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March  2015, 14(2): 657-676. doi: 10.3934/cpaa.2015.14.657

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

 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.
Citation: Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure and 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. [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. [3] S. Anita, Analysis and Control of Age-dependent Population Dynamics, Kluwer Academic Publishers, Netherlands, 2000. doi: 10.1007/978-94-015-9436-3. [4] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144. [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. [6] O. Arino, A survey of structured cell population dynamics, Acta Bio., 43 (1995), 3-25. [7] O. Arino and E. Sanchez, A survey of cellpopulation dynamics, J. Theor. Med., 1 (1997), 35-51. [8] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons, J. Frank. Inst., 297 (1974), 345-376. [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. [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. [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. [12] J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. [13] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [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. [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. [16] K-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [17] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [18] B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton, Cambridge Univ. Press, Cambridge, 1981. [19] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X. [20] Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel. J. Math., 70 (1990), 1-47. doi: 10.1007/BF02807217. [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. [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. [23] P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Diff. Equ., 2001 (2001), 1-35. [24] P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces, Diff. Int. Eq., 20 (2007), 197-239. [25] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Diff. Equ., 14 (2009), 1041-1084. [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. [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. [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. [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. [30] W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623. [31] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can, 191 (1975). [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. [33] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Int. Equ., 3(1990), 1035-1066. [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. [35] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.

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. [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. [3] S. Anita, Analysis and Control of Age-dependent Population Dynamics, Kluwer Academic Publishers, Netherlands, 2000. doi: 10.1007/978-94-015-9436-3. [4] W. Arendt, Vector valued Laplace transforms and Cauchy problems, Israel J. Math., 59 (1987), 327-352. doi: 10.1007/BF02774144. [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. [6] O. Arino, A survey of structured cell population dynamics, Acta Bio., 43 (1995), 3-25. [7] O. Arino and E. Sanchez, A survey of cellpopulation dynamics, J. Theor. Med., 1 (1997), 35-51. [8] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons, J. Frank. Inst., 297 (1974), 345-376. [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. [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. [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. [12] J. M. Cushing, The dynamics of hierarchical age-structured populations, J. Math. Biol., 32 (1994), 705-729. doi: 10.1007/BF00163023. [13] J. M. Cushing, An Introduction to Structured Population Dynamics, SIAM, Philadelphia, PA, 1998. doi: 10.1137/1.9781611970005. [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. [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. [16] K-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000. [17] J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. [18] B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton, Cambridge Univ. Press, Cambridge, 1981. [19] H. Kellermann and M. Hieber, Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180. doi: 10.1016/0022-1236(89)90116-X. [20] Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states, Israel. J. Math., 70 (1990), 1-47. doi: 10.1007/BF02807217. [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. [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. [23] P. Magal, Compact attractors for time-periodic age structured population models, Electron. J. Diff. Equ., 2001 (2001), 1-35. [24] P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces, Diff. Int. Eq., 20 (2007), 197-239. [25] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain, Adv. in Diff. Equ., 14 (2009), 1041-1084. [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. [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. [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. [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. [30] W. E. Ricker, Stock and recruitment, J. Fish. Res. Board Can., 11 (1954), 559-623. [31] W. E. Ricker, Computation and interpretation of biological studies of fish populations, Bull. Fish. Res. Board Can, 191 (1975). [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. [33] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators, Diff. Int. Equ., 3(1990), 1035-1066. [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. [35] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics, Marcell Dekker, New York, 1985.
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