ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)

For citation:

Nguyen B. H., Tsybulin V. G. Mathematical model of three competing populations and multistability of periodic regimes. Izvestiya VUZ. Applied Nonlinear Dynamics, 2023, vol. 31, iss. 3, pp. 316-333. DOI: 10.18500/0869-6632-003038, EDN: HHZEBK

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Language: 
Russian
Heading: 
Modeling of Global Processes. Nonlinear Dynamics and Humanities.
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-003038
EDN: 
HHZEBK

Mathematical model of three competing populations and multistability of periodic regimes

Autors: 
Nguyen Buu Hoang, Southern Federal University
Tsybulin Vyacheslav Georgievich, Southern Federal University
Abstract: 

Purpose of this work is to analyze oscillatory regimes in a system of nonlinear differential equations describing the competition of three non-antagonistic species in a spatially homogeneous domain.

Methods. Using the theory of cosymmetry, we establish a connection between the destruction of a two-parameter family of equilibria and the emergence of a continuous family of periodic regimes. With the help of a computational experiment in MATLAB, a search for limit cycles and an analysis of multistability were carried out.

Results. We studied dynamic scenarios for a system of three competing species for different coefficients of growth and interaction. For several combinations of parameters in a computational experiment, new continuous families of limit cycles (extreme multistability) are found. We establish bistability: the coexistence of isolated limit cycles, as well as a stationary solution and an oscillatory regime.

Conclusion. We found two scenarios for locating a family of limit cycles regarding a plane passing through three equilibria corresponding to the existence of only one species. Besides cycles lying in this plane, a family is possible with cycles intersecting this plane at two points. We can consider this case as an example of periodic processes leading to overpopulation and a subsequent decline in numbers. These results will further serve as the basis for the analysis of systems of competing populations in spatially heterogeneous areas.

