{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,3,7]],"date-time":"2025-03-07T05:13:50Z","timestamp":1741324430508,"version":"3.38.0"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T00:00:00Z","timestamp":1739750400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,2,17]],"date-time":"2025-02-17T00:00:00Z","timestamp":1739750400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Math. Biol."],"published-print":{"date-parts":[[2025,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We construct the traveling wave solutions of an FKPP growth process of two densities of particles, and prove that the critical traveling waves are locally stable in a space where the perturbations can grow exponentially at the back of the wave. The considered reaction\u2013diffusion system was introduced by Hannezo et al. (Cell 171(1):242\u2013255, 2017) in the context of branching morphogenesis: active, branching particles accumulate inactive particles, which do not react. Thus, the system features a continuum of steady state solutions, complicating the analysis. We adopt a result by Faye and Holzer (J Differ Equ 269(9):6559\u20136601, 2020) for proving the stability of the critical traveling waves, and modify the semi-group estimates to spaces with unbounded weights. We use a Feynman\u2013Kac formula to get an exponential a priori estimate for the tail of the PDE, a novel and simple approach.<\/jats:p>","DOI":"10.1007\/s00285-025-02189-x","type":"journal-article","created":{"date-parts":[[2025,2,18]],"date-time":"2025-02-18T00:30:21Z","timestamp":1739838621000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Convective stability of the critical waves of an FKPP-type model for self-organized growth"],"prefix":"10.1007","volume":"90","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1938-2590","authenticated-orcid":false,"given":"Florian","family":"Kreten","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,2,17]]},"reference":[{"key":"2189_CR1","first-page":"167","volume":"410","author":"J Alexander","year":"1990","unstructured":"Alexander J, Gardner R, Jones C (1990) A topological invariant arising in the stability analysis of travelling waves. Journal fuer die reine und angewandte Mathematik 410:167\u2013212","journal-title":"Journal fuer die reine und angewandte Mathematik"},{"issue":"2","key":"2189_CR2","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1007\/s00205-024-01961-5","volume":"248","author":"M Avery","year":"2024","unstructured":"Avery M (2024) Front selection in reaction\u2013diffusion systems via diffusive normal forms. Arch Ration Mech Anal 248(2):16. https:\/\/doi.org\/10.1007\/s00205-024-01961-5","journal-title":"Arch Ration Mech Anal"},{"issue":"2","key":"2189_CR3","doi-asserted-by":"publisher","first-page":"2206","DOI":"10.1137\/20M1343476","volume":"53","author":"M Avery","year":"2021","unstructured":"Avery M, Scheel A (2021) Asymptotic stability of critical pulled fronts via resolvent expansions near the essential spectrum. SIAM J Math Anal 53(2):2206\u20132242. https:\/\/doi.org\/10.1137\/20M1343476","journal-title":"SIAM J Math Anal"},{"key":"2189_CR4","doi-asserted-by":"publisher","first-page":"172","DOI":"10.1090\/cams\/8","volume":"2","author":"M Avery","year":"2022","unstructured":"Avery M, Scheel A (2022) Universal selection of pulled fronts. Commun Am Math Soc 2:172\u2013231. https:\/\/doi.org\/10.1090\/cams\/8","journal-title":"Commun Am Math Soc"},{"key":"2189_CR5","unstructured":"Avery M, Carter P, Rijk B, Scheel A (2023) Stability of coherent pattern formation through invasion in the Fitzhugh\u2013Nagumo system. Preprint available under arXiv:2310.08765 [math.AP]"},{"key":"2189_CR6","unstructured":"Barker B, Humpherys J, Zumbrun K (2015) STABLAB: a MATLAB-based numerical library for Evans function computation, available in the github repository under nonlinear-waves\/stablab"},{"issue":"2117","key":"2189_CR7","doi-asserted-by":"publisher","first-page":"20170184","DOI":"10.1098\/rsta.2017.0184","volume":"376","author":"B