{"title":"洛特卡-伏特拉混合系统解的终极边界性和永久性条件","authors":"A. V. Platonov","doi":"10.3103/s1066369x24700440","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In the paper, a generalized Lotka–Volterra-type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.</p>","PeriodicalId":46110,"journal":{"name":"Russian Mathematics","volume":"147 1","pages":""},"PeriodicalIF":0.5000,"publicationDate":"2024-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Conditions for Ultimate Boundedness of Solutions and Permanence for a Hybrid Lotka–Volterra System\",\"authors\":\"A. V. Platonov\",\"doi\":\"10.3103/s1066369x24700440\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<h3 data-test=\\\"abstract-sub-heading\\\">Abstract</h3><p>In the paper, a generalized Lotka–Volterra-type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.</p>\",\"PeriodicalId\":46110,\"journal\":{\"name\":\"Russian Mathematics\",\"volume\":\"147 1\",\"pages\":\"\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2024-09-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Russian Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.3103/s1066369x24700440\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Russian Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3103/s1066369x24700440","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Conditions for Ultimate Boundedness of Solutions and Permanence for a Hybrid Lotka–Volterra System
Abstract
In the paper, a generalized Lotka–Volterra-type system with switching is considered. The conditions for the ultimate boundedness of solutions and the permanence of the system are studied. With the aid of the direct Lyapunov method, the requirements for the switching law are established to guarantee the necessary dynamics of the system. An attractive compact invariant set is constructed in the phase space of the system, and a given region of attraction for this set is provided. A distinctive feature of the work is the use of a combination of two different Lyapunov functions, each of which plays its own special role in solving the problem.
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