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.
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Received: 
30.01.2023
Accepted: 
23.03.2023
Available online: 
27.04.2023
Published: 
31.05.2023
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