\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

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

Abstract Full Text(HTML) Related Papers Cited by
  • 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.

    Citation:

    \begin{equation} \\ \end{equation}
  • 加载中
  • [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.
  • 加载中
SHARE

Article Metrics

HTML views(1197) PDF downloads(516) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return