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

For citation:

Postnov D. E., Balanov A. G., Cherniakov V. I. Synchronization and chaos in population dynamics models. Izvestiya VUZ. Applied Nonlinear Dynamics, 1997, vol. 5, iss. 1, pp. 54-68. DOI: 10.18500/0869-6632-1997-5-1-54-68

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):
PDF icon 1997no1p054.pdf
(downloads: 0)
Language: 
Russian
Heading: 
Deterministic Chaos
Article type: 
Article
UDC: 
517.958:57
DOI: 
10.18500/0869-6632-1997-5-1-54-68

Synchronization and chaos in population dynamics models

Autors: 
Postnov Dmitrij Engelevich, Saratov State University
Balanov Aleksandr Gennadevich, Loughborough University
Cherniakov Vladislav Igorevich, Saratov State University
Abstract: 

We consider а microbiological system consisting of several bacteria—virus populations which are coupled via the flow of resources. The behavior of both continious time mathematical models and discrete time ones is discussed. The origin of complex
behavior due to properties of coupling between the oscillators with regular dynamics is illustrated. While studying the one-dimentional array of discrete time population models we reveal the mechanism of rise of spatial inhomogeneity of stationary oscillatory regimes characteristics.

Key words: 
-
Acknowledgments: 
The authors express their gratitude to Professor E. Mosekilde of the Technical University of Denmark, whose collaboration initiated this work.
Reference: 
  1. Мау RM. Biological populations with nonoverlapping generations - stable points, stable cycles, and chaos. Science 1974;186(4164):645-667. DOI: 10.1126/science.186.4164.645.
  2. May RM. Simple mathematical models with very complex dynamics. Nature. 1976;261:459-467. DOI: 10.1038/261459a0.
  3. May RM, Oster GF. Bifurcations and dynamical complexity in simple ecological models. American Naturalist. 1976;110(974):573-599. DOI: 10.1086/283092.
  4. Gilpin МЕ. Spiral chaos in а predator prey model. American Naturalist. 1979;113(2):306-308. DOI: 10.1086/283389.
  5. Scheffer M. Should we expect strange attractors behind plankton dynamics — and if so, should wе bother? J. Plankton Res. 1991;13:1291-1305.
  6. Doveri F, Scheffer M, Rinaldi S, Muratori S, Kuznetsov Y. Seasonality and chaos in а plankton—fish model. Theor. Population Biol. 1993;43:159-183.
  7. Liberoth M, Barfred M, Mosekilde E. Dynamics of a food web model of an aquatic ecosystem. Open Syst. Inf. Dyn. 1995;3:237-254. DOI: 10.1007/BF02228818.
  8. McCauley E, Murdoch WW. Cyclic and stable populations: plankton as a paradigm. American Naturalist. 1987;129(1):97-121. DOI: 10.1086/284624.
  9. Kaufman M, Thomas R. Model analysis of the bases of multistationarity in the humoral immune response. J. Theor. Biol. 1987;129(2):141-162. DOI: 10.1016/s0022-5193(87)80009-7.
  10. Layne SP, Spouge JL, Dembo M. Quantifying the infectivity of human immunodeficiency virus. Proc. Natl. Acad. Sci. USA. 1989;86(12):4644—4648. DOI: 10.1073/pnas.86.12.4644.
  11. Nowak M, May RM. AIDS pathogenesis: mathematical models of HIV and SIV infections. AIDS. 1993; 7 Suppl 1:S3-S18.
  12. Sturis J, Knudsen C, O’ Meara NM, Thomsen JS, Mosekilde E, Van Cauter E, Polonsky KS. Phase—locking regions in а forced model of slow insulin and glucose oscillations. Chaos. 1995;5(1):193-199. DOI: 10.1063/1.166068.
  13. Aron JL, Schwartz IВ. Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 1984;110(4):665-679. DOI: 10.1016/s0022-5193(84)80150-2.
  14. Schaffer WM. Order and chaos in ecological systems. Ecology 1985;66(1):93-106. DOI: 10.2307/1941309.
  15. Schaffer WM, Kot M. Chaos in ecological systems: The coals that Newcastle forgot. Trends Ecol. Evol. 1986;1(3):58-63. DOI: 10.1016/0169-5347(86)90018-2.
  16. Baier G, Thomsen JS, Mosekilde Е. Chaotic hierarchy in а model of competing populations. J. Theor.Biol. 1993;165(4):593—607. DOI: 10.1006/jtbi.1993.1209.
  17. Levin BR, Stewart FM, Chao L. Resource—limited growth, competition and predation: A model and experimental studies with bacteria and bacteriophage. American Naturalist. 1977;111(977):3-24. DOI: 10.1086/283134.
  18. Abarbanel HDI, Rabinovich МI, Selverston А, Bazhenov МV, Huerta R, Sushchik ММ, Rubchinskii LL. Synchronisation in neural networks. Phys. Usp. 1996;39(4):337-362. DOI: 10.1070/pu1996v039n04abeh000141.
  19. Моnod J. La technique de culture continue: Theorie et applications. Ann. Inst. Pasteur. 1950;79:390—410. (in French).
  20. Mosekilde E, Stranddorf H, Thomsen JS, Baier G. A hierarchy of complex behaviors in microbiological systems. In: Casti JL, Karlqvist A, editors. Cooperation and Conflict in General Evolutionary Processes. N.Y.: Wiley; 1994. P. 165-225.
  21. Arnold VI, Afraimovich VS, Ilyashenko YuS, Shilnokov LP. Bifurcation theory. Results of Science and Technology. Modern Problems of Mathematics. Fundamental Directions. 1986;5:5-218. (in Russian).
  22. Kevrekidis IG, Schmidt LD, Aris R. Some common features оf periodically forced reacting systems. Chemical Engeneering Science. 1986;41(5):1263-1276. DOI: 10.1016/0009-2509(86)87099-3.
  23. Kevrekidis IG, Aris R, Schmidt LD. Forcing аn entire bifurcation diagram: case studies in chemical oscillators. Physica D. 1986;23(1-3):391-395. DOI: 10.1016/0167-2789(86)90145-4.
  24. Taylor МА, Kevrekidis IG. Some common dynamic features of coupled reacting systems. Physica D. 1991;51(3):274-292. DOI: 10.1016/0167-2789(91)90239-6.
  25. Knudsen C, Sturis J, Thomsen JS. Generic bifurcation structures of Arnol’d tongues in forced oscillators. Phys. Rev A. 1991;44(6):3503-3510. DOI: 10.1103/PhysRevA.44.3503.
  26. Anishchenko VS. Dynamical Chaos — Models and Experiments. Singapore: World Scientific; 1995. 400 p.
  27. Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Synchronization оf chaos. Int. J. Bifurc. Chaos. 1992;2(3):633-644. DOI: 10.1142/S0218127492000756.
  28. Anishchenko VS, Vadivasova TE, Postnov DE, Safonova MA. Forced and mutual sinchronization of chaos. Soviet J. Commun. Tech. Electron. 1991;36(2):338-351. (in Russian).
  29. Anishchenko VS, Vadivasova TE, Postnov DE, Sosnovtzeva ОМ, Chua LO, Wu CW. Dynamics оf nonautonomous Chua’s circuit. Int. J. Bifurc. Chaos. 1995;5(6):1525-1540. DOI: 10.1142/S0218127495001162.
  30. Butenin NV, Neimark YuI, Fufaev NA. Introduction to the Theory of Nonlinear Oscillations. M.: Nauka; 1987. 384 p. (in Russian).
  31. Moon FC. Chaotic Vibrations: An Introduction for Applied Scientists and Engineers. N.Y.: Wiley; 1987. 309 p.
  32. Anishchenko VS, Postnov DE, Sosnovtseva QV, Khovanov IА. Dynamics оf the Chain of unidirectional coupled Chua’s circuit. Differential Equations: Bifurcations and Chaos. In: The Abstracts of the School-Conference. Inst. of Math., Ukrainian Ac. Sc. Kiev, 1994. P. 8.
Received: 
24.02.1997
Accepted: 
10.04.1997
Published: 
18.05.1997
  • 132 reads