Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator-prey model

Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator-prey model | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator-prey model Xun Cao This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4623523/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 19 Sep, 2024 Read the published version in Zeitschrift für angewandte Mathematik und Physik → Version 1 posted 7 You are reading this latest preprint version Abstract A reaction-diffusion Holling-Tanner predator-prey model with nonlocal prey competition involving \emph{purely spatial heat kernel} is investigated. \emph{The first bifurcation curve} is mathematically described, that is a piecewise smooth parameter curve of dividing the stability and instability of the coexistence equilibrium. The concepts of Turing/Hopf instability are extended to the higher codimension bifurcation instability, because the non-smooth points of the first bifurcation curve can be Bogdanov-Takens/Turing-Hopf/Hopf-Hopf instability point. Utilizing normal form method, spatiotemporal dynamics near $Z_2$ symmetric Bogdanov-Takens singularity are theoretically and numerically studied, including the stable coexistence of a pair of steady states with the shape of $\cos \frac{2x}{l}$ and a spatiotemporal staggered periodic solution with the shape of $\cos \omega t\cos \frac{2x}{l}$. It is found that the larger the spatial size of a habitat is, the more complex the distributions of a species can be, while too narrow or wide range of nonlocal interactions inhibit the formations of complex spatiotemporal patterns. Z2 symmetric Bogdanov-Takens bifurcation Tristable spatiotemporal patterns Nonlocal interactions Purely spatial heat kernel The first bifurcation curve Holling-Tanner predator-prey model Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Published Journal Publication published 19 Sep, 2024 Read the published version in Zeitschrift für angewandte Mathematik und Physik → Version 1 posted Editorial decision: Revision requested 12 Aug, 2024 Reviews received at journal 12 Aug, 2024 Reviewers agreed at journal 30 Jul, 2024 Reviewers invited by journal 28 Jun, 2024 Editor assigned by journal 28 Jun, 2024 Submission checks completed at journal 26 Jun, 2024 First submitted to journal 22 Jun, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4623523","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":322320835,"identity":"a1843f49-bdf4-49fc-b299-04aff338211a","order_by":0,"name":"Xun Cao","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAuUlEQVRIiWNgGAWjYPACmwQ2EMVDgpa0BDY2ErUcTmAgWgt/++Fj0rw7zufxyTcwPnjbxiBvTkiLxJm0ZGPeM7eLgQ5jNpzbxmC4s4GAFgOGHMPHvG23E9vYGNikedsYEgwOENLC/8bgMG/bOZAW9t/EaZEA23IAbAszUVokbjxLBnohGaglsVlyzjkJww2EtPD3Jx+TeNtmlzi/+fDBD2/KbOQJ2oIEGBtAthKvfhSMglEwCkYBbgAA7Tg4R4JpQGcAAAAASUVORK5CYII=","orcid":"","institution":"Harbin Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Xun","middleName":"","lastName":"Cao","suffix":""}],"badges":[],"createdAt":"2024-06-23 02:23:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4623523/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4623523/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s00033-024-02326-4","type":"published","date":"2024-09-19T15:57:59+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":65104110,"identity":"6ba59e84-9df4-4131-b0de-e648f759c3d7","added_by":"auto","created_at":"2024-09-23 16:11:44","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2044346,"visible":true,"origin":"","legend":"","description":"","filename":"Manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4623523/v1_covered_6574d49d-2d0e-4d83-91b0-17ed1090de5c.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Bogdanov-Takens bifurcation and multi-peak spatiotemporal staggered periodic patterns in a nonlocal Holling-Tanner predator-prey model","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"zeitschrift-fur-angewandte-mathematik-und-physik","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"zamp","sideBox":"Learn more about [Zeitschrift für angewandte Mathematik und Physik](http://link.springer.com/journal/33)","snPcode":"33","submissionUrl":"https://submission.nature.com/new-submission/33/3","title":"Zeitschrift für angewandte Mathematik und Physik","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Z2 symmetric Bogdanov-Takens bifurcation, Tristable spatiotemporal patterns, Nonlocal interactions, Purely spatial heat kernel, The first bifurcation curve, Holling-Tanner predator-prey model","lastPublishedDoi":"10.21203/rs.3.rs-4623523/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4623523/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"A reaction-diffusion Holling-Tanner predator-prey model with nonlocal prey competition involving \\emph{purely spatial heat kernel} is investigated. \\emph{The first bifurcation curve} is mathematically described, that is a piecewise smooth parameter curve of dividing the stability and instability of the coexistence equilibrium. The concepts of Turing/Hopf instability are extended to the higher codimension bifurcation instability, because the non-smooth points of the first bifurcation curve can be Bogdanov-Takens/Turing-Hopf/Hopf-Hopf instability point. Utilizing normal form method, spatiotemporal dynamics near $Z_2$ symmetric Bogdanov-Takens singularity are theoretically and numerically studied, including the stable coexistence of a pair of steady states with the shape of $\\cos \\frac{2x}{l}$ and a spatiotemporal staggered periodic solution with the shape of $\\cos \\omega t\\cos \\frac{2x}{l}$. 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