Mechanisms leading to bursting oscillations in the system of predator–prey communities coupled by migrations

Cover Page

Cite item

Full Text

Abstract

The purpose is to study the periodic regimes of the dynamics for two non-identical predator–prey communities coupled by migrations, associated with the partial synchronization of fluctuations in the abundance of communities. The combination of fluctuations in neighboring sites leads to the regimes that include both fast bursts (bursting oscillations) and slow oscillations (tonic spiking). These types of activity are characterized by a different ratio of synchronous and non-synchronous dynamics of communities in certain periods of time. In this paper, we describe scenarios of the transition between different types of burst activity. These types of dynamics differ from each other not so much in size, shape, and number of spikes in a burst, but in the order of these bursts relative to the slow-fast cycle. Methods. To study the proposed model, we use the bifurcation analysis methods of dynamic systems, as well as geometric methods based on the division of the full system into fast and slow equations (subsystems). Results. We showed that the dynamics of the first subsystem with a slow-fast limit cycle directly determines the dynamics of the second one with burst activity through a smooth dependence of regime on the number of predators and a non-smooth dependence on the number of prey. We constructed the invariant manifolds on which there are parts of dynamics with tonic (slow manifold) and burst (fast manifold) activity of the full system. Conclusion. We described the scenario for bursting with different waveforms, which are determined by the appearance of the fast invariant manifold and the location of its parts relative to the slow-fast cycle. The transitions between different types of burst are accompanied by a change in the oscillation period, the degree of synchronization, and, as a result, the dynamics becomes quasi-periodic when both communities are not synchronous with each other.

