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Kurilova E. V., Kulakov M. P., Frisman E. Y. Mechanisms leading to bursting oscillations in the system of predator–prey communities coupled by migrations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 2, pp. 143-169. DOI: 10.18500/0869-6632-003030, EDN: FGDHHM

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

Kurilova Ekaterina Viktorovna, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch
Kulakov Matvej Pavlovich, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch
Frisman Efim Yakovlevich, Institute for Complex Analysis of Regional Problems of Russian Academy of Sciences, Far Eastern Branch

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.

This work was carried out within the framework of the state targets of the Institute for Complex Analysis of Regional Problem of the Far Eastern Branch of the Russian Academy of Sciences
  1. Frisman YY, Kulakov MP, Revutskaya OL, Zhdanova OL, Neverova GP. The key approaches and review of current researches on dynamics of structured and interacting populations. Computer Research and Modeling. 2019;11(1):119–151 (in Russian). 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;220(7):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;91:270–285. DOI: 10.1016/j.chaos.2016.06.009.
  4. Jansen VAA. The dynamics of two diffusively coupled predator–prey populations. Theoretical Population Biology. 2001;59(2):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 SK. Chimera states in ecological network under weighted mean-field dispersal of species. Frontiers in Applied Mathematics and Statistics. 2019;5:15. 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;77(3):031920. DOI: 10.1103/PhysRevE.77.031920.
  8. Bakhanova YV, Kazakov AO, Korotkov AG. Spiral chaos in Lotka–Volterra like models. Middle Volga Mathematical Society Journal. 2017;19(2):13–24 (in Russian). DOI: 10.15507/2079- 6900.19.201701.013-024.
  9. Bakhanova YV, Kazakov AO, Korotkov AG, Levanova TA, Osipov GV. Spiral attractors as the root of a new type of «bursting activity» in the Rosenzweig–MacArthur model. The European Physical Journal Special Topics. 2018;227(7–9):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;91:92–107. 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;6(3):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;110(1):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;17(1):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;70(4):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;23(3):243–254. DOI: 10.14311/NNW.2013.23.015.
  16. Santos MS, Protachevicz PR, Iarosz KC, Caldas IL, Viana RL, Borges FS, Ren HP, Szezech Jr JD, Batista AM, Grebogi C. Spike-burst chimera states in an adaptive exponential integrate-and-fire neuronal network. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019;29(4):043106. DOI: 10.1063/1.5087129.
  17. Izhikevich EM. Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos. 2000;10(6):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;373(40):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;S141:374674. 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;19(2):1453–1464. DOI: 10.21595/jve.2017.18109.
  21. Makeeva AA, Dmitrichev AS, Nekorkin VI. Cycles-canards and torus-canards in a weakly inhomogeneous ensemble of FitzHugh–Nagumo neurons with excitatory synaptic couplings. Izvestiya VUZ. Applied Nonlinear Dynamics. 2020;28(5):524–546 (in Russian). DOI: 10.18500/ 0869-6632-2020-28-5-524-546.
  22. Holling CS. Some characteristics of simple types of predation and parasitism. The Canadian Entomologist. 1959;91(7):385–398. DOI: 10.4039/Ent91385-7.
  23. Holling CS. 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;97(S45):5–60. DOI: 10.4039/entm9745fv.
  24. Bazykin AD. Mathematical Biophysics of Interacting Populations. Moscow: Nauka; 1985. 181 p. (in Russian).
  25. Bazykin AD. 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 ML, MacArthur RH. Graphical representation and stability conditions of predator– prey interactions. The American Naturalist. 1963;97(895):209–223. DOI: 10.1086/282272.
  27. Rinaldi S, Muratori S. Slow-fast limit cycles in predator-prey models. Ecological Modelling. 1992;61(3–4):287–308. DOI: 10.1016/0304-3800(92)90023-8.
  28. Kulakov MP, Kurilova EV, Frisman EY. Synchronization and bursting activity in the model for two predator–prey systems coupled by predator migration. Mathematical Biology and Bioinformatics. 2019;14(2):588–611 (in Russian). DOI: 10.17537/2019.14.588.
  29. Kurilova EV, Kulakov MP, Frisman EY. Effects of synchronization by fluctuations in numbers of two predator–prey communities at saturation predator growth and limitation of the victim number. Information Science and Control Systems. 2015;(3(45)):24–34 (in Russian).
  30. Kurilova EV, Kulakov MP. Quasi-periodic dynamics in a model of predator–prey communities coupled by migration. Regional Problems. 2020;23(2):3–11 (in Russian). DOI: 10.31433/2618- 9593-2020-23-2-3-11.
  31. Dhooge A, Govaerts W, Kuznetsov YA, Meijer HGE, Sautois B. New features of the software MatCont for bifurcation analysis of dynamical systems. Mathematical and Computer Modelling of Dynamical Systems. 2008;14(2):147–175. DOI: 10.1080/13873950701742754.
  32. Benoit E, Callot JL, Diener F, Diener M. Chasse au canard. Collectanea Mathematica. 1981; 31–32:37–119 (in French).
  33. Arnold VI, Afraimovich VS, Ilyashenko YS, Shilnikov LP. Bifurcation theory and catastrophe theory. In: Dynamical Systems V. Vol. 5 of Encyclopaedia of Mathematical Sciences. Berlin: Springer-Verlag; 1994. P. 1–205.
  34. Ersoz EK, Desroches M, Mirasso CR, Rodrigues S. Anticipation via canards in excitable systems. Chaos: An Interdisciplinary Journal of Nonlinear Science. 2019;29(1):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;9(3):281–297. DOI: 10.1070/RD2004v009n03ABEH000281.
  36. Kolomiets ML, Shilnikov AL. Qualitative methods for case study of the Hindmarch–Rose model. Russian Journal of Nonlinear Dynamics. 2010;6(1):23–52 (in Russian). DOI: 10.20537/nd1001003.
  37. Hindmarsh JL, Rose RM. A model of neuronal bursting using three coupled first order differential equations. Proc. R. Soc. Lond. B. 1984;221(1222):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;11(3):939–962. DOI: 10.1137/110848931.
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