Hopf bifurcations of a diffusion model with a generaladvection and delay

Hopf bifurcations of a diffusion model with a generaladvection and delay | 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 Hopf bifurcations of a diffusion model with a generaladvection and delay Jingxiao Song, Shaofen Zou, Chengwei Ren This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-9028910/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract This paper investigates a class of reaction-diffusion population models definedon a bounded domain, which involves a general time-delayed per capita growthrate and a general advection term. By the Lyapunov-Schmidt reduction method,we establish the existence of spatially nonhomogeneous steady states when theparameter λ approaches the principal eigenvalue λ∗ of a non-self-adjoint ellip-tic operator. A detailed analysis of the characteristic equation further confirmsthe existence of Hopf bifurcations induced by large delays, which originate fromthese steady states. Specifically, we show that when λ approaches λ∗, the crit-ical delay value τλ,0 required for stability loss tends to infinity. Subsequently,by applying the center manifold reduction and the normal form theory, we as-certain the direction of these Hopf bifurcations and the stability of the resultingperiodic orbits. Finally, we use numerical simulations to illustrate the validity oftheoretical results where the growths are Logistic-type and weak Allee. Reaction-diffusion Advection Delay Hopf bifurcation Non-selfadjoint elliptic operator Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted 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-9028910","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":600528791,"identity":"bcf7e2df-e21d-4c14-a124-ea8db4ac0259","order_by":0,"name":"Jingxiao Song","email":"","orcid":"","institution":"Hunan University","correspondingAuthor":false,"prefix":"","firstName":"Jingxiao","middleName":"","lastName":"Song","suffix":""},{"id":600528792,"identity":"406364d9-bfd4-4bdd-8425-dc37dbdcadb4","order_by":1,"name":"Shaofen Zou","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA5ElEQVRIiWNgGAWjYDACCSCugDAZH0PoBCK0nIEwmY1J1sImTZQW+dnNDx8cqLhjL9/eY1ZduOMwAz97jgHDzx24tRjcOWZscODMM2bGnjNmt2eeOcwg2fPGgLH3DB4tEglm0h/bDrMxS+SY3eZtO8xgcCPHgJmxDY/DZqR/kzj47zAPG1BLMUiLPSEtDDdyzCQONhyW4AFqYQbbIkFAC9AZxQYHjh02kOA5VizN25bOI3HmWcHBXvwO2/jgQM1hYIg1b/zM22Ytx9+evPHBT3wOQwc8IOIACRpGwSgYBaNgFGABABp/TjWsnHe/AAAAAElFTkSuQmCC","orcid":"","institution":"Hunan University","correspondingAuthor":true,"prefix":"","firstName":"Shaofen","middleName":"","lastName":"Zou","suffix":""},{"id":600528793,"identity":"14502c78-da03-4c27-9ec3-6fe18eb8dfcd","order_by":2,"name":"Chengwei Ren","email":"","orcid":"","institution":"Hunan University","correspondingAuthor":false,"prefix":"","firstName":"Chengwei","middleName":"","lastName":"Ren","suffix":""}],"badges":[],"createdAt":"2026-03-04 10:10:20","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-9028910/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-9028910/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":105211059,"identity":"1edba0ea-72c6-4cdc-929f-ba5c98c3cedb","added_by":"auto","created_at":"2026-03-23 13:43:31","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":918951,"visible":true,"origin":"","legend":"","description":"","filename":"SZRJNS.pdf","url":"https://assets-eu.researchsquare.com/files/rs-9028910/v1_covered_b608fafb-51b0-4307-9c43-061e9aa61be1.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Hopf bifurcations of a diffusion model with a generaladvection and delay","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"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":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Reaction-diffusion, Advection, Delay, Hopf bifurcation, Non-selfadjoint elliptic operator","lastPublishedDoi":"10.21203/rs.3.rs-9028910/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-9028910/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"This paper investigates a class of reaction-diffusion population models definedon a bounded domain, which involves a general time-delayed per capita growthrate and a general advection term. 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