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Dynamics of nonlinear ecosystems under colored noise disturbances

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Abstract

Stochastic ecosystems of prey-predator type subjected to colored noises with broad-band spectra are investigated. Nonlinear models are considered for two different scenarios: one is the case of possible abundant prey supply and another is the case of possible large predator population. The stochastic averaging procedure is applied to obtain stationary probability solutions of the nonlinear systems. Two types of colored noise are considered: one is the low-pass filtered noise with the spectrum peak at zero frequency, and another is the randomized harmonic process with the spectrum peak at a nonzero frequency. For either type of the noises, the band width reflecting the level of the noise color can be adjusted using a single parameter. The analytical results are substantiated by those obtained from Monte Carlo simulations. It is found that the noise color has significant effects on the stationary state of the system. A narrower band width leads to a less stable system in the sense that the prey and predator populations deviate farther from the equilibrium point of the system without noise disturbances.

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References

  1. Lotka, A.J.: Elements of Physical Biology. William and Wilkins, Baltimore (1925)

    MATH  Google Scholar 

  2. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie d’animani conviventi. Mem. Acad. Lincei 2, 31–113 (1926)

    Google Scholar 

  3. Volterra, V.: Lecons sur la Theorie Mathematique de la Lutte Pour la Vie. Gauthiers-Vilars, Paris (1931)

    Google Scholar 

  4. May, R.M.: Stability and Complexity in Model Ecosystems. Oxford University Press, London (1973)

    Google Scholar 

  5. May, R.M.: Theoretical Ecology, Principles and Applications, 2nd edn. Sinauer Associates, Sunderland (1981)

    Google Scholar 

  6. Murray, J.D.: Mathematical Biology. Springer, New York (1993)

    Book  MATH  Google Scholar 

  7. Rosenzweig, M.L., MacArthue, R.H.: Graphical representation and stability conditions of predator–prey interactions. Am. Nat. 97, 205–223 (1963)

    Article  Google Scholar 

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

    Google Scholar 

  9. Arnold, L., Horthemke, W., Stucki, J.W.: The influence of external real and white noise on the Lotka–Volterra model. Biom. J. 21, 451–471 (1979)

    Article  MATH  Google Scholar 

  10. Rozenfeld, A.F., Tessone, C.J., Albano, E., Wio, H.S.: 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 

  11. Khasminskii, R.Z., Klebaner, F.C.: Long term behavior of solutions of the Lotka–Volterra system under small random perturbations. Ann. Appl. Probab. 11, 952–963 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dimentberg, M.F.: Lotka–Volterra system in a random environment. Phys. Rev. E 65, 036204 (2002)

    Article  MathSciNet  Google Scholar 

  13. Dimentberg, M.F.: Stochastic Lotka–Volterra system. In: Proceedings of IUTAM Symposium on Nonlinear Stochastic Dynamics, pp. 307–317. Kluwer Academic, Dordrecht (2003)

    Chapter  Google Scholar 

  14. Cai, G.Q., Lin, Y.K.: Stochastic analysis of the Lotka–Volterra model for ecosystems. 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. Wu, Y., Zhu, W.Q.: Stochastic analysis of a pulse-type prey–predator model. Phys. Rev. E 77, 041911 (2008)

    Article  MathSciNet  Google Scholar 

  17. Vaseur, D.A., Yodzis, P.: The color of environmental noise. Ecology 85, 1146–1152 (2004)

    Article  Google Scholar 

  18. Naess, A., Dimenberg, M.F., Gaidai, O.: Lotka–Volterra systems in environments with randomly disordered temporal periodicity. Phys. Rev. E 78, 021126 (2008)

    Article  MathSciNet  Google Scholar 

  19. Stratonovich, R.L.: Topics in the Theory of Random Noise, vol. 1. Gordon and Breach, New York (1963)

    Google Scholar 

  20. Khasminskii, R.Z.: A limit theorem for the solution of differential equations with random right hand sides. Theory Probab. Appl. 12, 144–147 (1966)

    Article  Google Scholar 

  21. Itô, K.: On stochastic differential equations. Mem. Am. Math. Soc. 4, 289–302 (1951)

    Google Scholar 

  22. Lin, Y.K., Cai, G.Q.: Probabilistic Structural Dynamics. McGraw-Hill, New York (2004)

    Google Scholar 

  23. Dimentberg, M.F.: Statistical Dynamics of Nonlinear and Time-Varying Systems. Wiley, New York (1988)

    MATH  Google Scholar 

  24. Wedig, W.V.: Analysis and simulation of nonlinear stochastic systems. In: Nonlinear Dynamics in Engineering Systems, pp. 337–344. Springer, Berlin (1989)

    Google Scholar 

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Correspondence to G. Q. Cai.

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Qi, L., Cai, G.Q. Dynamics of nonlinear ecosystems under colored noise disturbances. Nonlinear Dyn 73, 463–474 (2013). https://doi.org/10.1007/s11071-013-0801-3

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  • DOI: https://doi.org/10.1007/s11071-013-0801-3

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