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 Mat. Biolog. Bioinform.: Year: Volume: Issue: Page: Find

 Mat. Biolog. Bioinform., 2018, Volume 13, Issue 1, Pages 68–83 (Mi mbb328)

Mathematical Modeling

Phase multistability of dynamics modes of the Ricker model with periodic Malthusian parameter

K. V. Shlufmana, G. P. Neverovaab, E. Ya. Frismana

a Institute for complex analysis of regional problems of the Russian Academy of Sciences, Far Eastern branch
b Institute of Automation and Control Processes of the Russian Academy of Sciences, Far Eastern branch

Abstract: The paper investigates the phase multistability of dynamical modes of the Ricker model with 2-year periodic Malthusian parameter. It is shown that both the variable perturbation and the phase shift of the Malthusian parameter can lead to a phase shift or a change in the dynamic mode observed. The possibility of switches between different dynamic modes is due to multistability, since the model has two different stable 2-cycles. The first stable 2-cycle is the result of transcritical bifurcation and is synchronous to the oscillations of the Malthusian parameter. The second stable 2-cycle arises as a result of the tangent bifurcation and is asynchronous to the oscillations of the Malthusian parameter. This indicates that two-year fluctuations in the population size can be both synchronous and asynchronous to the fluctuations in the environment. The phase shift of the Malthusian parameter causes a phase shift in the stable 4-cycle of the first bifurcation series to one or even three elements of the 4-cycle. The phase shift to two elements of this 4-cycle is possible due to a change in the half-amplitude of the Malthusian parameter oscillation or the variable perturbation. At the same time, the longer period of the cycle, the more phases with their attraction basins it has, and the smaller the threshold values above which shift from the attraction basin to another one occur. As a result, in the case of cycles with long period (for example, 8-cycle) perturbations, that stable cycles with short period are able to "absorb", can cause different phase transitions, which significantly complicates the dynamics of the model trajectory and, as a consequence, the identification of the dynamic mode observed.

Key words: population dynamics, periodic Malthusian parameter, mathematical modeling, dynamic modes, multistability, phase multistability, attraction basins.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-04-00073_à Far Eastern Branch of the Russian Academy of Sciences 18-5-051

DOI: https://doi.org/10.17537/2018.13.68

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UDC: 574.34

Citation: K. V. Shlufman, G. P. Neverova, E. Ya. Frisman, “Phase multistability of dynamics modes of the Ricker model with periodic Malthusian parameter”, Mat. Biolog. Bioinform., 13:1 (2018), 68–83

Citation in format AMSBIB
\Bibitem{ShlNevFri18} \by K.~V.~Shlufman, G.~P.~Neverova, E.~Ya.~Frisman \paper Phase multistability of dynamics modes of the Ricker model with periodic Malthusian parameter \jour Mat. Biolog. Bioinform. \yr 2018 \vol 13 \issue 1 \pages 68--83 \mathnet{http://mi.mathnet.ru/mbb328} \crossref{https://doi.org/10.17537/2018.13.68 }