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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 & 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,, \emph{SIAM J. Appl. Math.}, 69 (2009), 1644. 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,, \emph{IEEE Trans. Autom. Contr.}, 43 (1998), 779. Google Scholar [3] S. Anita, Analysis and Control of Age-dependent Population Dynamics,, Kluwer Academic Publishers, (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. 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,, Birkh$\ddot{\mbox a}$user, (2001). doi: 10.1007/978-3-0348-5075-9. Google Scholar [6] O. Arino, A survey of structured cell population dynamics,, \emph{Acta Bio.}, 43 (1995), 3. Google Scholar [7] O. Arino and E. Sanchez, A survey of cellpopulation dynamics,, \emph{J. Theor. Med.}, 1 (1997), 35. Google Scholar [8] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons,, \emph{J. Frank. Inst.}, 297 (1974), 345. Google Scholar [9] A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model,, \emph{SIAM J. Appl. Math.}, 59 (1999), 1667. 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,, \emph{J. Math. Anal. Appl.}, 286 (2003), 435. 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,, \emph{J. Diff. Eq.}, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. Google Scholar [12] J. M. Cushing, The dynamics of hierarchical age-structured populations,, \emph{J. Math. Biol.}, 32 (1994), 705. doi: 10.1007/BF00163023. Google Scholar [13] J. M. Cushing, An Introduction to Structured Population Dynamics,, SIAM, (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,, \emph{J. Math. Anal. Appl.}, 341 (2008), 501. 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,, \emph{Fields Inst. Comm.}, 64 (2013), 353. 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, (2000). Google Scholar [17] J. K. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977). Google Scholar [18] B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton,, Cambridge Univ. Press, (1981). Google Scholar [19] H. Kellermann and M. Hieber, Integrated semigroups,, \emph{J. Funct. Anal.}, 84 (1989), 160. doi: 10.1016/0022-1236(89)90116-X. Google Scholar [20] Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states,, \emph{Israel. J. Math.}, 70 (1990), 1. doi: 10.1007/BF02807217. Google Scholar [21] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, \emph{Zeitschrift fur angewandte Math. Phys.}, 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. Google Scholar [22] P. Magal, Perturbation of a globally stable steady state and uniform persistence,, \emph{J. Dynam. Diff. Eq.}, 21 (2009), 1. doi: 10.1007/s10884-008-9127-0. Google Scholar [23] P. Magal, Compact attractors for time-periodic age structured population models,, \emph{Electron. J. Diff. Equ.}, 2001 (2001), 1. Google Scholar [24] P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces,, \emph{Diff. Int. Eq.}, 20 (2007), 197. Google Scholar [25] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, \emph{Adv. in Diff. Equ.}, 14 (2009), 1041. 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,, \emph{Mem. Amer. Math. Soc.}, 202 (2009). 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,, \emph{Proceedings of Royal Society A: Mathematical, 466 (2010), 965. doi: 10.1098/rspa.2009.0435. Google Scholar [28] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Springer, (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, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [30] W. E. Ricker, Stock and recruitment,, \emph{J. Fish. Res. Board Can.}, 11 (1954), 559. Google Scholar [31] W. E. Ricker, Computation and interpretation of biological studies of fish populations,, \emph{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,, \emph{J. Math. Biol.}, 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. Google Scholar [33] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, \emph{Diff. Int. Equ.}, 3 (1990), 1035. Google Scholar [34] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems,, \emph{J. Math. Anal. Appl.}, 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. Google Scholar [35] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (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,, \emph{SIAM J. Appl. Math.}, 69 (2009), 1644. 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,, \emph{IEEE Trans. Autom. Contr.