About the authors

Ekaterina Viktorovna Kurilova

Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

ul. Sholom-Aleikhema, 4, Birobidzhan, 679016, Russia

Matvej Pavlovich Kulakov

Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

ul. Sholom-Aleikhema, 4, Birobidzhan, 679016, Russia

Efim Yakovlevich Frisman

Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

ul. Sholom-Aleikhema, 4, Birobidzhan, 679016, Russia

References

  1. Фрисман Е. Я., Кулаков М. П., Ревуцкая О. Л., Жданова О. Л., Неверова Г. П. Основные направления и обзор современного состояния исследований динамики структурированных и взаимодействующих популяций // Компьютерные исследования и моделирование. 2019. Т. 11, № 1. С. 119–151. doi: 10.20537/2076-7633-2019-11-1-119-151.
  2. Mukhopadhyay B., Bhattacharyya R. Role of predator switching in an eco-epidemiological model with disease in the prey // Ecological Modelling. 2009. Vol. 220, no. 7. P. 931–939. doi: 10.1016/j.ecolmodel.2009.01.016.
  3. Saifuddin M., Biswas S., Samanta S., Sarkar S., Chattopadhyay J. Complex dynamics of an ecoepidemiological model with different competition coefficients and weak Allee in the predator // Chaos, Solitons & Fractals. 2016. Vol. 91. P. 270–285. doi: 10.1016/j.chaos.2016.06.009.
  4. Jansen V. A. A. The dynamics of two diffusively coupled predator–prey populations // Theoretical Population Biology. 2001. Vol. 59, no. 2. P. 119–131. doi: 10.1006/tpbi.2000.1506.
  5. Liu Y. The Dynamical Behavior of a Two Patch Predator-Prey Model. Theses, Dissertations, & Master Projects. Williamsburg: College of William and Mary, 2010. 46 p.
  6. Saha S., Bairagi N., Dana S. K. Chimera states in ecological network under weighted mean-field dispersal of species // Frontiers in Applied Mathematics and Statistics. 2019. Vol. 5. P. 15. doi: 10.3389/fams.2019.00015.
  7. Shen Y., Hou Z., Xin H. Transition to burst synchronization in coupled neuron networks // Physical Review E. 2008. Vol. 77, no. 3. P. 031920. doi: 10.1103/PhysRevE.77.031920.
  8. Баханова Ю. В., Казаков А. О., Коротков А. Г. Спиральный хаос в моделях типа Лотки– Вольтерры // Журнал Средневолжского математического общества. 2017. Т. 19, № 2. С. 13–24. doi: 10.15507/2079-6900.19.201701.013-024.
  9. Bakhanova Y. V., Kazakov A. O., Korotkov A. G., Levanova T. A., Osipov G. V. Spiral attractors as the root of a new type of «bursting activity» in the Rosenzweig–MacArthur model // The European Physical Journal Special Topics. 2018. Vol. 227, no. 7–9. P. 959–970. doi: 10.1140/epjst/e2018- 800025-6.
  10. Huang T., Zhang H. Bifurcation, chaos and pattern formation in a space- and time-discrete predator–prey system // Chaos, Solitons & Fractals. 2016. Vol. 91. P. 92–107. doi: 10.1016/j.chaos. 2016.05.009.
  11. Banerjee M., Mukherjee N., Volpert V. Prey-predator model with a nonlocal bistable dynamics of prey // Mathematics. 2018. Vol. 6, no. 3. P. 41. doi: 10.3390/math6030041.
  12. Yao Y., Song T., Li Z. Bifurcations of a predator–prey system with cooperative hunting and Holling III functional response // Nonlinear Dynamics. 2022. Vol. 110, no. 1. P. 915–932. doi: 10.1007/s11071-022-07653-7.
  13. Smirnov D. Revealing direction of coupling between neuronal oscillators from time series: Phase dynamics modeling versus partial directed coherence // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2007. Vol. 17, no. 1. P. 013111. doi: 10.1063/1.2430639.
  14. Dahasert N., Ozturk I., Kili¸c R. Experimental realizations of the HR neuron model with programmable hardware and synchronization applications // Nonlinear Dynamics. 2012. Vol. 70, no. 4. P. 2343–2358. doi: 10.1007/s11071-012-0618-5.
  15. Wang L., Liu S., Zeng Y. Diversity of firing patterns in a two-compartment model neuron: Using internal time delay as an independent variable // Neural Network World. 2013. Vol. 23, no. 3. P. 243–254. doi: 10.14311/NNW.2013.23.015.
  16. Santos M. S., Protachevicz P. R., Iarosz K. C., Caldas I. L., Viana R. L., Borges F. S., Ren H.-P., Szezech Jr. J. D., Batista A. M., Grebogi C. Spike-burst chimera states in an adaptive exponential integrate-and-fire neuronal network // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019. Vol. 29, no. 4. P. 043106. doi: 10.1063/1.5087129.
  17. Izhikevich E. M. Neural excitability, spiking and bursting // International Journal of Bifurcation and Chaos. 2000. Vol. 10, no. 6. P. 1171–1266. doi: 10.1142/S0218127400000840.
  18. Han X., Jiang B., Bi Q. Symmetric bursting of focus–focus type in the controlled Lorenz system with two time scales // Physics Letters A. 2009. Vol. 373, no. 40. P. 3643–3649. doi: 10.1016/j.physleta.2009.08.020.
