Skip to main content
SpringerLink
Log in

Dynamics of a prey-predator system under Poisson white noise excitation

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

Abstract

The classical Lotka-Volterra (LV) model is a well-known mathematical model for prey-predator ecosystems. In the present paper, the pulse-type version of stochastic LV model, in which the effect of a random natural environment has been modeled as Poisson white noise, is investigated by using the stochastic averaging method. The averaged generalized Itô stochastic differential equation and Fokker-Planck-Kolmogorov (FPK) equation are derived for prey-predator ecosystem driven by Poisson white noise. Approximate stationary solution for the averaged generalized FPK equation is obtained by using the perturbation method. The effect of prey self-competition parameter ɛ2 s on ecosystem behavior is evaluated. The analytical result is confirmed by corresponding Monte Carlo (MC) simulation.

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. May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton, NJ (1973)

    Google Scholar 

  2. Smith, J.M.: Models in Ecology. Cambridge University Press, Cambridge, UK (1974)

    MATH  Google Scholar 

  3. Berryman, A.A.: Population Cycles: The Case for Trophic Interactions. Oxford University Press, London (2002)

    Google Scholar 

  4. Haken, H.: Synergetics (3rd edn). Springer-Verlag, Berlin (2002)

    Google Scholar 

  5. Neal, D.: Introduction to Population Biology. Cambridge University Press, Cambridge, UK (2004)

    Google Scholar 

  6. Lotka, A.J.: Undamped oscillations derived from the law of mass action. J. Am. Chem. Soc. 42, 1595–1599 (1920)

    Article  Google Scholar 

  7. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei 2, 31–113 (1926)

    Google Scholar 

  8. Volterra, V.: Variations and fluctuations of the number of individuals in animal species living together. In: Chapman, R.N. ed. Animal Ecology, McGraw-Hill, New York, 409–448 (1931)

    Google Scholar 

  9. Rosenzweig, M.L., MacArthue, R.H.: Graphical representation and stability conditions of predator-prey interactions. American Nature 97, 205–223 (1963)

    Article  Google Scholar 

  10. Bazykin, A.D.: Nonlinear Dynamics of Interacting Populations. World Scientific, Singapore (1998)

    Google Scholar 

  11. MacDonald, N.: Time delay in prey-predator models. Mathematical Biosciences 28, 321–330 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Hornfeldt, B.: Delayed density dependence as a determinant of vole cycles. Ecology 75, 791–806 (1994)

    Article  Google Scholar 

  13. Turchin, P.: Complex Population Dynamics: A Theoretical/Empirical Synthesis. Princeton University Press, Princeton, NJ (2003)

    Google Scholar 

  14. Cai, G.Q., Lin, Y.K.: Stochastic analysis of the Lotka-Volterra model for ecosystem. Phys. Rev. E 70, 041910 (2004)

    Article  Google Scholar 

  15. Cai, G.Q., Lin, Y.K.: Stochastic analysis of time-delayed ecosystems. Phys. Rev. E 76, 041913 (2007)

    Article  Google Scholar 

  16. Rozenfeld, A.F., Tessone, C.J., Albano, E., et al.: On the influence of noise on the critical and oscillatory behavior of a predator-prey model: Coherent stochastic resonance at the proper frequency of the system. Phys. Lett. A 280, 45–52 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Reichenbach, T., Mobilia, M., Frey, E.: Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. Phys. Rev. E 74, 051907 (2006)

    Article  MathSciNet  Google Scholar 

  18. Wu, Y., Zhu, W.Q.: Stochastic analysis of a pulse-type preypredator model. Phys. Rev. E 77, 041911 (2008)

    Article  MathSciNet  Google Scholar 

  19. Rauch, E.M., Sayama, H., Bar-Yam, Y.: Relationship between measures of fitness and time scale in evolution. Phys. Rev. Lett 88, 228101 (2002)

    Article  Google Scholar 

  20. Mankin, R., Laas, T., Sauga, A., et al.: Colored-noise-induced hopf bifurcations in predator-prey communities. Phys. Rev. E 74, 021101 (2006)

    Article  MathSciNet  Google Scholar 

  21. Xu, Y., Duan, J., Xu, W.: An averaging principle for stochastic dynamical systems with Lévy noise. Physica D 240, 1395–1401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Xu, Y., Gu, R., Zhang, H., et al.: Stochastic bifurcations in a bistable Duffing-Van der Pol oscillator with colored noise. Phys. Rev. E 83, 056215 (2011)

    Article  Google Scholar 

  23. Zeng, Y., Zhu, W.Q.: Stochastic averaging of quasi linear systems driven by Poisson white noise. Probab. Engng. Mech. 25, 99–107 (2010)

    Article  Google Scholar 

  24. Zeng, Y., Zhu, W.Q.: Generalized Fokker-Planck-Kolmogorov equation for response of nonlinear dynamical systems to non-Gaussian random excitation. Int. J. Non-linear Mech. 45, 572–586 (2010)

    Article  Google Scholar 

  25. Di Paola, M., Falsone, G.: Stochastic dynamics of non-linear systems driven by non-normal delta-correlated processes. J. Appl. Mech. 60, 141–148 (1993)

    Article  MATH  Google Scholar 

  26. Di Paola, M., Falsone, G.: Itô and Stratonovich integrals for delta-correlated processes. Probab. Engng. Mech. 8, 197–208 (1993)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics, State Key Lab of Fluid Power Transmission and Control, Zhejiang University, 310027, Hangzhou, China

    Shan-Shan Pan & Wei-Qiu Zhu

Authors
  1. Shan-Shan Pan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei-Qiu Zhu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei-Qiu Zhu.

Additional information

The project was supported by the National Natural Science Foundation of China (10932009, 11072212, 11272279, and 11321202).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pan, SS., Zhu, WQ. Dynamics of a prey-predator system under Poisson white noise excitation. Acta Mech Sin 30, 739–745 (2014). https://doi.org/10.1007/s10409-014-0069-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-014-0069-y

Keywords

Search

Navigation