{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,2]],"date-time":"2026-10-02T02:32:02Z","timestamp":1790908322408,"version":"4.1.0"},"reference-count":28,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T00:00:00Z","timestamp":1746403200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T00:00:00Z","timestamp":1746403200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100000086","name":"Directorate for Mathematical and Physical Sciences","doi-asserted-by":"publisher","award":["DMS-2105816"],"award-info":[{"award-number":["DMS-2105816"]}],"id":[{"id":"10.13039\/100000086","id-type":"DOI","asserted-by":"publisher"}]},{"name":"European Research Council","award":["01071417"],"award-info":[{"award-number":["01071417"]}]},{"DOI":"10.13039\/501100000923","name":"Australian Research Council","doi-asserted-by":"publisher","award":["DP190102545"],"award-info":[{"award-number":["DP190102545"]}],"id":[{"id":"10.13039\/501100000923","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2025,6]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We consider a Gatenby\u2013Gawlinski-type model of invasive tumors in the presence of an Allee effect. We describe the construction of bistable one-dimensional traveling fronts using singular perturbation techniques in different parameter regimes corresponding to tumor interfaces with, or without, an acellular gap. By extending the front as a planar interface, we perform a stability analysis to long wavelength perturbations transverse to the direction of front propagation and derive a simple stability criterion for the front in two spatial dimensions. In particular we find that in general the presence of the acellular gap indicates transversal instability of the associated planar front, which can lead to complex interfacial dynamics such as the development of finger-like protrusions and\/or different invasion speeds.<\/jats:p>","DOI":"10.1007\/s00285-025-02209-w","type":"journal-article","created":{"date-parts":[[2025,5,5]],"date-time":"2025-05-05T14:00:53Z","timestamp":1746453653000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Deformations of acid-mediated invasive tumors in a model with Allee effect"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1884-9988","authenticated-orcid":false,"given":"Paul","family":"Carter","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Arjen","family":"Doelman","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Peter","family":"van Heijster","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel","family":"Levy","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Philip","family":"Maini","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erin","family":"Okey","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Paige","family":"Yeung","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,5,5]]},"reference":[{"issue":"3","key":"2209_CR1","doi-asserted-by":"publisher","first-page":"227","DOI":"10.1038\/nrc2329","volume":"8","author":"AR Anderson","year":"2008","unstructured":"Anderson AR, Quaranta V (2008) Integrative mathematical oncology. Nat Rev Cancer 8(3):227\u2013234","journal-title":"Nat Rev Cancer"},{"key":"2209_CR2","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.physd.2018.09.003","volume":"388","author":"R Bastiaansen","year":"2019","unstructured":"Bastiaansen R, Doelman A (2019) The dynamics of disappearing pulses in a singularly perturbed reaction-diffusion system with parameters that vary in time and space. Phys D Nonlinear Phenomena 388:45\u201372","journal-title":"Phys D Nonlinear Phenomena"},{"issue":"6","key":"2209_CR3","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1007\/s00332-023-09963-5","volume":"33","author":"E Byrnes","year":"2023","unstructured":"Byrnes E, Carter P, Doelman A, Liu L (2023) Large amplitude radially symmetric spots and gaps in a dryland ecosystem model. J Nonlinear Sci 33(6):107","journal-title":"J Nonlinear Sci"},{"issue":"3","key":"2209_CR4","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1137\/23M1610872","volume":"84","author":"P Carter","year":"2024","unstructured":"Carter P (2024) A stabilizing effect of advection on planar interfaces in singularly perturbed reaction-diffusion equations. SIAM J Appl Math 84(3):1227\u20131253","journal-title":"SIAM J Appl Math"},{"key":"2209_CR5","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2022.133596","volume":"444","author":"P Carter","year":"2023","unstructured":"Carter P, Doelman A, Lilly K, Obermayer