Анализ стохастической чувствительности тьюринговских паттернов в распределенных системах реакции-диффузии
Ключевые слова:
модель реакции-диффузии, неустойчивость Тьюринга, самоорганизация, стохастическая чувствительность
Аннотация
В данной работе исследуется распределенная стохастическая модель Брюсселятора с диффузией. Мы показываем, что в зоне неустойчивости Тьюринга генерируется множество устойчивых пространственно неоднородных структур. Влияние случайного шума на стохастическую динамику вблизи этих структур анализируется прямым численным моделированием. Изучены шумовые переходы между сосуществующими паттернами. Стохастическая чувствительность модели определяется как среднеквадратичное отклонение от исходной неискаженной модели. Показано, что стохастическая чувствительность пространственно неоднородна и существенно отличается для сосуществующих структур. Обсуждается зависимость стохастической чувствительности от изменения коэффициентов диффузии и интенсивности шума.
Литература
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2. Wang X., Lutscher F. Turing patterns in a predator-prey model with seasonality // Journal of Mathematical Biology. 2019. Vol. 78. P. 711-737.
https://doi.org/10.1007/s00285-018-1289-8
3. Yuan S., Xu Ch., Zhang T. Spatial dynamics in a predator-prey model with herd behavior // Chaos. 2013. Vol. 23. No. 3. P. 033102.
https://doi.org/10.1063/1.4812724
4. Valenti D., Tranchina L., Brai M., Caruso A., Cosentino C., Spagnolo B. Environmental metal pollution considered as noise: Effects on the spatial distribution of benthic foraminifera in two coastal marine areas of Sicily (Southern Italy) // Ecological Modeling. 2008. Vol. 213. Issues 3-4. P. 449-462.
https://doi.org/10.1016/j.ecolmodel.2008.01.023
5. Morales M.A., Fernández-Cervantes I., Agustín-Serrano R., Anzo A., Sampedro M.P. Patterns formation in ferrofluids and solid dissolutions using stochastic models with dissipative dynamics // The European Physical Journal B. 2016. Vol. 89.
https://doi.org/10.1140/epjb/e2016-70344-7
6. Kuramoto Y. Chemical oscillations, waves, and turbulence. Berlin-Heidelberg: Springer, 1984.
https://doi.org/10.1007/978-3-642-69689-3
7. Turing A.M. The chemical basis of morphogenesis // Philosophical Transactions of the Royal Society of London. Series B. Biological Sciences. 1952. Vol. 237. P. 37-72.
https://doi.org/10.1098/rstb.1952.0012
8. Gambino G., Lombardo M.C., Sammartino M., Sciacca V. Turing pattern formation in the Brusselator system with nonlinear diffusion // Physical Review E. 2013. Vol. 88. Issue 4. P. 042925.
https://doi.org/10.1103/PhysRevE.88.042925
9. Kolinichenko A.P., Ryashko L.B. Modality analysis of patterns in reaction-diffusion systems with random perturbations // Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. 2019. Vol. 53. P. 73-82.
https://doi.org/10.20537/2226-3594-2019-53-07
10. Zheng Q., Wang Z., Shen J., Iqbal H.M.A. Turing bifurcation and pattern formation of stochastic reaction-diffusion system // Advances in Mathematical Physics. 2017. Vol. 2017.
https://doi.org/10.1155/2017/9648538
11. George N.B., Unni V.R., Raghunathan M., Sujith R.I. Pattern formation during transition from combustion noise to thermoacoustic instability via intermittency, Journal of Fluid Mechanics, 2018, vol. 849, pp. 615-644.
https://doi.org/10.1017/jfm.2018.427
12. Biancalani T., Jafarpour F., Goldenfeld N. Giant amplification of noise in fluctuation-induced pattern formation // Physical Review Letters. 2017. Vol. 118. Issue 1. P. 018101.
https://doi.org/10.1103/PhysRevLett.118.018101
13. Engblom S. Stochastic simulation of pattern formation in growing tissue: A multilevel approach // Bulletin of Mathematical Biology. 2019. Vol. 81. P. 3010-3023.
https://doi.org/10.1007/s11538-018-0454-y
14. Horsthemke W., Lefever R. Noise-induced transitions. Berlin-Heidelberg: Springer, 1984.
https://doi.org/10.1007/3-540-36852-3
15. Anishchenko V.S., Astakhov V.V., Neiman A.B., Vadivasova T.E., Schimansky-Geier L. Nonlinear dynamics of chaotic and stochastic systems. Berlin-Heidelberg: Springer, 2007.
https://doi.org/10.1007/978-3-540-38168-6
16. Bashkirtseva I., Ryashko L., Slepukhina E. Stochastic generation and deformation of toroidal oscillations in neuron model // International Journal of Bifurcation and Chaos. 2018. Vol. 28. No. 6. P. 1850070.
https://doi.org/10.1142/S0218127418500700
17. Ryashko L. Sensitivity analysis of the noise-induced oscillatory multistability in Higgins model of glycolysis // Chaos. 2018. Vol. 28. Issue 3. P. 033602.
https://doi.org/10.1063/1.4989982
18. Bashkirtseva I., Ryashko L. Stochastic sensitivity and method of principal directions in excitability analysis of the Hodgkin-Huxley model // International Journal of Bifurcation and Chaos. 2019. Vol. 29. No. 13. P. 1950186.
https://doi.org/10.1142/S0218127419501864
19. Kolinichenko A., Ryashko L. Multistability and stochastic phenomena in the distributed Brusselator model // Journal of Computational and Nonlinear Dynamics. 2020. Vol. 15. No. 1.
https://doi.org/10.1115/1.4045405
20. Sauer T. Numerical solution of stochastic differential equations in finance // Handbook of Computational Finance. Berlin-Heidelberg: Springer, 2012. P. 529-550.
https://doi.org/10.1007/978-3-642-17254-0_19
21. Neuenkirch A., Szölgyenyi M., Szpruch L. An adaptive Euler-Maruyama scheme for stochastic differential equations with discontinuous drift and its convergence analysis // SIAM Journal on Numerical Analysis. 2019. Vol. 57. No. 1. P. 378-403.
https://doi.org/10.1137/18M1170017
Поступила в редакцию
2020-03-15
Опубликована 2020-05-20
Опубликована 2020-05-20