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Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model
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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" /> <meta name="citation_title" content="Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model" /> <meta name="citation_doi" content="10.5281/zenodo.1080010" /> <meta name="citation_keywords" content="bifurcation" /> <meta name="citation_keywords" content="chaos" /> <meta name="citation_keywords" content="dynamics of populations" /> <meta name="citation_keywords" content="fractals" /> <meta name="citation_abstract_html_url" content="https://zenodo.org/records/1080010" /> <meta property="og:title" content="Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model" /> <meta property="og:description" content="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" /> <meta property="og:url" content="https://zenodo.org/records/1080010" /> <meta property="og:site_name" content="Zenodo" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@zenodo_org" /> <meta name="twitter:title" content="Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model" /> <meta name="twitter:description" content="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" /> <meta name="citation_pdf_url" content="https://zenodo.org/records/1080010/files/12841.pdf"/> <link rel="alternate" type="application/pdf" href="https://zenodo.org/records/1080010/files/12841.pdf"> <link rel="canonical" href="https://zenodo.org/records/1080010"> <title>Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model</title> <link rel="shortcut icon" type="image/x-icon" href="/static/favicon.ico"/> <link rel="apple-touch-icon" sizes="120x120" href="/static/apple-touch-icon-120.png"/> <link rel="apple-touch-icon" sizes="152x152" href="/static/apple-touch-icon-152.png"/> <link rel="apple-touch-icon" sizes="167x167" href="/static/apple-touch-icon-167.png"/> <link rel="apple-touch-icon" sizes="180x180" href="/static/apple-touch-icon-180.png"/> <link rel="stylesheet" href="/static/dist/css/3526.0d9b3c8be998e2e93a52.css" /> <!-- HTML5 shim and Respond.js for IE8 support of HTML5 elements and media queries --> <!--[if lt IE 9]> <script src="https://oss.maxcdn.com/html5shiv/3.7.2/html5shiv.min.js"></script> <script src="https://oss.maxcdn.com/respond/1.4.2/respond.min.js"></script> <![endif]--> </head> <body data-invenio-config='{"isMathJaxEnabled": "//cdnjs.cloudflare.com/ajax/libs/mathjax/3.2.2/es5/tex-mml-chtml.js?config=TeX-AMS-MML_HTMLorMML"}' itemscope itemtype="http://schema.org/WebPage" data-spy="scroll" data-target=".scrollspy-target"> <a id="skip-to-main" class="ui button primary ml-5 mt-5 skip-link" href="#main">Skip to main</a> <!--[if lt IE 8]> <p class="browserupgrade">You are using an <strong>outdated</strong> browser. 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Dumachev</span></a> </li> <li class="creatibutor-wrap separated"> <a class="ui creatibutor-link" href="/search?q=metadata.creators.person_or_org.name%3A%22Vladimir+A.+Rodin%22" > <span class="creatibutor-name">Vladimir A. Rodin</span></a> </li> </ul> </div> </div> </div> </section> </section> <section id="description" class="rel-mt-2 rich-input-content" aria-label="Record description"> <h2 id="description-heading" class="sr-only">Description</h2> <div style="word-wrap: break-word;"> <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</p> </div> </section> <section id="record-files" class="rel-mt-2 rel-mb-3" aria-label="Files" ><h2 id="files-heading">Files</h2> <div class="ui accordion panel mb-10 open" href="#files-preview-accordion-panel"> <h3 class="active title panel-heading open m-0"> <div role="button" id="files-preview-accordion-trigger" aria-controls="files-preview-accordion-panel" aria-expanded="true" tabindex="0" class="trigger" aria-label="File preview" > <span id="preview-file-title">12841.pdf</span> <i class="angle right icon" aria-hidden="true"></i> </div> </h3> <div role="region" id="files-preview-accordion-panel" aria-labelledby="files-preview-accordion-trigger" class="active content preview-container pt-0 open" > <div> <iframe