Barker","year":"2018","unstructured":"Barker B, Humpherys J, Lyng G, Lytle J (2018) Evans function computation for the stability of travelling waves. Philos Trans R Soc A Math Phys Eng Sci 376(2117):20170184. https:\/\/doi.org\/10.1098\/rsta.2017.0184","journal-title":"Philos Trans R Soc A Math Phys Eng Sci"},{"key":"2189_CR8","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-322-96657-5_1","volume-title":"Invariant manifolds for semilinear partial differential equations","author":"P Bates","year":"1989","unstructured":"Bates P, Jones C (1989) Invariant manifolds for semilinear partial differential equations. Vieweg & Teubner Verlag, Wiesbaden, pp 1\u201338. https:\/\/doi.org\/10.1007\/978-3-322-96657-5_1"},{"key":"2189_CR9","doi-asserted-by":"crossref","unstructured":"Bovier A, Hartung L (2022) The speed of invasion in an advancing population. Preprint available under arXiv:2204.11072 [math.PR]","DOI":"10.1007\/s00285-023-01989-3"},{"key":"2189_CR10","volume-title":"Elementary differential equations","author":"W Boyce","year":"2017","unstructured":"Boyce W, DiPrima R, Meade D (2017) Elementary differential equations, 11th edn. Wiley, Hoboken","edition":"11"},{"key":"2189_CR11","doi-asserted-by":"publisher","DOI":"10.1090\/memo\/0285","author":"M Bramson","year":"1983","unstructured":"Bramson M (1983) Convergence of solutions of the Kolmogorov equation to travelling waves. Mem Am Math Soc. https:\/\/doi.org\/10.1090\/memo\/0285","journal-title":"Mem Am Math Soc"},{"key":"2189_CR12","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4471-0049-2","volume-title":"Essential mathematical biology. Springer undergraduate mathematics series","author":"N Britton","year":"2003","unstructured":"Britton N (2003) Essential mathematical biology. Springer undergraduate mathematics series, vol 1. Springer, London. https:\/\/doi.org\/10.1007\/978-1-4471-0049-2"},{"key":"2189_CR13","doi-asserted-by":"publisher","first-page":"4780","DOI":"10.1103\/PhysRevLett.77.4780","volume":"77","author":"J Cardy","year":"1996","unstructured":"Cardy J, T\u00e4uber U (1996) Theory of branching and annihilating random walks. Phys Rev Lett 77:4780\u20134783. https:\/\/doi.org\/10.1103\/PhysRevLett.77.4780","journal-title":"Phys Rev Lett"},{"key":"2189_CR14","doi-asserted-by":"publisher","first-page":"142","DOI":"10.1007\/978-1-4612-5929-9","volume-title":"Applications of centre manifold theory. Applied mathematical sciences","author":"J Carr","year":"1982","unstructured":"Carr J (1982) Applications of centre manifold theory. Applied mathematical sciences, vol 35, 1st edn. Springer, New York, p 142. https:\/\/doi.org\/10.1007\/978-1-4612-5929-9","edition":"1"},{"issue":"4","key":"2189_CR15","doi-asserted-by":"publisher","first-page":"137","DOI":"10.1007\/s00526-019-1576-2","volume":"58","author":"A Ducrot","year":"2019","unstructured":"Ducrot A, Giletti T, Matano H (2019) Spreading speeds for multidimensional reaction\u2013diffusion systems of the prey\u2013predator type. Calc Var Partial Differ Equ 58(4):137. https:\/\/doi.org\/10.1007\/s00526-019-1576-2","journal-title":"Calc Var Partial Differ Equ"},{"issue":"1","key":"2189_CR16","doi-asserted-by":"publisher","first-page":"13","DOI":"10.1007\/s00033-018-1048-0","volume":"70","author":"G Faye","year":"2018","unstructured":"Faye G, Holzer M (2018) Asymptotic stability of the critical fisher-kpp front using pointwise estimates. Z Angew Math Phys 70(1):13. https:\/\/doi.org\/10.1007\/s00033-018-1048-0","journal-title":"Z Angew Math Phys"},{"issue":"9","key":"2189_CR17","doi-asserted-by":"publisher","first-page":"6559","DOI":"10.1016\/j.jde.2020.05.012","volume":"269","author":"G Faye","year":"2020","unstructured":"Faye G, Holzer M (2020) Asymptotic stability of the critical pulled front in a Lotka\u2013Volterra competition model. J Differ Equ 269(9):6559\u20136601. https:\/\/doi.org\/10.1016\/j.jde.2020.05.012","journal-title":"J