Pattern Formation from Nonlocal Conflict Memory in a PDE-ODE Model

Pattern Formation from Nonlocal Conflict Memory in a PDE-ODE 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 Pattern Formation from Nonlocal Conflict Memory in a PDE-ODE Model Shu Li, Binxiang Dai, Hao Wang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-8151446/v1 This work is licensed under a CC BY 4.0 License Status: Under Revision Version 1 posted 5 You are reading this latest preprint version Abstract Many territorial animals shape their movement decisions using memory of past conflicts, yet the population-level consequences of such nonlocal conflict memory remain poorly understood. We propose and analyze a PDE-ODE hybrid model for a single population moving in response to spatial memory of territorial conflicts. The population density satisfies a diffusion equation with delayed, nonlocal advection generated by a convolution of a conflict variable, while the conflict intensity evolves according to a local ODE encoding a warning mechanism with decay and resetting. Linearization about the positive homogeneous steady state yields a non-self-adjoint operator whose spectrum reduces to a countable family of characteristic equations indexed by Fourier modes. For Gaussian and Laplacian perception kernels, the steady state is stable for all perceptual radii in the memoryless case, whereas a top-hat kernel admits diffusion-driven (Turing) instability below a critical radius, producing stationary spatial patterns. With positive memory delay, we prove the occurrence of Hopf bifurcations for all kernels, and for the top-hat kernel, we further identify codimension-two Turing-Hopf and double-Hopf points that organize more intricate spatiotemporal dynamics. Numerical simulations confirm the analytical thresholds and illustrate how conflict frequency, memory decay, and perceptual range jointly regulate aggregation, segregation, and population cycles driven by nonlocal conflict memory. PDE-ODE coupled model nonlocal memory warning mechanisms spatiotemporal dynamics Turing Hopf bifurcations Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Revision Version 1 posted Editorial decision: Revision requested 12 May, 2026 Reviewers invited by journal 26 Apr, 2026 Editor assigned by journal 28 Nov, 2025 Submission checks completed at journal 28 Nov, 2025 First submitted to journal 19 Nov, 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-8151446","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":629764995,"identity":"efc25e64-b004-40af-9910-7731c8963d18","order_by":0,"name":"Shu Li","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"prefix":"","firstName":"Shu","middleName":"","lastName":"Li","suffix":""},{"id":629764996,"identity":"c2841238-f710-4d65-92a9-45894551ef85","order_by":1,"name":"Binxiang Dai","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAs0lEQVRIiWNgGAWjYBACAxBKqIBwJEjQcoZkLYxtpGgxl0je+OHhPDt7gwPMB2/zMNjlEdRi2XOsWCJxW3LihgNsydY8DMnFhB12vMcAqOVAgsEBHjNpHoYDiQ0EtRzmMf6ROOcA0GH834jUcrzHTCKx4QDjhgM8bMRpAfqlzCLhWHLizMNsxpZzDJIJawGG2OabP2rs7PmONz+88abCjrAWBGAGu5N49aNgFIyCUTAK8AAANYw6BL4rzBYAAAAASUVORK5CYII=","orcid":"","institution":"Central South University","correspondingAuthor":true,"prefix":"","firstName":"Binxiang","middleName":"","lastName":"Dai","suffix":""},{"id":629764997,"identity":"f1620e46-d841-43f3-b7c1-71bd472c6389","order_by":2,"name":"Hao Wang","email":"","orcid":"","institution":"University of Alberta","correspondingAuthor":false,"prefix":"","firstName":"Hao","middleName":"","lastName":"Wang","suffix":""}],"badges":[],"createdAt":"2025-11-19 06:08:37","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-8151446/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-8151446/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":108181805,"identity":"54a19b69-d690-4bd3-8825-fbdaaa7fd4a1","added_by":"auto","created_at":"2026-04-30 08:58:56","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1838215,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-8151446/v1_covered_e574680f-e28f-4883-abdf-c1446a022c27.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Pattern Formation from Nonlocal Conflict Memory in a PDE-ODE Model","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"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":"PDE-ODE coupled model, nonlocal memory, warning mechanisms, spatiotemporal dynamics, Turing, Hopf bifurcations","lastPublishedDoi":"10.21203/rs.3.rs-8151446/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-8151446/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Many territorial animals shape their movement decisions using memory of past conflicts, yet the population-level consequences of such nonlocal conflict memory remain poorly understood. We propose and analyze a PDE-ODE hybrid model for a single population moving in response to spatial memory of territorial conflicts. The population density satisfies a diffusion equation with delayed, nonlocal advection generated by a convolution of a conflict variable, while the conflict intensity evolves according to a local ODE encoding a warning mechanism with decay and resetting. Linearization about the positive homogeneous steady state yields a non-self-adjoint operator whose spectrum reduces to a countable family of characteristic equations indexed by Fourier modes. For Gaussian and Laplacian perception kernels, the steady state is stable for all perceptual radii in the memoryless case, whereas a top-hat kernel admits diffusion-driven (Turing) instability below a critical radius, producing stationary spatial patterns. With positive memory delay, we prove the occurrence of Hopf bifurcations for all kernels, and for the top-hat kernel, we further identify codimension-two Turing-Hopf and double-Hopf points that organize more intricate spatiotemporal dynamics. 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