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Takens–Bogdanov singularity for age structured models
School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
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.
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. |
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