{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T09:28:52Z","timestamp":1780478932972,"version":"3.54.1"},"reference-count":50,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T00:00:00Z","timestamp":1747872000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T00:00:00Z","timestamp":1747872000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100000266","name":"Engineering and Physical Sciences Research Council","doi-asserted-by":"publisher","award":["EP\/V047469\/1"],"award-info":[{"award-number":["EP\/V047469\/1"]}],"id":[{"id":"10.13039\/501100000266","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-2051568"],"award-info":[{"award-number":["DMS-2051568"]}],"id":[{"id":"10.13039\/100000001","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":"Abstract<\/jats:title>\n The Hilbert number H<\/jats:italic>(n<\/jats:italic>) is defined as the maximum number of limit cycles of a planar autonomous system of ordinary differential equations (ODEs) with right-hand sides containing polynomials of degree at most \n \n $$n \\in {{\\mathbb {N}}}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The dynamics of chemical reaction systems with two chemical species can be (under mass-action kinetics) described by such planar autonomous ODEs, where n<\/jats:italic> is equal to the maximum order of the chemical reactions in the system. Analogues of the Hilbert number\u00a0H<\/jats:italic>(n<\/jats:italic>) for three different classes of chemical reaction systems are investigated: (i) chemical systems with reactions up to the n<\/jats:italic>-th order; (ii) systems with up to n<\/jats:italic>-molecular chemical reactions; and (iii) weakly reversible chemical reaction networks. In each case (i), (ii) and (iii), the question on the number of limit cycles is considered. Lower bounds on the modified Hilbert numbers are provided for both algebraic and non-algebraic limit cycles. Furthermore, given a general algebraic curve \n \n $$h(x,y)=0$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> of degree \n \n $$n_h \\in {{\\mathbb {N}}}$$<\/jats:tex-math>\n \n \n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n \u2208<\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and containing one or more ovals in the positive quadrant, a chemical system is constructed which has the oval(s) as its stable algebraic limit cycle(s). The ODEs describing the dynamics of the constructed chemical system contain polynomials of degree at most \n \n $$n=2\\,n_h+1.$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n =<\/mml:mo>\n 2<\/mml:mn>\n \n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n +<\/mml:mo>\n 1<\/mml:mn>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Considering \n \n $$n_h \\ge 4,$$<\/jats:tex-math>\n \n \n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n \u2265<\/mml:mo>\n 4<\/mml:mn>\n ,<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> the algebraic curve \n \n $$h(x,y)=0$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> can contain multiple closed components with the maximum number of ovals given by Harnack\u2019s curve theorem as \n \n $$1+(n_h-1)(n_h-2)\/2$$<\/jats:tex-math>\n \n \n 1<\/mml:mn>\n +<\/mml:mo>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n -<\/mml:mo>\n 1<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \n (<\/mml:mo>\n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n -<\/mml:mo>\n 2<\/mml:mn>\n )<\/mml:mo>\n <\/mml:mrow>\n \/<\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, which is equal to 4 for \n \n $$n_h=4.$$<\/jats:tex-math>\n \n \n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n 4<\/mml:mn>\n .<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> Algebraic curve \n \n $$h(x,y)=0$$<\/jats:tex-math>\n \n \n h<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n y<\/mml:mi>\n )<\/mml:mo>\n =<\/mml:mo>\n 0<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> with \n \n $$n_h=4$$<\/jats:tex-math>\n \n \n \n n<\/mml:mi>\n h<\/mml:mi>\n <\/mml:msub>\n =<\/mml:mo>\n 4<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula> and the maximum number of four ovals is used to construct a chemical system which has four stable algebraic limit cycles.