Steady-state probability characteristics of Verhulst and Hongler models with multiplicative white Poisson noise

  • Alexander A. DubkovEmail author
  • Anna A. Kharcheva
Regular Article


Based on recently obtained from Kolmogorov–Feller equation exact analytical results for the steady-state probability density function of nonlinear dynamical systems driven by white Poisson noise with exponentially distributed amplitudes of pulses we analyze some models of ecology and genetics. Specifically, we find the steady-state probability distribution of the population density in the framework of well-known Verhulst equation with fluctuating population mortality in the form of Poisson sequence with unipolar pulses, leading to an abrupt decrease in population density at random times. As shown, the most probable value of the population density tends to zero with increasing the mean rate of pulses, that is, to an extinction of biological population in perspective. Further, we consider the stochastic Hongler equation which can serve as an approximate model of genetic selection. In the case of multiplicative white Poisson noise having bipolar exponentially distributed amplitudes of pulses we observe noise-induced transition to bimodality (through the trimodal phase) in the steady-state probability distribution with an increase in the mean frequency of pulses. We also discovered a new phenomenon, namely, a direct transition from unimodality to trimodality with a change in the noise intensity, which could not be detected in the framework of the Gaussian perturbation.

Graphical abstract


Statistical and Nonlinear Physics 


Author contribution statement

A.A. Dubkov performed most of the calculations and wrote the manuscript. A.A. Kharcheva helped with calculations, graphing, discussing the results obtained and revising the manuscript.


  1. 1.
    B.V. Gnedenko,The Theory of Probability (MIR Publishers, Moscow, 1969) Google Scholar
  2. 2.
    C. Van den Broeck, J. Stat. Phys. 31, 467 (1983) ADSCrossRefGoogle Scholar
  3. 3.
    J.M. Sancho, M. San Miguel, L. Pesquera, M.A. Rodriguez, Physica A 142, 532 (1987) ADSCrossRefGoogle Scholar
  4. 4.
    J. Łuczka, R. Bartussek, P. Hänggi, Europhys. Lett. 31, 431 (1995) ADSCrossRefGoogle Scholar
  5. 5.
    R. Zygadło, Phys. Lett. A 329, 459 (2004) ADSMathSciNetCrossRefGoogle Scholar
  6. 6.
    E. Daly, A. Porporato, Phys. Rev. E 73, 026108 (2006) ADSCrossRefGoogle Scholar
  7. 7.
    A.A. Dubkov, O.V. Rudenko, S.N. Gurbatov, Phys. Rev. E 93, 062125 (2016) ADSCrossRefGoogle Scholar
  8. 8.
    O.V. Rudenko, A.A. Dubkov, S.N. Gurbatov, Dokl. Math. 94, 476 (2016) MathSciNetCrossRefGoogle Scholar
  9. 9.
    A.A. Dubkov, B. Spagnolo, Fluct. Noise Lett. 5, L267 (2005) CrossRefGoogle Scholar
  10. 10.
    V.I. Klyatskin,Dynamics of Stochastic Systems (Amsterdam, Netherlands. Elsevier, 2005) Google Scholar
  11. 11.
    D.R. Cox, V. Isham, Adv. Appl. Prob. 18, 558 (1986) CrossRefGoogle Scholar
  12. 12.
    J.M.G. Vilar, J.M. Rubi, Sci. Rep. 8, 887 (2018) ADSCrossRefGoogle Scholar
  13. 13.
    R. Zygadło, Phys. Rev. E 54, 5964 (1996) ADSCrossRefGoogle Scholar
  14. 14.
    R. Zygadło, Phys. Rev. E 77, 021130 (2008) ADSCrossRefGoogle Scholar
  15. 15.
    A. Dubkov, Acta Phys. Pol. B 43, 935 (2012) Google Scholar
  16. 16.
    A.A. Dubkov, B. Spagnolo, Eur. Phys. J. B 65, 361 (2008) ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    M. Bologna, H. Calisto, Eur. Phys. J. B 83, 409 (2011) ADSCrossRefGoogle Scholar
  18. 18.
    A.A. Dubkov, A.A. Kharcheva, Phys. Rev. E 89, 052146 (2014) ADSCrossRefGoogle Scholar
  19. 19.
    W. Horsthemke, R. Lefever,Noise-Induced Transitions: Theory and Applications in Physics, Chemistry, and Biology (Springer-Verlag, Berlin, 1984) Google Scholar
  20. 20.
    M.O. Hongler, Helv. Phys. Acta 52, 280 (1979) MathSciNetGoogle Scholar
  21. 21.
    O.A. Chichigina, A.A. Dubkov, D. Valenti, B. Spagnolo, Phys. Rev. E 84, 021134 (2011) ADSCrossRefGoogle Scholar
  22. 22.
    M. Abramowitz, I.A. Stegun,Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables (Dover Publications, Inc., New York, 1972) Google Scholar

Copyright information

© EDP Sciences / Società Italiana di Fisica / Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Radiophysics Department, Lobachevsky State UniversityNizhni NovgorodRussia

Personalised recommendations