{"title":"Criteria for internal fixed points existence of discrete dynamic Lotka–Volterra systems with homogeneous tournaments","authors":"D. Eshmamatova, M. Tadzhieva, R. Ganikhodzhaev","doi":"10.18500/0869-6632-003012","DOIUrl":null,"url":null,"abstract":"Purpose of the work is to study the dynamics of the asymptotic behavior of trajectories of discrete Lotka–Volterra dynamical systems with homogeneous tournaments operating in an arbitrary (𝑚 − 1)-dimensional simplex. It is known that a dynamic system is an object or a process for which the concept of a state is uniquely defined as a set of certain quantities at a given time, and a law describing the evolution of initial state over time is given. Mainly in questions of population genetics, biology, ecology, epidemiology and economics, systems of nonlinear differential equations describing the evolution of the process under study often arise. Since the Lotka–Volterra equations often arise in life phenomena, the main purpose of the work is to study the trajectories of discrete dynamical Lotka–Volterra systems using elements of graph theory. Methods. In the paper cards of fixed points are constructed for quadratic Lotka–Volterra mappings, that allow describing the dynamics of the systems under consideration. Results. Using cards of fixed points of a discrete dynamical system, criteria for the existence of fixed points with odd nonzero coordinates are given in a particular case, and these results on the location of fixed points of Lotka–Volterra systems are generalized accordingly in the case of an arbitrary simplex. The main results are theorems 5–9, which allow us to describe the dynamics of these systems arising in a number of genetic, epidemiological and ecological models. Conclusion. The results obtained in the paper give a detailed description of the dynamics of the trajectories of Lotka–Volterra maps with homogeneous tournaments. The map of fixed points highlights a specific area in the simplex that is most important and interesting for studying the dynamics of these maps. The results obtained are applicable in environmental problems, for example, to describe and study the cycle of biogens.","PeriodicalId":41611,"journal":{"name":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","volume":"24 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Izvestiya Vysshikh Uchebnykh Zavedeniy-Prikladnaya Nelineynaya Dinamika","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.18500/0869-6632-003012","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"PHYSICS, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 2
Abstract
Purpose of the work is to study the dynamics of the asymptotic behavior of trajectories of discrete Lotka–Volterra dynamical systems with homogeneous tournaments operating in an arbitrary (𝑚 − 1)-dimensional simplex. It is known that a dynamic system is an object or a process for which the concept of a state is uniquely defined as a set of certain quantities at a given time, and a law describing the evolution of initial state over time is given. Mainly in questions of population genetics, biology, ecology, epidemiology and economics, systems of nonlinear differential equations describing the evolution of the process under study often arise. Since the Lotka–Volterra equations often arise in life phenomena, the main purpose of the work is to study the trajectories of discrete dynamical Lotka–Volterra systems using elements of graph theory. Methods. In the paper cards of fixed points are constructed for quadratic Lotka–Volterra mappings, that allow describing the dynamics of the systems under consideration. Results. Using cards of fixed points of a discrete dynamical system, criteria for the existence of fixed points with odd nonzero coordinates are given in a particular case, and these results on the location of fixed points of Lotka–Volterra systems are generalized accordingly in the case of an arbitrary simplex. The main results are theorems 5–9, which allow us to describe the dynamics of these systems arising in a number of genetic, epidemiological and ecological models. Conclusion. The results obtained in the paper give a detailed description of the dynamics of the trajectories of Lotka–Volterra maps with homogeneous tournaments. The map of fixed points highlights a specific area in the simplex that is most important and interesting for studying the dynamics of these maps. The results obtained are applicable in environmental problems, for example, to describe and study the cycle of biogens.
期刊介绍:
Scientific and technical journal Izvestiya VUZ. Applied Nonlinear Dynamics is an original interdisciplinary publication of wide focus. The journal is included in the List of periodic scientific and technical publications of the Russian Federation, recommended for doctoral thesis publications of State Commission for Academic Degrees and Titles at the Ministry of Education and Science of the Russian Federation, indexed by Scopus, RSCI. The journal is published in Russian (English articles are also acceptable, with the possibility of publishing selected articles in other languages by agreement with the editors), the articles data as well as abstracts, keywords and references are consistently translated into English. First and foremost the journal publishes original research in the following areas: -Nonlinear Waves. Solitons. Autowaves. Self-Organization. -Bifurcation in Dynamical Systems. Deterministic Chaos. Quantum Chaos. -Applied Problems of Nonlinear Oscillation and Wave Theory. -Modeling of Global Processes. Nonlinear Dynamics and Humanities. -Innovations in Applied Physics. -Nonlinear Dynamics and Neuroscience. All articles are consistently sent for independent, anonymous peer review by leading experts in the relevant fields, the decision to publish is made by the Editorial Board and is based on the review. In complicated and disputable cases it is possible to review the manuscript twice or three times. The journal publishes review papers, educational papers, related to the history of science and technology articles in the following sections: -Reviews of Actual Problems of Nonlinear Dynamics. -Science for Education. Methodical Papers. -History of Nonlinear Dynamics. Personalia.