VARIETIES OF APERIODIC DYNAMICS IN THE EVENT-DRIVEN POPULATION MODELS
Abstract
The paper proposes a population model with the event-step structure, which includes continuous and discrete components. The dynamics of a hybrid system is analyzed in a computing environment based on the numerical solution of the sequence of Cauchy problems for the system of differential equations of generations decrease. We examine the dynamics of the functional iteration, which has two local extrema and characterizes the impermanence of the fish reproduction effectiveness. A transi-tional aperiodic regime is established with the possibility of attracting the trajectory to two attractors. After the bifurcation of disappearance of two nontrivial stationary points, an interval attractor arises for which a boundary crisis is possible.
About the Author
A. Y. Perevaryukha
Санкт-Петербургский институт информатики и автоматизации РАН
Russian Federation
Russian Federation
References
1. Вул, Е.Б. Универсальность Фейгенбаума и термодинамический формализм / Е.Б. Вул, Я.Г. Синай, К.М. Ханин // Успехи математических наук. – 1984. – Т. 39, вып. 3. – С. 3–37.
2. Фейгенбаум, М. Универсальность в поведении нелинейных систем / М. Фейгенбаум // Успехи физических наук. – 1983. – Т. 141, вып. 2. – С. 343–374.
3. Переварюха, А.Ю. Интерпретация поведения моделей динамики биоресурсов и момен-тальная хаотизация в новой модели / А.Ю. Переварюха // Нелинейный мир. – 2012. – № 4. – С. 255–261.
4. Singer, D. Stable orbits and bifurcations of the maps on the interval / D. Singer // SIAM jour-nal of applied math. – 1978. – Vol. 35. – P. 260–268.
5. Vellekoop, М. On intervals, transitivity = chaos / М. Vellekoop, R. Berglund // The Ameri-can Mathematical Monthly. – 1994. – Vol. 101, № 4. – P. 353–355.
6. Allee, W.C. Studies in animal aggregations: mass protection against colloidal silver among goldfishes / W.C. Allee, E Bowen // Journal of Experimental Zoology. – 1932. – № 2. – P. 185–207.
7. Dynamical complexities in the Leslie-Gower predator-prey model as consequences of the Allee effect on prey / E. Gonzalez-Olivares [et al.] // Appl. Math. Modell. – 2011. – Vol. 35. – P. 366–381.
8. Yongli, C. Spatiotemporal complexity of a Leslie–Gower predator-prey model with the weak Allee effect / C. Yongli, Zh. Caidi, W. Weiming // Journal of Applied Mathematics. – 2013. – Vol. 2013, Article ID 535746.
9. Ricker, W. Stock and recruitment / W. Ricker // Journal Fisheries research board of Canada. – 1954. – Vol. 11, № 5. – P. 559–623.
10. Paar, V. Sensitive dependence of lifetimes of chaotic transient on numerical accuracy for a model with dry friction and frequency dependent driving amplitude / V. Paar, N. Pavin // Modern Physics Letters B. – 1996. – Vol. 10, № 4 & 5. – P. 153–159.
11. Grebogi, C. Chaotic attractors in crisis / C. Grebogi, E. Ott, J.A. Yorke // Physical Review Letters. – 1982. – Vol. 48, № 22. – P. 1507–1510.
12. Grebogi, C. Chaos, strange attractors and fractal basin boundaries in nonlinear dynamics / C. Grebogi, E. Ott, J.A. Yorke // Science. – 1987. – Vol. 238, № 4827. – P. 632–638.
13. Minto, C. Blanchard Survival variability and population density in fish populations / C. Minto, R.A. Myers // Nature. – 2008. – Vol. 452. – P. 344–348.
14. Еремеева, Е.Ф. Теория этапности развития и её значение в рыбоводстве / Е.Ф. Ереме-ева, А.И. Смирнов // Теоретические основы рыбоводства. – М. : Наука, 1965. – С. 129–138.
Review
For citations:
Perevaryukha A.Y. VARIETIES OF APERIODIC DYNAMICS IN THE EVENT-DRIVEN POPULATION MODELS. Informatics. 2014;(2):58-65. (In Russ.)
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