E, Rao S (2023) Criteria for the (in)stability of planar interfaces in singularly perturbed 2-component reaction\u2013diffusion equations. Phys D Nonlinear Phenomena 444:133596","journal-title":"Phys D Nonlinear Phenomena"},{"key":"2209_CR6","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/s002850000067","volume":"42","author":"M Chaplain","year":"2001","unstructured":"Chaplain M, Ganesh M, Graham I (2001) Spatio-temporal pattern formation on spherical surfaces: numerical simulation and application to solid tumour growth. J Math Biol 42:387\u2013423","journal-title":"J Math Biol"},{"key":"2209_CR7","doi-asserted-by":"crossref","unstructured":"Crossley RM, Painter KJ, Lorenzi T, Maini PK, Baker RE (2024) Phenotypic switching mechanisms determine the structure of cell migration into extracellular matrix under the \u2018go-or-grow\u2019hypothesis. Math Biosci, 109240","DOI":"10.1016\/j.mbs.2024.109240"},{"issue":"2","key":"2209_CR8","doi-asserted-by":"publisher","first-page":"1325","DOI":"10.1007\/s10884-021-10003-7","volume":"34","author":"PN Davis","year":"2022","unstructured":"Davis PN, van Heijster P, Marangell R, Rodrigo MR (2022) Traveling wave solutions in a model for tumor invasion with the acid-mediation hypothesis. J Dyn Different Equ 34(2):1325\u20131347","journal-title":"J Dyn Different Equ"},{"key":"2209_CR9","unstructured":"Doedel EJ, Champneys AR, Dercole F, Fairgrieve TF, Kuznetsov YA, Oldeman B, Paffenroth R, Sandstede B, Wang X, Zhang C (2007) Auto-07p: continuation and bifurcation software for ordinary differential equations"},{"key":"2209_CR10","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/j.mbs.2009.04.001","volume":"220","author":"A Fasano","year":"2009","unstructured":"Fasano A, Herrero M, Rodrigo M (2009) Slow and fast invasion waves in a model of acid-mediated tumour growth. Math Biosci 220:45\u201356","journal-title":"Math Biosci"},{"issue":"4","key":"2209_CR11","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.122.048101","volume":"122","author":"C Fernandez-Oto","year":"2019","unstructured":"Fernandez-Oto C, Tzuk O, Meron E (2019) Front instabilities can reverse desertification. Phys Rev Lett 122(4):048101","journal-title":"Phys Rev Lett"},{"key":"2209_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.nonrwa.2021.103387","volume":"63","author":"T Gallay","year":"2022","unstructured":"Gallay T, Mascia C (2022) Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity. Nonlinear Anal Real World Appl 63:103387","journal-title":"Nonlinear Anal Real World Appl"},{"issue":"24","key":"2209_CR13","first-page":"5745","volume":"56","author":"RA Gatenby","year":"1996","unstructured":"Gatenby RA, Gawlinski ET (1996) A reaction-diffusion model of cancer invasion. Cancer Res 56(24):5745\u20135753","journal-title":"Cancer Res"},{"issue":"3","key":"2209_CR14","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pcbi.1009844","volume":"18","author":"P Gerlee","year":"2022","unstructured":"Gerlee P, Altrock P, Malik A, Krona C, Nelander S (2022) Autocrine signaling can explain the emergence of allee effects in cancer cell populations. PLoS Computation Biol 18(3):e1009844","journal-title":"PLoS Computation Biol"},{"key":"2209_CR15","first-page":"176","volume":"227","author":"AB Holder","year":"2014","unstructured":"Holder AB, Rodrigo MR, Herrero MA (2014) A model for acid-mediated tumour growth with nonlinear acid production term. Appl Math Comput 227:176\u2013198","journal-title":"Appl Math Comput"},{"issue":"8","key":"2209_CR16","doi-asserted-by":"publisher","DOI":"10.1371\/journal.pbio.3000399","volume":"17","author":"K Johnson","year":"2019","unstructured":"Johnson K, Howard G, Mo W, Strasser M, Ernesto E, Lima B, Huang S, Brock A (2019) Cancer cell population growth kinetics at low densities deviate from the exponential growth model and suggest an allee effect. PLoS Biol 17(8):e3000399","journal-title":"PLoS Biol"},{"key":"2209_CR17","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1007\/s11538-024-01286-0","volume":"86","author":"D Katsaounis","year":"2024","unstructured":"Katsaounis D, Harbour N, Williams T, Chaplain M, Sfakianakis N (2024) A genuinely hybrid, multiscale 3d cancer invasion and metastasis modelling framework. Bull Math Biol 86:64","journal-title":"Bull Math Biol"},{"issue":"5","key":"2209_CR18","doi-asserted-by":"publisher","first-page":"371","DOI":"10.1038\/nrc3712","volume":"14","author":"KS Korolev","year":"2014","unstructured":"Korolev KS, Xavier JB, Gore