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

For citation:

Bashkirtseva I. A., Karpenko L. V., Ryashko L. B. Stochastic sensitivity of limit cycles for «predator – two preys» model. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 6, pp. 42-64. DOI: 10.18500/0869-6632-2010-18-6-42-64

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):
(downloads: 146)
Article type: 

Stochastic sensitivity of limit cycles for «predator – two preys» model

Bashkirtseva Irina Adolfovna, Ural Federal University named after the first President of Russia B.N.Yeltsin
Karpenko Larisa Vladimirovna, Ural Federal University named after the first President of Russia B.N.Yeltsin
Ryashko Lev Borisovich, Ural Federal University named after the first President of Russia B.N.Yeltsin

We consider the population dynamics model «predator – two preys». A deterministic stability of limit cycles of this three-dimensional model in a period doubling bifurcations zone at the transition from an order to chaos is investigated. Stochastic sensitivity of cycles for additive and parametrical random disturbances is analyzed with the help of stochastic sensitivity function technique. Thin effects of stochastic influences are demonstrated. Growth of stochastic sensitivity of cycles for period doubling under transition from order to chaos is shown. For the index of sensitivity growth the universality low is established.

  1. Kolmogorov AN. Qualitative study of mathematical models of population dynamics. In: Problems of cybernetics. Moscow: Nauka. 1972;25:100–106. (in Russian).
  2. Svirezhev YuM,  Logofet DO. Stability of Biological Communities. Moscow: Mir; 1983.
  3. Bazykin AD. Mathematical biophysics of interacting populations. Moscow: Nauka; 1985. 181 p. (in Russian).
  4. Romanovsky YuM, Stepanova NV, Chernavsky DS. Mathematical biophysics. Moscow: Nauka; 1984. 304 p. (in Russian).
  5. Turchin P. Complex population dynamics: a theoretical/empirical synthesis. Princeton University Press; 2003.
  6. Morozov A, Petrovskii S, Li BL. Bifurcations and chaos in a predator-prey system with the Allee effect. Proc. Royal Soc. London Series B–Biol. Sci. 2004;271:1407–1414. DOI: 10.1098/rspb.2004.2733.
  7. Krivan V. Optimal foraging and predator-prey dynamics. Theoretical Population Biology. 1996;49(3):265–290. DOI: 10.1006/tpbi.1996.0014.
  8. Arneodo A, Coullet P, Tresser C. Occurrence of strange attractors in three dimensional Volterra equations. Phys. Lett. A. 1980;79(4):259–263. DOI: 10.1016/0375-9601(80)90342-4.
  9. Xiao D, Li W. Limit cycles for the competitive three dimensional Lotka–Volterra system. J. Diff. Eqns. 2000;164(1):1–15. DOI: 10.1006/jdeq.1999.3729.
  10. Aponina EA, Aponin YuM, Bazykin AD. Analysis of complex dynamic behavior in the model «predator-two preys». Problems of Ecological Monitoring and Modelling of Ecosystem. Leningrad: Gidrometeoizdat. 1982;5:163–180 (in Russian). 
  11. Feigenbaum MJ. Quantitative universality for a class of nonlinear transformations. J. of Stat. Phys. 1978;19(1):25–52. DOI: 10.1007/BF01020332.
  12. Lorenz EN. Deterministic nonperiodic flow. J. Atm. Sci. 1963;20(2):130–141. DOI: 10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2.
  13. Rossler OE. An equation for continuous chaos. Phys. Lett. A. 1976;57(5):397–398. DOI: 10.1016/0375-9601(76)90101-8.
  14. Chua LO, Komuro M, Matsumoto T. The double scroll family. IEEE Trans. Circuits Syst. 1986;CAS-33(11):1072–1118. DOI: 10.1109/TCS.1986.1085869.
  15. Anishchenko VS. Complex Oscillations in Simple Systems. Moscow: Nauka; 1990. (in Russian).
  16. Arnold L. Random Dynamical Systems. Springer-Verlag; 1998.
  17. Blank ML. Finite-dimensional stochastic attractors of infinite-dimensional dynamical systems. Funct. Anal. Appl. 1986;20(2):128–130.
