{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T16:26:26Z","timestamp":1774542386901,"version":"3.50.1"},"reference-count":46,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T00:00:00Z","timestamp":1684281600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006752","name":"Universidade do Porto","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006752","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Qual. Theory Dyn. Syst."],"published-print":{"date-parts":[[2023,9]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of <jats:italic>Susceptible<\/jats:italic> individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a <jats:italic>codimension two singularity<\/jats:italic> in the parameter space <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\\mathcal {R}_0, p)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>R<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\mathcal {R}_0$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>R<\/mml:mi>\n                    <mml:mn>0<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> is the <jats:italic>basic reproduction number<\/jats:italic> and <jats:italic>p<\/jats:italic> is the proportion of <jats:italic>Susceptible<\/jats:italic> individuals successfully vaccinated at birth. We exhibit explicitly the <jats:italic>Hopf<\/jats:italic>, <jats:italic>transcritical<\/jats:italic>, <jats:italic>Belyakov<\/jats:italic>, <jats:italic>heteroclinic<\/jats:italic> and <jats:italic>saddle-node bifurcation<\/jats:italic> curves unfolding the <jats:italic>singularity<\/jats:italic>. The two parameters <jats:inline-formula><jats:alternatives><jats:tex-math>$$(\\mathcal {R}_0, p)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>R<\/mml:mi>\n                      <mml:mn>0<\/mml:mn>\n                    <\/mml:msub>\n                    <mml:mo>,<\/mml:mo>\n                    <mml:mi>p<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease persists and we illustrate our main result with numerical simulations, emphasizing the role of the parameters.\n<\/jats:p>","DOI":"10.1007\/s12346-023-00802-2","type":"journal-article","created":{"date-parts":[[2023,5,17]],"date-time":"2023-05-17T15:01:56Z","timestamp":1684335716000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["SIR Model with Vaccination: Bifurcation Analysis"],"prefix":"10.1007","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7709-1631","authenticated-orcid":false,"given":"Jo\u00e3o P. S.","family":"Maur\u00edcio de Carvalho","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8182-9889","authenticated-orcid":false,"given":"Alexandre A.","family":"Rodrigues","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,5,17]]},"reference":[{"key":"802_CR1","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4939-9828-9","volume-title":"Mathematical Models in Epidemiology. Texts in Applied Mathematics","author":"F Brauer","year":"2019","unstructured":"Brauer, F., Castillo-Chavez, C., Feng, Z.: Mathematical Models in Epidemiology. Texts in Applied Mathematics, vol. 69. Springer, New York (2019). https:\/\/doi.org\/10.1007\/978-1-4939-9828-9"},{"key":"802_CR2","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1137\/S0036144500371907","volume":"42","author":"HW Hethcote","year":"2000","unstructured":"Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev. 42, 599\u2013653 (2000). https:\/\/doi.org\/10.1137\/S0036144500371907","journal-title":"SIAM Rev."