Key words: 
Volterra model
nonlinear differential equations
competition
family of limit cycles
multistability
Acknowledgments: 
The authors are grateful to the referee for careful reading and stimulating comments. The work was carried out at the Southern Federal University with the support of the Russian Science Foundation, grant No. 23-21-00221.
Reference: 
  1. Svirezhev IM, Logofet DO. Resilience of Biological Communities. Moscow: Nauka; 1978. 352 p. (in Russian).
  2. Murray JD. Mathematical Biology. I. An Introduction. New York: Springer; 2002. 551 p. DOI: 10.1007/b98868.
  3. Bazykin AD. Nonlinear Dynamics of Interacting Populations. Singapore: World Scientific; 1998. 216 p. DOI: 10.1142/2284.
  4. Rubin A, Riznichenko G. Mathematical Biophysics. New York: Springer; 2014. 273 p. DOI: 10. 1007/978-1-4614-8702-9.
  5. 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.
  6. Lotka AJ. Elements of Physical Biology. Philadelphia, Pennsylvania: Williams & Wilkins; 1925. 495 p.
  7. Volterra V. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Memoria della Reale Accademia Nazionale dei Lincei. 1926;2:31–113 (in Italian).
  8. May RM, Leonard WJ. Nonlinear aspects of competition between three species. SIAM Journal on Applied Mathematics. 1975;29(2):243–253. DOI: 10.1137/0129022.
  9. Chia-Wei C, Lih-Ing W, Sze-Bi H. On the asymmetric May–Leonard model of three competing species. SIAM Journal on Applied Mathematics. 1998;58(1):211–226. DOI: 10.1137/S003613999 4272060.
  10. Antonov V, Dolicanin D, Romanovski VG, Toth J. Invariant planes and periodic oscillations in the May–Leonard asymmetric model. MATCH Communications in Mathematical and in Computer Chemistry. 2016;76(2):455–474.
  11. van der Hoff Q, Greeff JC, Fay TH. Defining a stability boundary for three species competition models. Ecological Modelling. 2009;220(20):2640–2645. DOI: 10.1016/j.ecolmodel.2009.07.027.
  12. Hou Z, Baigent S. Heteroclinic limit cycles in competitive Kolmogorov systems. Discrete & Continuous Dynamical Systems. 2013;33(9):4071–4093. DOI: 10.3934/dcds.2013.33.4071.
  13. Zeeman EC, Zeeman ML. An n-dimensional competitive Lotka–Volterra system is generically determined by the edges of its carrying simplex. Nonlinearity. 2002;15(6):2019–2032. DOI: 10. 1088/0951-7715/15/6/312. 
  14. Zeeman EC, Zeeman ML. From local to global behavior in competitive Lotka-Volterra systems. Transactions of the American Mathematical Society. 2003;355(2):713–734. DOI: 10.1090/s0002- 9947-02-03103-3.
  15. Chen X, Jiang J, Niu L. On Lotka–Volterra equations with identical minimal intrinsic growth rate. SIAM Journal on Applied Dynamical Systems. 2015;14(3):1558–1599. DOI: 10.1137/15M1006878.
  16. Jiang J, Liang F. Global dynamics of 3D competitive Lotka-Volterra equations with the identical intrinsic growth rate. Journal of Differential Equations. 2020;268(6):2551–2586. DOI: 10.1016/ j.jde.2019.09.039.
  17. Nguyen BH, Ha DT, Tsybulin VG. Multistability for system of three competing species. Computer Research and Modeling. 2022;14(6):1325–1342 (in Russian). DOI: 10.20537/2076-7633-2022-14- 6-1325-1342.
  18. Yudovich VI. Bifurcations under perturbations violating cosymmetry. Doklady Physics. 2004;49(9): 522–526. DOI: 10.1134/1.1810578.
  19. Yudovich VI. Secondary cycle of equilibria in a system with cosymmetry, its creation by bifurcation and impossibility of symmetric treatment of it. Chaos. 1995;5(2):402–411. DOI: 10.1063/1.166110.
  20. Epifanov AV, Tsybulin VG. Modeling of oscillatory scenarios of the coexistence of competing populations. Biophysics. 2016;61(4):696–704. DOI: 10.1134/S0006350916040072.
  21. Epifanov AV, Tsybulin VG. Regarding the dynamics of cosymmetric predator – prey systems. Computer Research and Modeling. 2017;9(5):799–813 (in Russian). DOI: 10.20537/2076-7633- 2017-9-5-799-813.
  22. Ha TD, Tsybulin VG. Multi-stable scenarios for differential equations describing the dynamics of a predators and preys system. Computer Research and Modeling. 2020;12(6):1451–1466 (in Russian). DOI: 10.20537/2076-7633-2020-12-6-1451-1466.
  23. Fay TH, Greeff JC. A three species competition model as a decision support tool. Ecological Modelling. 2008;211(1–2):142–152. DOI: 10.1016/j.ecolmodel.2007.08.023.
  24. Bashkirtseva IA, Karpenko LV, Ryashko LB. Stochastic sensitivity of limit cycles for «predator – two preys» model. Izvestiya VUZ. Applied Nonlinear Dynamics. 2010;18(6):42–64 (in Russian). DOI: 10.18500/0869-6632-2010-18-6-42-64.
  25. Abramova EP, Ryazanova TV. Dynamic regimes of the stochastic “prey – predatory” model with competition and saturation. Computer Research and Modeling. 2019;11(3):515–531 (in Russian). DOI: 10.20537/2076-7633-2019-11-3-515-531.
  26. Bayliss A, Nepomnyashchy AA, Volpert VA. Mathematical modeling of cyclic population dynamics. Physica D: Nonlinear Phenomena. 2019;394:56–78. DOI: 10.1016/j.physd.2019.01.010.
  27. Frischmuth K, Budyansky AV, Tsybulin VG. Modeling of invasion on a heterogeneous habitat: taxis and multistability. Applied Mathematics and Computation. 2021;410:126456. DOI: 10.1016/ j.amc.2021.126456.
  28. Budyansky AV, Tsybulin VG. Modeling the dynamics of populations in a heterogeneous environment: Invasion and multistability. Biophysics. 2022;67(1):146–152. DOI: 10.1134/S0006350922010043.
  29. Ha TD, Tsybulin VG. Multistability for a mathematical model of the dynamics of predators and preys in a heterogeneous area. Contemporary Mathematics. Fundamental Directions. 2022;68(3): 509–521 (in Russian). DOI: 10.22363/2413-3639-2022-68-3-509-521.
  30. Pontryagin LS. Ordinary Differential Equations. USA: Addison-Wesley; 1962. 304 p. DOI: 10.1016/ C2013-0-01692-1.
  31. Waugh I, Illingworth S, Juniper M. Matrix-free continuation of limit cycles for bifurcation analysis of large thermoacoustic systems. Journal of Computational Physics. 2013;240:225–247. DOI: 10.1016/j.jcp.2012.12.034.
Received: 
30.01.2023
Accepted: 
23.03.2023
Available online: 
27.04.2023
Published: 
31.05.2023
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