<\/jats:p>","DOI":"10.1007\/s00285-025-02221-0","type":"journal-article","created":{"date-parts":[[2025,5,22]],"date-time":"2025-05-22T08:47:05Z","timestamp":1747903625000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Planar chemical reaction systems with algebraic and non-algebraic limit cycles"],"prefix":"10.1007","volume":"90","author":[{"given":"Gheorghe","family":"Craciun","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8470-3763","authenticated-orcid":false,"given":"Radek","family":"Erban","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,5,22]]},"reference":[{"key":"2221_CR1","doi-asserted-by":"crossref","first-page":"398","DOI":"10.3166\/ejc.15.398-406","volume":"15","author":"D Angeli","year":"2009","unstructured":"Angeli D (2009) A tutorial on chemical reaction network dynamics. Eur J Control 15:398\u2013406","journal-title":"Eur J Control"},{"key":"2221_CR2","doi-asserted-by":"crossref","unstructured":"Banaji M, Boros B, Hofbauer J (2024) Oscillations in three-reaction quadratic mass-action systems. Stud Appl Math 152(1):249\u2013278","DOI":"10.1111\/sapm.12639"},{"key":"2221_CR3","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF02403068","volume":"24","author":"I Bendixson","year":"1901","unstructured":"Bendixson I (1901) Sur les courbes d\u00e9finies par des \u00e9quations diff\u00e9rentielles. Acta Math 24:1\u201388","journal-title":"Acta Math"},{"issue":"42","key":"2221_CR4","first-page":"1","volume":"2022","author":"B Boros","year":"2022","unstructured":"Boros B, Hofbauer J (2022) Limit cycles in mass-conserving deficiency-one mass-action systems. Electron J Qualitative Theory Diff Equ 2022(42):1\u201318","journal-title":"Electron J Qualitative Theory Diff Equ"},{"key":"2221_CR5","volume":"72","author":"B Boros","year":"2023","unstructured":"Boros B, Hofbauer J (2023) Some minimal bimolecular mass-action systems with limit cycles. Nonlinear Anal: Real World Appl 72:103839","journal-title":"Nonlinear Anal: Real World Appl"},{"issue":"Suppl 1","key":"2221_CR6","doi-asserted-by":"crossref","first-page":"175","DOI":"10.1007\/s10884-021-10051-z","volume":"36","author":"B Boros","year":"2024","unstructured":"Boros B, Hofbauer J (2024) Oscillations in planar deficiency-one mass-action systems. J Dyn Diff Equ 36(Suppl 1):175\u2013197","journal-title":"J Dyn Diff Equ"},{"issue":"4","key":"2221_CR7","doi-asserted-by":"crossref","first-page":"1936","DOI":"10.1137\/19M1303034","volume":"80","author":"B Boros","year":"2020","unstructured":"Boros B, Craciun G, Yu P (2020) Weakly reversible mass-action systems with infinitely many positive steady states. SIAM J Appl Math 80(4):1936\u20131946","journal-title":"SIAM J Appl Math"},{"key":"2221_CR8","doi-asserted-by":"crossref","DOI":"10.1016\/j.chaos.2023.113937","volume":"175","author":"Y Carvalho","year":"2023","unstructured":"Carvalho Y, Da Cruz L, Gouveia L (2023) New lower bound for the Hilbert number in low degree Kolmogorov systems. Chaos, Solitons Fractals 175:113937","journal-title":"Chaos, Solitons Fractals"},{"key":"2221_CR9","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.jde.2004.01.003","volume":"200","author":"J Chavarriga","year":"2004","unstructured":"Chavarriga J, Llibre J, Sorolla J (2004) Algebraic limit cycles of degree 4 for quadratic systems. J Diff Equ 200:206\u2013244","journal-title":"J Diff Equ"},{"key":"2221_CR10","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1023\/A:1013171019668","volume":"88","author":"C Christopher","year":"2001","unstructured":"Christopher C (2001) Polynomial vector fields with prescribed algebraic limit cycles. Geometriae Dedicata 88:255\u2013258","journal-title":"Geometriae Dedicata"},{"key":"2221_CR11","volume-title":"Limit cycles of differential equations","author":"C Christopher","year":"2007","unstructured":"Christopher C, Li C (2007) Limit cycles of differential equations. Birkh\u00e4user, Basel"},{"issue":"1","key":"2221_CR12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1137\/17M1129076","volume":"3","author":"G Craciun","year":"2019","unstructured":"Craciun G (2019) Polynomial dynamical systems, reaction networks, and toric differential inclusions. SIAM J Appl Algebra Geom 3(1):87\u2013106","journal-title":"SIAM J Appl Algebra Geom"},{"issue":"1","key":"2221_CR13","doi-asserted-by":"crossref","first-page":"244","DOI":"10.1007\/s10910-007-9307-x","volume":"44","author":"G Craciun","year":"2008","unstructured":"Craciun G, Pantea C (2008) Identifiability of chemical reaction networks. J Math Chem 44(1):244\u2013259","journal-title":"J Math Chem"},{"issue":"1","key":"2221_CR14","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1137\/19M1244494","volume":"80","author":"G