Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model

<!DOCTYPE html> <html lang="en" dir="ltr"> <head>  <script async src="https://www.googletagmanager.com/gtag/js?id=G-P63WKM1TM1"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-P63WKM1TM1'); </script>  <script type="text/javascript" > (function(m,e,t,r,i,k,a){m[i]=m[i]||function(){(m[i].a=m[i].a||[]).push(arguments)}; m[i].l=1*new Date(); for (var j = 0; j < document.scripts.length; j++) {if (document.scripts[j].src === r) { return; }} k=e.createElement(t),a=e.getElementsByTagName(t)[0],k.async=1,k.src=r,a.parentNode.insertBefore(k,a)}) (window, document, "script", "https://mc.yandex.ru/metrika/tag.js", "ym"); ym(55165297, "init", { clickmap:false, trackLinks:true, accurateTrackBounce:true, webvisor:false }); </script> <noscript><div><img src="https://mc.yandex.ru/watch/55165297" style="position:absolute; left:-9999px;" alt="" /></div></noscript>    <title>Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model</title> <meta name="description" content="Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model"> <meta name="keywords" content="bifurcation, chaos, dynamics of populations, fractals"> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <meta name="citation_title" content="Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model"> <meta name="citation_author" content="Vladislav N. Dumachev"> <meta name="citation_author" content="Vladimir A. Rodin"> <meta name="citation_publication_date" content="2009/07/24"> <meta name="citation_journal_title" content="International Journal of Computer and Information Engineering"> <meta name="citation_volume" content="3"> <meta name="citation_issue" content="7"> <meta name="citation_firstpage" content="1803"> <meta name="citation_lastpage" content="1805"> <meta name="citation_pdf_url" content="https://publications.waset.org/12841/pdf"> <link href="https://cdn.waset.org/favicon.ico" type="image/x-icon" rel="shortcut icon"> <link href="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/css/bootstrap.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/plugins/fontawesome/css/all.min.css" rel="stylesheet"> <link href="https://cdn.waset.org/static/css/site.css?v=150220211555" rel="stylesheet"> </head> <body> <header> <div class="container"> <nav class="navbar navbar-expand-lg navbar-light"> <a class="navbar-brand" href="https://waset.org"> <img src="https://cdn.waset.org/static/images/wasetc.png" alt="Open Science Research Excellence" title="Open Science Research Excellence" /> </a> <button class="d-block d-lg-none navbar-toggler ml-auto" type="button" data-toggle="collapse" data-target="#navbarMenu" aria-controls="navbarMenu" aria-expanded="false" aria-label="Toggle navigation"> <span class="navbar-toggler-icon"></span> </button> <div class="w-100"> <div class="d-none d-lg-flex flex-row-reverse"> <form method="get" action="https://waset.org/search" class="form-inline my-2 my-lg-0"> <input class="form-control mr-sm-2" type="search" placeholder="Search Conferences" value="" name="q" aria-label="Search"> <button class="btn btn-light my-2 my-sm-0" type="submit"><i class="fas fa-search"></i></button> </form> </div> <div class="collapse navbar-collapse mt-1" id="navbarMenu"> <ul class="navbar-nav ml-auto align-items-center" id="mainNavMenu"> <li class="nav-item"> <a class="nav-link" href="https://waset.org/conferences" title="Conferences in 2024/2025/2026">Conferences</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/disciplines" title="Disciplines">Disciplines</a> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/committees" rel="nofollow">Committees</a> </li> <li class="nav-item dropdown"> <a class="nav-link dropdown-toggle" href="#" id="navbarDropdownPublications" role="button" data-toggle="dropdown" aria-haspopup="true" aria-expanded="false"> Publications </a> <div class="dropdown-menu" aria-labelledby="navbarDropdownPublications"> <a class="dropdown-item" href="https://publications.waset.org/abstracts">Abstracts</a> <a class="dropdown-item" href="https://publications.waset.org">Periodicals</a> <a class="dropdown-item" href="https://publications.waset.org/archive">Archive</a> </div> </li> <li class="nav-item"> <a class="nav-link" href="https://waset.org/page/support" title="Support">Support</a> </li> </ul> </div> </div> </nav> </div> </header> <main> <div class="container mt-4"> <div class="row"> <div class="col-md-9 mx-auto"> <form method="get" action="https://publications.waset.org/search"> <div id="custom-search-input"> <div class="input-group"> <i class="fas fa-search"></i> <input