Differ Equ"},{"issue":"1","key":"2189_CR18","doi-asserted-by":"publisher","first-page":"53","DOI":"10.1016\/0022-0396(79)90152-9","volume":"31","author":"N Fenichel","year":"1979","unstructured":"Fenichel N (1979) Geometric singular perturbation theory for ordinary differential equations. J Differ Equ 31(1):53\u201398. https:\/\/doi.org\/10.1016\/0022-0396(79)90152-9","journal-title":"J Differ Equ"},{"issue":"4","key":"2189_CR19","doi-asserted-by":"publisher","first-page":"355","DOI":"10.1111\/j.1469-1809.1937.tb02153.x","volume":"7","author":"R Fisher","year":"1937","unstructured":"Fisher R (1937) The wave of advance of advantageous genes. Ann Eugen 7(4):355\u2013369. https:\/\/doi.org\/10.1111\/j.1469-1809.1937.tb02153.x","journal-title":"Ann Eugen"},{"issue":"3","key":"2189_CR20","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1088\/0951-7715\/7\/3\/003","volume":"7","author":"T Gallay","year":"1994","unstructured":"Gallay T (1994) Local stability of critical fronts in nonlinear parabolic partial differential equations. Nonlinearity 7(3):741\u2013764. https:\/\/doi.org\/10.1088\/0951-7715\/7\/3\/003","journal-title":"Nonlinearity"},{"issue":"3","key":"2189_CR21","doi-asserted-by":"publisher","first-page":"3275","DOI":"10.1137\/21M1451038","volume":"56","author":"L Garenaux","year":"2024","unstructured":"Garenaux L (2024) Nonlinear convective stability of a critical pulled front undergoing a turing bifurcation at its back: a case study. SIAM J Math Anal 56(3):3275\u20133325. https:\/\/doi.org\/10.1137\/21M1451038","journal-title":"SIAM J Math Anal"},{"issue":"1","key":"2189_CR22","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1512\/iumj.2009.58.3497","volume":"58","author":"A Ghazaryan","year":"2009","unstructured":"Ghazaryan A (2009) Nonlinear stability of high Lewis number combustion fronts. Indiana Univ Math J 58(1):181\u2013212","journal-title":"Indiana Univ Math J"},{"issue":"5","key":"2189_CR23","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1051\/mmnp\/20138503","volume":"8","author":"A Ghazaryan","year":"2013","unstructured":"Ghazaryan A, Latushkin Y, Schecter S (2013) Stability of traveling waves in partly parabolic systems. Math Model Nat Phenom 8(5):31\u201347. https:\/\/doi.org\/10.1051\/mmnp\/20138503","journal-title":"Math Model Nat Phenom"},{"issue":"5","key":"2189_CR24","doi-asserted-by":"publisher","first-page":"1279","DOI":"10.1112\/plms.12270","volume":"119","author":"L Girardin","year":"2019","unstructured":"Girardin L, Lam K (2019) Invasion of open space by two competitors: spreading properties of monostable two-species competition\u2013diffusion systems. Proc Lond Math Soc 119(5):1279\u20131335","journal-title":"Proc Lond Math Soc"},{"key":"2189_CR25","doi-asserted-by":"publisher","first-page":"462","DOI":"10.1007\/978-1-4612-1140-2","volume-title":"Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences","author":"J Guckheimer","year":"1983","unstructured":"Guckheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Applied mathematical sciences, vol 1, 1st edn. Springer, New York, p 462. https:\/\/doi.org\/10.1007\/978-1-4612-1140-2","edition":"1"},{"issue":"9","key":"2189_CR26","doi-asserted-by":"publisher","first-page":"512","DOI":"10.1111\/dgd.12570","volume":"60","author":"E Hannezo","year":"2018","unstructured":"Hannezo E, Simons B (2018) Statistical theory of branching morphogenesis. Dev Growth Differ 60(9):512\u2013521. https:\/\/doi.org\/10.1111\/dgd.12570","journal-title":"Dev Growth Differ"},{"issue":"1","key":"2189_CR27","doi-asserted-by":"publisher","first-page":"242","DOI":"10.1016\/j.cell.2017.08.026","volume":"171","author":"E Hannezo","year":"2017","unstructured":"Hannezo E, Scheele C, Moad M, Drogo N, Heer R, Sampogna R, Rheenen J, Simons B (2017) A unifying theory of branching morphogenesis. Cell 171(1):242\u2013255. https:\/\/doi.org\/10.1016\/j.cell.2017.08.026","journal-title":"Cell"},{"issue":"2","key":"2189_CR28","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1007\/s10884-015-9440-3","volume":"27","author":"J