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Takens–Bogdanov singularity for age structured models

Research was partially supported by NSFC, the Fundamental Research Funds for the Central Universities, and Laboratory of Mathematics and Complex Systems, Ministry of Education

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  • The main purpose of this article is to derive a easily feasible method for the determination of Takens–Bogdanov singularity in age structured models. We present a SIR epidemic model with age structure as an example to illustrate the theoretical results.

    Mathematics Subject Classification: 34K18, 35K90, 37L10, 37G10.


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  • [1] V. I. Arnol'd, Geometrical Methods in the Theory of Ordinary Differential Equations, Grundlehren der Mathematischen Wissenschaften, 250, Springer-Verlag, New York-Berlin, 1983.
    [2] R. I. Bogdanov, Bifurcations of a limit cycle for a family of vector fields on the plane, Selecta Math. Soviet, 1 (1981), 373-388. 
    [3] R. I. Bogdanov, Versal deformations of a singular point on the plane in the case of zero eigenvalues, Funct. Anal. i Priloežn, 9 (1975), 63. 
    [4] J. Z. Cao and R. Yuan, Bogdanov-Takens bifurcation for neutral functional differential equations, Electronic Journal of Differential Equations, 2013 (2013), 12 pp.
    [5] S.-N. Chow and J. K. Hale, Methods of Bifurcation Theory, Grundlehren der Mathematischen Wissenschaften, 251, Springer-Verlag, New York-Berlin, 1982.
    [6] S.-N. ChowC. Z. Li and  D. WangNormal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
    [7] J. X. ChuA. DucrotP. Magal and S. G. Ruan, Hopf bifurcation in a size structured population dynamic model with random growth, J. Differ. Equ., 247 (2009), 956-1000.  doi: 10.1016/j.jde.2009.04.003.
    [8] J. X. Chu and P. Magal, Hopf bifurcation for a size structured model with resting phase, Discrete Contin. Dyn. Syst., 33 (2013), 4891-4921.  doi: 10.3934/dcds.2013.33.4891.
    [9] J. X. ChuP. Magal and R. Yuan, Hopf bifurcation for a maturity structured population dynamic model, J. Nonlinear Sci., 21 (2011), 521-562.  doi: 10.1007/s00332-010-9091-9.
    [10] J. M. Cushing, An Introduction to Structured Population Dynamics, CBMS-NSF Regional Conference Series in Applied Mathematics, 71, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.
    [11] A. DucrotZ. Liu and P. Magal, Essential growth rate for bounded linear perturbation of non densely defined Cauchy problems, J. Math. Anal. Appl., 341 (2008), 501-518.  doi: 10.1016/j.jmaa.2007.09.074.
    [12] F. Dumortier, R. Roussarie, J. Sotomayor and H. Zoladek, Bifurcations of Planar Vector Fields: Nilpotent Singularities and Abelian Integrals, Lecture Notes in Math, 1480, Springer-Verlag, Berlin, 1991. doi: 10.1007/BFb0098353.
    [13] T. Faria, Bifurcation aspects for some delayed population models with diffusion, in Differential Equations with Applications to Biology, Fields Inst. Commun., Amer. Math. Soc., Providence, RI, 21 (1999), 143–158.
    [14] T. Faria, Normal form and Hopf bifurcation for partial differential equations with delays, Transactions of the American Mathematical Society, 352 (2000), 2217-2238.  doi: 10.1090/S0002-9947-00-02280-7.
    [15] T. Faria, Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 254 (2001), 433-463.  doi: 10.1006/jmaa.2000.7182.
    [16] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Applied Mathematical Sciences, 42, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-1140-2.
    [17] J. K. Hale, L. T. Magalh$\widetilde{a}$es and W. M. Oliva, Dynamics in Infinite Dimensions, Applied Math. Sciences, 47, Springer-Verlag, New York, 2002. doi: 10.1007/b100032.
    [18] M. Iannelli, Mathematical Theory of Age-Structured Population Dynamics, Appl. Math. Monographs C. N. R., Vol. 7, Giadini Editori e Stampatori, Pisa, 1994.
    [19] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998.
    [20] W. M. LiuH. W. Hethcote and S. A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol., 25 (1987), 359-380.  doi: 10.1007/BF00277162.
    [21] W. M. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.
    [22] Z. H. Liu and N. W. Li, Stability and bifurcation in a predator-prey model with age structure and delays, J. Nonlinear Sci., 25 (2015), 937-957.  doi: 10.1007/s00332-015-9245-x.
    [23] Z. H. LiuP. Magal and S. G. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift fur Angewandte Mathematik und Physik, 62 (2011), 191-222.  doi: 10.1007/s00033-010-0088-x.
    [24] Z. H. LiuP. Magal and S. G. Ruan, Normal forms for semilinear equations with non-dense domain with applications to age structured models, J. Differential Equations, 257 (2014), 921-1011.  doi: 10.1016/j.jde.2014.04.018.
    [25] Z. H. LiuP. Magal and S. G. Ruan, Oscillations in age-structured models of consumer-resource mutualisms, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 537-555.  doi: 10.3934/dcdsb.2016.21.537.
    [26] Z. H. Liu, P. Magal and D. M. Xiao, Bogdanov-Takens bifurcation in a predator-prey model, Zeitschrift fur Angewandte Mathematik und Physik, 67 (2016), Art. 137, 29 pp. doi: 10.1007/s00033-016-0724-1.
    [27] Z. H. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Science China Mathematics, 60 (2017), 1371-1398.  doi: 10.1007/s11425-016-0371-8.
    [28] Z. H. Liu and R. Yuan, The effect of diffusion for a predator-prey system with nonmonotonic functional response, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 14 (2004), 4309-4316.  doi: 10.1142/S0218127404011867.
    [29] P. Magal and S. G. Ruan, Center manifolds for semilinear equations with non-dense domain and applications on Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009).  doi: 10.1090/S0065-9266-09-00568-7.
    [30] F. Takens, Forced oscillations and bifurcations, Comm. Math. Inst. Rijksuniv. Utrecht, Math. Inst. Rijksuniv. Utrecht, Utrecht, (1974), 1-59. 
    [31] F. Takens, Singularities of vector fields, Inst. Hautes Études Sci. Publ. Math., (1974), 47-100. 
    [32] H. Tang and Z. H. Liu, Hopf bifurcation for a predator-prey model with age structure, Applied Mathematical Modelling, 40 (2016), 726-737.  doi: 10.1016/j.apm.2015.09.015.
    [33] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066. 
    [34] H. R. Thieme, Differentiability of convolutions, integrated semigroups of bounded semi-variation, and the inhomogeneous Cauchy problem, J. Evol. Equ., 8 (2008), 283-305.  doi: 10.1007/s00028-007-0355-2.
    [35] H. R. Thieme, "Integrated semigroups" and integrated solutions to abstract Cauchy problems, J. Math. Anal. Appl., 152 (1990), 416-447.  doi: 10.1016/0022-247X(90)90074-P.
    [36] Z. Wang and Z. H. Liu, Hopf bifurcation of an age-structured compartmental pest-pathogen model, J. Math. Anal. Appl., 385 (2012), 1134-1150.  doi: 10.1016/j.jmaa.2011.07.038.
    [37] G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Monographs and Textbooks in Pure and Applied Mathematics, 89, Marcel Dekker, Inc., New York, 1985.
    [38] Y. X. Xu and M. Y. Huang, Homoclinic orbits and Hopf bifurcations in delay differential systems with T-B singularity, J. Differential Equations, 244 (2008), 582-598.  doi: 10.1016/j.jde.2007.09.003.
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