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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 |
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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