Mathematical Analysis of a Delayed Predator–Prey System with Prey Refuge: Bifurcations and Stability Thresholds

Mathematical Analysis of a Delayed Predator–Prey System with Prey Refuge: Bifurcations and Stability Thresholds | 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 Mathematical Analysis of a Delayed Predator–Prey System with Prey Refuge: Bifurcations and Stability Thresholds Ahmed Buseri Ashine This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8452898/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 study investigates the complex interplay between prey refuge and time delay in a predator–prey system governed by a Holling Type II functional response. Building upon the work of Elmojtaba et al. [ 22 ], we establish that while increasing environmental carrying capacity triggers a "Paradox of Enrichment" via Hopf bifurcation and discrete time delays induce destabilizing limit-cycle oscillations, the utilization of a constant prey refuge acts as a vital regulatory "buffer." Analytical proofs and numerical simulations demonstrate that the system transitions from predator extinction to stable coexistence through a transcritical bifurcation, and that increasing the refuge constant can effectively counteract the destabilizing effects of both enrichment and gestation lags. Ultimately, our findings suggest that maintaining physical refuges is essential for ecosystem resilience, as they expand the region of asymptotic stability and prevent species extinction in otherwise oscillatory environments. predator–prey prey refuge time delay transcritical bifurcation Hopf bifurcation Paradox of Enrichment numerical simulation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 11 Apr, 2026 Reviews received at journal 11 Apr, 2026 Reviewers agreed at journal 10 Feb, 2026 Reviewers invited by journal 16 Jan, 2026 Editor assigned by journal 04 Jan, 2026 Submission checks completed at journal 30 Dec, 2025 First submitted to journal 26 Dec, 2025 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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Thresholds","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":"boundary-value-problems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bovp","sideBox":"Learn more about [Boundary Value Problems](http://boundaryvalueproblems.springeropen.com)","snPcode":"13661","submissionUrl":"https://submission.nature.com/new-submission/13661/3","title":"Boundary Value Problems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":true},"keywords":"predator–prey, prey refuge, time delay, transcritical bifurcation, Hopf bifurcation, Paradox of Enrichment, numerical simulation","lastPublishedDoi":"10.21203/rs.3.rs-8452898/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8452898/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates the complex interplay between prey refuge and time delay in a predator\u0026ndash;prey system governed by a Holling Type II functional response. Building upon the work of Elmojtaba et al. [\u003cspan citationid=\"CR22\" class=\"CitationRef\"\u003e22\u003c/span\u003e], we establish that while increasing environmental carrying capacity triggers a \"Paradox of Enrichment\" via Hopf bifurcation and discrete time delays induce destabilizing limit-cycle oscillations, the utilization of a constant prey refuge acts as a vital regulatory \"buffer.\" Analytical proofs and numerical simulations demonstrate that the system transitions from predator extinction to stable coexistence through a transcritical bifurcation, and that increasing the refuge constant can effectively counteract the destabilizing effects of both enrichment and gestation lags. Ultimately, our findings suggest that maintaining physical refuges is essential for ecosystem resilience, as they expand the region of asymptotic stability and prevent species extinction in otherwise oscillatory environments.\u003c/p\u003e","manuscriptTitle":"Mathematical Analysis of a Delayed Predator–Prey System with Prey Refuge: Bifurcations and Stability Thresholds","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2026-01-20 12:35:36","doi":"10.21203/rs.3.rs-8452898/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-04-11T18:41:11+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2026-04-11T05:59:12+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"75432219502595300946366525491966193442","date":"2026-02-10T07:08:11+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2026-01-16T11:40:37+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2026-01-04T15:02:28+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-12-31T01:28:43+00:00","index":"","fulltext":""},{"type":"submitted","content":"Boundary Value Problems","date":"2025-12-26T07:01:30+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"boundary-value-problems","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bovp","sideBox":"Learn more about [Boundary Value Problems](http://boundaryvalueproblems.springeropen.com)","snPcode":"13661","submissionUrl":"https://submission.nature.com/new-submission/13661/3","title":"Boundary Value Problems","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"e91938ff-ebb0-4cc0-b9d9-077cee39a901","owner":[],"postedDate":"January 20th, 2026","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-04-27T17:23:12+00:00","versionOfRecord":[],"versionCreatedAt":"2026-01-20 12:35:36","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-8452898","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-8452898","identity":"rs-8452898","version":["v1"]},"buildId":"XKTyCvWXoU3ODBz1xrDgd","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}