Craciun","year":"2020","unstructured":"Craciun G, Jin J, Yu P (2020) An efficient characterization of complex-balanced, detailed-balanced, and weakly reversible systems. SIAM J Appl Math 80(1):183\u2013205","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"2221_CR15","doi-asserted-by":"crossref","DOI":"10.1103\/PhysRevE.91.042111","volume":"91","author":"A Duncan","year":"2015","unstructured":"Duncan A, Liao S, Vejchodsk\u1ef3 T, Erban R, Grima R (2015) Noise-induced multistability in chemical systems: discrete versus continuum modeling. Phys Rev E 91(4):042111","journal-title":"Phys Rev E"},{"issue":"5","key":"2221_CR16","doi-asserted-by":"crossref","first-page":"865","DOI":"10.1017\/S0956792520000492","volume":"32","author":"G Enciso","year":"2021","unstructured":"Enciso G, Erban R, Kim J (2021) Identifiability of stochastically modelled reaction networks. Eur J Appl Math 32(5):865\u2013887","journal-title":"Eur J Appl Math"},{"key":"2221_CR17","volume-title":"Stochastic Modelling of Reaction-Diffusion Processes","author":"R Erban","year":"2020","unstructured":"Erban R, Chapman SJ (2020) Stochastic Modelling of Reaction-Diffusion Processes, vol 60. Cambridge University Press, Cambridge"},{"key":"2221_CR18","doi-asserted-by":"crossref","first-page":"76","DOI":"10.1007\/s11538-023-01170-3","volume":"85","author":"R Erban","year":"2023","unstructured":"Erban R, Kang H (2023) Chemical systems with limit cycles. Bull Math Biol 85:76","journal-title":"Bull Math Biol"},{"issue":"3","key":"2221_CR19","doi-asserted-by":"crossref","first-page":"984","DOI":"10.1137\/080731360","volume":"70","author":"R Erban","year":"2009","unstructured":"Erban R, Chapman SJ, Kevrekidis I, Vejchodsky T (2009) Analysis of a stochastic chemical system close to a SNIPER bifurcation of its mean-field model. SIAM J Appl Math 70(3):984\u20131016","journal-title":"SIAM J Appl Math"},{"issue":"4","key":"2221_CR20","first-page":"351","volume":"35","author":"C Escher","year":"1979","unstructured":"Escher C (1979) Models of chemical reaction systems with exactly evaluable limit cycle oscillations. Zeitschrift f\u00fcr Physik B Condensed Matter 35(4):351\u2013361","journal-title":"Zeitschrift f\u00fcr Physik B Condensed Matter"},{"issue":"1","key":"2221_CR21","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1007\/BF01295082","volume":"40","author":"C Escher","year":"1980","unstructured":"Escher C (1980) Global stability analysis of open two-variable quadratic mass-action systems with elliptical limit cycles. Zeitschrift f\u00fcr Physik B Condensed Matter 40(1):137\u2013141","journal-title":"Zeitschrift f\u00fcr Physik B Condensed Matter"},{"issue":"4","key":"2221_CR22","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1002\/bbpc.19800840419","volume":"84","author":"C Escher","year":"1980","unstructured":"Escher C (1980) Models of chemical reaction systems with exactly evaluable limit cycle oscillations and their bifurcation behaviour. Berichte der Bunsengesellschaft f\u00fcr physikalische Chemie 84(4):387\u2013391","journal-title":"Berichte der Bunsengesellschaft f\u00fcr physikalische Chemie"},{"issue":"3","key":"2221_CR23","doi-asserted-by":"crossref","first-page":"337","DOI":"10.1016\/0301-0104(81)87009-7","volume":"63","author":"C Escher","year":"1981","unstructured":"Escher C (1981) Bifurcation and coexistence of several limit cycles in models of open two-variable quadratic mass-action systems. Chem Phys 63(3):337\u2013348","journal-title":"Chem Phys"},{"key":"2221_CR24","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1007\/BF00255665","volume":"49","author":"M Feinberg","year":"1972","unstructured":"Feinberg M (1972) Complex balancing in general kinetic systems. Arch Ration Mech Anal 49:187\u2013194","journal-title":"Arch Ration Mech Anal"},{"key":"2221_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-030-03858-8","volume-title":"Foundations of chemical reaction network theory","author":"M Feinberg","year":"2019","unstructured":"Feinberg M (2019) Foundations of chemical reaction network theory. Springer, New York"},{"issue":"1","key":"2221_CR26","first-page":"79","volume":"17","author":"D Frank-Kamenetsky","year":"1943","unstructured":"Frank-Kamenetsky D, Salnikov I (1943) On the possibility of auto-oscillation in homogeneous chemical systems with quadratic autocatalysis. Zhurnal Fizicheskoi Khimii (J Phys Chem) 