#### 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

# Mathematical model of three competing populations and multistability of periodic regimes

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.

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