Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect

Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect | 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 Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect Zhenshu Wen, Huilin Jia, Yonghui Xia This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-6627657/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 8 You are reading this latest preprint version Abstract This study investigates the bifurcation dynamics driven by the interaction between a generalist predator and a weak Allee effect in a Leslie-type model through a rigorous theoretical analysis. We reveal that the interplay between the generalist predator (parameter e) and the weak Allee effect (parameter β) leads to a rich bifurcation structure. Specifically: (1) when β = e, the model exhibits a nilpotent cusp of codimension 2, accompanied by degenerate Bogdanov-Takens and Hopf bifurcations of codimension 2; (2) when β ≠ e, the nilpotent cusp can reach codimension 3, and the model undergoes degenerate Bogdanov-Takens and Hopf bifurcations of codimension up to 3. Nearly all conditions guaranteeing these dynamics are derived explicitly and precisely. Our results demonstrate that the generalist predator and weak Allee effect not only enrich the system’s dynamical behaviors and bifurcation patterns but may also lead to prey extinction for certain positive initial densities. Furthermore, smaller values of e favor the coexistence of predator and prey, while larger values of e are detrimental to the prey population. Numerical simulations, including prey extinction and the coexistence of multiple steady states (e.g., homoclinic orbits, limit cycles, and double limit cycles), provide intuitive illustrations of the theoretical findings. This work advances the understanding of higher-codimension bifurcation dynamics in ecological systems. continue to represent an active area of research focus due to their ability to capture fundamental ecological processes while remaining mathematically tractable [1, and h represents the quality of the prey as food for the predator. The study of predator-prey dynamics provides crucial insights into the conditions governing species coexistence and extinction, offering valuable guidance for ecosystem management and conservation policymaking. These models demonstrate that system outcomes are highly sensitive to various biological factors, particularly the functional response characterizing predator feeding behavior. The different functional response types (Holling type I, II, III, IV, and so on) are known to generate distinct dynamical regimes, ranging from stable equilibria to complex oscillatory patterns, each with important ecological implications [15, 16, 17, 18, 19, 20, 21]. Note that in the model (1), the predator is called a specialist predator, since it only relies on a single prey species to survive and will die out in absence of this prey. While in some situations, especially when the prey is severely scarce, the predator can turn to other alternative prey species for food and can persist by switching to other food sources, 2020 MSC: 34C07; 34C25; 34D15; 37G15 Leslie–type model generalist predator weak Allee effect degenerate Bogdanov–Takens bifurcation Hopf bifurcation Coexistence Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 18 Mar, 2026 Reviews received at journal 24 Jul, 2025 Reviewers agreed at journal 04 Jun, 2025 Reviewers agreed at journal 23 May, 2025 Reviewers invited by journal 23 May, 2025 Editor assigned by journal 11 May, 2025 Submission checks completed at journal 11 May, 2025 First submitted to journal 09 May, 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. 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-6627657","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":462024643,"identity":"bf7ea766-33f7-4d1e-b248-52e5144c23b5","order_by":0,"name":"Zhenshu Wen","email":"","orcid":"","institution":"Huaqiao University","correspondingAuthor":false,"prefix":"","firstName":"Zhenshu","middleName":"","lastName":"Wen","suffix":""},{"id":462024644,"identity":"35ea5797-6085-4f7f-aeec-8851b6443170","order_by":1,"name":"Huilin Jia","email":"","orcid":"","institution":"Huaqiao University","correspondingAuthor":false,"prefix":"","firstName":"Huilin","middleName":"","lastName":"Jia","suffix":""},{"id":462024645,"identity":"18a2ebac-1dca-409b-8dab-55eb606d7408","order_by":2,"name":"Yonghui Xia","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAArUlEQVRIiWNgGAWjYDACZgYGCQYGGx5+/gbStKTJSM44QIJFQC2HbQwaEohUbnCc+eGNDzXneQwYDjB++JhDhBbJZjZjyxnHbvOYMzcwS87cRoQWfmYGM2nehts8lg0H2Jh5idHCxsz+DajlHI/BgQQitfAz84BsOUCCFslmnmKgX5J5JGccbCbOLwbnj28EhpidPT9/88EPH4nRggQYG0hTPwpGwSgYBaMANwAAjZQvuzeDiKAAAAAASUVORK5CYII=","orcid":"","institution":"Foshan University","correspondingAuthor":true,"prefix":"","firstName":"Yonghui","middleName":"","lastName":"Xia","suffix":""}],"badges":[],"createdAt":"2025-05-09 10:23:26","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-6627657/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-6627657/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":83534961,"identity":"198fce0b-41a5-4a86-96cc-1bc61cf74193","added_by":"auto","created_at":"2025-05-28 06:15:40","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1900370,"visible":true,"origin":"","legend":"","description":"","filename":"WJX510.pdf","url":"https://assets-eu.researchsquare.com/files/rs-6627657/v1_covered_cd62110b-c83d-49ab-abd2-11308ffccce7.