title="Preview" class="preview-iframe" id="preview-iframe" name="preview-iframe" src="/records/1080010/preview/12841.pdf?include_deleted=0" > </iframe> </div> </div> </div> <div class="ui accordion panel mb-10 open" href="#files-list-accordion-panel"> <h3 class="active title panel-heading open m-0"> <div role="button" id="files-list-accordion-trigger" aria-controls="files-list-accordion-panel" aria-expanded="true" tabindex="0" class="trigger"> Files <small class="text-muted"> (947.7 kB)</small> <i class="angle right icon" aria-hidden="true"></i> </div> </h3> <div role="region" id="files-list-accordion-panel" aria-labelledby="files-list-accordion-trigger" class="active content pt-0"> <div> <table class="ui striped table files fluid open"> <thead> <tr> <th>Name</th> <th>Size</th> <th class> <a role="button" class="ui compact mini button right floated archive-link" href="https://zenodo.org/api/records/1080010/files-archive"> <i class="file archive icon button" aria-hidden="true"></i> Download all </a> </th> </tr> </thead> <tbody> <tr> <td class="ten wide"> <div> <a href="/records/1080010/files/12841.pdf?download=1">12841.pdf</a> </div> <small class="ui text-muted font-tiny">md5:877e8f356bc4b82a335ecb4b55d7b5ad <div class="ui icon inline-block" data-tooltip="This is the file fingerprint (checksum), which can be used to verify the file integrity."> <i class="question circle checksum icon"></i> </div> </small> </td> <td>947.7 kB</td> <td class="right aligned"> <span> <a role="button" class="ui compact mini button preview-link" href="/records/1080010/preview/12841.pdf?include_deleted=0" target="preview-iframe" data-file-key="12841.pdf"> <i class="eye icon" aria-hidden="true"></i>Preview </a> <a role="button" class="ui compact mini button" href="/records/1080010/files/12841.pdf?download=1"> <i class="download icon" aria-hidden="true"></i>Download </a> </span> </td> </tr> </tbody> </table> </div> </div> </div> </section> <section id="additional-details" class="rel-mt-2" aria-label="Additional record details"> <h2 id="record-details-heading">Additional details</h2> <div class="ui divider"></div> <div class="ui fluid accordion padded grid rel-mb-1"> <div class="active title sixteen wide mobile four wide tablet three wide computer column"> <h3 class="ui header"> <div id="references-accordion-trigger" role="button" tabindex="0" aria-expanded="true" aria-controls="references-panel" class="trigger" > <i class="caret right icon" aria-hidden="true"></i>References </div> </h3> </div> <div id="references-panel" role="region" aria-labelledby="references-accordion-trigger" class="active content sixteen wide mobile twelve wide tablet thirteen wide computer column" > <ul class="ui bulleted list details-list"> <li class="item">Computer and nonlinear phenomena. M.: Nauka, 1988.</li> <li class="item">E.I.Skaletskaja, E.J.Frisman, A.P.Shapiro, Discrete models of the dynamics of numbers of populations and craft optimisation. M.: Nauka. 1979.</li> <li class="item">A.P.Shapiro, About stability of cannibalistic populations.Mathemematical modelling of populational ecological processes. Vladivostok, 1987, p. 106-112.</li> <li class="item">A.P.Kuznetsov, S.P.Kuznetsov, Critical dynamics of one-dimensional displays. Izvestie vuzov. PND, 1993. Vol.</li> <li class="item">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.</li> <li class="item">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.</li> <li class="item">T. Brecar, L. Lender, Differentiated sprouts and accidents. M.: ir,1984.</li> <li class="item">V.A. Rodin, V.N. Dumachev, Evolution of antagonistically interacting populations based on the two-dimensional Verhulst-Pearl model. Mathematical modeling. 2005. Vol.17, p. 11-22.</li> </ul> </div> </div> <div class="ui divider"></div> </section> <section id="citations-search" data-record-pids='{"doi": {"client": "datacite", "identifier": "10.5281/zenodo.1080010", "provider": "datacite"}, "oai": {"identifier": "oai:zenodo.org:1080010", "provider": "oai"}}' data-record-parent-pids='{"doi": {"client": "datacite", "identifier": "10.5281/zenodo.1080009", "provider": "datacite"}}' data-citations-endpoint="https://zenodo-broker.web.cern.ch/api/relationships" aria-label="Record citations" class="rel-mb-1" > </section> </article> <aside class="sixteen wide tablet five wide computer column sidebar" aria-label="Record details"> <section id="metrics" aria-label="Metrics" class="ui segment rdm-sidebar sidebar-container"> <div class="ui tiny two statistics rel-mt-1"> <div class="ui statistic"> <div class="value">56</div> <div class="label"> <i aria-hidden="true" class="eye icon"></i> Views </div> </div> <div class="ui statistic"> <div class="value">36</div> 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Vol.17, p. 11-22."}], "resource_type": {"id": "publication-article", "title": {"de": "Zeitschriftenartikel", "en": "Journal article"}}, "rights": [{"description": {"en": "The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited."}, "icon": "cc-by-icon", "id": "cc-by-4.0", "props": {"scheme": "spdx", "url": "https://creativecommons.org/licenses/by/4.0/legalcode"}, "title": {"en": "Creative Commons Attribution 4.0 International"}}], "subjects": [{"subject": "bifurcation"}, {"subject": "chaos"}, {"subject": "dynamics of populations"}, {"subject": "fractals"}], "title": "Behavior of Solutions of the System of Recurrence Equations Based on the Verhulst-Pearl Model", "version": "12841"}, "parent": {"access": {"owned_by": {"user": "32148"}, "settings": {"accept_conditions_text": null, "allow_guest_requests": false, "allow_user_requests": false, "secret_link_expiration": 0}}, "communities": {"default": "a59dd046-9a86-4a47-97ce-b51f1bb8fc3f", "entries": [{"access": {"member_policy": "open", "members_visibility": "public", "record_submission_policy": "open", "review_policy": "open", "visibility": "public"}, "children": {"allow": false}, "created": "2017-05-31T21:24:26.028360+00:00", "custom_fields": {}, "deletion_status": {"is_deleted": false, "status": "P"}, "id": "a59dd046-9a86-4a47-97ce-b51f1bb8fc3f", "links": {}, "metadata": {"curation_policy": "", "description": "", "page": "", "title": "World Academy of Science, Engineering and Technology"}, "revision_id": 0, "slug": "waset", "updated": "2017-11-15T12:37:31.935319+00:00"}], "ids": ["a59dd046-9a86-4a47-97ce-b51f1bb8fc3f"]}, "id": "1080009", "pids": {"doi": {"client": "datacite", "identifier": "10.5281/zenodo.1080009", "provider": "datacite"}}}, "pids": {"doi": {"client": "datacite", "identifier": "10.5281/zenodo.1080010", "provider": "datacite"}, "oai": {"identifier": "oai:zenodo.org:1080010", "provider": "oai"}}, "revision_id": 8, "stats": {"all_versions": {"data_volume": 34116372.0, "downloads": 36, "unique_downloads": 36, "unique_views": 56, "views": 57}, "this_version": {"data_volume": 33168695.0, "downloads": 35, "unique_downloads": 35, "unique_views": 55, "views": 56}}, "status": "published", "ui": {"access_status": {"description_l10n": "The record and files are publicly accessible.", "embargo_date_l10n": null, "icon": "unlock", "id": "open", "message_class": "", "title_l10n": "Open"}, "created_date_l10n_long": "January 18, 2018", "creators": {"affiliations": [], "creators": [{"person_or_org": {"family_name": "Vladislav N. Dumachev", "name": "Vladislav N. Dumachev", "type": "personal"}}, {"person_or_org": {"family_name": "Vladimir A. Rodin", "name": "Vladimir A. Rodin", "type": "personal"}}]}, "custom_fields": {}, "description_stripped": "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", "is_draft": false, "languages": [{"id": "eng", "title_l10n": "English"}], "publication_date_l10n_long": "July 24, 2009", "publication_date_l10n_medium": "Jul 24, 2009", "resource_type": {"id": "publication-article", "title_l10n": "Journal article"}, "rights": [{"description_l10n": "The Creative Commons Attribution license allows re-distribution and re-use of a licensed work on the condition that the creator is appropriately credited.", "icon": "cc-by-icon", "id": "cc-by-4.0", "props": {"scheme": "spdx", "url": "https://creativecommons.org/licenses/by/4.0/legalcode"}, "title_l10n": "Creative Commons Attribution 4.0 International"}], "updated_date_l10n_long": "August 2, 2024", "version": "12841"}, "updated": "2024-08-02T23:13:29.985730+00:00", "versions": {"index": 1, "is_latest": true}}' 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