Multiple Time-Scale Analysis for Double Hopf Bifurcations inMemory-Driven Nonlocal Diffusion Systems

Multiple Time-Scale Analysis for Double Hopf Bifurcations inMemory-Driven Nonlocal Diffusion Systems | 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 Multiple Time-Scale Analysis for Double Hopf Bifurcations inMemory-Driven Nonlocal Diffusion Systems Yong Wang, Mengbo Han, Weihua Jiang, Yuqun Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9068840/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract This paper presents a systematic investigation of the spatiotemporal dynamics of a predator–prey model incorporating memory-dependent diffusion and nonlocal interactions. First, the conditions for the emergence of Turing instability under the combined effects of memory diffusion and nonlocality are derived, thereby theoretically elucidating the intrinsic mechanisms of spatial pattern formation and the key factors influencing it.Next, rigorous existence conditions and criteria for Turing bifurcation, Hopf bifurcation, and double Hopf bifurcation are established.Moreover, a major innovation of this work lies in the development of a systematic computational framework for the normal forms of double Hopf bifurcations in reaction–diffusion systems with coupled memory diffusion and nonlocal effects, based on a multiscale method. The coefficients of the normal forms are explicitly expressed in terms of the parameters of the original system. On this basis, several typical spatiotemporal dynamical modes associated with the original system are further identified and characterized.The results demonstrate that nonlocal interactions significantly promote the formation of spatially heterogeneous structures, whereas memory diffusion effectively suppresses classical Fickian diffusion–induced Turing instability. Under the combined influence of these two mechanisms, the system can exhibit complex spatiotemporal behaviors, including stable spatially inhomogeneous periodic solutions and quasiperiodic solutions. Spatial memory Nonlocal effects Double Hopf bifurcation Multiple time scale Spatiotemporal pattern Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Reviewers agreed at journal 18 Apr, 2026 Reviewers agreed at journal 15 Apr, 2026 Reviewers agreed at journal 02 Apr, 2026 Reviewers invited by journal 01 Apr, 2026 Editor assigned by journal 13 Mar, 2026 Submission checks completed at journal 13 Mar, 2026 First submitted to journal 09 Mar, 2026 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. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. 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-9068840","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":618326030,"identity":"0cd04986-7409-4354-a8c1-ef1f1bf70f2c","order_by":0,"name":"Yong Wang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3ElEQVRIiWNgGAWjYNCCCgZmBuYDQAYb0VrOGDAzsCWQooWxzYCBeC0GN5KfPfw67w+7wTGgyz6UHWbgn91ASEuaubHsNgNmg2NsCYwzzh1mkLhzgICW2wlm0pIgLfd7DJh52w4zGEgkENKS/k1acg7IFv4PzH+J05JjJvmxAaSFh4GZkRgtkvfflEkzHDNmljzGZnCw51w6j8QNAlr4zhzfJvmjRi6Z7xjzwwc/yqzl+GcQ0KJwgIGBmYeBIRnEAbIZePCrBwL5BmBM/mBgsCOochSMglEwCkYuAABhNEIeOHAvYgAAAABJRU5ErkJggg==","orcid":"","institution":"Tianjin University of Finance and Economics","correspondingAuthor":true,"prefix":"","firstName":"Yong","middleName":"","lastName":"Wang","suffix":""},{"id":618326031,"identity":"a9cc0355-f939-4d18-8338-9a0e07ce892b","order_by":1,"name":"Mengbo Han","email":"","orcid":"","institution":"Tianjin University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Mengbo","middleName":"","lastName":"Han","suffix":""},{"id":618326033,"identity":"34925ba2-bd14-456b-9c0b-53e1e6b2c55f","order_by":2,"name":"Weihua Jiang","email":"","orcid":"","institution":"Harbin Institute of Technology","correspondingAuthor":false,"prefix":"","firstName":"Weihua","middleName":"","lastName":"Jiang","suffix":""},{"id":618326034,"identity":"04b3013c-1ed8-42af-8e55-d992f97f0de3","order_by":3,"name":"Yuqun Wang","email":"","orcid":"","institution":"Tianjin University of Finance and Economics","correspondingAuthor":false,"prefix":"","firstName":"Yuqun","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2026-03-09 06:09:19","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9068840/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9068840/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":106404622,"identity":"4e7ec3e0-fada-4a90-926b-f69e84401d96","added_by":"auto","created_at":"2026-04-08 09:16:23","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":2254719,"visible":true,"origin":"","legend":"","description":"","filename":"MemoryDrivenNonlocalDiffusion.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9068840/v1_covered_fbe5807c-fbc4-419a-8fb8-89709ca26a68.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Multiple Time-Scale Analysis for Double Hopf Bifurcations inMemory-Driven Nonlocal Diffusion Systems","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-nonlinear-science","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jnls","sideBox":"Learn more about [Journal of Nonlinear Science](http://link.springer.com/journal/332)","snPcode":"332","submissionUrl":"https://submission.nature.com/new-submission/332/3","title":"Journal of Nonlinear Science","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Spatial memory, Nonlocal effects, Double Hopf bifurcation, Multiple time scale, Spatiotemporal pattern","lastPublishedDoi":"10.21203/rs.3.rs-9068840/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9068840/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper presents a systematic investigation of the spatiotemporal dynamics of a predator–prey model incorporating memory-dependent diffusion and nonlocal interactions. 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