},{"key":"802_CR3","doi-asserted-by":"crossref","unstructured":"Bonyah, E.: Al Basir, F., Ray, S.: Hopf bifurcation in a mathematical model of tuberculosis with delay. In: Manchanda, P., Lozi, R., Siddiqi, A. (eds.) Mathematical Modelling, Optimization, Analytic and Numerical Solutions, in: Industrial and Applied Mathematics, pp. 301\u2013311. Springer, Singapore (2020)","DOI":"10.1007\/978-981-15-0928-5_14"},{"key":"802_CR4","doi-asserted-by":"publisher","first-page":"711","DOI":"10.1007\/s11071-020-05757-6","volume":"101","author":"K Rajagopal","year":"2020","unstructured":"Rajagopal, K., Hasanzadeh, N., Parastesh, F., Hamarash, I.I., Jafari, S., Hussain, I.: A fractional-order model for the novel coronavirus (COVID-19) outbreak. Nonlinear Dyn. 101, 711\u2013718 (2020). https:\/\/doi.org\/10.1007\/s11071-020-05757-6","journal-title":"Nonlinear Dyn."},{"key":"802_CR5","doi-asserted-by":"publisher","first-page":"713","DOI":"10.1126\/science.abb5659","volume":"368","author":"S Cobey","year":"2020","unstructured":"Cobey, S.: Modeling infectious disease dynamics. Science 368, 713\u2013714 (2020). https:\/\/doi.org\/10.1126\/science.abb5659","journal-title":"Science"},{"key":"802_CR6","doi-asserted-by":"publisher","first-page":"55","DOI":"10.1098\/rspa.1932.0171","volume":"138","author":"WO Kermack","year":"1932","unstructured":"Kermack, W.O., McKendrick, A.G.: Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc. R. Soc. Lond. 138, 55\u201383 (1932). https:\/\/doi.org\/10.1098\/rspa.1932.0171","journal-title":"Proc. R. Soc. Lond."},{"key":"802_CR7","doi-asserted-by":"publisher","unstructured":"Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. In: Mathematical Models in Medicine. Springer, Berlin, pp 1\u201315 (1976). https:\/\/doi.org\/10.1007\/978-3-642-93048-5_1","DOI":"10.1007\/978-3-642-93048-5_1"},{"key":"802_CR8","doi-asserted-by":"publisher","first-page":"471","DOI":"10.1007\/s002850050175","volume":"39","author":"FA Milner","year":"1999","unstructured":"Milner, F.A., Pugliese, A.: Periodic solutions: a robust numerical method for an S-I-R model of epidemics. J. Math. Biol. 39, 471\u2013492 (1999). https:\/\/doi.org\/10.1007\/s002850050175","journal-title":"J. Math. Biol."},{"key":"802_CR9","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/j.physd.2022.133268","volume":"434","author":"JPS Maur\u00edcio de Carvalho","year":"2022","unstructured":"Maur\u00edcio de Carvalho, J.P.S., Rodrigues, A.A.P.: Strange attractors in a dynamical system inspired by a seasonally forced SIR model. Phys. D 434, 12 (2022). https:\/\/doi.org\/10.1016\/j.physd.2022.133268","journal-title":"Phys. D"},{"key":"802_CR10","doi-asserted-by":"publisher","first-page":"1655","DOI":"10.1007\/s00285-017-1130-9","volume":"75","author":"PG Barrientos","year":"2017","unstructured":"Barrientos, P.G., Rodr\u00edguez, J.A., Ruiz-Herrera, A.: Chaotic dynamics in the seasonally forced SIR epidemic model. J. Math. Biol. 75, 1655\u20131668 (2017). https:\/\/doi.org\/10.1007\/s00285-017-1130-9","journal-title":"J. Math. Biol."},{"key":"802_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.chaos.2021.111275","volume":"151","author":"JPS Maur\u00edcio de Carvalho","year":"2021","unstructured":"Maur\u00edcio de Carvalho, J.P.S., Moreira-Pinto, B.: A fractional-order model for CoViD-19 dynamics with reinfection and the importance of quarantine. Chaos Solitons Fractals 151, 111275 (2021). https:\/\/doi.org\/10.1016\/j.chaos.2021.111275","journal-title":"Chaos Solitons Fractals"},{"key":"802_CR12","doi-asserted-by":"publisher","first-page":"128498","DOI":"10.1016\/j.physleta.2022.128498","volume":"454","author":"A d\u2019Onofrio","year":"2022","unstructured":"d\u2019Onofrio, A., Duarte, J., Janu\u00e1rio, C., Martins, N.: A SIR forced model with interplays with the external world and periodic internal contact interplays. Phys. Lett. A 454, 128498 (2022). https:\/\/doi.org\/10.1016\/j.physleta.2022.128498","journal-title":"Phys. Lett. A"},{"key":"802_CR13","doi-asserted-by":"publisher","first-page":"S5","DOI":"10.1038\/nm1209","volume":"11","author":"SA Plotkin","year":"2005","unstructured":"Plotkin, S.A.: Vaccines: past, present and future. Nat. Med. 11, S5\u2013S11 (2005). https:\/\/doi.org\/10.1038\/nm1209","journal-title":"Nat. Med."