17(1):79\u201386","journal-title":"Zhurnal Fizicheskoi Khimii (J Phys Chem)"},{"key":"2221_CR27","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1007\/s12346-023-00746-7","volume":"22","author":"A Gasull","year":"2023","unstructured":"Gasull A, Giacomini H (2023) Number of limit cycles for planar systems with invariant algebraic curves. Qualitative Theory Dyn Syst 22:44","journal-title":"Qualitative Theory Dyn Syst"},{"key":"2221_CR28","doi-asserted-by":"publisher","DOI":"10.1090\/proc\/17116","author":"A Gasull","year":"2024","unstructured":"Gasull A, Santana P (2024) A note on Hilbert 16th problem. Proceedings of the American Mathematical Society. https:\/\/doi.org\/10.1090\/proc\/17116","journal-title":"Proceedings of the American Mathematical Society"},{"issue":"25","key":"2221_CR29","doi-asserted-by":"crossref","first-page":"2340","DOI":"10.1021\/j100540a008","volume":"81","author":"D Gillespie","year":"1977","unstructured":"Gillespie D (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340\u20132361","journal-title":"J Phys Chem"},{"issue":"1","key":"2221_CR30","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1515\/ans-2017-6033","volume":"18","author":"J Gin\u00e9","year":"2018","unstructured":"Gin\u00e9 J, Llibre J, Valls C (2018) The cubic polynomial differential systems with two circles as algebraic limit cycles. Adv Nonlinear Stud 18(1):183\u2013193","journal-title":"Adv Nonlinear Stud"},{"key":"2221_CR31","first-page":"363","volume":"30","author":"V H\u00e1rs","year":"1981","unstructured":"H\u00e1rs V, T\u00f3th J (1981) On the inverse problem of reaction kinetics. Qualitative Theory Diff Equ 30:363\u2013379","journal-title":"Qualitative Theory Diff Equ"},{"key":"2221_CR32","doi-asserted-by":"crossref","first-page":"3609","DOI":"10.1016\/j.jde.2009.01.038","volume":"246","author":"C Li","year":"2009","unstructured":"Li C, Liu C, Yang J (2009) A cubic system with thirteen limit cycles. J Diff Equ 246:3609\u20133619","journal-title":"J Diff Equ"},{"issue":"6","key":"2221_CR33","doi-asserted-by":"crossref","first-page":"1401","DOI":"10.1016\/j.jde.2009.11.023","volume":"248","author":"J Llibre","year":"2010","unstructured":"Llibre J, Ram\u00edrez R, Sadovskaia N (2010) On the 16th Hilbert problem for algebraic limit cycles. J Diff Equ 248(6):1401\u20131409","journal-title":"J Diff Equ"},{"issue":"3\u20134","key":"2221_CR34","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1016\/S0898-1221(02)00161-X","volume":"44","author":"N Lloyd","year":"2002","unstructured":"Lloyd N, Pearson J, S\u00e1ez E, Sz\u00e1nt\u00f3 I (2002) A cubic Kolmogorov system with six limit cycles. Comput Math Appl 44(3\u20134):445\u2013455","journal-title":"Comput Math Appl"},{"key":"2221_CR35","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/j.physd.2005.07.014","volume":"210","author":"C Muratov","year":"2005","unstructured":"Muratov C, Vanden-Eijnden E, E W (2005) Self-induced stochastic resonance in excitable systems. Physica D 210:227\u2013240","journal-title":"Physica D"},{"issue":"10","key":"2221_CR36","doi-asserted-by":"crossref","first-page":"1658","DOI":"10.3390\/math8101658","volume":"8","author":"I Nagy","year":"2020","unstructured":"Nagy I, Romanovski VG, T\u00f3th J (2020) Two nested limit cycles in two-species reactions. Mathematics 8(10):1658","journal-title":"Mathematics"},{"key":"2221_CR37","volume-title":"Differential Equations and Dynamical Systems","author":"L Perko","year":"2013","unstructured":"Perko L (2013) Differential Equations and Dynamical Systems, vol 7. Springer, New York"},{"key":"2221_CR38","unstructured":"Plesa T (2024) Mapping dynamical systems into chemical reactions. arXiv:2406.03473"},{"issue":"10","key":"2221_CR39","doi-asserted-by":"crossref","first-page":"1884","DOI":"10.1007\/s10910-016-0656-1","volume":"54","author":"T Plesa","year":"2016","unstructured":"Plesa T, Vejchodsk\u00fd T, Erban R (2016) Chemical reaction systems with a homoclinic bifurcation: an inverse problem. J Math Chem 54(10):1884\u20131915","journal-title":"J Math Chem"},{"issue":"144","key":"2221_CR40","doi-asserted-by":"crossref","first-page":"20180199","DOI":"10.1098\/rsif.2018.0199","volume":"15","author":"T Plesa","year":"2018","unstructured":"Plesa T, Zygalakis K, Anderson D, Erban R (2018) Noise control for molecular computing. J Royal Soc Interface 15(144):20180199","journal-title":"J Royal Soc