Classification of Character of Rest Points of theLotka-Volterra Operator Acting in S4
Classification of Character of Rest Points of theLotka-Volterra Operator Acting in S4
DOI:
https://doi.org/10.56143/ujmcs.v1i1.12Ключевые слова:
skew-symmetric matrix; boundary point; interior point; fixed point; partially oriented graph; eigenvalue; saddle point; attractor; repeller.Аннотация
In the paper, we consider the problem of studying the trajectories of points under the action of the Lotka-Volterra operator. In short, the goal is to study the dynamics of trajectories of interior points by finding fixed points and studying the Jacobian spectrum at these points of the operator in question. It turned out that in a number of applied problems, there are Lotka-Volterra mappings of exactly this type, and the points of the simplex in this case are considered as the states of the system under study. In this case, the simplex-preserving mapping defines the discrete law of evolution of the given system. Starting from a certain starting point, we can consider the sequence that determines the evolution of this point. The work explicitly shows sets of limit points for positive and negative trajectories, which in turn describe in applied problems the beginning and the end of the evolutionary process, respectively.
Библиографические ссылки
[1] Devaney R.L.: A First Course in Chaotic Dynamical Systems. Theory and Experiment. Second edition. Taylor and Francis Group, 2020. - 329 p.
[2] Holmgren R.A. : A first Course in Discrete Dynamical Systems. Springer Verlag, 1994. - 225 p.
[3] Brin M., Stuck G. : Introduction to Dynamical Systems. Cambridge University Press, 2004. - 254 p.
[4] Subathra G. and Jayalalitha G. : Fixed Points-Julia sets. Advances in Mathematics: Scientific Journal, 2020. - 9. – P. 6759–6763.
[5] Eshmamatova D.B., Tadzhieva M.A., Ganikhodzhaev R.N.: Criteria for internal fixed points existence of discrete dynamic Lotka-Volterra systems with homogeneous tournaments. Izvestiya Vysshikh Uchebnykh Zavedeniy. Prikladnaya Dinamika. 2023. – 30. – No. 6. – P. 702-716. https://doi.org/10.18500/0869-6632-003012
[6] Eshmamatova D. B., Tadzhieva M.A., Ganikhodzhaev R.N.: Criteria for the Existence of Internal Fixed Points of Lotka-Volterra Quadratic Stochastic Mappings with Homogeneous Tournaments Acting in an (m-1)-Dimensional Simplex. Journal of Applied Nonlinear Dynamics. 2023. – 12 – No. 4. – P. 679-688. https://doi.org/10.5890/JAND.2023.12.004
[7] Tadzhieva M.A., Eshmamatova D. B., Ganikhodzhaev R.N.: Volterra-Type Quadratic Stochastic Operators with a Homogeneous Tournament. Journal of Mathematical Sciences. 2024. – 278. No. 3. – P. 546-556. DOI: 10.1007/s10958-024-06937-0
[8] Eshmamatova D. B., Tadzhieva M.A., Ganikhodzhaev R.N.: Degenerate cases in Lotka-Volterra systems. AIP Conference Proceedings. 2023. – 2781. – P. 020034-1-8. DOI: 10.1063/5.0144887
[9] Eshmamatova D. B.: Discrete Analogue of the SIR Model. AIP Conference Proceedings. 2023. – 2781. – P. 020024-1-8.
DOI: 10.1063/5.0144884
[10] Eshmamatova D.B., Seytov Sh. J., Narziev N. B.: Basins of Fixed Points for the Composition of the Lotka-Volterra Mappings and Their
Classification. Lobachevskii Journal of Mathematics. 2023. – 44. No. 2. – P. 558-569. DOI: 10.1134/S1995080223020142
[11] Eshmamatova D. B., Seytov Sh. J.: Discrete Dynamical Systems of Lotka Volterra and Their Applications on the Modeling of the Biogen Cycle in Ecosystem. Lobachevskii Journal of Mathematics. 2023. – 44. No. 4. – P. 1462-1476. DOI: 10.1134/S1995080223040248
[12] Harari F., Palmer E.: Perechisleniye grafov [Enumeration of graphs] (monograph). M.: Mir. – 324 p. [in Russian].
