Vestnik КRAUNC. Fiz.-Mat. Nauki. 2025. vol. 52. no. 3. P. 111 – 130. ISSN 2079-6641

MATHEMATICAL MODELING
https://doi.org/10.26117/2079-6641-2025-52-3-111-130
Research Article
Full text in Russian
MSC 35B45, 35K20, 35K57, 35K59

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A Two-Component Competition Model With Two Different Free Boundaries

R. T. Zunnunov^{\ast}, M. S. Rasulov^{\ast 1,3}, R. I. Parovik^4

^1V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, 9 University str., 100174, Uzbekistan
^2Branch of the Russian State University of Oil and Gas (National Research University) named after I. M. Gubkin in Tashkent, Mirzo-Ulugbek district, Durmon Yuli str., 34, Tashkent, 100125, Uzbekistan
^3Tashkent State University of Economics, Tashkent, 49 Islom Karimov str., 100066, Uzbekistan
^4Institute of Cosmophysical Research and Radio Wave Propagation, Far Eastern Branch of the Russian Academy of Sciences, 7 Mirnaya srt., 684034, Paratunka, Russia

Abstract. This paper investigates the dynamics of a competitive Lotka-Volterra system containing two free boundaries, where each boundary models the propagation front of one of the two competing species. A free boundary problem is considered for a system of quasilinear parabolic equations with nonlinear convective terms. The paper first establishes a priori estimates of the H¨older norms to solve the problem. Based on these a priori estimates, the existence and uniqueness of the solution are proven. Next, an implicit finitedifference scheme is used to find a numerical solution to the problem, which characterizes the densities of the two competing populations. Using the Python programming language, the obtained solutions are visualized, and graphs of the free boundary dynamics are constructed. From an application perspective, the free boundary problem for the Lotka-Volterra diffusion system is a mathematical model describing predatorprey propagation in a population with a dynamic boundary of the domain of existence. This problem arises when one of the populations (for example, a predator) influences the boundaries of the range of its prey, or when the boundaries of the range are formed under the influence of external factors, and the diffusion itself occurs in this system.

Key words: model; free boundaries; system of quasilinear parabolic equations; a priori estimates; existence and uniqueness of solutions; numerical algorithm; Python

Received: 13.11.2025; Revised: 20.11.2025; Accepted: 22.11.2025; First online: 23.11.2025

For citation. Zunnunov R. T., Rasulov M. S., Parovik R. I. A two-component competition model with two different free boundaries. Vestnik KRAUNC. Fiz.-mat. nauki. 2025, 52: 3, 111-130. EDN: CDFHIP. https://doi.org/10.26117/2079-6641-2025-52-3-111-130.

Funding. The work was carried out within the framework of the agreement between IKIR FEB RAS and the
V.I. Romanovsky Institute of Mathematics (Tashkent, Uzbekistan) No. 1117 dated 04.28.2022 (0469/01/22 NTMI) on mathematical research.

Competing interests. There are no conflicts of interest regarding authorship and publication.

Contribution and Responsibility. All authors contributed to this article. Authors are solely responsible for providing the final version of the article in print. The final version of the manuscript was approved by all authors.

^{\ast}Correspondence: E-mail: rasulovms@bk.ru

The content is published under the terms of the Creative Commons Attribution 4.0 International License

© Zunnunov R. T., Rasulov M. S., Parovik R. I., 2025

© Institute of Cosmophysical Research and Radio Wave Propagation, 2025 (original layout, design, compilation)

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Information about the authors

Zunnunov Rakhimjon Temirbekovich – D.Sci. (Phys. Math.), Leading Researcher, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, ORCID 0000-0001-9392-5464

Rasulov Mirojiddin Sobirjonovich – PhD (Phys. Math.), Senior Researcher, V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences, Tashkent, Uzbekistan, ORCID 0000-0003-0704-6012.

Parovik Roman Ivanovich – DSc (Phys. Math.), Professor, Leading Researcher at the Laboratory of Modeling Physical Processes, Institute of Cosmophysical Research and Radio Wave Propagation FEB RAS, S. Paratunka, Russia, ORCID 0000-0002-1576-1860.

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