{"title":"Convergence conditions for continuous and discrete models of population dynamics","authors":"A. Aleksandrov","doi":"10.21638/11701/spbu10.2022.401","DOIUrl":null,"url":null,"abstract":"Some classes of continuous and discrete generalized Volterra models of population dynamics are considered. It is supposed that there are relationships of the type \"symbiosis\", \"compensationism\" or \"neutralism\" between any two species in a biological community. The objective of the work is to obtain conditions under which the investigated models possess the convergence property. This means that the studying system admits a bounded solution that is globally asimptotically stable. To determine the required conditions, the V. I. Zubov's approach and its discrete-time counterpart are used. Constructions of Lyapunov functions are proposed, and with the aid of these functions, the convergence problem for the considered models is reduced to the problem of the existence of positive solutions for some systems of linear algebraic inequalities. In the case where parameters of models are almost periodic functions, the fulfilment of the derived conditions implies that limiting bounded solutions are almost periodic, as well. An example is presented illustrating the obtained theoretical conclusions.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":"39 1","pages":""},"PeriodicalIF":0.3000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.21638/11701/spbu10.2022.401","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, INTERDISCIPLINARY APPLICATIONS","Score":null,"Total":0}
引用次数: 0
Abstract
Some classes of continuous and discrete generalized Volterra models of population dynamics are considered. It is supposed that there are relationships of the type "symbiosis", "compensationism" or "neutralism" between any two species in a biological community. The objective of the work is to obtain conditions under which the investigated models possess the convergence property. This means that the studying system admits a bounded solution that is globally asimptotically stable. To determine the required conditions, the V. I. Zubov's approach and its discrete-time counterpart are used. Constructions of Lyapunov functions are proposed, and with the aid of these functions, the convergence problem for the considered models is reduced to the problem of the existence of positive solutions for some systems of linear algebraic inequalities. In the case where parameters of models are almost periodic functions, the fulfilment of the derived conditions implies that limiting bounded solutions are almost periodic, as well. An example is presented illustrating the obtained theoretical conclusions.
研究了种群动力学的连续和离散广义Volterra模型。在一个生物群落中,任何两个物种之间都存在着“共生”、“补偿”或“中性”的关系。本文的目的是得到所研究的模型具有收敛性的条件。这意味着所研究的系统承认一个全局渐近稳定的有界解。为了确定所需的条件,使用了V. I. Zubov方法及其离散时间对应方法。利用Lyapunov函数的构造,将所考虑的模型的收敛问题简化为一类线性代数不等式系统正解的存在性问题。在模型参数几乎是周期函数的情况下,所导出的条件的满足意味着极限有界解也几乎是周期的。最后给出了一个算例来说明所得的理论结论。
期刊介绍:
The journal is the prime outlet for the findings of scientists from the Faculty of applied mathematics and control processes of St. Petersburg State University. It publishes original contributions in all areas of applied mathematics, computer science and control. Vestnik St. Petersburg University: Applied Mathematics. Computer Science. Control Processes features articles that cover the major areas of applied mathematics, computer science and control.