},{"key":"802_CR14","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1038\/nrmicro2668","volume":"9","author":"SA Plotkin","year":"2011","unstructured":"Plotkin, S.A., Plotkin, S.L.: The development of vaccines: how the past led to the future. Nat. Rev. Microbiol. 9, 889\u2013893 (2011). https:\/\/doi.org\/10.1038\/nrmicro2668","journal-title":"Nat. Rev. Microbiol."},{"key":"802_CR15","doi-asserted-by":"publisher","first-page":"842","DOI":"10.1016\/j.amc.2006.06.074","volume":"184","author":"OD Makinde","year":"2007","unstructured":"Makinde, O.D.: Adomian decomposition approach to a SIR epidemic model with constant vaccination strategy. Appl. Math. Comput. 184, 842\u2013848 (2007). https:\/\/doi.org\/10.1016\/j.amc.2006.06.074","journal-title":"Appl. Math. Comput."},{"key":"802_CR16","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1007\/s40435-022-00969-7","volume":"11","author":"P Saha","year":"2022","unstructured":"Saha, P., Ghosh, U.: Complex dynamics and control analysis of an epidemic model with non-monotone incidence and saturated treatment. Int. J. Dyn. Control 11, 301\u2013323 (2022). https:\/\/doi.org\/10.1007\/s40435-022-00969-7","journal-title":"Int. J. Dyn. Control"},{"key":"802_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s12591-019-00486-8","volume":"2019","author":"JK Ghosh","year":"2019","unstructured":"Ghosh, J.K., Ghosh, U., Biswas, M.H.A., Sarkar, S.: Qualitative analysis and optimal control strategy of an SIR model with saturated incidence and treatment. Differ. Equ. Dyn. Syst. 2019, 1\u201315 (2019). https:\/\/doi.org\/10.1007\/s12591-019-00486-8","journal-title":"Differ. Equ. Dyn. Syst."},{"key":"802_CR18","doi-asserted-by":"publisher","first-page":"1123","DOI":"10.1016\/S0092-8240(98)90005-2","volume":"60","author":"B Shulgin","year":"1998","unstructured":"Shulgin, B., Stone, L., Agur, Z.: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1123\u20131148 (1998). https:\/\/doi.org\/10.1016\/S0092-8240(98)90005-2","journal-title":"Bull. Math. Biol."},{"key":"802_CR19","doi-asserted-by":"publisher","first-page":"532","DOI":"10.1186\/s13662-019-2447-z","volume":"2019","author":"A Elazzouzi","year":"2019","unstructured":"Elazzouzi, A., Alaoui, A.L., Tilioua, M., Tridane, A.: Global stability analysis for a generalized delayed SIR model with vaccination and treatment. Adv. Differ. Equ. 2019, 532 (2019). https:\/\/doi.org\/10.1186\/s13662-019-2447-z","journal-title":"Adv. Differ. Equ."},{"key":"802_CR20","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/S0895-7177(00)00040-6","volume":"31","author":"L Stone","year":"2000","unstructured":"Stone, L., Shulgin, B., Agur, Z.: Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Model. 31, 207\u2013215 (2000). https:\/\/doi.org\/10.1016\/S0895-7177(00)00040-6","journal-title":"Math. Comput. Model."},{"key":"802_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cnsns.2022.106482","volume":"111","author":"A Algaba","year":"2022","unstructured":"Algaba, A., Dom\u00ednguez-Moreno, M.C., Merino, M., Rodr\u00edguez-Luis, A.J.: Double-zero degeneracy and heteroclinic cycles in a perturbation of the Lorenz system. Commun. Nonlinear Sci. Numer. Simul. 