Стохастическая чувствительность квазипериодических и хаотических аттракторов дискретной модели Лотки-Вольтерры
Ключевые слова:
популяционная динамика, стохастическая чувствительность, хаос, замкнутая инвариантная кривая
Аннотация
Целью исследования, представленного в данной статье, является анализ возможных динамических режимов детерминированной и стохастической модели Лотки-Вольтерры. В зависимости от двух параметров системы строится карта режимов. Изучаются параметрические зоны существования устойчивых равновесий, циклов, замкнутых инвариантных кривых, а также хаотических аттракторов. Описываются бифуркации удвоения периода, Неймарка-Саккера и кризиса. Демонстрируется сложная форма бассейнов притяжения нерегулярных аттракторов (замкнутой инвариантной кривой и хаоса). Помимо детерминированной системы подробно изучается стохастическая, описывающая влияние внешнего случайного воздействия. Здесь ключевым является нахождение чувствительности таких сложных аттракторов, как замкнутая инвариантная кривая и хаос. В случае хаоса дан алгоритм нахождения критических линий, описывающих границу хаотического аттрактора. Опираясь на найденную функцию стохастической чувствительности, строятся доверительные полосы, позволяющие описать разброс случайных состояний вокруг детерминированного аттрактора.
Литература
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https://doi.org/10.1142/2252
https://doi.org/10.1007/s11071-019-05202-3
2. Khan A.Q. Bifurcations of a two-dimensional discrete-time predator-prey model // Advances in Difference Equations. 2019. Vol. 2019.
https://doi.org/10.1186/s13662-019-1995-6
3. Pal S., Pal N., Chattopadhyay J. Hunting cooperation in a discrete-time predator-prey system // International Journal of Bifurcation and Chaos. 2018. Vol. 28. No. 7. P. 1850083.
https://doi.org/10.1142/S0218127418500839
4. Zhao M., Li C.P., Wang J.L. Complex dynamic behaviors of a discrete-time predator-prey system // Journal of Applied Analysis and Computation. 2017. Vol. 7. No. 2. P. 478-500.
https://doi.org/10.11948/2017030
5. Saratchandran P.P., Ajithprasad K.C., Harikrishnan K.P. Numerical exploration of the parameter plane in a discrete predator-prey model // Ecological Complexity. 2015. Vol. 21. P. 112-119.
https://doi.org/10.1016/j.ecocom.2014.11.010
6. ReniSagayaRaj M., George Maria Selvam A., Dhineshbabu R. Stability in a discrete nonlinear prey-predator model with functional response // International Journal of Emerging Technologies in Computational and Applied Sciences. 2014. Vol. 7. No. 2. P. 190-193.
7. ReniSagayaRaj M., George Maria Selvam A., Meganathan M. Dynamics in a discrete prey-predator system // International Journal of Engineering Research and Development. 2013. Vol. 6. No. 5. P. 1-5.
8. Elsadany A.A.E., El-Metwally H.A., Elabbasy E.M., Agiza H.N. Chaos and bifurcation of a nonlinear discrete prey-predator system // Computational Ecology and Software. 2012. Vol. 2. No. 3. P. 169-180.
http://www.iaees.org/publications/journals/ces/articles/2012-2(3)/2012-2(3).asp
9. Ren J.L., Yu L.P., Siegmund S. Bifurcations and chaos in a discrete predator-prey model with Crowley-Martin functional response // Nonlinear Dynamics. 2017. Vol. 90. No. 1. P. 19-41.
https://doi.org/10.1007/s11071-017-3643-6
10. Avrutin V., Gardini L., Sushko I., Tramontana F. Continuous and discontinuous piecewise-smooth one-dimensional maps: Invariant sets and bifurcation structures. Singapore: World Scientific, 2019.