Hendricks","year":"2015","unstructured":"Hendricks J, Humpherys J, Lyng G, Zumbrun K (2015) Stability of viscous weak detonation waves for Majda\u2019s model. J Dyn Differ Equ 27(2):237\u2013260. https:\/\/doi.org\/10.1007\/s10884-015-9440-3","journal-title":"J Dyn Differ Equ"},{"key":"2189_CR29","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0089647","volume-title":"Geometric theory of semilinear parabolic equations. Lecture notes in mathematics","author":"D Henry","year":"1981","unstructured":"Henry D (1981) Geometric theory of semilinear parabolic equations. Lecture notes in mathematics, 1st edn. Springer, Heidelberg. https:\/\/doi.org\/10.1007\/BFb0089647","edition":"1"},{"key":"2189_CR30","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1506-6","volume-title":"Basic theory of ordinary differential equations. Universitext","author":"P Hsieh","year":"1999","unstructured":"Hsieh P, Sibuya Y (1999) Basic theory of ordinary differential equations. Universitext, 1st edn. Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4612-1506-6","edition":"1"},{"key":"2189_CR31","doi-asserted-by":"publisher","first-page":"44","DOI":"10.1007\/BFb0095239","volume-title":"Geometric singular perturbation theory","author":"C Jones","year":"1995","unstructured":"Jones C (1995) Geometric singular perturbation theory. Springer, Berlin, pp 44\u2013118. https:\/\/doi.org\/10.1007\/BFb0095239"},{"key":"2189_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-04339-4","volume-title":"The algorithmic beauty of seaweeds, sponges and corals. The virtual laboratory","author":"JK Kaandorp","year":"2001","unstructured":"Kaandorp JK (2001) The algorithmic beauty of seaweeds, sponges and corals. The virtual laboratory, 1st edn. Springer, Heidelberg. https:\/\/doi.org\/10.1007\/978-3-662-04339-4","edition":"1"},{"key":"2189_CR33","first-page":"470","volume-title":"Brownian motion and stochastic calculus. Graduate texts in mathematics","author":"I Karatzas","year":"1991","unstructured":"Karatzas I, Shreve S (1991) Brownian motion and stochastic calculus. Graduate texts in mathematics, 2nd edn. Springer, New York, p 470","edition":"2"},{"issue":"3","key":"2189_CR34","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1038\/nrm2357","volume":"9","author":"E Karsenti","year":"2008","unstructured":"Karsenti E (2008) Self-organization in cell biology: a brief history. Nat Rev Mol Cell Biol 9(3):255\u2013262. https:\/\/doi.org\/10.1038\/nrm2357","journal-title":"Nat Rev Mol Cell Biol"},{"issue":"2","key":"2189_CR35","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1016\/0022-5193(71)90051-8","volume":"30","author":"E Keller","year":"1971","unstructured":"Keller E, Segel L (1971) Traveling bands of chemotactic bacteria: a theoretical analysis. J Theor Biol 30(2):235\u2013248. https:\/\/doi.org\/10.1016\/0022-5193(71)90051-8","journal-title":"J Theor Biol"},{"issue":"6","key":"2189_CR36","first-page":"1","volume":"1","author":"A Kolmogorov","year":"1937","unstructured":"Kolmogorov A, Petrovskii I, Piscunov N (1937) A study of the equation of diffusion with increase in the quantity of matter, and its application to a biological problem. Byul. Moskovskogo Gos. Univ. 1(6):1\u201325","journal-title":"Byul. Moskovskogo Gos. Univ."},{"issue":"6","key":"2189_CR37","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1007\/s00285-022-01753-z","volume":"84","author":"F Kreten","year":"2022","unstructured":"Kreten F (2022) Traveling waves of an fkpp-type model for self-organized growth. J Math Biol 84(6):42. https:\/\/doi.org\/10.1007\/s00285-022-01753-z","journal-title":"J Math Biol"},{"issue":"2","key":"2189_CR38","doi-asserted-by":"publisher","first-page":"181","DOI":"10.1083\/jcb.200108110","volume":"155","author":"T Misteli","year":"2001","unstructured":"Misteli T (2001) The concept of self-organization in cellular architecture. J Cell