111, 106482 (2022). https:\/\/doi.org\/10.1016\/j.cnsns.2022.106482","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"802_CR22","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/s10955-021-02811-4","volume":"184","author":"AAP Rodrigues","year":"2021","unstructured":"Rodrigues, A.A.P.: Dissecting a resonance wedge on heteroclinic bifurcations. J. Stat. Phys. 184, 25 (2021). https:\/\/doi.org\/10.1007\/s10955-021-02811-4","journal-title":"J. Stat. Phys."},{"key":"802_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1006\/jdeq.2002.4177","volume":"185","author":"K Yagasaki","year":"2002","unstructured":"Yagasaki, K.: Melnikov\u2019s method and codimension-two bifurcations in forced oscillations. J. Differ. Equ. 185, 1\u201324 (2002). https:\/\/doi.org\/10.1006\/jdeq.2002.4177","journal-title":"J. Differ. Equ."},{"key":"802_CR24","doi-asserted-by":"publisher","first-page":"2235","DOI":"10.1007\/s00285-019-01342-7","volume":"78","author":"J Duarte","year":"2019","unstructured":"Duarte, J., Janu\u00e1rio, C., Martins, N., Rogovchenko, S., Rogovchenko, Y.: Chaos analysis and explicit series solutions to the seasonally forced SIR epidemic model. J. Math. Biol. 78, 2235\u20132258 (2019). https:\/\/doi.org\/10.1007\/s00285-019-01342-7","journal-title":"J. Math. Biol."},{"key":"802_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-3978-7","volume-title":"Elements of Applied Bifurcation Theory. Applied Mathematical Sciences","author":"YA Kuznetsov","year":"2004","unstructured":"Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory. Applied Mathematical Sciences, vol. 112. Springer, New York (2004). https:\/\/doi.org\/10.1007\/978-1-4757-3978-7"},{"key":"802_CR26","doi-asserted-by":"publisher","first-page":"1482","DOI":"10.1016\/j.chaos.2006.04.022","volume":"34","author":"Y Jin","year":"2007","unstructured":"Jin, Y., Wang, W., Xiao, S.: An SIRS model with a nonlinear incidence rate. Chaos Solitons Fractals 34, 1482\u20131497 (2007). https:\/\/doi.org\/10.1016\/j.chaos.2006.04.022","journal-title":"Chaos Solitons Fractals"},{"key":"802_CR27","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1016\/j.matcom.2023.01.023","volume":"208","author":"J Zhang","year":"2023","unstructured":"Zhang, J., Qiao, Y.: Bifurcation analysis of an SIR model considering hospital resources and vaccination. Math. Comput. Simul. 208, 157\u2013185 (2023). https:\/\/doi.org\/10.1016\/j.matcom.2023.01.023","journal-title":"Math. Comput. Simul."},{"key":"802_CR28","doi-asserted-by":"publisher","first-page":"1662","DOI":"10.1016\/j.jde.2014.05.030","volume":"257","author":"C Shan","year":"2014","unstructured":"Shan, C., Zhu, H.: Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds. J. Differ. Equ. 257, 1662\u20131688 (2014). https:\/\/doi.org\/10.1016\/j.jde.2014.05.030","journal-title":"J. Differ. Equ."},{"key":"802_CR29","doi-asserted-by":"publisher","first-page":"1794","DOI":"10.1137\/040604947","volume":"65","author":"ME Alexander","year":"2005","unstructured":"Alexander, M.E., Moghadas, S.M.: Bifurcation analysis of an SIRS epidemic model with generalized incidence. SIAM J. Appl. Math. 65, 1794\u20131816 (2005)","journal-title":"SIAM J. Appl. Math."},{"key":"802_CR30","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s00285-022-01787-3","volume":"85","author":"Q Pan","year":"2022","unstructured":"Pan, Q., Huang, J., Wang, H.: An SIRS model with nonmonotone incidence and saturated treatment in a changing environment. J. Math. Biol. 85, 23 (2022). https:\/\/doi.org\/10.1007\/s00285-022-01787-3","journal-title":"J. Math. Biol."