Cross-diffusion induced spatio-temporal patterns in Schnakenberg reaction-diffusion model

Cross-diffusion induced spatio-temporal patterns in Schnakenberg reaction-diffusion 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 Cross-diffusion induced spatio-temporal patterns in Schnakenberg reaction-diffusion model Rui Yang This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1501523/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 4 You are reading this latest preprint version Abstract Schnakenberg system is a typical mathematical model to describe activator-depleted kinetics. In this paper, by introducing linear cross-diffusion into Schnakenberg system, we derive cross-diffusion-driven Turing instability conditions. It has been revealed that it is no longer necessary to have long-range inhibition and short-range activation for Turing instability with the help of cross-diffusion. Then the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which help us to derive parameter space more specific where certain patterns such as hexagon-like pattern, stripe-like pattern and the coexistence pattern will emerge. Furthermore, the numerical simulations in both Turing instability region and Turing-Hopf region provide an indication of the wealth of patterns that the system can exhibit. Besides, different initial conditions are employed to help better understanding the complex patterns. Schnakenberg model Cross-diffusion Turing instability Amplitude equations. Full Text Cite Share Download PDF Status: Under Review Version 1 posted Reviews received at journal 05 Apr, 2022 Reviewers invited by journal 05 Apr, 2022 Editor assigned by journal 04 Apr, 2022 First submitted to journal 03 Apr, 2022 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-1501523","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":96279574,"identity":"29942b87-33cf-42a9-bb37-e12acbe4228b","order_by":0,"name":"Rui Yang","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAz0lEQVRIiWNgGAWjYPACCTkGZhDNRoIWYwZmZtK0MCQ2MBCrxeD42cOvK9ss0ue38x9g+FB2mIF/dgMBLWfy0izPtknkbjjMzMA449xhBok7BwhoOZBjZtgI0gL0CzNv22EGA4kEAlrOvwFrSZdvBmr5S5SWGznGD4FaEhiADmNmJEaL5I03ZowN5yQMgX4xONhzLp1H4gYBLXznc4w/NpTVycv3H3z44EeZtRz/DAJaFA4wsEnAOAeAmAe/eiCQB0bhB4KqRsEoGAWjYGQDAMvCP81TUHaVAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0000-0001-7725-2538","institution":"Shanghai Institute of Technology","correspondingAuthor":true,"prefix":"","firstName":"Rui","middleName":"","lastName":"Yang","suffix":""}],"badges":[],"createdAt":"2022-03-29 12:01:40","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1501523/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1501523/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":20075218,"identity":"98e1bb08-7177-4628-99c7-959e5fef6484","added_by":"auto","created_at":"2022-04-07 16:08:27","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":3340217,"visible":true,"origin":"","legend":"","description":"","filename":"crossdiffusion.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1501523/v1_covered.pdf"}],"financialInterests":"","formattedTitle":"Cross-diffusion induced spatio-temporal patterns in Schnakenberg reaction-diffusion model","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1501523/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"Schnakenberg model, Cross-diffusion, Turing instability, Amplitude equations.","lastPublishedDoi":"10.21203/rs.3.rs-1501523/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1501523/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"Schnakenberg system is a typical mathematical model to describe activator-depleted kinetics. In this paper, by introducing linear cross-diffusion into Schnakenberg system, we derive cross-diffusion-driven Turing instability conditions. It has been revealed that it is no longer necessary to have long-range inhibition and short-range activation for Turing instability with the help of cross-diffusion. Then the multiple scales method is applied to obtain the amplitude equations at the critical value of Turing bifurcation, which help us to derive parameter space more specific where certain patterns such as hexagon-like pattern, stripe-like pattern and the coexistence pattern will emerge. Furthermore, the numerical simulations in both Turing instability region and Turing-Hopf region provide an indication of the wealth of patterns that the system can exhibit. Besides, different initial conditions are employed to help better understanding the complex patterns.","manuscriptTitle":"Cross-diffusion induced spatio-temporal patterns in Schnakenberg reaction-diffusion model","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2022-04-07 16:08:21","doi":"10.21203/rs.3.rs-1501523/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"editorInvitedReview","content":"","date":"2022-04-06T01:42:22+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2022-04-05T11:44:01+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2022-04-04T08:46:26+00:00","index":"","fulltext":""},{"type":"submitted","content":"Nonlinear Dynamics","date":"2022-04-03T10:00:56+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"nonlinear-dynamics","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"nody","sideBox":"Learn more about [Nonlinear Dynamics](https://www.springer.com/journal/11071)","snPcode":"11071","submissionUrl":"https://submission.nature.com/new-submission/11071/3","title":"Nonlinear Dynamics","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"stoa","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"db07b38d-6780-48ea-87b4-dc6db14b0254","owner":[],"postedDate":"April 7th, 2022","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"under-review","subjectAreas":[],"tags":[],"updatedAt":"2022-07-06T07:07:08+00:00","versionOfRecord":[],"versionCreatedAt":"2022-04-07 16:08:21","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-1501523","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-1501523","identity":"rs-1501523","version":["v1"]},"buildId":"J0_U0BvcaRcwD8yVFaRlm","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}