Interface"},{"issue":"5","key":"2221_CR41","doi-asserted-by":"crossref","first-page":"887","DOI":"10.1017\/S0956792518000517","volume":"30","author":"T Plesa","year":"2019","unstructured":"Plesa T, Erban R, Othmer H (2019) Noise-induced mixing and multimodality in reaction networks. Eur J Appl Math 30(5):887\u2013911","journal-title":"Eur J Appl Math"},{"key":"2221_CR42","doi-asserted-by":"crossref","unstructured":"Plesa T, Vejchodsk\u00fd T, Erban R (2017) Test models for statistical inference: Two-dimensional reaction systems displaying limit cycle bifurcations and bistability. In: Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology, pp. 3\u201327. Springer","DOI":"10.1007\/978-3-319-62627-7_1"},{"key":"2221_CR43","doi-asserted-by":"crossref","first-page":"1621","DOI":"10.1063\/1.444861","volume":"78","author":"G P\u00f3ta","year":"1983","unstructured":"P\u00f3ta G (1983) Two-component bimolecular systems cannot have limit cycles: a complete proof. J Chem Phys 78:1621\u20131622","journal-title":"J Chem Phys"},{"issue":"1","key":"2221_CR44","doi-asserted-by":"crossref","first-page":"331","DOI":"10.1088\/1361-6544\/aae94d","volume":"32","author":"R Prohens","year":"2018","unstructured":"Prohens R, Torregrosa J (2018) New lower bounds for the Hilbert numbers using reversible centers. Nonlinearity 32(1):331","journal-title":"Nonlinearity"},{"key":"2221_CR45","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1007\/BF01012599","volume":"13","author":"V Reddy","year":"1975","unstructured":"Reddy V (1975) On the existence of the steady state in the stochastic Volterra-Lotka model. J Stat Phys 13:61\u201364","journal-title":"J Stat Phys"},{"key":"2221_CR46","doi-asserted-by":"crossref","first-page":"222","DOI":"10.1016\/S0007-4497(03)00019-8","volume":"127","author":"B Schuman","year":"2003","unstructured":"Schuman B, T\u00f3th J (2003) No limit cycle in two species second order kinetics. Bull des Sci Math 127:222\u2013230","journal-title":"Bull des Sci Math"},{"issue":"2","key":"2221_CR47","first-page":"153","volume":"23","author":"S Shi","year":"1980","unstructured":"Shi S (1980) A concrete example of the existence of four limit cycles for plane quadratic systems. Sci Sinica 23(2):153\u2013158","journal-title":"Sci Sinica"},{"key":"2221_CR48","volume-title":"Differential equations, dynamical systems, and linear algebra","author":"S Smale","year":"1974","unstructured":"Smale S, Hirsch M (1974) Differential equations, dynamical systems, and linear algebra, vol 60. Academic Press, New York"},{"issue":"8","key":"2221_CR49","doi-asserted-by":"crossref","first-page":"4164","DOI":"10.1063\/1.1680609","volume":"59","author":"JJ Tyson","year":"1973","unstructured":"Tyson JJ, Light JC (1973) Properties of two-component bimolecular and trimolecular chemical reaction systems. J Chem Phys 59(8):4164\u20134173","journal-title":"J Chem Phys"},{"issue":"6\u20137","key":"2221_CR50","first-page":"733","volume":"58","author":"PY Yu","year":"2018","unstructured":"Yu PY, Craciun G (2018) Mathematical analysis of chemical reaction systems. Israel J Chem 58(6\u20137):733\u2013741","journal-title":"Israel J Chem"}],"container-title":["Journal of Mathematical Biology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02221-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00285-025-02221-0\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00285-025-02221-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,31]],"date-time":"2025-05-31T07:38:47Z","timestamp":1748677127000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00285-025-02221-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,5,22]]},"references-count":50,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2025,6]]}},"alternative-id":["2221"],"URL":"https:\/\/doi.org\/10.1007\/s00285-025-02221-0","relation":{},"ISSN":["0303-6812","1432-1416"],"issn-type":[{"value":"0303-6812","type":"print"},{"value":"1432-1416","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,5,22]]},"assertion":[{"value":"7 June 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 February 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"11 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 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":"Authors have no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}},{"value":"Not applicable.","order":3,"name":"Ethics","group":{"name":"EthicsHeading","label":"Ethical approval"}}],"article-number":"64"}}