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

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Ganikhodzhaev R. N., Eshmamatova D. B., Muminov U. R., Masharipov S. I. Degenerate cases in discrete Lotka – Volterra dynamical systems. Izvestiya VUZ. Applied Nonlinear Dynamics, 2025, vol. 33, iss. 2, pp. 165-183. DOI: 10.18500/0869-6632-003153, EDN: JIIRXF

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Language: 
Russian
Heading: 
Applied Problems of Nonlinear Oscillation and Wave Theory
Article type: 
Article
UDC: 
530.182
DOI: 
10.18500/0869-6632-003153
EDN: 
JIIRXF

Degenerate cases in discrete Lotka – Volterra dynamical systems

Autors: 
Ganikhodzhaev Rasul Nabiyevich, National University of Uzbekistan named after Mirzo Ulugbek
Eshmamatova Dilfuza Bakhromovna, Toshkent davlat transport universiteti
Muminov Ulugbek Rahimjonovich, National University of Uzbekistan named after Mirzo Ulugbek
Masharipov Sirojiddin Ismoyiljon ugli, National University of Uzbekistan named after Mirzo Ulugbek
Abstract: 

The purpose of the work is to study the asymptotic behavior of trajectories of interior points of discrete Lotka–Volterra dynamical systems with degenerate skew-symmetric matrices operating in two-dimensional and three-dimensional simplexes. It turned out that in a number of applied problems, the Lotka–Volterra mappings of this type arise and the simplex points in this case are considered as the state of the system under study. In this case, the mapping preserving the simplex determines the discrete law of evolution of this system. For an arbitrary starting point, we can construct a sequence — an orbit that determines its evolution. And if in this case the mapping in question is an automorphism, we can define both a positive and a negative orbit for the point in question. At the same time, the limiting sets of positive and negative orbits are of particular interest.

Methods. It is known that for Lotka–Volterra mappings it is possible to define limit sets, which in the case of non-degenerate mappings consist of a single point. In this paper, we define these sets for degenerate Lotka–Volterra mappings by constructing the Lyapunov function and applying Jacobian spectrum analysis. It should be noted that these sets allow us to describe the dynamics of the systems under consideration.

Results. Taking into account that the considered mappings are automorphisms, using the Lyapunov functions and applying the analysis of the Jacobian spectrum, sets of limit points of both positive and negative trajectories are constructed and it is proved that in the degenerate case they are infinite. It is also shown that partially oriented graphs can be constructed for degenerate mappings.

Conclusion. Degenerate cases of Lotka–Volterra mappings have not been considered by other authors before us. These mappings are interesting because they can be considered as discrete models of epidemiological situations, in particular, for studying the course of airborne viral infections. The results obtained in this work provide a detailed description of the dynamics of the trajectories of Lotka–Volterra mappings with degenerate matrices. In addition, partially oriented graphs were constructed for the systems under consideration in order to visually represent the dynamics of epidemiological situations.
 

Key words: 
Lotka–Volterra mapping
orbit
Lyapunov function
partially–oriented graph
Acknowledgments: 
D.B.Eshmamatova, U.R.Muminov and S.I.Masharipov express their gratitude to their scientific supervisor, Professor R.N.Ganikhodzhaev for the idea and formulation of the considered problem.
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Received: 
15.07.2023
Accepted: 
14.09.2024
Available online: 
07.12.2024
Published: 
31.03.2025
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