},{"key":"802_CR31","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1186\/s13662-018-1675-y","volume":"2018","author":"J Li","year":"2018","unstructured":"Li, J., Teng, Z.: Bifurcations of an SIRS model with generalized non-monotone incidence rate. Adv. Differ. Equ. 2018, 1\u201321 (2018). https:\/\/doi.org\/10.1186\/s13662-018-1675-y","journal-title":"Adv. Differ. Equ."},{"key":"802_CR32","doi-asserted-by":"publisher","first-page":"11628","DOI":"10.3934\/mbe.2022541","volume":"19","author":"AK Misra","year":"2022","unstructured":"Misra, A.K., Maurya, J., Sajid, M.: Modeling the effect of time delay in the increment of number of hospital beds to control an infectious disease. Math. Biosci. Eng. 19, 11628\u201311656 (2022). https:\/\/doi.org\/10.3934\/mbe.2022541","journal-title":"Math. Biosci. Eng."},{"key":"802_CR33","doi-asserted-by":"publisher","first-page":"1859","DOI":"10.1016\/j.jde.2019.03.005","volume":"267","author":"M Lu","year":"2019","unstructured":"Lu, M., Huang, J., Ruan, S., Yu, P.: Bifurcation analysis of an SIRS epidemic model with a generalized nonmonotone and saturated incidence rate. J. Differ. Equ. 267, 1859\u20131898 (2019). https:\/\/doi.org\/10.1016\/j.jde.2019.03.005","journal-title":"J. Differ. Equ."},{"key":"802_CR34","doi-asserted-by":"publisher","first-page":"63","DOI":"10.1016\/j.chaos.2017.03.047","volume":"99","author":"J Li","year":"2017","unstructured":"Li, J., Teng, Z., Wang, G., Zhang, L., Hu, C.: Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment. Chaos Solitons Fractals 99, 63\u201371 (2017). https:\/\/doi.org\/10.1016\/j.chaos.2017.03.047","journal-title":"Chaos Solitons Fractals"},{"key":"802_CR35","doi-asserted-by":"publisher","first-page":"207","DOI":"10.1016\/S0895-7177(00)00040-6","volume":"31","author":"L Stone","year":"2000","unstructured":"Stone, L., Shulgin, B.: Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Model. 31, 207\u2013215 (2000). https:\/\/doi.org\/10.1016\/S0895-7177(00)00040-6","journal-title":"Math. Comput. Model."},{"key":"802_CR36","doi-asserted-by":"publisher","first-page":"1039","DOI":"10.1016\/S0895-7177(02)00257-1","volume":"36","author":"Z Lu","year":"2002","unstructured":"Lu, Z., Chi, X., Chen, L.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. Comput. Model. 36, 1039\u20131057 (2002). https:\/\/doi.org\/10.1016\/S0895-7177(02)00257-1","journal-title":"Math. Comput. Model."},{"key":"802_CR37","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/S0898-1221(99)00206-0","volume":"38","author":"XA Zhang","year":"1999","unstructured":"Zhang, X.A., Chen, L.: The periodic solution of a class of epidemic models. Comput. Math. Appl. 38, 61\u201371 (1999). https:\/\/doi.org\/10.1016\/S0898-1221(99)00206-0","journal-title":"Comput. Math. Appl."},{"key":"802_CR38","unstructured":"Jones, J.H.: Notes on $$\\cal{R}_0$$. California: Department of Anthropological Sciences 323, 19 pages (2007)"},{"key":"802_CR39","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s11538-020-00713-2","volume":"82","author":"SW Park","year":"2020","unstructured":"Park, S.W., Bolker, B.M.: A note on observation processes in epidemic models. Bull. Math. Biol. 82, 1\u20138 (2020)","journal-title":"Bull. Math. Biol."},{"key":"802_CR40","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1155\/2011\/527610","volume":"2011","author":"J Li","year":"2011","unstructured":"Li, J., Blakeley, D., Smith, R.J.: The failure of $$\\cal{R} _0$$. Comput. Math. Methods Med. 2011, 17 (2011). https:\/\/doi.org\/10.1155\/2011\/527610","journal-title":"Comput. Math. Methods Med."},{"key":"802_CR41","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1007\/s00220-003-0902-9","volume":"240","author":"Q Wang","year":"2003","unstructured":"Wang, Q., Young, L.S.: Strange attractors in periodically-kicked limit cycles and Hopf bifurcations. Commun. Math. Phys. 240, 509\u2013529 (2003). https:\/\/doi.org\/10.1007\/s00220-003-0902-9","journal-title":"Commun. Math. Phys."},{"key":"802_CR42","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-1140-2","volume-title":"Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences","author":"J Guckenheimer","year":"1983","unstructured":"Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, New York (1983). https:\/\/doi.org\/10.1007\/978-1-4612-1140-2"},{"key":"802_CR43","series-title":"Springer Undergraduate Texts in Mathematics and Technology (SUMAT)","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-71278-9","volume-title":"Modeling of Curves and Surfaces with MATLAB","author":"V Rovenski","year":"2010","unstructured":"Rovenski, V.: Modeling of Curves and Surfaces with MATLAB. Springer Undergraduate Texts in Mathematics and Technology (SUMAT), Springer, New York (2010)"},{"key":"802_CR44","doi-asserted-by":"publisher","first-page":"451","DOI":"10.1016\/j.mbs.2007.02.006","volume":"209","author":"GAK van Voorn","year":"2007","unstructured":"van Voorn, G.A.K., Hemerik, L., Boer, M.P., Kooi, B.W.: Heteroclinic orbits indicate overexploitation in predator\u2013prey systems with a strong Allee effect. Math. Biosci. 209, 451\u2013469 (2007). https:\/\/doi.org\/10.1016\/j.mbs.2007.02.006","journal-title":"Math. Biosci."},{"key":"802_CR45","doi-asserted-by":"publisher","first-page":"3449","DOI":"10.1016\/j.camwa.2011.08.061","volume":"62","author":"E Gonz\u00e1lez-Olivares","year":"2011","unstructured":"Gonz\u00e1lez-Olivares, E., Gonz\u00e1lez-Ya\u00f1ez, B., Lorca, J.M., Rojas-Palma, A., Flores, J.D.: Consequences of double Allee effect on the number of limit cycles in a predator\u2013prey model. Comput. Math. Appl. 62, 3449\u20133463 (2011). https:\/\/doi.org\/10.1016\/j.camwa.2011.08.061","journal-title":"Comput. Math. Appl."},{"key":"802_CR46","doi-asserted-by":"publisher","first-page":"1643","DOI":"10.1007\/s10884-020-09858-z","volume":"34","author":"AAP Rodrigues","year":"2022","unstructured":"Rodrigues, A.A.P.: Unfolding a Bykov attractor: from an attracting torus to strange attractors. J. Dyn. Differ. Equ. 34, 1643\u20131677 (2022). https:\/\/doi.org\/10.1007\/s10884-020-09858-z","journal-title":"J. Dyn. Differ. Equ."}],"container-title":["Qualitative Theory of Dynamical Systems"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12346-023-00802-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12346-023-00802-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12346-023-00802-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T12:28:36Z","timestamp":1687177716000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12346-023-00802-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,5,17]]},"references-count":46,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2023,9]]}},"alternative-id":["802"],"URL":"https:\/\/doi.org\/10.1007\/s12346-023-00802-2","relation":{},"ISSN":["1575-5460","1662-3592"],"issn-type":[{"value":"1575-5460","type":"print"},{"value":"1662-3592","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,5,17]]},"assertion":[{"value":"29 November 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"22 April 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 May 2023","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declaration"}},{"value":"The authors declare that they have no conflict of interest. Jo\u00e3o Carvalho and Alexandre Rodrigues have equally contributed to this work.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"105"}}