Biol 155(2):181\u2013185","journal-title":"J Cell Biol"},{"issue":"4","key":"2189_CR39","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1051\/mmnp\/20094401","volume":"4","author":"HG Othmer","year":"2009","unstructured":"Othmer HG, Painter K, Umulis D, Xue C (2009) The intersection of theory and application in elucidating pattern formation in developmental biology. Math Model Nat Phenom 4(4):3\u201382. https:\/\/doi.org\/10.1051\/mmnp\/20094401","journal-title":"Math Model Nat Phenom"},{"key":"2189_CR40","doi-asserted-by":"publisher","first-page":"162","DOI":"10.1016\/j.jtbi.2018.06.019","volume":"481","author":"K Painter","year":"2019","unstructured":"Painter K (2019) Mathematical models for chemotaxis and their applications in self-organisation phenomena. J Theor Biol 481:162\u2013182. https:\/\/doi.org\/10.1016\/j.jtbi.2018.06.019","journal-title":"J Theor Biol"},{"issue":"2","key":"2189_CR41","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1007\/BF02101705","volume":"164","author":"R Pego","year":"1994","unstructured":"Pego R, Weinstein M (1994) Asymptotic stability of solitary waves. Commun Math Phys 164(2):305\u2013349. https:\/\/doi.org\/10.1007\/BF02101705","journal-title":"Commun Math Phys"},{"issue":"2","key":"2189_CR42","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1006\/jdeq.2000.3984","volume":"176","author":"V Rottsch\u00e4fer","year":"2001","unstructured":"Rottsch\u00e4fer V, Wayne C (2001) Existence and stability of traveling fronts in the extended Fisher\u2013Kolmogorov equation. J Differ Equ 176(2):532\u2013560. https:\/\/doi.org\/10.1006\/jdeq.2000.3984","journal-title":"J Differ Equ"},{"key":"2189_CR43","doi-asserted-by":"publisher","first-page":"983","DOI":"10.1016\/S1874-575X(02)80039-X","volume-title":"Handbook of dynamical systems. Handbook of dynamical systems","author":"B Sandstede","year":"2002","unstructured":"Sandstede B (2002) Chapter 18 - stability of travelling waves. In: Fiedler B (ed) Handbook of dynamical systems. Handbook of dynamical systems, vol 2. Elsevier Science, Amsterdam, pp 983\u20131055. https:\/\/doi.org\/10.1016\/S1874-575X(02)80039-X"},{"issue":"3","key":"2189_CR44","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1016\/0001-8708(76)90098-0","volume":"22","author":"D Sattinger","year":"1976","unstructured":"Sattinger D (1976) On the stability of waves of nonlinear parabolic systems. Adv Math 22(3):312\u2013355. https:\/\/doi.org\/10.1016\/0001-8708(76)90098-0","journal-title":"Adv Math"},{"key":"2189_CR45","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-5037-9","volume-title":"Dynamics of evolutionary equations. Applied mathematical sciences","author":"G Sell","year":"2002","unstructured":"Sell G, You Y (2002) Dynamics of evolutionary equations. Applied mathematical sciences, 1st edn. Springer, New York. https:\/\/doi.org\/10.1007\/978-1-4757-5037-9","edition":"1"},{"key":"2189_CR46","doi-asserted-by":"publisher","first-page":"634","DOI":"10.1007\/978-1-4612-0873-0","volume-title":"Shock waves and reaction\u2013diffusion equations. Grundlehren der mathematischen Wissenschaften","author":"J Smoller","year":"1994","unstructured":"Smoller J (1994) Shock waves and reaction\u2013diffusion equations. Grundlehren der mathematischen Wissenschaften, vol 258, 2nd edn. Springer, New York, p 634. https:\/\/doi.org\/10.1007\/978-1-4612-0873-0","edition":"2"},{"issue":"1","key":"2189_CR47","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1016\/S0167-2789(97)00141-3","volume":"113","author":"S Tobias","year":"1998","unstructured":"Tobias S, Proctor M, Knobloch E (1998) Convective and absolute instabilities of fluid flows in finite geometry. Phys D 113(1):43\u201372. https:\/\/doi.org\/10.1016\/S0167-2789(97)00141-3","journal-title":"Phys D"},{"issue":"641","key":"2189_CR48","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1098\/rstb.1952.0012","volume":"237","author":"AM Turing","year":"1952","unstructured":"Turing AM (1952) The