American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Refined blow-up results for nonlinear fourth order differential equations
  • CPAA Home
  • This Issue
  • Next Article
    Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain
March  2015, 14(2): 657-676. doi: 10.3934/cpaa.2015.14.657

Hopf bifurcation in an age-structured population model with two delays

Xianlong Fu 1, , Zhihua Liu 2, and  Pierre Magal 3,

1. 

Department of Mathematics, Shanghai Key Laboratory of PMMP, East China Normal University, Shanghai, 200241
2. 

School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China
3. 

Institut de Mathématiques de Bordeaux, UMR CNRS 5251, INRIA Bordeaux sud-ouest, EPI Anubis, UFR Sciences de la Vie, Université Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux

Received  March 2014 Revised  August 2014 Published  December 2014

This paper is devoted to the study of an age-structured population system with Riker type birth function. Two time lag factors is considered for the model. One lag lies in the birth process and the another is in the birth function. We investigate some dynamical properties of the equation by using integrated semigroup theory, through which we obtain some conditions of asymptotical stability and Hopf bifurcation occurring at positive steady state for the system. The obtained results show how the two delays affect these dynamical properties.
Keywords: integrated semigroup, asymptotical stability, Population system, Hopf bifurcation., age-structured.
Mathematics Subject Classification: 34G20, 37G15, 35B32, 35K55, 92D2.
Citation: Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657
References:
[1]

A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: 10.1137/080723673.  Google Scholar

[2]

L. J. S. Allen and D. B. Thrasher, The effects of vaccination in an age-dependent model for varicella and herpes zoster, IEEE Trans. Autom. Contr., 43 (1998), 779-789. Google Scholar

[3]

S. Anita, Analysis and Control of Age-dependent Population Dynamics, Kluwer Academic Publishers, Netherlands, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar

[4]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[6]

O. Arino, A survey of structured cell population dynamics, Acta Bio., 43 (1995), 3-25. Google Scholar

[7]

O. Arino and E. Sanchez, A survey of cellpopulation dynamics, J. Theor. Med., 1 (1997), 35-51. Google Scholar

[8]

D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons, J. Frank. Inst., 297 (1974), 345-376. Google Scholar

[9]

A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model, SIAM J. Appl. Math., 59 (1999), 1667-1685. doi: 10.1137/S0036139997331239.  Google Scholar

[10]

A. Calsina and M. Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452. doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar

[11]

J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Diff. Eq., 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003.  Google Scholar

[12]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023.  Google Scholar

[13]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar

[14]

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.  Google Scholar

[15]

A. Ducrotc, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial diffrential equations with time delay, Fields Inst. Comm., 64 (2013), 353-390. doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

[16]

K-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[17]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[18]

B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton, Cambridge Univ. Press, Cambridge, 1981.  Google Scholar

[19]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[20]

Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel. J. Math., 70 (1990), 1-47. doi: 10.1007/BF02807217.  Google Scholar

[21]

Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift fur angewandte Math. Phys., 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x.  Google Scholar

[22]

P. Magal, Perturbation of a globally stable steady state and uniform persistence, J. Dynam. Diff. Eq., 21 (2009), 1-20. doi: 10.1007/s10884-008-9127-0.  Google Scholar

[23]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Diff. Equ., 2001 (2001), 1-35.  Google Scholar

[24]

P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces, Diff. Int. Eq., 20 (2007), 197-239.  Google Scholar

[25]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Diff. Equ., 14 (2009), 1041-1084.  Google Scholar

[26]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), No. 951. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[27]

P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proceedings of Royal Society A: Mathematical, Physical and Engineering Sciences, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435.  Google Scholar

[28]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[29]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[30]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623. Google Scholar

[31]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can, 191 (1975). Google Scholar

[32]

Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574. doi: 10.1007/s00285-010-0381-5.  Google Scholar

[33]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Int. Equ., 3(1990), 1035-1066.  Google Scholar

[34]

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.  Google Scholar

[35]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.  Google Scholar

show all references

References:
[1]

A. S. Ackleh and K. Deng, A nonautonomous juvenile-adult model: Well-posedness and long-time behavior via a comparison principle, SIAM J. Appl. Math., 69 (2009), 1644-1661. doi: 10.1137/080723673.  Google Scholar

[2]

L. J. S. Allen and D. B. Thrasher, The effects of vaccination in an age-dependent model for varicella and herpes zoster, IEEE Trans. Autom. Contr., 43 (1998), 779-789. Google Scholar

[3]

S. Anita, Analysis and Control of Age-dependent Population Dynamics, Kluwer Academic Publishers, Netherlands, 2000. doi: 10.1007/978-94-015-9436-3.  Google Scholar

[4]

W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144.  Google Scholar

[5]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Birkhäuser, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.  Google Scholar

[6]

O. Arino, A survey of structured cell population dynamics, Acta Bio., 43 (1995), 3-25. Google Scholar

[7]

O. Arino and E. Sanchez, A survey of cellpopulation dynamics, J. Theor. Med., 1 (1997), 35-51. Google Scholar

[8]

D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons, J. Frank. Inst., 297 (1974), 345-376. Google Scholar

[9]

A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model, SIAM J. Appl. Math., 59 (1999), 1667-1685. doi: 10.1137/S0036139997331239.  Google Scholar

[10]

A. Calsina and M. Sanchón, Stability and instability of equilibria of an equation of size structured population dynamics, J. Math. Anal. Appl., 286 (2003), 435-452. doi: 10.1016/S0022-247X(03)00464-5.  Google Scholar

[11]