  19. Gu H., Pan B., Xu J. Bifurcation scenarios of neural firing patterns across two separated chaotic regions as indicated by theoretical and biological experimental models // Abstract and Applied Analysis. 2013. Vol. S141. P. 374674. doi: 10.1155/2013/374674.
  20. Chen J., Li X., Hou J., Zuo D. Bursting oscillation and bifurcation mechanism in fractional-order Brusselator with two different time scales // Journal of Vibroengineering. 2017. Vol. 19, no. 2. P. 1453–1464. doi: 10.21595/jve.2017.18109.
  21. Макеева А. А., Дмитричев А. С., Некоркин В. И. Циклы-утки и торы-утки в слабонеоднородном ансамбле нейронов ФитцХью-Нагумо с возбуждающими связями // Известия вузов. ПНД. 2020. Т. 28, № 5. С. 524–546. doi: 10.18500/0869-6632-2020-28-5-524-546.
  22. Holling C. S. Some characteristics of simple types of predation and parasitism // The Canadian Entomologist. 1959. Vol. 91, no. 7. P. 385–398. doi: 10.4039/Ent91385-7.
  23. Holling C. S. The functional response of predators to prey density and its role in mimicry and population regulation // The Memoirs of the Entomological Society of Canada. 1965. Vol. 97, no. S45. P. 5–60. doi: 10.4039/entm9745fv.
  24. Базыкин А. Д. Математическая биофизика взаимодействующих популяций. М.: Наука, 1985. 181 с.
  25. Bazykin A. D. Nonlinear Dynamics of Interacting Populations. World Scientific Series on Nonlinear Science Series A: Vol. 11. New–Jersey, London, Hong Kong: World Scientific, 1998. 216 p. doi: 10.1142/2284.
  26. Rosenzweig M. L., MacArthur R. H. Graphical representation and stability conditions of predator– prey interactions // The American Naturalist. 1963. Vol. 97, no. 895. P. 209–223. DOI: 10.1086/ 282272.
  27. Rinaldi S., Muratori S. Slow-fast limit cycles in predator-prey models // Ecological Modelling. 1992. Vol. 61, no. 3–4. P. 287–308. doi: 10.1016/0304-3800(92)90023-8.
  28. Кулаков М. П., Курилова Е. В., Фрисман Е. Я. Синхронизация, тоническая и пачечная динамика в модели двух сообществ «хищник–жертва», связанных миграциями хищника // Математическая биология и биоинформатика. 2019. Т. 14, № 2. С. 588–611. doi: 10.17537/2019.14.588.
  29. Курилова Е. В., Кулаков М. П., Фрисман Е. Я. Последствия синхронизации колебаний численностей в двух взаимодействующих сообществах типа «хищник–жертва» при насыщении хищника и лимитировании численности жертвы // Информатика и системы управления. 2015. № 3(45). С. 24–34.
  30. Курилова Е. В., Кулаков М. П. Квазипериодические режимы динамики в модели миграционно связанных сообществ «Хищник–жертва» // Региональные проблемы. 2020. Т. 23, № 2. С. 3–11. doi: 10.31433/2618-9593-2020-23-2-3-11.
  31. Dhooge A., Govaerts W., Kuznetsov Y. A., Meijer H. G. E., Sautois B. New features of the software MatCont for bifurcation analysis of dynamical systems // Mathematical and Computer Modelling of Dynamical Systems. 2008. Vol. 14, no. 2. P. 147–175. doi: 10.1080/13873950701742754.
  32. Benoit E., Callot J. L., Diener F., Diener M. Chasse au canard // Collectanea Mathema–tica. 1981. Vol. 31–32. P. 37–119.
  33. Арнольд В. И., Афраймович В. С., Ильяшенко Ю. С., Шильников Л. П. Теория бифуркаций // Динамические системы – 5. Итоги науки и техники. Современные проблемы математики. Фундаментальные направления. Т. 5. М.: ВИНИТИ, 1986. С. 5–218.
  34. Ersoz E. K., Desroches M., Mirasso C. R., Rodrigues S. Anticipation via canards in excitable systems // Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019. Vol. 29, no. 1. P. 013111. doi: 10.1063/1.5050018.
  35. Shilnikov A., Cymbalyuk G. Homoclinic bifurcations of periodic orbits on a route from tonic spiking to bursting in neuron models // Regular and Chaotic Dynamics. 2004. Vol. 9, no. 3. P. 281–297. doi: 10.1070/RD2004v009n03ABEH000281.
  36. Коломиец М. Л., Шильников А. Л. Методы качественной теории для модели Хиндмарш–Роуз // Нелинейная динамика. 2010. Т. 6, № 1. С. 23–52. doi: 10.20537/nd1001003.
  37. Hindmarsh J. L., Rose R. M. A model of neuronal bursting using three coupled first order differential equations // Proc. R. Soc. Lond. B. 1984. Vol. 221, no. 1222. P. 87–102. DOI: 10.1098/ rspb.1984.0024.
  38. Linaro D., Champneys A., Desroches M., Storace M. Codimension-two homoclinic bifurcations underlying spike adding in the Hindmarsh–Rose burster // SIAM Journal on Applied Dynamical Systems. 2012. Vol. 11, no. 3. P. 939–962. doi: 10.1137/110848931.

This website uses cookies

You consent to our cookies if you continue to use our website.

About Cookies