Document Zbl 1105.65122

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M. A.</a>; <a href="/authors/el-saka.hala-a-a" title="Author Profile">El-Saka, H. A. A.</a></div> <h2 class="title"> <strong>Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models.</strong> <i>(English)</i> <a class="label nowrap" href="/1105.65122">Zbl 1105.65122</a> </h2> <div class="source"> <a href="/serials/132" title="Journal Profile">J. Math. Anal. Appl.</a> <a href="/?q=in%3A187998" title="Articles in this Issue">325, No. 1, 542-553 (2007)</a>. </div> <div class="abstract">Summary: We are concerned with the fractional-order predator-prey model and the fractional-order rabies model. Existence and uniqueness of solutions are proved. The stability of equilibrium points are studied. Numerical solutions of these models are given. An example is given where the equilibrium point is a centre for the integer order system but locally asymptotically stable for its fractional-order counterpart.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A5078945">Cited in <strong>1</strong> Review</a></div><div class="clear"><a href="/?q=rf%3A5078945">Cited in <strong>286</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A65R20" title="MSC2020">65R20</a> </td> <td class="space"> Numerical methods for integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D30" title="MSC2020">92D30</a> </td> <td class="space"> Epidemiology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A45J05" title="MSC2020">45J05</a> </td> <td class="space"> Integro-ordinary differential equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A26A33" title="MSC2020">26A33</a> </td> <td class="space"> Fractional derivatives and integrals </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apredator-prey+system">predator-prey system</a>; <a href="/?q=ut%3Arabies+system">rabies system</a>; <a href="/?q=ut%3Afractional-order+differential+equations">fractional-order differential equations</a>; <a href="/?q=ut%3Aequilibrium+points">equilibrium points</a>; <a href="/?q=ut%3Astability">stability</a>; <a href="/?q=ut%3Apredictor-corrector+method">predictor-corrector method</a>; <a href="/?q=ut%3Anumerical+examples">numerical examples</a></div> <div class="software"> <h3>Software:</h3><a href="/software/24209">FracPECE</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1105.65122" data-ciurl="/ci/05078945" data-biburl="/bibtex/05078945.bib" data-amsurl="/amsrefs/05078945.bib" data-xmlurl="/xml/05078945.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/05078945.pdf" title="Zbl 1105.65122 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.jmaa.2006.01.087" aria-label="DOI for “Equilibrium points, stability and numerical solutions of fractional-order predator-prey and rabies models”" title="10.1016/j.jmaa.2006.01.087">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Ahmed, E.; El-Sayed, A. M.A.; El-Mesiry, E. M.; El-Saka, H. A.A., Numerical solution for the fractional replicator equation, IJMPC, 16, 7, 1-9 (2005) · <a href="/1080.65536" class="nowrap">Zbl 1080.65536</a></td> </tr><tr> <td>[2]</td> <td class="space">Diethelm, K.; Freed, A., On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity, (Keil, F.; Mackens, W.; Voß, H.; Werther, J., Scientific Computing in Chemical Engineering II—Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties (1999), Springer: Springer Heidelberg), 217-224</td> </tr><tr> <td>[3]</td> <td class="space">Diethelm, K.; Freed, A., The FracPECE subroutine for the numerical solution of differential equations of fractional order, (Heinzel, S.; Plesser, T., Forschung und wissenschaftliches Rechnen 1998 (1999), Gesellschaft für Wisseschaftliche Datenverarbeitung: Gesellschaft für Wisseschaftliche Datenverarbeitung Göttingen), 57-71</td> </tr><tr> <td>[4]</td> <td class="space">Diethelm, K., Predictor-corrector strategies for single- and multi-term fractional differential equations, (Lipitakis, E. 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Algorithms, 36, 31-52 (2004) · <a href="/1055.65098" class="nowrap">Zbl 1055.65098</a></td> </tr><tr> <td>[8]</td> <td class="space">Diethelm, K.; Ford, N. J., Multi-order fractional differential equations and their numerical solution, Appl. Math. Comput., 154, 621-640 (2004) · <a href="/1060.65070" class="nowrap">Zbl 1060.65070</a></td> </tr><tr> <td>[9]</td> <td class="space">El-Mesiry, E. M.; El-Sayed, A. M.A.; El-Saka, H. A.A., Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput., 160, 3, 683-699 (2005) · <a href="/1062.65073" class="nowrap">Zbl 1062.65073</a></td> </tr><tr> <td>[10]</td> <td class="space">El-Sayed, A. M.A., Fractional differential-difference equations, J. Fract. Calc., 10, 101-106 (1996) · <a href="/0888.34060" class="nowrap">Zbl 0888.34060</a></td> </tr><tr> <td>[11]</td> <td class="space">El-Sayed, A. M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal., 33, 2, 181-186 (1998) · <a href="/0934.34055" class="nowrap">Zbl 0934.34055</a></td> </tr><tr> <td>[12]</td> <td class="space">El-Sayed, A. M.A.; Gaafar, F. M., Fractional order differential equations with memory and fractional-order relaxation-oscillation model, (PU.M.A) Pure Math. Appl., 12 (2001) · <a href="/0895.34003" class="nowrap">Zbl 0895.34003</a></td> </tr><tr> <td>[13]</td> <td class="space">El-Sayed, A. M.A.; El-Mesiry, E. M.; El-Saka, H. A.A., Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. Appl. Math., 23, 1, 33-54 (2004) · <a href="/1213.34025" class="nowrap">Zbl 1213.34025</a></td> </tr><tr> <td>[14]</td> <td class="space">El-Sayed, A. M.A.; Gaafar, F. M.; Hashem, H. H., On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations, Math. Sci. Res. J., 8, 11, 336-348 (2004) · <a href="/1068.45008" class="nowrap">Zbl 1068.45008</a></td> </tr><tr> <td>[15]</td> <td class="space">Gorenflo, R.; Mainardi, F., Fractional calculus: Integral and differential equations of fractional order, (Carpinteri, A.; Mainardi, F., Fractals and Fractional Calculus in Continuum Mechanics (1997), Springer: Springer Wien), 223-276 · <a href="/1438.26010" class="nowrap">Zbl 1438.26010</a></td> </tr><tr> <td>[16]</td> <td class="space">D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., Vol. 2, Lille, France 963, 1996; D. Matignon, Stability results for fractional differential equations with applications to control processing, Computational Eng. in Sys. Appl., Vol. 2, Lille, France 963, 1996</td> </tr><tr> <td>[17]</td> <td class="space">Podlubny, I.; El-Sayed, A. M.A., On Two Definitions of Fractional Calculus (1996), Slovak Academy of Science, Institute of Experimental Phys., UEF-03-96</td> </tr><tr> <td>[18]</td> <td class="space">Podlubny, I., Fractional Differential Equations (1999), Academic Press · <a href="/0918.34010" class="nowrap">Zbl 0918.34010</a></td> </tr><tr> <td>[19]</td> <td class="space">Stanislavsky, A. A., Memory effects and macroscopic manifestation of randomness, Phys. Rev. E, 61, 4752 (2000)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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