J. Chu, A. Ducrot, P. Magal and S. Ruan, Hopf bifurcation in a size-structured population dynamic model with random growth, J. Diff. Eq., 247 (2009), 956-1000. doi: 10.1016/j.jde.2009.04.003.  Google Scholar

[12]

J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023.  Google Scholar

[13]

J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005.  Google Scholar

[14]

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.  Google Scholar

[15]

A. Ducrotc, P. Magal and S. Ruan, Projectors on the generalized eigenspaces for partial diffrential equations with time delay, Fields Inst. Comm., 64 (2013), 353-390. doi: 10.1007/978-1-4614-4523-4_14.  Google Scholar

[16]

K-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.  Google Scholar

[17]

J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[18]

B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton, Cambridge Univ. Press, Cambridge, 1981.  Google Scholar

[19]

H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X.  Google Scholar

[20]

Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel. J. Math., 70 (1990), 1-47. doi: 10.1007/BF02807217.  Google Scholar

[21]

Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems, Zeitschrift fur angewandte Math. Phys., 62 (2011), 191-222. doi: 10.1007/s00033-010-0088-x.  Google Scholar

[22]

P. Magal, Perturbation of a globally stable steady state and uniform persistence, J. Dynam. Diff. Eq., 21 (2009), 1-20. doi: 10.1007/s10884-008-9127-0.  Google Scholar

[23]

P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Diff. Equ., 2001 (2001), 1-35.  Google Scholar

[24]

P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces, Diff. Int. Eq., 20 (2007), 197-239.  Google Scholar

[25]

P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Diff. Equ., 14 (2009), 1041-1084.  Google Scholar

[26]

P. Magal and S. Ruan, Center manifold theorem for semilinear equations with non-dense domain and applications to Hopf bifurcation in age structured models, Mem. Amer. Math. Soc., 202 (2009), No. 951. doi: 10.1090/S0065-9266-09-00568-7.  Google Scholar

[27]

P. Magal and S. Ruan, Sustained oscillations in an evolutionary epidemiological model of influenza A drift, Proceedings of Royal Society A: Mathematical, Physical and Engineering Sciences, 466 (2010), 965-992. doi: 10.1098/rspa.2009.0435.  Google Scholar

[28]

J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations, Springer, Berlin, 1986. doi: 10.1007/978-3-662-13159-6.  Google Scholar

[29]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar

[30]

W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623. Google Scholar

[31]

W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can, 191 (1975). Google Scholar

[32]

Y. Su, S. Ruan and J. Wei, Periodicity and synchronization in blood-stage malaria infection, J. Math. Biol., 63 (2011), 557-574. doi: 10.1007/s00285-010-0381-5.  Google Scholar

[33]

H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Int. Equ., 3(1990), 1035-1066.  Google Scholar

[34]

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.  Google Scholar

[35]

G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.  Google Scholar

[1]

Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046
[2]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[3]

Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264
[4]

Guangrui Li, Ming Mei, Yau Shu Wong. Nonlinear stability of traveling wavefronts in an age-structured reaction-diffusion population model. Mathematical Biosciences & Engineering, 2008, 5 (1) : 85-100. doi: 10.3934/mbe.2008.5.85
[5]

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
[6]

Yingli Pan, Ying Su, Junjie Wei. Bistable waves of a recursive system arising from seasonal age-structured population models. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 511-528. doi: 10.3934/dcdsb.2018184
[7]

Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195
[8]

Z.-R. He, M.-S. Wang, Z.-E. Ma. Optimal birth control problems for nonlinear age-structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 589-594. doi: 10.3934/dcdsb.2004.4.589
[9]

Diène Ngom, A. Iggidir, Aboudramane Guiro, Abderrahim Ouahbi. An observer for a nonlinear age-structured model of a harvested fish population. Mathematical Biosciences & Engineering, 2008, 5 (2) : 337-354. doi: 10.3934/mbe.2008.5.337
[10]

Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409
[11]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[12]

Piotr Gwiazda, Karolina Kropielnicka, Anna Marciniak-Czochra. The Escalator Boxcar Train method for a system of age-structured equations. Networks & Heterogeneous Media, 2016, 11 (1) : 123-143. doi: 10.3934/nhm.2016.11.123
[13]

Peixuan Weng. Spreading speed and traveling wavefront of an age-structured population diffusing in a 2D lattice strip. Discrete & Continuous Dynamical Systems - B, 2009, 12 (4) : 883-904. doi: 10.3934/dcdsb.2009.12.883
[14]

Sebastian Aniţa, Ana-Maria Moşsneagu. Optimal harvesting for age-structured population dynamics with size-dependent control. Mathematical Control & Related Fields, 2019, 9 (4) : 607-621. doi: 10.3934/mcrf.2019043
[15]

Frédérique Billy, Jean Clairambault, Franck Delaunay, Céline Feillet, Natalia Robert. Age-structured cell population model to study the influence of growth factors on cell cycle dynamics. Mathematical Biosciences & Engineering, 2013, 10 (1) : 1-17. doi: 10.3934/mbe.2013.10.1
[16]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[17]

Ming Mei, Yau Shu Wong. Novel stability results for traveling wavefronts in an age-structured reaction-diffusion equation. Mathematical Biosciences & Engineering, 2009, 6 (4) : 743-752. doi: 10.3934/mbe.2009.6.743
[18]

Yicang Zhou, Paolo Fergola. Dynamics of a discrete age-structured SIS models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 841-850. doi: 10.3934/dcdsb.2004.4.841
[19]

Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641
[20]

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

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (93)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences