{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T07:33:15Z","timestamp":1778830395347,"version":"3.51.4"},"reference-count":27,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2009,1,13]],"date-time":"2009-01-13T00:00:00Z","timestamp":1231804800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J Nonlinear Sci"],"published-print":{"date-parts":[[2009,6]]},"DOI":"10.1007\/s00332-008-9036-8","type":"journal-article","created":{"date-parts":[[2009,1,12]],"date-time":"2009-01-12T17:09:17Z","timestamp":1231780157000},"page":"301-339","source":"Crossref","is-referenced-by-count":12,"title":["Spikes for the Gierer\u2013Meinhardt System with\u00a0Discontinuous Diffusion\u00a0Coefficients"],"prefix":"10.1007","volume":"19","author":[{"given":"Juncheng","family":"Wei","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Matthias","family":"Winter","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2009,1,13]]},"reference":[{"key":"9036_CR1","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/0895-7177(93)90025-T","volume":"17","author":"D.L. Benson","year":"1993","unstructured":"Benson, D.L., Maini, P.K., Sherratt, J.A.: Analysis of pattern formation in reaction diffusion models with spatially inhomogeneous diffusion coefficients. Math. Comput. Model. 17, 29\u201334 (1993a)","journal-title":"Math. Comput. Model."},{"key":"9036_CR2","doi-asserted-by":"crossref","first-page":"365","DOI":"10.1007\/BF02460888","volume":"55","author":"D.L. Benson","year":"1993","unstructured":"Benson, D.L., Sherratt, J.A., Maini, P.K.: Diffusion driven instability in an inhomogeneous domain. Bull. Math. Biol. 55, 365\u2013384 (1993b)","journal-title":"Bull. Math. Biol."},{"key":"9036_CR3","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1007\/s002850050135","volume":"37","author":"D.L. Benson","year":"1998","unstructured":"Benson, D.L., Maini, P.K., Sherratt, J.A.: Unravelling the Turing bifurcation using spatially varying diffusion coefficients. J. Math. Biol. 37, 381\u2013387 (1998)","journal-title":"J. Math. Biol."},{"key":"9036_CR4","doi-asserted-by":"crossref","first-page":"R686","DOI":"10.1016\/S0960-9822(06)00356-3","volume":"7","author":"S.S. Blair","year":"1997","unstructured":"Blair, S.S.: Limb development: Marginal fringe benefits. Curr. Biol. 7, R686\u2013R690 (1997)","journal-title":"Curr. Biol."},{"key":"9036_CR5","doi-asserted-by":"crossref","first-page":"R570","DOI":"10.1016\/j.cub.2004.07.015","volume":"14","author":"S.S. Blair","year":"2004","unstructured":"Blair, S.S.: Developmental biology: Notching the hindbrain. Curr. Biol. 14, R570\u2013R572 (2004)","journal-title":"Curr. Biol."},{"key":"9036_CR6","doi-asserted-by":"crossref","first-page":"3350","DOI":"10.1063\/1.873048","volume":"5","author":"A.H. Boozer","year":"1998","unstructured":"Boozer, A.H.: Equations for studies of feedback stabilization. Phys. Plasmas 5, 3350 (1998)","journal-title":"Phys. Plasmas"},{"key":"9036_CR7","doi-asserted-by":"crossref","first-page":"245","DOI":"10.4310\/MAA.2001.v8.n2.a3","volume":"8","author":"E.N. Dancer","year":"2001","unstructured":"Dancer, E.N.: On stability and Hopf bifurcations for chemotaxis systems. Methods Appl. Anal. 8, 245\u2013256 (2001)","journal-title":"Methods Appl. Anal."},{"key":"9036_CR8","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1007\/BF00289234","volume":"12","author":"A. Gierer","year":"1972","unstructured":"Gierer, A., Meinhardt, H.: A theory of biological pattern formation. Kybernetik (Berlin) 12, 30\u201339 (1972)","journal-title":"Kybernetik (Berlin)"},{"key":"9036_CR9","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/S0167-2789(00)00206-2","volume":"150","author":"D. Iron","year":"2001","unstructured":"Iron, D., Ward, M.J., Wei, J.: The stability of spike solutions to the one-dimensional Gierer\u2013Meinhardt model. Physica D 150, 25\u201362 (2001)","journal-title":"Physica D"},{"key":"9036_CR10","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1146\/annurev.cellbio.17.1.189","volume":"17","author":"K.D. Irvine","year":"2001","unstructured":"Irvine, K.D., Rauskolb, C.: Boundaries in development: Formation and function. Annu. Rev. Cell Dev. Biol. 17, 189\u2013214 (2001)","journal-title":"Annu. Rev. Cell Dev. Biol."},{"key":"9036_CR11","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1093\/imammb\/9.3.197","volume":"9","author":"P.K. Maini","year":"1992","unstructured":"Maini, P.K., Benson, D.L., Sherratt, J.A.: Pattern formation in reaction-diffusion models with spatially inhomogeneous diffusion coefficients. IMA J. Math. Appl. Med. Biol. 9, 197\u2013213 (1992)","journal-title":"IMA J. Math. Appl. Med. Biol."},{"key":"9036_CR12","doi-asserted-by":"crossref","first-page":"375","DOI":"10.1016\/0012-1606(83)90175-6","volume":"96","author":"H. Meinhardt","year":"1983","unstructured":"Meinhardt, H.: Cell determination boundaries as organizing regions for secondary embryonic fields. Dev. Biol. 96, 375\u2013385 (1983)","journal-title":"Dev. Biol."},{"key":"9036_CR13","doi-asserted-by":"crossref","first-page":"339","DOI":"10.1103\/PhysRevE.51.339","volume":"51","author":"I. Mercader","year":"1995","unstructured":"Mercader, I., Net, M., Knobloch, E.: Binary fluid convection in a cylinder. Phys. Rev. E 51, 339\u2013350 (1995)","journal-title":"Phys. Rev. E"},{"key":"9036_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/b98868","volume-title":"Mathematical Biology I\/II","author":"J. Murray","year":"2002","unstructured":"Murray, J.: Mathematical Biology I\/II. Springer, Berlin (2002)"},{"key":"9036_CR15","doi-asserted-by":"crossref","first-page":"037104","DOI":"10.1063\/1.2778553","volume":"17","author":"Y. Nishiura","year":"2007","unstructured":"Nishiura, Y., Teramoto, T., Yuan, X., Ueda, K.-I.: Dynamics of traveling pulses in heterogeneous media. Chaos 17, 037104 (2007)","journal-title":"Chaos"},{"key":"9036_CR16","first-page":"80","volume":"181","author":"K. Page","year":"2003","unstructured":"Page, K., Maini, P.K., Monk, N.A.M.: Pattern formation in spatially heterogeneous Turing reaction-diffusion models. Physica\u00a0D 181, 80\u2013101 (2003)","journal-title":"Physica\u00a0D"},{"key":"9036_CR17","first-page":"95","volume":"202","author":"K. Page","year":"2005","unstructured":"Page, K., Maini, P.K., Monk, N.A.M.: Complex pattern formation in reaction-diffusion systems with spatially varying parameters. Physica\u00a0D 202, 95\u2013115 (2005)","journal-title":"Physica\u00a0D"},{"key":"9036_CR18","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1098\/rstb.1952.0012","volume":"237","author":"A.M. Turing","year":"1952","unstructured":"Turing, A.M.: The chemical basis of morphogenesis. Philos. Trans. R. Soc. Lond. B 237, 37\u201372 (1952)","journal-title":"Philos. Trans. R. Soc. Lond. B"},{"key":"9036_CR19","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s00332-002-0531-z","volume":"13","author":"M.J. Ward","year":"2003","unstructured":"Ward, M.J., Wei, J.: Hopf bifurcations and oscillatory instabilities of solutions for the one-dimensional Gierer\u2013Meinhardt model. J. Nonlinear Sci. 13, 209\u2013264 (2003)","journal-title":"J. Nonlinear Sci."