Two-cycles of the Ricker model with the periodic Malthusian parameter: stability and multistability

K. V. Shlufman ; G. P. Neverova ; E. Ya. Frisman

Russian journal of nonlinear dynamics, Tome 12 (2016) no. 4, pp. 553-565.

Résumé

This paper investigates the emergence and stability of 2-cycles for the Ricker model with the 2-year periodic Malthusian parameter. It is shown that the stability loss of the trivial solution occurs through the transcritical bifurcation resulting in a stable 2-cycle. The subsequent tangent bifurcation leads to the appearance of two new 2-cycles: stable and unstable ones. As a result, there is multistability. It is shown that the coexistence of two different stable 2-cycles is possible in a narrow area of the parameter space. Further stability loss of the 2-cycles occurs according to the Feigenbaum scenario.

Détail

Keywords: recurrence equation, Ricker model, periodic Malthusian parameter, stability, multistability.
Mots-clés : bifurcation

Bibliographie

[1] Ricker W. E., Computation and interpretation of biological statistics of fish populations, Bulletin of the Fisheries Research Board of Canada, 191, Department of the Environment Fisheries and Marine Service, Ottawa, 1975, 400 pp.

[2] Sacker R. J., von Bremen H. F., “Global asymptotic stability in the Jia Li model for genetically altered mosquitos”, Difference Equations and Discrete Dynamical Systems:Proc. of the 9th Internat. Conf. on Difference Equations and Applications (University of Southern California, Los Angeles, Calif., 2004), 87–100 | MR | Zbl

[3] Sacker R. J., von Bremen H. F., “Some stability results in a model for genetically altered mosquitoes”, Modeling and Control of Autonomous Decision Support Based Systems:Proc. of the 13th Internat. Workshop on Dynamics and Control (Wiesensteig, Germany, 2005), 301–308

[4] Sacker R. J., von Bremen H. F., “A conjecture on the stability of the periodic solutions of Ricker's equation with periodic parameters”, Appl. Math. Comput., 217:3 (2010), 1213–1219 | MR | Zbl

[5] Zhou Zh., Zou X., “Stable periodic solutions in a discrete logistic equation”, Appl. Math. Lett., 16:2 (2003), 165–171 | DOI | MR | Zbl

[6] Kon R., “Attenuant cycles of population models with periodic carrying capacity”, J. Difference Equ. Appl., 11:4–5 (2005), 423–430 | DOI | MR | Zbl

[7] Sacker R. J., “A note on periodic Ricker maps”, J. Difference Equ. Appl., 13:1 (2007), 89–92 | DOI | MR | Zbl

[8] Shapiro A. P., Luppov S. P., Recurrence equations in the theory of population biology, Nauka, Moscow, 1983 (Russian) | MR

[9] Skaletskaya E. I., Frisman E. Ya., Shapiro A. P., Discrete models of the dynamics of population size and optimization of harvest, Nauka, Moscow, 1979 (Russian)

[10] Ashikhmina E. V., Izrailsky Yu. G., Frisman E. Ya., “Dynamics of the Ricker model with periodical parameter variation”, Vestn. Dalnevost. Otd. RAN, 2004, no. 5, 19–28 (Russian)

[11] Ashikhmina E. V., Frisman E. Ya., Izrailsky Yu. G., “Harvest optimization for Rikker's population, when environment parameters limiting population growth change cyclically”, Dal'nevost. Mat. Zh., 4:1 (2003), 127–133 (Russian)

[12] Kuznetsov S. P., Dynamical chaos, 2nd ed., Fizmatlit, Moscow, 2006 (Russian)

[13] Kuznetsov A. P., Sedova J. V., “Bifurcation of three- and four-dimensional maps: Universal properties”, Izv. Vyssh. Uchebn. Zaved. Prikl. Nelin. Dinam., 20:5 (2012), 26–43 (Russian) | Zbl

[14] Lasunskii A. V., “On cycles of a discrete periodic logistic equation”, Tr. Inst. Mat. i Mekh. UrO RAN, 16:2 (2010), 154–157 (Russian)

[15] Lasunskii A. V., “On the period of solutions of a discrete periodic logistic equation”, Tr. KNC RAN. Ser. 5. Matem. Model. i Inform. Technol., 3 (2012), 44–48 (Russian)

[16] Elaydi S. N., Luís R., Oliveira H., “Towards a theory of periodic difference equations and its application to population dynamics”, Dynamics, Games and Science, v. 1, Springer Proc. Math., 1, eds. M. M. Peixoto et al., Springer, Heidelberg, 2011, 287–321 | DOI | MR | Zbl

[17] Elaydi S. N., Sacker R. J., “Basin of attraction of periodic orbits of maps in the real line”, J. Difference Equ. Appl., 10:10 (2004), 881–888 | DOI | MR | Zbl

[18] Elaydi S. N., Yakubu A., “Open problems and conjectures basins of attraction of stable cycles”, J. Difference Equ. Appl., 8:8 (2002), 755–760 | DOI | MR

[19] AlSharawi Z., Angelos J., Elaydi S., Rakesh L., “An extension of Sharkovsky's theorem to periodic difference equations”, J. Math. Anal. Appl., 316:1 (2006), 128–141 | DOI | MR | Zbl

[20] Li J., “Simple mathematical models for interacting wild and transgenic mosquito populations”, Math. Biosci., 189:1 (2004), 39–59 | DOI | MR | Zbl

[21] Li J., “Heterogeneity in modelling of mosquito populations with transgenic mosquitoes”, J. Difference Equ. Appl., 11:4–5 (2005), 443–457 | MR | Zbl

[22] Li J., “Discrete-time models with mosquitoes carrying genetically-modified bacteria”, Math. Biosci., 240:1 (2012), 35–44 | DOI | MR | Zbl

[23] Li J., “Stage-structured models for interacting wild and sterile mosquitoes”, Journal of Shanghai Normal University (Natural Science. Mathematics), 43:5 (2014), 511–523

[24] Chow Sh.-N., Li Ch. Zh., Wang D., Normal forms and bifurcation of planar vector fields, Cambridge Univ. Press, Cambridge, 1994, viii+472 pp. | MR | Zbl