  18. Scheutzow M. Comparison of various concepts of a random attractor: A case study. Arch. Math. 2002;78:233–240. DOI: 10.1007/s00013-002-8241-1.
  19. Schmalfuss B. The random attractor of the stochastic Lorenz system. ZAMP. 1997;48:951–975. DOI: 10.1007/s000330050074.
  20. Billings L, Schwartz IB. Exciting chaos with noise: unexpected dynamics in epidemic outbreaks. J. Math. Biol. 2002;44(1):31–48. DOI: 10.1007/s002850100110.
  21. Schenk-Hoppe KR. Bifurcations of the randomly perturbed logistic map. Discussion Paper No 353, University of Bielefeld: Department of Economics; 1997.
  22. Sieber M, Malchow H, Schimansky-Geier L. Constructive effects of environmental noise in an excitable prey-predator plankton system with infected prey. Ecological Complexity. 2007;4:223–233. DOI: 10.1016/j.ecocom.2007.06.005.
  23. Pontryagin LS, Andronov AA, Witt AA. On the statistical treatment of dynamical systems. JETP. 1933;3:165–180 (in Russian).
  24. Stratonovich RL. Selected questions of the theory of fluctuations in radio engineering. Moscow: Sov. Radio; 1961. 558 p. (in Russian).
  25. Anishchenko VS, Astakhov VV, Vadivasova TE, Neiman AB, Strelkova GI, Schimansky-Geier L. Nonlinear Effects in Chaotic and Stochastic Systems. Izhevsk: Institute of Computer Sciences; 2003.
  26. McDonnell MD, Stocks NG, Pearce CEM, Abbott D. Stochastic resonance: From Suprathreshold Stochastic Resonance to Stochastic Signal Quantization. Cambridge University Press; 2008.
  27. Horsthemke W, Lefever R. Noise-Induced Transitions. Moscow: Mir; 1987. 400 p. (in Russian).
  28. Gassmann F. Noise-induced chaos-order transitions. Phys. Rev. E. 1997;55(33):2215–2221. DOI: 10.1103/PhysRevE.55.2215.
  29. Gao JB, Hwang SK, Liu JM. When can noise induce chaos? Phys. Rev. Lett. 1999;82(6):1132–1135. DOI: 10.1103/PhysRevLett.82.1132.
  30. Bashkirtseva IA, Ryashko LB. Sensitivity analysis of the stochastically and periodically forced Brusselator. Physica A. 2000;278:126–139. DOI: 10.1016/S0378-4371(99)00453-7.
  31. Fedotov S, Bashkirtseva I, Ryashko L. Stochastic dynamo model for subcritical transition. Phys. Rev. E. 2006;73(6):066307. DOI: 10.1103/PhysRevE.73.066307.
  32. Venttsel’ AD, Freidlin MI. Fluctuations in Dynamical Systems under the Action of Small Random Perturbations. Moscow: Nauka; 1979. (in Russian).
  33. Bashkirtseva IA, Ryashko LB. The quasi-potential method in the analysis of the sensitivity of self-oscillations to stochastic perturbations. Izvestiya VUZ. Applied Nonlinear Dynamics. 1998;6(5):19–27 (in Russian).
  34. Bashkirtseva IA, Ryashko LB. Quasipotential method in local stability analysis of the stochastically forces limit cycles.
  35. Bashkirceva IA, Karpenko LV, Ryashko LB. Analysis of attractors for stochastically forced «predator–prey» model. Izvestiya VUZ. Applied Nonlinear Dynamics. 2009;17(2):37–53 (in Russian). DOI: 10.18500/0869-6632-2009-17-2-37-53.
  36. Ito K. On stochastic differential equations. Matematika. 1957;1(1):78–116. (in Russian).
  37. Bashkirtseva IA, Ryashko L. B. Stochastic sensitivity of 3D-cycles. Mathematics and computers in simulation. 2004;66(1):55–67. DOI: 10.1016/j.matcom.2004.02.021.
  38. Hofbauer J, Sigmund K. On the stabilizing effect of predators and competitors on ecological communities. J. Math. Biol. 1989;27(5):537–548. DOI: 10.1007/BF00288433.
  39. Paine RT. Food web complexity and species diversity. Amer. Natur. 1966;100:65–75. DOI: 10.1086/282400.
  40. Vance RR. Predation and resource partitioning in one predator-two prey model communities. Amer. Natur. 1978;112:797–813. DOI: 10.1086/283324.
  41. Schuster G. Deterministic Chaos. An Introduction. Moscow: Mir; 1988. (in Russian).
Short text (in English):
(downloads: 115)