}, 43 (1998), 779. Google Scholar [3] S. Anita, Analysis and Control of Age-dependent Population Dynamics,, Kluwer Academic Publishers, (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. 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,, Birkh$\ddot{\mbox a}$user, (2001). doi: 10.1007/978-3-0348-5075-9. Google Scholar [6] O. Arino, A survey of structured cell population dynamics,, \emph{Acta Bio.}, 43 (1995), 3. Google Scholar [7] O. Arino and E. Sanchez, A survey of cellpopulation dynamics,, \emph{J. Theor. Med.}, 1 (1997), 35. Google Scholar [8] D. M. Auslander, G. F. Oster and C. B. Huffaker, Dynamics of interacting populatons,, \emph{J. Frank. Inst.}, 297 (1974), 345. Google Scholar [9] A. Calsina and J. Saldana, Global dynamics and optimal life history of a structured population model,, \emph{SIAM J. Appl. Math.}, 59 (1999), 1667. 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,, \emph{J. Math. Anal. Appl.}, 286 (2003), 435. 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,, \emph{J. Diff. Eq.}, 247 (2009), 956. doi: 10.1016/j.jde.2009.04.003. Google Scholar [12] J. M. Cushing, The dynamics of hierarchical age-structured populations,, \emph{J. Math. Biol.}, 32 (1994), 705. doi: 10.1007/BF00163023. Google Scholar [13] J. M. Cushing, An Introduction to Structured Population Dynamics,, SIAM, (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,, \emph{J. Math. Anal. Appl.}, 341 (2008), 501. 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,, \emph{Fields Inst. Comm.}, 64 (2013), 353. 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, (2000). Google Scholar [17] J. K. Hale, Theory of Functional Differential Equations,, Springer-Verlag, (1977). Google Scholar [18] B. D. Hassard, N. D. Kazarinoff and Y. H.Wan, Theory and Applications of Hopf Bifurcaton,, Cambridge Univ. Press, (1981). Google Scholar [19] H. Kellermann and M. Hieber, Integrated semigroups,, \emph{J. Funct. Anal.}, 84 (1989), 160. doi: 10.1016/0022-1236(89)90116-X. Google Scholar [20] Y. Kifer, Principal eigenvalues, topological pressure, and stochastic stability of equilibrium states,, \emph{Israel. J. Math.}, 70 (1990), 1. doi: 10.1007/BF02807217. Google Scholar [21] Z. Liu, P. Magal and S. Ruan, Hopf bifurcation for non-densely defined Cauchy problems,, \emph{Zeitschrift fur angewandte Math. Phys.}, 62 (2011), 191. doi: 10.1007/s00033-010-0088-x. Google Scholar [22] P. Magal, Perturbation of a globally stable steady state and uniform persistence,, \emph{J. Dynam. Diff. Eq.}, 21 (2009), 1. doi: 10.1007/s10884-008-9127-0. Google Scholar [23] P. Magal, Compact attractors for time-periodic age structured population models,, \emph{Electron. J. Diff. Equ.}, 2001 (2001), 1. Google Scholar [24] P. Magal and S. Ruan, On integrated semigroups and age structured models in $L^p$ spaces,, \emph{Diff. Int. Eq.}, 20 (2007), 197. Google Scholar [25] P. Magal and S. Ruan, On semilinear Cauchy problems with non-dense domain,, \emph{Adv. in Diff. Equ.}, 14 (2009), 1041. 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,, \emph{Mem. Amer. Math. Soc.}, 202 (2009). 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,, \emph{Proceedings of Royal Society A: Mathematical, 466 (2010), 965. doi: 10.1098/rspa.2009.0435. Google Scholar [28] J. A. J. Metz and O. Diekmann, The Dynamics of Physiologically Structured Populations,, Springer, (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, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar [30] W. E. Ricker, Stock and recruitment,, \emph{J. Fish. Res. Board Can.}, 11 (1954), 559. Google Scholar [31] W. E. Ricker, Computation and interpretation of biological studies of fish populations,, \emph{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,, \emph{J. Math. Biol.}, 63 (2011), 557. doi: 10.1007/s00285-010-0381-5. Google Scholar [33] H. R. Thieme, Semiflows generated by Lipschitz perturbations of non-densely defined operators,, \emph{Diff. Int. Equ.}, 3 (1990), 1035. Google Scholar [34] H. R. Thieme, Integrated semigroups and integrated solutions to abstract Cauchy problems,, \emph{J. Math. Anal. Appl.}, 152 (1990), 416. doi: 10.1016/0022-247X(90)90074-P. Google Scholar [35] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics,, Marcell Dekker, (1985). Google Scholar
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