Метод функции стохастической чувствительности в анализе кусочно-гладкой модели популяционной динамики
Ключевые слова:
кусочно-гладкие отображения, популяционная динамика, стохастическая чувствительность
Аннотация
Работа посвящена применению метода функции стохастической чувствительности к аттракторам кусочно-гладкого одномерного отображения, описывающего динамику численности популяции. Первым этапом исследования является параметрический анализ возможных режимов детерминированной модели: определение зон существования устойчивых равновесий и хаотических аттракторов. Для определения параметрических границ хаотического аттрактора применяется теория критических точек. В случае, когда на систему оказывает влияние случайное воздействие, на основе техники функции стохастической чувствительности дается описание разброса случайных состояний вокруг равновесия и хаотического аттрактора. Проводится сравнительный анализ влияния параметрического и аддитивного шума на аттракторы системы. С помощью техники доверительных интервалов изучаются вероятностные механизмы вымирания популяции под действием шума. Анализируются изменения параметрических границ существования популяции под действием случайного возмущения.
Литература
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2. Lande R., Engen S., Saether B.-E. Stochastic population dynamics in ecology and conservation. Oxford University Press, 2003.
https://doi.org/10.1093/acprof:oso/9780198525257.001.0001
3. Chesson P. Stochastic population models // Ecological Heterogeneity. Ecological Studies (Analysis and Synthesis). New York: Springer, 1991. Vol. 86. P. 123-143.
https://doi.org/10.1007/978-1-4612-3062-5_7
4. Gadrich T., Katriel G. A mechanistic stochastic Ricker model: analytical and numerical investigations // International Journal of Bifurcation and Chaos. 2016. Vol. 26. No. 4. 1650067.
https://doi.org/10.1142/S021812741650067X
5. May R.M. Simple mathematical models with very complicated dynamics // Nature. 1976. Vol. 261. No. 5560. P. 459-467.
https://doi.org/10.1038/261459a0
6. Murray J.D. Discrete population models for a single species // Interdisciplinary Applied Mathematics. New York: Springer, 1993. P. 44-78.
https://doi.org/10.1007/978-0-387-22437-4_2
7. Higgins R.J., Kent C.M., Kocic V.L., Kostrov Y. Dynamics of a nonlinear discrete population model with jumps // Applicable Analysis and Discrete Mathematics. 2015. Vol. 9. No. 2. P. 245-270.
https://doi.org/10.2298/AADM150930019H
8. Avrutin V., Sushko I. A gallery of bifurcation scenarios in piecewise smooth 1d map // Global Analysis of Dynamic Models in Economics and Finance. Berlin-Heidelberg: Springer, 2012. P. 369-395.
https://doi.org/10.1007/978-3-642-29503-4_14
9. Sushko I., Gardini L., Avrutin V. Nonsmooth one-dimensional maps: some basic concepts and definitions // Journal of Difference Equations and Applications. 2016. Vol. 22. No. 12. P. 1816-1870.
https://doi.org/10.1080/10236198.2016.1248426
10. Avrutin V., Gardini L., Schanz M., Sushko I. Bifurcations of chaotic attractors in one-dimensional piecewise smooth maps // International Journal of Bifurcation and Chaos. 2014. Vol. 24. No. 8. 1440012.
https://doi.org/10.1142/S0218127414400124
11. Laurea M.B., Champneys A.R., Budd C.J., Kowalczyk P. Piecewise-smooth dynamical systems. London: Springer, 2008.
https://doi.org/10.1007/978-1-84628-708-4
12. Bashkirtseva I. Crises, noise, and tipping in the Hassell population model // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2018. Vol. 28. No. 3. 033603.
https://doi.org/10.1063/1.4990007
13. Bashkirtseva I. Preventing noise-induced extinction in discrete population models // Discrete Dynamics in Nature and Society. 2017. Vol. 2017. P. 1-10.
https://doi.org/10.1155/2017/9610609
14. Bashkirtseva I., Ryashko L. Stochastic sensitivity analysis of noise-induced order-chaos transitions in discrete-time systems with tangent and crisis bifurcations // Physica A: Statistical Mechanics and its Applications. 2017. Vol. 467. P. 573-584.
https://doi.org/10.1016/j.physa.2016.09.048
15. Bashkirtseva I., Nasyrova V., Ryashko L. Analysis of noise effects in a map-based neuron model with Canard-type quasiperiodic oscillations // Communications in Nonlinear Science and Numerical Simulation. 2018. Vol. 63. P. 261-270.
https://doi.org/10.1016/j.cnsns.2018.03.015
16. Bashkirtseva I., Nasyrova V., Ryashko L. Noise-induced bursting and chaos in the two-dimensional Rulkov model // Chaos, Solitons and Fractals. 2018. Vol. 110. P. 76-81.
https://doi.org/10.1016/j.chaos.2018.03.011
17. Bashkirtseva I.A., Ryashko L.B. Stochastic sensitivity of regular and multi-band chaotic attractors in discrete systems with parametric noise // Physics Letters A. 2017. Vol. 381. No. 37. P. 3203-3210.
https://doi.org/10.1016/j.physleta.2017.08.017
18. Bashkirtseva I., Ryashko L. Approximating chaotic attractors by period-three cycles in discrete stochastic systems // International Journal of Bifurcation and Chaos. 2015. Vol. 25. No. 10. 1550138.
https://doi.org/10.1142/S0218127415501382
Поступила в редакцию
2019-04-04
Опубликована 2019-05-20
Опубликована 2019-05-20