J (2014) Turning ecology and evolution against cancer. Nat Rev Cancer 14(5):371\u2013380","journal-title":"Nat Rev Cancer"},{"issue":"6","key":"2209_CR19","doi-asserted-by":"publisher","first-page":"1473","DOI":"10.1088\/0951-7715\/14\/6\/304","volume":"14","author":"M Krupa","year":"2001","unstructured":"Krupa M, Szmolyan P (2001) Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14(6):1473","journal-title":"Nonlinearity"},{"key":"2209_CR20","doi-asserted-by":"publisher","first-page":"R1","DOI":"10.1088\/0951-7715\/23\/1\/R01","volume":"23","author":"J Lowengrub","year":"2010","unstructured":"Lowengrub J, Frieboes H, Jin F, Chuang Y, Li X, Macklin P, Wise S, Cristini V (2010) Nonlinear modelling of cancer: bridging the gap between cells and tumours. Nonlinearity 23:R1\u2013R91","journal-title":"Nonlinearity"},{"key":"2209_CR21","doi-asserted-by":"publisher","first-page":"461","DOI":"10.1016\/j.jtbi.2010.08.028","volume":"267","author":"N Martin","year":"2010","unstructured":"Martin N, Gaffney E, Gatenby R, Maini P (2010) Tumour\u2013stromal interactions in acid-mediated invasion: a mathematical model. J Theoret Biol 267:461\u2013470","journal-title":"J Theoret Biol"},{"key":"2209_CR22","doi-asserted-by":"publisher","DOI":"10.1242\/dev.185231","volume":"147","author":"R McLennan","year":"2020","unstructured":"McLennan R, McKinney M, Teddy J, Morrison J, Kasemeier-Kulesa J, Ridenour D, Manthe C, Giniunaite R, Robinson M, Baker R, Maini P, Kulesa P (2020) Neural crest cells bulldoze through the microenvironment using aquaporin 1 to stabilize filopodia. Development 147:dev185231","journal-title":"Development"},{"issue":"4\u20135","key":"2209_CR23","first-page":"209","volume":"16","author":"A Perumpanani","year":"1996","unstructured":"Perumpanani A, Sherratt J, Norbury J, Byrne H (1996) Biological inferences from a mathematical model for malignant invasion. Invasion Metastasis 16(4\u20135):209\u20132021","journal-title":"Invasion Metastasis"},{"key":"2209_CR24","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1016\/j.jtbi.2015.12.024","volume":"394","author":"L Sewalt","year":"2016","unstructured":"Sewalt L, Harley K, van Heijster P, Balasuriya S (2016) Influences of allee effects in the spreading of malignant tumours. J Theoret Biol 394:77\u201392","journal-title":"J Theoret Biol"},{"key":"2209_CR25","doi-asserted-by":"publisher","first-page":"15","DOI":"10.1007\/s11538-019-00675-0","volume":"82","author":"M Strobl","year":"2020","unstructured":"Strobl M, Krause A, Damaghi M, Gillies R, Anderson A, Maini P (2020) Mix and match: phenotypic coexistence as a key facilitator of cancer invasion. Bull Math Biol 82:15","journal-title":"Bull Math Biol"},{"issue":"1","key":"2209_CR26","doi-asserted-by":"publisher","first-page":"85","DOI":"10.1006\/jtbi.2001.2522","volume":"216","author":"S Turner","year":"2002","unstructured":"Turner S, Sherratt JA (2002) Intercellular adhesion and cancer invasion: a discrete simulation using the extended potts model. J Theoret Biol 216(1):85\u2013100","journal-title":"J Theoret Biol"},{"key":"2209_CR27","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1007\/s00109-008-0303-9","volume":"86","author":"A Verkman","year":"2008","unstructured":"Verkman A, Hara-Chikuma M, Papadopoul M (2008) Aquaporins\u2014new players in cancer biology. J Mol Med 86:523\u2013529","journal-title":"J Mol Med"},{"key":"2209_CR28","doi-asserted-by":"publisher","DOI":"10.1142\/8973","volume-title":"Dynamics of cancer: mathematical foundations of oncology","author":"D Wodarz","year":"2014","unstructured":"Wodarz D, Komarova N (2014) Dynamics of cancer: mathematical foundations of oncology. World Scientific, Singapore"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02209-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-025-02209-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02209-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T07:38:37Z","timestamp":1748677117000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-025-02209-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,5]]},"references-count":28,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["2209"],"URL":"https:\/\/doi.org\/10.1007\/s00285-025-02209-w","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,5]]},"assertion":[{"value":"29 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"28 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare that they have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"55"}}