya prey-predator model under the influence of parametric stochastic forcing is considered. Due to Allee effect, this conceptual population model even in the deterministic case demonstrates both local and global bifurcations with the change of predator mortality. It is shown that random noise can transform system dynamics from the regime of coexistence, in equilibrium or periodic modes, to the extinction of both species. Geometry of attractors and separatrices, dividing basins of attraction, plays an important role in understanding the probabilistic mechanisms of these stochastic phenomena. Parametric analysis of noise-induced extinction is carried out on the base of the direct numerical simulation and new analytical stochastic sensitivity functions technique taking into account the arrangement of attractors and separatrices.

Keywords

Statistical and Nonlinear Physics 
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References

  1. 1.
    A.D. Bazykin, Nonlinear Dynamics of Interacting Populations (World Scientific, 1998)Google Scholar
  2. 2.
    F. Brauer, C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology (Springer-Verlag, 2001), TAM 40Google Scholar
  3. 3.
    P. Turchin, Complex Population Dynamics: a Theoretical/Empirical Synthesis (Princeton University Press, 2003)Google Scholar
  4. 4.
    M. Rietkerk, S.C. Dekker, P.C. de Ruiter, J. van de Koppel, Science 305, 1926 (2004)Google Scholar
  5. 5.
    R. Lande, S. Engen, B.-E. Saether, Stochastic Population Dynamics in Ecology and Conservation (Oxford University Press, 2003)Google Scholar
  6. 6.
    L. Ridolfi, P. D’Odorico, F. Laio, Noise-Induced Phenomena in the Environmental Sciences (Cambridge University Press, 2011)Google Scholar
  7. 7.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer, 1984)Google Scholar
  8. 8.
    V.S. Anishchenko, V.V. Astakhov, A.B. Neiman, T.E. Vadivasova, L. Schimansky-Geier, Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development (Springer-Verlag, 2007)Google Scholar
  9. 9.
    B. Lindner, J. Garcia-Ojalvo, A. Neiman, L. Schimansky-Geier, Phys. Rep. 392, 321 (2004)ADSCrossRefGoogle Scholar
  10. 10.
    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998)ADSCrossRefGoogle Scholar
  11. 11.
    M.D. McDonnell, N.G. Stocks, C.E.M. Pearce, D. Abbott, Stochastic Resonance: from Suprathreshold Stochastic Resonance to Stochastic Signal Quantization (Cambridge University Press, 2008)Google Scholar
  12. 12.
    J.B. Gao, S.K. Hwang, J.M. Liu, Phys. Rev. Lett. 82, 1132 (1999)ADSCrossRefGoogle Scholar
  13. 13.
    K. Matsumoto, I. Tsuda, J. Stat. Phys. 31, 87 (1983)ADSMathSciNetCrossRefGoogle Scholar
  14. 14.
    Y.-C. Lai, T. Tel, Transient Chaos: Complex Dynamics on Finite Time Scales (Springer, 2011)Google Scholar
  15. 15.
    L.J.S. Allen, An Introduction to the Stochastic Process With Applications to Biology (Pearson Education, 2003)Google Scholar
  16. 16.
    B. Spagnolo, D. Valenti, A. Fiasconaro, Math. Biosci. Eng. 1, 185 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Hänggi, ChemPhysChem 3, 285 (2002)CrossRefGoogle Scholar
  18. 18.
    S. Petrovskii, A. Morozov, H. Malchow, M. Sieber, Eur. Phys. J. B 78, 253 (2010)ADSCrossRefGoogle Scholar
  19. 19.
    Q. He, M. Mobilia, U.C. Täuber, Eur. Phys. J. B 82, 97 (2011)ADSCrossRefGoogle Scholar
  20. 20.
    J. Müller, C. Kuttler, Methods and Models in Mathematical Biology: Deterministic and Stochastic Approaches (Springer, 2015)Google Scholar
  21. 21.
    D. Valenti et al., arXiv:1511.07266v2, to be published in Math. Model. Nat. Phenom. (2016)
  22. 22.
    S. Kraut, U. Feudel, Phys. Rev. E 66, 015207 (2002)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    A.L. Kawczyński, B. Nowakowski, Phys. Chem. Chem. Phys. 10, 289 (2008)CrossRefGoogle Scholar
  24. 24.
    I. Bashkirtseva, A.B. Neiman, L. Ryashko, Phys. Rev. E 87, 052711 (2013)ADSCrossRefGoogle Scholar
  25. 25.
    D.V. Alexandrov, I. Bashkirtseva, L. Ryashko, Eur. Phys. J. B 89, 62 (2016)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    G.A. van Voorn, L. Hemerik, M.P. Boer, B.W. Kooi, Math Biosci. 209, 451 (2007)MathSciNetCrossRefGoogle Scholar
  27. 27.
    W.Z. Lidicker Jr., Open Ecol. J. 3, 71 (2010)CrossRefGoogle Scholar
  28. 28.
    B. Dennis, Oikos 96, 389 (2002)CrossRefGoogle Scholar
  29. 29.
    J.M. Drake, D.M. Lodge, Biological Invasions 8, 365 (2006).CrossRefGoogle Scholar
  30. 30.
    G.-Q. Sun, Z. Jin, L. Li, Q.-X. Liu, J. Biol. Phys. 35, 185 (2009).CrossRefGoogle Scholar
  31. 31.
    I. Bashkirtseva, L. Ryashko, Chaos 21, 047514 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    E.M. Hart, L. Avilés, PLoS One 9, e110049 (2014)CrossRefGoogle Scholar
  33. 33.
    C. Kurrer, K. Schulten, Physica D 50, 311 (1991)ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    M.I. Freidlin, A.D. Wentzell, Random Perturbations of Dynamical Systems (Springer, 1984)Google Scholar
  35. 35.
    I. Bashkirtseva, G. Chen, L. Ryashko, Chaos 22, 033104 (2012)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    L. Ryashko, I. Bashkirtseva, Phys. Rev. E 83, 061109 (2011)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Ural Federal UniversityEkaterinburgRussia

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