American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A two-group age of infection epidemic model with periodic behavioral changes
  • DCDS-B Home
  • This Issue
  • Next Article
    Poxvirus, red and grey squirrel dynamics: Is the recovery of a common predator affecting system equilibria? Insights from a predator-prey ecoepidemic model
June  2020, 25(6): 2041-2056. doi: 10.3934/dcdsb.2019201

Takens–Bogdanov singularity for age structured models

Zhihua Liu and  Rong Yuan

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China

Received  January 2019 Revised  March 2019 Published  June 2020 Early access  September 2019

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

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.

Keywords: Takens–Bogdanov singularity, age structured model, epidemic model.
Mathematics Subject Classification: 34K18, 35K90, 37L10, 37G10.
Citation: Zhihua Liu, Rong Yuan. Takens–Bogdanov singularity for age structured models. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2041-2056. doi: 10.3934/dcdsb.2019201
References:
[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. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[7]

J. X. Chu, A. Ducrot, P. 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. Chu, P. 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. Ducrot, Z. 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. Liu, H. 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. Liu, S. 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. Liu, P. 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. Liu, P. 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. Liu, P. 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.

show all references

References:
[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. Chow, C. Z. Li and D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, Cambridge, 1994.  doi: 10.1017/CBO9780511665639.
[7]

J. X. Chu, A. Ducrot, P. 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. Chu, P. 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. Ducrot, Z. 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. Liu, H. 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. Liu, S. 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. Liu, P. 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. Liu, P. 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. Liu, P. 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.

[1]

Min Lu, Chuang Xiang, Jicai Huang. Bogdanov-Takens bifurcation in a SIRS epidemic model with a generalized nonmonotone incidence rate. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3125-3138. doi: 10.3934/dcdss.2020115
[2]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[3]

Hao Kang, Qimin Huang, Shigui Ruan. Periodic solutions of an age-structured epidemic model with periodic infection rate. Communications on Pure and Applied Analysis, 2020, 19 (10) : 4955-4972. doi: 10.3934/cpaa.2020220
[4]

Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69
[5]

Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107
[6]

Dandan Sun, Yingke Li, Zhidong Teng, Tailei Zhang. Stability and Hopf bifurcation in an age-structured SIR epidemic model with relapse. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022141
[7]

Fadia Bekkal-Brikci, Khalid Boushaba, Ovide Arino. Nonlinear age structured model with cannibalism. Discrete and Continuous Dynamical Systems - B, 2007, 7 (2) : 201-218. doi: 10.3934/dcdsb.2007.7.201
[8]

Jicai Huang, Sanhong Liu, Shigui Ruan, Xinan Zhang. Bogdanov-Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1041-1055. doi: 10.3934/cpaa.2016.15.1041
[9]

Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva. An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences & Engineering, 2010, 7 (1) : 123-147. doi: 10.3934/mbe.2010.7.123
[10]

Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929
[11]

Yanxia Dang, Zhipeng Qiu, Xuezhi Li. Competitive exclusion in an infection-age structured vector-host epidemic model. Mathematical Biosciences & Engineering, 2017, 14 (4) : 901-931. doi: 10.3934/mbe.2017048
[12]

Majid Gazor, Mojtaba Moazeni. Parametric normal forms for Bogdanov--Takens singularity; the generalized saddle-node case. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 205-224. doi: 10.3934/dcds.2015.35.205
[13]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[14]

Fred Brauer. A model for an SI disease in an age - structured population. Discrete and Continuous Dynamical Systems - B, 2002, 2 (2) : 257-264. doi: 10.3934/dcdsb.2002.2.257
[15]

Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641
[16]

Janet Dyson, Eva Sánchez, Rosanna Villella-Bressan, Glenn F. Webb. An age and spatially structured model of tumor invasion with haptotaxis. Discrete and Continuous Dynamical Systems - B, 2007, 8 (1) : 45-60. doi: 10.3934/dcdsb.2007.8.45
[17]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2245-2270. doi: 10.3934/dcdsb.2019226
[18]

Xun Cao, Xianyong Chen, Weihua Jiang. Bogdanov-Takens bifurcation with $ Z_2 $ symmetry and spatiotemporal dynamics in diffusive Rosenzweig-MacArthur model involving nonlocal prey competition. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3747-3785. doi: 10.3934/dcds.2022031
[19]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[20]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete and Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (506)
  • HTML views (393)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy