Document Zbl 07922168

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Math. Anal. Appl.</a> <a href="/?q=in%3A518419" title="Articles in this Issue">541, No. 1, Article ID 128678, 20 p. (2025)</a>. </div> <div class="abstract">This paper is concerned with the spreading speeds and traveling waves associated with a two-species reaction-diffusion Lotka-Volterra model with nonlinear competition. First, value ranges for the individual spreading speeds of each species are established via comparison with classical Fisher-KPP equations. It is then studied whether spreading is single- or multiple-speed, the selection mechanism of the spreading speed being also investigated.<br class="zbmathjax-paragraph">It is determined that multiple spreading speeds cannot occur and the (single) spreading speed coincides with the minimal speed of the traveling waves. Sufficient conditions for the linear selection of the minimal speed of the traveling waves are then established via linearization, followed by a comparison argument. Conversely, a novel, general condition for the nonlinear selection of the minimal wave speed in terms of the relation between the decay rate and the wave speed is established via a limiting argument for lower solutions. All theoretical results are then validated via comprehensive numerical simulations.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A12667">Paul Georgescu (Iaşi)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A92D25" title="MSC2020">92D25</a> </td> <td class="space"> Population dynamics (general) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35C07" title="MSC2020">35C07</a> </td> <td class="space"> Traveling wave solutions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A35K57" title="MSC2020">35K57</a> </td> <td class="space"> Reaction-diffusion equations </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aspreading+speed">spreading speed</a>; <a href="/?q=ut%3Atraveling+waves">traveling waves</a>; <a href="/?q=ut%3Alinear+selection">linear selection</a>; <a href="/?q=ut%3Anonlinear+selection">nonlinear selection</a>; <a href="/?q=ut%3Aminimal+wave+speed">minimal wave speed</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 07922168" data-ciurl="/ci/07922168" data-biburl="/bibtex/07922168.bib" data-amsurl="/amsrefs/07922168.bib" data-xmlurl="/xml/07922168.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07922168.pdf" title="Zbl 07922168 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.2024.128678" aria-label="DOI for “Spreading speed and traveling wave for a reaction-diffusion system with nonlinear competition”" title="10.1016/j.jmaa.2024.128678">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">Fang, J.; Zhao, X. Q., Travelling waves for monotone semiflows with weak compactness, SIAM J. Math. Anal., 46, 3678-3704, 2014 · <a href="/1315.35051" class="nowrap">Zbl 1315.35051</a></td> </tr><tr> <td>[2]</td> <td class="space">Gilpin, M. E.; Ayala, F. J., Global models of growth and competition, Proc. Natl. Acad. Sci., 70, 12, 3590-3593, 1973 · <a href="/0272.92016" class="nowrap">Zbl 0272.92016</a></td> </tr><tr> <td>[3]</td> <td class="space">Girardin, L., The effect of random dispersal on competitive exclusion-A review, Math. Biosci., 318, Article 108271 pp., 2019 · <a href="/1437.92097" class="nowrap">Zbl 1437.92097</a></td> </tr><tr> <td>[4]</td> <td class="space">Hamel, F.; Nadin, G., Spreading properties and complex dynamics for monostable reaction-diffusion equations, Commun. Partial Differ. Equ., 37, 511-537, 2012 · <a href="/1254.35024" class="nowrap">Zbl 1254.35024</a></td> </tr><tr> <td>[5]</td> <td class="space">Lewis, M. A.; Li, B.; Weinberger, H. 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X., Existence of traveling wave solutions of parabolic-parabolic Chemotaxis systems, Nonlinear Anal., Real World Appl., 42, 93-119, 2018 · <a href="/1441.37101" class="nowrap">Zbl 1441.37101</a></td> </tr><tr> <td>[14]</td> <td class="space">Uchiyama, K., The behavior of solutions of some non-linear di usion equations for large time, J. Math. Kyoto Univ., 18, 453-508, 1978 · <a href="/0408.35053" class="nowrap">Zbl 0408.35053</a></td> </tr><tr> <td>[15]</td> <td class="space">Wang, M. X.; Zhang, Q. Y.; Zhao, X. Q., Traveling waves and spreading properties for a reaction-diffusion competition model with seasonal succession, Nonlinearity, 35, 134-169, 2022 · <a href="/1479.35206" class="nowrap">Zbl 1479.35206</a></td> </tr><tr> <td>[16]</td> <td class="space">Weinberger, H. F.; Lewis, M. A.; Li, B., Analysis of linear determinacy for spread in cooperative models, J. Math. 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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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