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Global dynamics of an age-structured model with relapse
Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions
1. | School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China |
2. | Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France |
3. | CNRS, IMB, UMR 5251, F-33400 Talence, France |
In this article we study a class of delay differential equations with infinite delay in weighted spaces of uniformly continuous functions. We focus on the integrated semigroup formulation of the problem and so doing we provide a spectral theory. As a consequence we obtain a local stability result and a Hopf bifurcation theorem for the semiflow generated by such a problem.
References:
[1] |
M. Adimy,
Bifurcation de Hopf locale par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 311 (1990), 423-428.
|
[2] |
M. Adimy,
Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177 (1993), 125-134.
doi: 10.1006/jmaa.1993.1247. |
[3] |
M. Adimy and O. Arino,
Bifurcation de Hopf globale pour des équations à retard par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 317 (1993), 767-772.
|
[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, Birkhä user, Basel, 2001. |
[6] |
O. Arino and E. Sanchez, A theory of linear delay differential equations in infinite dimensional spaces, Delay Differential Equations and Applications, 285–346, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_8. |
[7] |
P. Auger and A. Ducrot,
A model of fishery with fish stock involving delay equations, Phi. Trans. Roy. Soc. A, 367 (2009), 4907-4922.
doi: 10.1098/rsta.2009.0147. |
[8] |
E. Bocchi, On the return to equilibrium problem for axisymmetric floating structures in shallow water, Submitted, https://hal.archives-ouvertes.fr/hal-01971965. |
[9] |
F. E. Browder,
On the spectral theory of elliptic differential operators, Math. Ann., 142 (1961), 22-130.
doi: 10.1007/BF01343363. |
[10] |
O. Diekmann, P. Getto and M. Gyllenberg,
Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069.
doi: 10.1137/060659211. |
[11] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Function-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. |
[12] |
O. Diekmann and M. Gyllenberg,
Equations with infinite delay: Blending the abstract and the concrete, J. Differential Equations, 252 (2012), 819-851.
doi: 10.1016/j.jde.2011.09.038. |
[13] |
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. |
[14] |
A. Ducrot, Z. Liu and P. Magal,
Projectors on the generalized eigenspaces for neutral functional differential equations in $L^p$ spaces, Canadian Journal of Mathematics, 62 (2010), 74-93.
doi: 10.4153/CJM-2010-005-2. |
[15] |
K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[16] |
K. Ezzinbi and M. Adimy, The basic theory of abstract semilinear functional differential equations with non-dense domain, Delay Differential Equations and Applications, 347–407, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_9. |
[17] |
M. V. S. Frasson and S. M. Verduyn Lunel,
Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121.
doi: 10.1007/s00020-003-1155-x. |
[18] |
S. A. Gourley, G. Rost and H. Thieme,
Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46 (2014), 1160-1184.
doi: 10.1137/120878197. |
[19] |
J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971. |
[20] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[21] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988. |
[22] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. |
[23] |
J. K. Hale and J. Kato,
Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
|
[24] |
B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcaton, London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge, 1981.
![]() ![]() |
[25] |
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991.
doi: 10.1007/BFb0084432. |
[26] |
Y. Hino, S. Murakami, T. Naito and N. V. Minh,
A variation-of-constants formula for abstract functional differential equations in phase space, J. Differential Equations, 179 (2002), 336-355.
doi: 10.1006/jdeq.2001.4020. |
[27] |
M. A. Kaashoek and S. M. Verduyn Lunel,
Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517.
doi: 10.1090/S0002-9947-1992-1155350-0. |
[28] |
F. Kappel, Linear autonomous functional differential equations, Delay Differential Equations and Applications, 41–139, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_3. |
[29] |
H. Kellermann and M. Hieber,
Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
doi: 10.1016/0022-1236(89)90116-X. |
[30] |
Z. Liu, P. Magal and S. Ruan,
Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups, Journal of Differential Equations, 244 (2008), 1784-1809.
doi: 10.1016/j.jde.2008.01.007. |
[31] |
Z. Liu, P. Magal and S. Ruan,
Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.
doi: 10.1007/s00033-010-0088-x. |
[32] |
Z. Liu, P. Magal and S. 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. |
[33] |
P. Magal,
Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations, 2001 (2001), 1-35.
|
[34] |
P. Magal and S. Ruan,
On integrated semigroups and age structured models in Lp spaces, Differential and Integral Equations, 20 (2007), 197-139.
|
[35] |
P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models, Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. |
[36] |
P. Magal and S. Ruan,
On semilinear cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.
|
[37] |
P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201, Springer International Publishing, 2018. |
[38] |
P. Magal and X.-Q. Zhao,
Global attractors in uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[39] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[40] |
H. Matsunaga, S. Murakami, Y. Nagabuchi and N. Van Minh,
Center manifold theorem and stability for integral equations with infinite delay, Funkcialaj Ekvacioj, 58 (2015), 87-134.
doi: 10.1619/fesi.58.87. |
[41] |
C. C. McCluskey,
Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[42] |
G. Rost and J. Wu,
SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402.
doi: 10.3934/mbe.2008.5.389. |
[43] |
S. Ruan and G. S. K. Wolkowicz,
Bifurcation analysis of a chemostat model with a distributed delay, Journal of Mathematical Analysis and Applications, 204 (1996), 786-812.
