ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Muzychuk O. V. Relaxation оf probability characteristics оf «beast - sacrifice» system in the meduim with Markoff fluctuations of parameters. Izvestiya VUZ. Applied Nonlinear Dynamics, 2002, vol. 10, iss. 1, pp. 170-180. DOI: 10.18500/0869-6632-2002-10-1-170-180

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
Article type: 

Relaxation оf probability characteristics оf «beast - sacrifice» system in the meduim with Markoff fluctuations of parameters

Muzychuk Oleg Vladimirovich, Nizhny Novgorod State University of Architecture

Well-known Lottky - Volterra scheme for self-regulated «beast - sacrifice» associations with Gaussian Markoff fluctuations of trophical coefficient was considered. Numerical solution of the system of relaxation equations, closured by special way, for population number mean values and dispersions was obtained. On this base the evolution оf model probability distributions was constructed. Though we used the approximative method оf analysis, for considering stochastic system it leads to exact results for stationary probability characteristics for both δ-correlated and quasi-static parameter’s fluctuations. For the last situation the relaxation of probability characteristics was investigated by numerical averaging оf stochastic equations.

Such analysis gave us the possibility to detect essential difference in results of the influence оf wide-band оr low-frequency parameter’s fluctuations оn statistical dynamics of the system.

Key words: 
This work was supported by grants RFBR № 99-02-17544, № 00-15-96620 and Ministry of Education of the Russian Federation № E00-3.5-216.
  1. Rabinovich MI, Trubetskov DI. Introduction to the Theory of Oscillations and Waves. Moscow: Nauka; 1984. 432 p. (in Russian).
  2. Svirezhev YM, Logofet DO. Stability of Biological Communities. MIR Publishers; 1983. 319 p.
  3. Svirezhev YM. Nonlinear Waves, Dissipative Structures and Disasters in Ecology. Moscow: Nauka; 1987. 368 p. (in Russian).
  4. Dimentberg MF. Exact solutions of the Fokker—Planck— Kolmogorov equation for certain multidimensional dynamic systems. Journal of Applied Mathematics and Mechanics. 1983;47(4):458-460. DOI: 10.1016/0021-8928(83)90082-5.
  5. Muzychuk OV. Probabilistic characteristics of the predator-victim system with randomly changing parameters. Izvestiya VUZ. Applied Nonlinear Dynamics. 1997;5(2-3):80-86 (in Russian).
  6. Feller V. An Introduction to Probability Theory and Its Applications. 2nd ed. Vol. 1. Wiley; 1957. 527 p. Vol. 2. Wiley; 1971. 704 p.
  7. Klyatskin VI. Stochastic Equations and Waves in Randomly Inhomogeneous Media. Moscow: Nauka; 1980. 336 p. (in Russian).
  8. Muzychuk OV. Analytic-numerical construction of nonstationary probability distributions for one class of nonlinear stochastic systems. Radiophysics and Quantum Electronics. 2000;43(9):742-748. DOI: 10.1023/A:1004842310933.
  9. Muzychuk OV. Relaxation of probabilistic characteristics of dynamic systems described by the Verhulst equation with “pink” noise. Izvestiya VUZ. Applied Nonlinear Dynamics. 2000;8(5):36-42 (in Russian).
  10. Malakhov AN. Cumulant Analysis of Non-Gaussian Random Processes and Their Transformations. Moscow: Sovetskoe Radio; 1978. 376 p. (in Russian).
Available online: