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Volume 13   Issue 1   Year 2018
Konstantin V. Shlufman, Galina P. Neverova, Efim Ya. Frisman

Phase Multistability of Dynamics Modes of the Ricker Model with Periodic Malthusian Parameter

Mathematical Biology & Bioinformatics. 2018;13(1):68-83.

doi: 10.17537/2018.13.68.



  1. Anishchenko V.S., Astakhov V.V., Nikolaev V.V., Shabunin A.V. Chaotic synchronization in a network of symmetrically coupled oscillators. Journal of Communications Technology and Electronics. 2000;45(2):179-185.
  2. Bezruchko B.P., Prokhorov M.D., Seleznev Ye.P. Oscillation types, multistability, and basins of attractors in symmetrically coupled period-doubling systems. Izvestiya VUZ. Applied Nonlinear Dynamics. 2002;10(4):47-67 (in Rus.).
  3. Smirnov D.A., Sidak Ε.V., Bezruchko B.P. Statistical properties of phase synchronization coefficient estimator. Izvestiya VUZ. Applied Nonlinear Dynamics. 2008;16(2):111-121 (in Rus.).
  4. Koblyanskiy S, Shabunin A., Astakhov V. Forced synchronization of periodic oscillations in a system with phase multistability. Nelineinaya Dinamika (Russian Journal of Nonlinear Dynamics). 2010;6(2):277-289. doi: 10.20537/nd1002004
  5. Kuznetsov A.P., Savin A.V., Sedova Y.V., Tyuryukina L.V. Bifurcation of Images. Saratov: Press Center Ltd “Nauka”, 2012. 196 p. (in Rus.).
  6. Koronovskiy A.A., Trubetskov D.I. Nonlinear Dynamics in Action: How Ideas of Nonlinear Dynamics Penetrate the Ecology, Economy and Social Sciences. Saratov, 2002. 324 p. (in Rus.).
  7. Bazykin A.D. Nonlinear Dynamics of Interacting Populations. Eds. Khibnik A.I., Krauskopf B. 1998. 216 p. (World Scientific Series on Nonlinear Science. Series A, Vol. 11). doi: 10.1142/2284
  8. Wilmshurst J.F., Greer R., Henry J.D. Correlated cycles of snowshoe hares and Dall’s sheep lambs. Can. J. Zool. 2006;84:736-743. doi: 10.1139/z06-051
  9. Elmhagen B., Hellström P., Angerbjörn A., Kindberg J. Changes in vole and lemming fluctuations in northern Sweden 1960-2008 revealed by fox dynamics. Annales Zoologici Fennici. 2011;48(3):167-179. doi: 10.5735/086.048.0305
  10. Kulakov M.P., Neverova G.P., Frisman E.Ya. Multistability in dynamic models of migration coupled populations with an age structure. Nelineinaya Dinamika (Russian Journal of Nonlinear Dynamics). 2014;10(4):407-425. doi: 10.20537/nd1404002
  11. Kurilova E.V., Kulakov M.P., Frisman E.Ya. Effects of synchronization by fluctuations in numbers of two predator-prey communities at saturation predator growth and limitation of the victim number. Information Science and Control Systems. 2015(3):24-34 (in Russ.).
  12. Krebs C.J., Kielland K., Bryant J., O’Donoghue M., Doyle F., McIntyre C., DiFolco D., Berg N., Carriere S., Boonstra R. et al. Synchrony in the showshoe hare (Lepus Americanus) cycle in northwestern North America, 1970-2012. Can. J. Zool. 2013;91(8):562-572. doi: 10.1139/cjz-2013-0012
  13. Petrovskaya N., Petrovskii S. Catching ghosts with a coarse net: use and abuse of spatial sampling data in detecting synchronization. Journal of the Royal Society Interface. 2017;14(127):20160855. doi: 10.1098/rsif.2016.0855
  14. Henson S.M., Cushing J.M, Costantino R.F., Dennis B., Desharnais R.A. Phase switching in population cycles. Proc. R. Soc. Lond. B. 1998;265:2229-2234. doi: 10.1098/rspb.1998.0564
  15. Chernyavsky F.B., Lazutkin A.N. Tsikly lemmingov i polevok na Severe (Cycles of Lemmings and Voles in the North). Institute of Biological Problems of the North, Far-Eastern Branch of RAS, 2004. 150 p. (in Rus.).
  16. Zhigal'skiĭ OA. Structure of the bank vole (Myodes glareolus) population cycles in the center and periphery of its distribution area. Izvestiia RAN. Ser. Biol. (Biology Bulletin). 2011;6:733-746 (in Rus.).
  17. Kausrud K.L., Mysterud A., Steen H., Vik J.O., Østbye E., Cazelles B., Framstad E., Eikeset A.M., Mysterud I., Solhøy T., Stenseth N. Chr. Linking climate change to lemming cycles. Nature. 2008;456:93-97. doi: 10.1038/nature07442
  18. Henttonen H., Wallgren H. Small rodent dynamics and communities in the birch forest zone of northern Fennoscandia. In: Nordic mountain birch ecosystems. Ch. 22. Ed. Wielgolaski F.E. New York and London: UNESCO, Paris and Parthenon Publishing Group, 2001:262-278.
  19. Frisman E.Ya., Neverova G.P., Kulakov M.P., Zhigalskii O.A. Changing the dynamic modes in populations with short life cycle: mathematical modeling and simulation. Mathematical Biology and Bioinformatics. 2014;9(2):414-429 (in Russ.). doi: 10.17537/2014.9.414
  20. Revutskaya O.L., Kulakov M.P., Neverova G.P., Frisman E.Ya. The sex ratio influence on the dynamics of structured population. Mathematical Biology and Bioinformatics. 2017;12(2):237-255 (in Russ.). doi: 10.17537/2017.12.237
  21. Kaev A.M. Temporal structure of pink salmon Oncorhynchus gorbuschamigratory flow to the Okhotsk Sea. Izvestia TINRO (Bulletin of the Far East Scientific Research Fisheries Center). 2002;1-3:904-920 (in Rus.).
  22. Shlufman K.V., Neverova G.P., Frisman E.Ya. Two-cycles of the Ricker model with the periodic Malthusian parameter: stability and multistability. Nelineinaya Dinamika (Russian Journal of Nonlinear Dynamics). 2016;12(4):553-565. doi: 10.20537/nd1604001
  23. Shlufman K.V., Neverova G.P., Frisman E.Ya. Dynamic modes of the Ricker model with periodic Malthusian parameter. Nelineinaya Dinamika (Russian Journal of Nonlinear Dynamics). 2017;13(3):363-380. doi: 10.20537/nd1703005
  24. Revutskaya O.L. Regional’nye problemy (Regional Problems). 2012;15(2):5-11 (in Russ.).
  25. Nedorezov L.V., Sadykova D.L. Dynamics of larch bud moth populations: application of Moran-Ricker models with time lag. Ecological Modelling. 2015;297:26-32. doi: 10.1016/j.ecolmodel.2014.11.003
  26. Neverova G.P., Yarovenko I.P., Frisman E.Y. Dynamics of populations with delayed density dependent birth rate regulation. Ecological Modelling. 2016;340:64-73. doi: 10.1016/j.ecolmodel.2016.09.005
  27. Ashikhmina E.V., Frisman E.Ya., Skaletskaya E.I., Kulikov A.N. Mathematical model for dynamics of the number of pelt products from the local population of mantchurian squirrels. Ecological Modelling. 1985;30(1-2):145-156. doi: 10.1016/0304-3800(85)90040-7
  28. Revutskaya O.L. Analysis of the squirrel population dynamics dependence on the forage reserve (by the example of the Jewish Autonomous Region). Regional’nye problemy (Regional Problems). 2010;13(2):37-44 (in Russ.).
  29. Zhou Z., Zou X. Stable periodic solutions in a discrete logistic equation. Appl. Math. Lett. 2003;16:165-171. doi: 10.1016/S0893-9659(03)80027-7
  30. Elaydi S., Sacker R. Basin of attraction of periodic orbits of maps on the real line. Journal of Difference Equations and Applications. 2004;10:881-888. doi: 10.1080/10236190410001731443
  31. Ashikhmina E.V., Izrailsky Yu.G., Frisman E.Ya. Dynamics of the Ricker model with periodical parameter variation. Vestnik of the Far East Branch of the Russian Academy of Sciences. 2004(5):9-28 (in Russ.).
  32. Kon R. Attenuant cycles of population models with periodic carrying capacity. J. Difference Eq. Appl. 2005. V.11. Ή4-5:23-430. doi: 10.1080/10236190412331335472
  33. AlSharawi Z., Angelos J., Elaydi S., Rakesh L. An extension of Sharkovsky's theorem to periodic difference equations. Journal of Mathematical Analysis and Applications. 2006;316:128-141. doi: 10.1016/j.jmaa.2005.04.059
  34. AlSharawi Z., Angelos J., Elaydi S. Existence and stability of periodic orbits of periodic difference equations with delays. International Journal of Bifurcation and Chaos. 2008;18(1):203-217. doi: 10.1142/S0218127408020239
  35. Sacker R.J. A note on periodic Ricker maps. Journal of Difference Equations and Applications. 2007;13(1):89-92. doi: 10.1080/10236190601008752
  36. Elaydi S.N., Luis R., Oliveira H. Towards a theory of periodic difference equations and its application to population dynamics. In: Dynamics, Games and Science I. Berlin, Heidelberg: Springer, 2011:287-321. doi: 10.1007/978-3-642-11456-4_19
  37. Sacker R.J., Hubertus F. A conjecture on the stability of the periodic solutions of Ricker’s equation with periodic parameters. Applied Mathematics and Computation. 2010;217(3):1213-1219. doi: 10.1016/j.amc.2010.05.049
  38. Ricker W.E. Metody otsenki i interpretatsii biologicheskikh pokazatelei populiatsii ryb. Moskow, 1979. 408 p. (Translation of: Ricker W.E. Computation and interpretation of biological statistics of fish populations. Vol. 191. Ottawa: Department of the Environment, Fisheries and Marine Service, 1975. 382 p.).
  39. Li Ch., Chou Sh.-N., Van D. Normal Forms and Bifurcations of Vector Fields on the Plane. Moscow, MTsNMO, 2005 (in Rus.).
  40. Skaletskaya E.I., Frisman E.Ya., Shapiro A.P. Discrete Models of Population Dynamics and Optimization of Industry. Moscow. Nauka, 1979. 168 p. (in Rus.).
Table of Contents Original Article
Math. Biol. Bioinf.
doi: 10.17537/2018.13.68
published in Russian

Abstract (rus.)
Abstract (eng.)
Full text (rus., pdf)


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