chemical basis of morphogenesis philosophical. Trans R Soc Lond Ser B Biol Sci 237(641):37\u201372. https:\/\/doi.org\/10.1098\/rstb.1952.0012","journal-title":"Trans R Soc Lond Ser B Biol Sci"},{"key":"2189_CR49","doi-asserted-by":"publisher","DOI":"10.3792\/pjaa.53.225","author":"K Uchiyama","year":"1977","unstructured":"Uchiyama K (1977) The behavior of solutions of the equation of Kolmogorov\u2013Petrovsky\u2013Piskunov. Proc Jpn Acad Ser A Math Sci. https:\/\/doi.org\/10.3792\/pjaa.53.225","journal-title":"Proc Jpn Acad Ser A Math Sci"},{"issue":"5","key":"2189_CR50","doi-asserted-by":"publisher","first-page":"1810","DOI":"10.1137\/S0036139901386417","volume":"62","author":"F Varas","year":"2002","unstructured":"Varas F, Vega J (2002) Linear stability of a plane front in solid combustion at large heat of reaction. SIAM J Appl Math 62(5):1810\u20131822","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"2189_CR51","doi-asserted-by":"publisher","first-page":"267","DOI":"10.1007\/978-1-4471-0049-2","volume":"6","author":"V Volpert","year":"2009","unstructured":"Volpert V, Petrovskii S (2009) Reaction\u2013diffusion waves in biology. Phys Life Rev 6(4):267\u2013310. https:\/\/doi.org\/10.1007\/978-1-4471-0049-2","journal-title":"Phys Life Rev"},{"issue":"1747","key":"2189_CR52","doi-asserted-by":"publisher","first-page":"20170103","DOI":"10.1098\/rstb.2017.0103","volume":"373","author":"R Wedlich-S\u00f6ldner","year":"2018","unstructured":"Wedlich-S\u00f6ldner R, Betz T (2018) Self-organization: the fundament of cell biology. Philos Trans R Soc B Biol Sci 373(1747):20170103. https:\/\/doi.org\/10.1098\/rstb.2017.0103","journal-title":"Philos Trans R Soc B Biol Sci"},{"issue":"11\u201312","key":"2189_CR53","doi-asserted-by":"publisher","first-page":"1889","DOI":"10.1080\/03605309208820907","volume":"17","author":"J Xin","year":"1992","unstructured":"Xin J (1992) Multidimensional stability of traveling waves in a bistable reaction\u2013diffusion equation, 1. Commun Partial Differ Equ 17(11\u201312):1889\u20131899. https:\/\/doi.org\/10.1080\/03605309208820907","journal-title":"Commun Partial Differ Equ"},{"issue":"3","key":"2189_CR54","doi-asserted-by":"publisher","first-page":"741","DOI":"10.1512\/iumj.1998.47.1604","volume":"47","author":"K Zumbrun","year":"1998","unstructured":"Zumbrun K, Howard P (1998) Pointwise semigroup methods and stability of viscous shock waves. Indiana Univ Math J 47(3):741\u2013871","journal-title":"Indiana Univ Math J"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02189-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-025-02189-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02189-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,3,6]],"date-time":"2025-03-06T12:51:22Z","timestamp":1741265482000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-025-02189-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,17]]},"references-count":54,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,3]]}},"alternative-id":["2189"],"URL":"https:\/\/doi.org\/10.1007\/s00285-025-02189-x","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"type":"print","value":"0303-6812"},{"type":"electronic","value":"1432-1416"}],"subject":[],"published":{"date-parts":[[2025,2,17]]},"assertion":[{"value":"20 May 2023","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"21 October 2024","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 January 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 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 numerical analysis of the spectrum in Appendix\u00a0 was performed via STABLAB (Barker et\u00a0al. ). The simulations of the Reaction\u2013Diffusion System () were done with Wolfram Mathematica. The code can be accessed upon request.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Numerical analysis"}},{"value":"The author has no conflict of interest to declare.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"33"}}