},{"key":"9036_CR20","doi-asserted-by":"crossref","first-page":"1297","DOI":"10.1137\/S0036139900375112","volume":"62","author":"M. Ward","year":"2002","unstructured":"Ward, M., McInerney, D., Houston, P., Gavaghan, D., Maini, P.: The dynamics and pinning of a spike for a reaction-diffusion system. SIAM J. Appl. Math. 62, 1297\u20131328 (2002)","journal-title":"SIAM J. Appl. Math."},{"key":"9036_CR21","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1017\/S0956792599003770","volume":"10","author":"J. Wei","year":"1999","unstructured":"Wei, J.: On single interior spike solutions of Gierer\u2013Meinhardt system: uniqueness and spectrum estimates. Eur. J. Appl. Math. 10, 353\u2013378 (1999)","journal-title":"Eur. J. Appl. Math."},{"key":"9036_CR22","doi-asserted-by":"crossref","first-page":"1241","DOI":"10.1137\/S0036141098347237","volume":"30","author":"J. Wei","year":"1999","unstructured":"Wei, J., Winter, M.: On the two-dimensional Gierer\u2013Meinhardt system with strong coupling. SIAM J. Math. Anal. 30, 1241\u20131263 (1999)","journal-title":"SIAM J. Math. Anal."},{"key":"9036_CR23","doi-asserted-by":"crossref","first-page":"478","DOI":"10.1006\/jdeq.2001.4019","volume":"178","author":"J. Wei","year":"2000","unstructured":"Wei, J., Winter, M.: Spikes for the two-dimensional Gierer\u2013Meinhardt system: The strong coupling case. J. Differ. Equ. 178, 478\u2013518 (2000)","journal-title":"J. Differ. Equ."},{"key":"9036_CR24","doi-asserted-by":"crossref","first-page":"415","DOI":"10.1007\/s00332-001-0380-1","volume":"11","author":"J. Wei","year":"2001","unstructured":"Wei, J., Winter, M.: Spikes for the two-dimensional Gierer\u2013Meinhardt system: The weak coupling case. J.\u00a0Nonlinear Sci. 11, 415\u2013458 (2001)","journal-title":"J.\u00a0Nonlinear Sci."},{"key":"9036_CR25","doi-asserted-by":"crossref","unstructured":"Wei, J., Winter, M.: On the Gierer\u2013Meinhardt system with precursors. Discrete Contin. Dyn. Syst. Ser. A (2009, to appear)","DOI":"10.3934\/dcds.2009.25.363"},{"key":"9036_CR26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/S0022-5193(69)80016-0","volume":"25","author":"L. Wolpert","year":"1969","unstructured":"Wolpert, L.: Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol. 25, 1\u201347 (1969)","journal-title":"J. Theor. Biol."},{"key":"9036_CR27","doi-asserted-by":"crossref","first-page":"961","DOI":"10.1242\/dev.109.4.961","volume":"109","author":"L. Wolpert","year":"1990","unstructured":"Wolpert, L., Hornbruch, A.: Double anterior chick limb buds and models for cartilage rudiment specification. Development 109, 961\u2013966 (1990)","journal-title":"Development"}],"container-title":["Journal of Nonlinear Science"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-008-9036-8.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00332-008-9036-8\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00332-008-9036-8","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,9]],"date-time":"2024-03-09T00:08:27Z","timestamp":1709942907000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00332-008-9036-8"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1,13]]},"references-count":27,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2009,6]]}},"alternative-id":["9036"],"URL":"https:\/\/doi.org\/10.1007\/s00332-008-9036-8","relation":{},"ISSN":["0938-8974","1432-1467"],"issn-type":[{"value":"0938-8974","type":"print"},{"value":"1432-1467","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1,13]]}}}