type="text" class="search-query" name="q" placeholder="Author, Title, Abstract, Keywords" value=""> <input type="submit" class="btn_search" value="Search"> </div> </div> </form> </div> </div> <div class="row mt-3"> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Commenced</strong> in January 2007</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Frequency:</strong> Monthly</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Edition:</strong> International</div> </div> </div> <div class="col-sm-3"> <div class="card"> <div class="card-body"><strong>Paper Count:</strong> 33093</div> </div> </div> </div> <div class="card publication-listing mt-3 mb-3"> <h5 class="card-header" style="font-size:.9rem">Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model</h5> <div class="card-body"> <p class="card-text"><strong>Authors:</strong> <a href="https://publications.waset.org/search?q=Vladislav%20N.%20Dumachev">Vladislav N. Dumachev</a>, <a href="https://publications.waset.org/search?q=Vladimir%20A.%20Rodin"> Vladimir A. Rodin</a> </p> <p class="card-text"><strong>Abstract:</strong></p> By utilizing the system of the recurrence equations, containing two parameters, the dynamics of two antagonistically interconnected populations is studied. The following areas of the system behavior are detected: the area of the stable solutions, the area of cyclic solutions occurrence, the area of the accidental change of trajectories of solutions, and the area of chaos and fractal phenomena. The new two-dimensional diagram of the dynamics of the solutions change (the fractal cabbage) has been obtained. In the cross-section of this diagram for one of the equations the well-known Feigenbaum tree of doubling has been noted.Keywordsbifurcation, chaos, dynamics of populations, fractals <iframe src="https://publications.waset.org/12841.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=bifurcation" title="bifurcation">bifurcation</a>, <a href="https://publications.waset.org/search?q=chaos" title=" chaos"> chaos</a>, <a href="https://publications.waset.org/search?q=dynamics%20of%20populations" title=" dynamics of populations"> dynamics of populations</a>, <a href="https://publications.waset.org/search?q=fractals" title=" fractals"> fractals</a> </p> <p class="card-text"><strong>Digital Object Identifier (DOI):</strong> <a href="https://doi.org/10.5281/zenodo.1080010" target="_blank">doi.org/10.5281/zenodo.1080010</a> </p> <a href="https://publications.waset.org/12841/behavior-of-solutions-of-the-system-of-recurrence-equations-based-on-the-verhulst-pearl-model" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/12841/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/12841/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/12841/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/12841/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/12841/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/12841/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/12841/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/12841/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/12841/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/12841/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/12841.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">1277</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[1] Computer and nonlinear phenomena. M.: Nauka, 1988. <br>[2] E.I.Skaletskaja, E.J.Frisman, A.P.Shapiro, Discrete models of the dynamics of numbers of populations and craft optimisation. M.: Nauka. 1979. <br>[3] A.P.Shapiro, About stability of cannibalistic populations.Mathemematical modelling of populational ecological processes. Vladivostok, 1987, p. 106-112. <br>[5] A.P.Kuznetsov, S.P.Kuznetsov, Critical dynamics of one-dimensional displays. Izvestie vuzov. PND, 1993. Vol. <br>[6] S.P.Kuznetsov, Dynamics of two single-directed and connected Feigenbaum systems at a hyperchaos threshold. Analysis of the renormalization group. Izvie vuzov. Radio physics. 1990. Vol.33. 7. <br>[7] V.I. Arnold, A.N. Varchenko, S.M. Gusein-zade, Features of differentiated mapping. Classification of critical points of caustic and wave fronts. M.: Nauka. 1982. <br>[8] T. Brecar, L. Lender, Differentiated sprouts and accidents. M.: ir,1984. <br>[9] V.A. Rodin, V.N. Dumachev, Evolution of antagonistically interacting populations based on the two-dimensional Verhulst-Pearl model. Mathematical modeling. 2005. 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Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model