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect","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":"Leslie–type model, generalist predator, weak Allee effect, degenerate Bogdanov–Takens bifurcation, Hopf bifurcation, Coexistence","lastPublishedDoi":"10.21203/rs.3.rs-6627657/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-6627657/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eThis study investigates the bifurcation dynamics driven by the interaction between a generalist predator and a weak Allee effect in a Leslie-type model through a rigorous theoretical analysis. We reveal that the interplay between the generalist predator (parameter e) and the weak Allee effect (parameter β) leads to a rich bifurcation structure. Specifically: (1) when β = e, the model exhibits a nilpotent cusp of codimension 2, accompanied by degenerate Bogdanov-Takens and Hopf bifurcations of codimension 2; (2) when β ≠ e, the nilpotent cusp can reach codimension 3, and the model undergoes degenerate Bogdanov-Takens and Hopf bifurcations of codimension up to 3. Nearly all conditions guaranteeing these dynamics are derived explicitly and precisely. Our results demonstrate that the generalist predator and weak Allee effect not only enrich the system’s dynamical behaviors and bifurcation patterns but may also lead to prey extinction for certain positive initial densities. Furthermore, smaller values of e favor the coexistence of predator and prey, while larger values of e are detrimental to the prey population. Numerical simulations, including prey extinction and the coexistence of multiple steady states (e.g., homoclinic orbits, limit cycles, and double limit cycles), provide intuitive illustrations of the theoretical findings. This work advances the understanding of higher-codimension bifurcation dynamics in ecological systems.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003econtinue to represent an active area of research focus due to their ability to capture fundamental ecological processes while remaining mathematically tractable [1, and h represents the quality of the prey as food for the predator. The study of predator-prey dynamics provides crucial insights into the conditions governing species coexistence and extinction, offering valuable guidance for ecosystem management and conservation policymaking. These models demonstrate that system outcomes are highly sensitive to various biological factors, particularly the functional response characterizing predator feeding behavior. The different functional response types (Holling type I, II, III, IV, and so on) are known to generate distinct dynamical regimes, ranging from stable equilibria to complex oscillatory patterns, each with important ecological implications [15, 16, 17, 18, 19, 20, 21]. Note that in the model (1), the predator is called a specialist predator, since it only relies on a single prey species to survive and will die out in absence of this prey. While in some situations, especially when the prey is severely scarce, the predator can turn to other alternative prey species for food and can persist by switching to other food sources,\u003c/p\u003e\n\u003cp\u003e\u003cem\u003e2020 MSC:\u003c/em\u003e 34C07; 34C25; 34D15; 37G15\u003c/p\u003e","manuscriptTitle":"Bifurcation dynamics in a Leslie-type model: interaction between generalist predator and weak Allee effect","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2025-05-28 06:07:33","doi":"10.21203/rs.3.rs-6627657/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"decision","content":"Revision requested","date":"2026-03-18T14:00:27+00:00","index":"","fulltext":""},{"type":"editorInvitedReview","content":"","date":"2025-07-24T14:33:10+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"267884330732657249437782633503744972460","date":"2025-06-04T07:14:27+00:00","index":"hide","fulltext":""},{"type":"reviewerAgreed","content":"9146712672583806913797488344430181666","date":"2025-05-23T12:54:33+00:00","index":"hide","fulltext":""},{"type":"reviewersInvited","content":"","date":"2025-05-23T10:58:37+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2025-05-12T01:33:21+00:00","index":"","fulltext":""},{"type":"checksComplete","content":"","date":"2025-05-12T01:31:17+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of Nonlinear Science","date":"2025-05-09T10:19:27+00:00","index":"","fulltext":""}],"status":"published","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}}],"origin":"","ownerIdentity":"f602ffe3-0f43-4aec-bf12-226d9d8da08c","owner":[],"postedDate":"May 28th, 2025","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2026-05-01T10:38:13+00:00","versionOfRecord":[],"versionCreatedAt":"2025-05-28 06:07:33","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-6627657","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-6627657","identity":"rs-6627657","version":["v1"]},"buildId":"8U1c8b4HqxoKbykW_rLl7","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}