https://doi.org/10.1142/8285
11. Allen L.J.S., Fagan J.F., Högnäs G., Fagerholm H. Population extinction in discrete-time stochastic population models with an Allee effect // Journal of Difference Equations and Applications. 2005. Vol. 11. Issue 4-5. P. 273-293.
https://doi.org/10.1080/10236190412331335373
12. Bashkirtseva I.A. Stochastic sensitivity synthesis in discrete-time systems with parametric noise // IFAC-PapersOnLine. 2018. Vol. 51. No. 32. P. 610-614.
https://doi.org/10.1016/j.ifacol.2018.11.491
13. Bashkirtseva I., Tsvetkov I. Impact of the parametric noise on map-based dynamical systems // AIP Conference Proceedings. 2018. Vol. 2025. Issue 1. 040004.
https://doi.org/10.1063/1.5064888
14. Bashkirtseva I., Ekaterinchuk E., Ryashko L. Analysis of noise-induced transitions in a generalized logistic model with delay near Neimark-Sacker bifurcation // Journal of Physics A: Mathematical and Theoretical. 2017. Vol. 50. No. 27. P. 275102.
https://doi.org/10.1088/1751-8121/aa734b
15. Bashkirtseva I., Ryashko L. Noise-induced shifts in the population model with a weak Allee effect // Physica A: Statistical Mechanics and its Applications. 2018. Vol. 491. P. 28-36.
https://doi.org/10.1016/j.physa.2017.08.157
16. Bashkirtseva I., Ryashko L. Stochastic sensitivity analysis of noise-induced extinction in the Ricker model with delay and Allee effect // Bulletin of Mathematical Biology. 2018. Vol. 80. No. 6. P. 1596-1614.
https://doi.org/10.1007/s11538-018-0422-6
17. Jungeilges J., Ryazanova T. Transitions in consumption behaviors in a peer-driven stochastic consumer network // Chaos, Solitons and Fractals. 2019. Vol. 128. P. 144-154.
https://doi.org/10.1016/j.chaos.2019.07.042
18. Belyaev A.V., Ryazanova T.V. The stochastic sensitivity function method in analysis of the piecewise-smooth model of population dynamics // Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. 2019. Vol. 53. P. 36-47.
https://doi.org/10.20537/2226-3594-2019-53-04
19. Belyaev A., Ryazanova T. Stochastic sensitivity of attractors for a piecewise smooth neuron model // Journal of Difference Equations and Applications. 2019. Vol. 25. No. 9-10. P. 1468-1487.
https://doi.org/10.1080/10236198.2019.1678596
20. Bashkirtseva I.A., Ryashko L.B. Stochastic sensitivity analysis of the attractors for the randomly forced Ricker model with delay // Physics Letters A. 2014. Vol. 378. No. 48. P. 3600-3606.
https://doi.org/10.1016/j.physleta.2014.10.022
21. Bashkirtseva I.A., Ryashko L.B. Stochastic sensitivity analysis of chaotic attractors in 2D non-invertible maps // Chaos, Solitons and Fractals. 2019. Vol. 126. P. 78-84.
https://doi.org/10.1016/j.chaos.2019.05.032
22. Zhao Ming, Xuan Zuxing, Li Cuiping. Dynamics of a discrete-time predator-prey system // Advances in Difference Equations. 2016. Vol. 2016. No. 1.
https://doi.org/10.1186/s13662-016-0903-6
23. Bischi G-I., Stefanini L., Gardini L. Synchronization, intermittency and critical curves in a duopoly game // Mathematics and Computers in Simulation. 1998. Vol. 44. No. 6. P. 559-585.
https://doi.org/10.1016/S0378-4754(97)00100-6
24. Mira C., Gardini L., Barugola A., Cathala J-C. Chaotic dynamics in two-dimensional noninvertible maps. Singapore: World Scientific, 1996.
https://doi.org/10.1142/2252
Поступила в редакцию
2020-04-01
Опубликована 2020-05-20
Опубликована 2020-05-20