Skip to main content
SpringerLink
Log in

Verhulst model with Lévy white noise excitation

  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

The transient dynamics of the Verhulst model perturbed by arbitrary non-Gaussian white noise is investigated. Based on the infinitely divisible distribution of the Lévy process we study the nonlinear relaxation of the population density for three cases of white non-Gaussian noise: (i) shot noise; (ii) noise with a probability density of increments expressed in terms of Gamma function; and (iii) Cauchy stable noise. We obtain exact results for the probability distribution of the population density in all cases, and for Cauchy stable noise the exact expression of the nonlinear relaxation time is derived. Moreover starting from an initial delta function distribution, we find a transition induced by the multiplicative Lévy noise, from a trimodal probability distribution to a bimodal probability distribution in asymptotics. Finally we find a nonmonotonic behavior of the nonlinear relaxation time as a function of the Cauchy stable noise intensity.

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. W. Horsthemke, R. Lefever, Noise-Induced Transitions: Theory and Applications in Physics, Chemistry and Biology (Springer-Verlag, Berlin, 1984)

    MATH  Google Scholar 

  2. M. Eigen, P. Schuster, The Hypercycle: A Principle of Natural Self-Organization (Springer, Berlin, 1979)

    Google Scholar 

  3. A. Morita, J. Chem. Phys. 76, 4191 (1982)

    Article  ADS  Google Scholar 

  4. S. Ciuchi, F. de Pasquale, B. Spagnolo, Phys. Rev. E 47, 3915 (1993)

    Article  ADS  Google Scholar 

  5. J.H. Mathis, T.R. Kiffe, Stochastic Population Models: A Compartmental Perspective (Springer-Verlag, Berlin, 1984)

    Google Scholar 

  6. M. Eigen, Naturwissenschaften 58, 465 (1971)

    Article  ADS  Google Scholar 

  7. L. Acedo, Physica A 370, 613 (2006)

    Article  ADS  Google Scholar 

  8. Bao-Quan Ai, Xian-Ju Wang, Guo-Tao Liu, Liang-Gang Liu, Phys. Rev. E 67, 022903–1 (2003)

    Article  ADS  Google Scholar 

  9. G. DeRise, J.A. Adam, J. Phys. A: Math. Gen. 23, L727S (1990)

    Article  ADS  MathSciNet  Google Scholar 

  10. S. Ciuchi, F. de Pasquale, B. Spagnolo, Phys. Rev. E 54, 706 (1996)

    Article  ADS  Google Scholar 

  11. K.J. McNeil, D.F. Walls, J. Stat. Phys. 10, 439 (1974)

    Article  ADS  Google Scholar 

  12. H. Ogata, Phys. Rev. A 28, 2296 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  13. F. Schlögl, Z. Phys. 253, 147 (1972)

    Article  ADS  Google Scholar 

  14. S. Chaturvedi, C.W. Gardiner, D.F. Walls, Phys. Lett. A 57, 404 (1976)

    Article  ADS  Google Scholar 

  15. C.W. Gardiner, S. Chaturvedi, J. Stat. Phys. 17, 429 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  16. V. Bouché, J. Phys. A: Math. Gen. 15, 1841 (1982)

    Article  ADS  Google Scholar 

  17. H.K. Leung, J. Chem. Phys. 86, 6847 (1987)

    Article  ADS  Google Scholar 

  18. A.K. Das, Can. J. Phys. 61, 1046 (1983)

    MATH  ADS  Google Scholar 

  19. R. Herman, E.W. Montroll, Proc. Natl. Acad. Sci. USA 69, 3019 (1972)

    Article  ADS  Google Scholar 

  20. E.W. Montroll, Proc. Natl. Acad. Sci. USA 75, 4633 (1978)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  21. H.K. Leung, Phys. Rev. A 37, 1341 (1988)

    Article  ADS  Google Scholar 

  22. P.J. Jackson, C.J. Lambert, R. Mannella, P. Martano, P.V.E. McClintock, N.G. Stocks, Phys. Rev. A 40, 2875 (1989)

    Article  ADS  Google Scholar 

  23. K. Binder, Phys. Rev. B 8, 3423 (1973)

    Article  ADS  Google Scholar 

  24. J. Golec, S. Sathananthan, Math. Comput. Modell. 38, 585 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. R. Mannella, C.J. Lambert, N.G. Stocks, P.V.E. McClintock, Phys. Rev. A 41, 3016 (1990)

    Article  ADS  Google Scholar 

  26. H. Calisto, M. Bologna, Phys. Rev. E 75, 050103–1(R) (2007)

    Article  ADS  Google Scholar 

  27. M. Suzuki, K. Kaneko, S. Takesue, Prog. Theor. Phys. 67, 1756 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. M. Suzuki, S. Takesue, F. Sasagawa, Prog. Theor. Phys. 68, 98 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  29. L. Brenig, N. Banai, Physica D 5, 208 (1982)

    Article  ADS  MathSciNet  Google Scholar 

  30. J. Makino, A. Morita, Progr. Theor. Phys. 73, 1268 (1985)

    Article  ADS  Google Scholar 

  31. A. Morita, J. Makino, Phys. Rev. A 34, 1595 (1986)

    Article  ADS  Google Scholar 

  32. A.A. Dubkov, B. Spagnolo, Fluct. Noise Lett. 5, L267 (2005)

    Article  MathSciNet  Google Scholar 

  33. A.A. Dubkov, B. Spagnolo, V.V. Uchaikin, Intern. J. Bifurcation and Chaos (2008), in press

  34. W. Feller, An Introduction to Probability Theory and its Applications (John Wiley & Sons, Inc., New York, 1971), Vol. 2

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Radiophysics Department, Nizhniy Novgorod State University, 23 Gagarin Ave., 603950, Nizhniy Novgorod, Russia

    A. A. Dubkov

  2. Dipartimento di Fisica e Tecnologie Relative, Group of Interdisciplinary Physics, Università di Palermo and CNISM-INFM, Unità di Palermo, Viale delle Scienze, 90128, Palermo, Italy

    B. Spagnolo

Authors
  1. A. A. Dubkov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Spagnolo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Dubkov.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Dubkov, A.A., Spagnolo, B. Verhulst model with Lévy white noise excitation. Eur. Phys. J. B 65, 361–367 (2008). https://doi.org/10.1140/epjb/e2008-00337-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1140/epjb/e2008-00337-0

PACS

Search

Navigation