doi: 10.1006/jmaa.1996.0468. |
[44] |
W. R. Ruess,
Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403.
doi: 10.1090/S0002-9947-09-04833-8. |
[45] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. |
[46] |
H. R. Thieme,
Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.
|
[47] |
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. |
[48] |
H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics–Molecules, Cells and Man (Houston, TX, 1995), 691–711, Ser. Math. Biol. Med., 6, World Sci. Publishing, River Edge, NJ, 1997. |
[49] |
C. C. Travis and G. F. Webb,
Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.
doi: 10.1090/S0002-9947-1974-0382808-3. |
[50] |
C. C. Travis and G. F. Webb,
Existence, stability, and compactness in the $\alpha-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143.
doi: 10.2307/1998809. |
[51] |
H.-O. Walther,
Differential equations with locally bounded delay, Journal of Differential Equations, 252 (2012), 3001-3039.
doi: 10.1016/j.jde.2011.11.004. |
[52] |
G. F. Webb,
Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Differential Equations, 20 (1976), 71-89.
doi: 10.1016/0022-0396(76)90097-8. |
[53] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. |
[54] |
G. F. Webb,
An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.
doi: 10.1090/S0002-9947-1987-0902796-7. |
show all references
References:
[1] |
M. Adimy,
Bifurcation de Hopf locale par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 311 (1990), 423-428.
|
[2] |
M. Adimy,
Integrated semigroups and delay differential equations, J. Math. Anal. Appl., 177 (1993), 125-134.
doi: 10.1006/jmaa.1993.1247. |
[3] |
M. Adimy and O. Arino,
Bifurcation de Hopf globale pour des équations à retard par des semi-groupes intégrés, C. R. Acad. Sci. Paris Sér. I, 317 (1993), 767-772.
|
[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, Birkhä user, Basel, 2001. |
[6] |
O. Arino and E. Sanchez, A theory of linear delay differential equations in infinite dimensional spaces, Delay Differential Equations and Applications, 285–346, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_8. |
[7] |
P. Auger and A. Ducrot,
A model of fishery with fish stock involving delay equations, Phi. Trans. Roy. Soc. A, 367 (2009), 4907-4922.
doi: 10.1098/rsta.2009.0147. |
[8] |
E. Bocchi, On the return to equilibrium problem for axisymmetric floating structures in shallow water, Submitted, https://hal.archives-ouvertes.fr/hal-01971965. |
[9] |
F. E. Browder,
On the spectral theory of elliptic differential operators, Math. Ann., 142 (1961), 22-130.
doi: 10.1007/BF01343363. |
[10] |
O. Diekmann, P. Getto and M. Gyllenberg,
Stability and bifurcation analysis of Volterra functional equations in the light of suns and stars, SIAM J. Math. Anal., 39 (2007), 1023-1069.
doi: 10.1137/060659211. |
[11] |
O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations, Function-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, 1995. |
[12] |
O. Diekmann and M. Gyllenberg,
Equations with infinite delay: Blending the abstract and the concrete, J. Differential Equations, 252 (2012), 819-851.
doi: 10.1016/j.jde.2011.09.038. |
[13] |
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. |
[14] |
A. Ducrot, Z. Liu and P. Magal,
Projectors on the generalized eigenspaces for neutral functional differential equations in $L^p$ spaces, Canadian Journal of Mathematics, 62 (2010), 74-93.
doi: 10.4153/CJM-2010-005-2. |
[15] |
K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Springer-Verlag, New York, 2000. |
[16] |
K. Ezzinbi and M. Adimy, The basic theory of abstract semilinear functional differential equations with non-dense domain, Delay Differential Equations and Applications, 347–407, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_9. |
[17] |
M. V. S. Frasson and S. M. Verduyn Lunel,
Large time behaviour of linear functional differential equations, Integral Equations Operator Theory, 47 (2003), 91-121.
doi: 10.1007/s00020-003-1155-x. |
[18] |
S. A. Gourley, G. Rost and H. Thieme,
Uniform persistence in a model for bluetongue dynamics, SIAM J. Math. Anal., 46 (2014), 1160-1184.
doi: 10.1137/120878197. |
[19] |
J. K. Hale, Functional Differential Equations, Springer-Verlag, New York, 1971. |
[20] |
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977. |
[21] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988. |
[22] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, Applied Mathematical Sciences, 99. Springer-Verlag, New York, 1993. |
[23] |
J. K. Hale and J. Kato,
Phase space for retarded equations with infinite delay, Funkcial. Ekvac., 21 (1978), 11-41.
|
[24] |
B. D. Hassard, N. D. Kazarinoff and Y.-H. Wan, Theory and Applications of Hopf Bifurcaton, London Mathematical Society Lecture Note Series, vol. 41. Cambridge University Press, Cambridge, 1981.
![]() ![]() |
[25] |
Y. Hino, S. Murakami and T. Naito, Functional Differential Equations with Infinite Delay, Lecture Notes in Math., vol. 1473, Springer-Verlag, Berlin, Heidelberg, New York, 1991.
doi: 10.1007/BFb0084432. |
[26] |
Y. Hino, S. Murakami, T. Naito and N. V. Minh,
A variation-of-constants formula for abstract functional differential equations in phase space, J. Differential Equations, 179 (2002), 336-355.
doi: 10.1006/jdeq.2001.4020. |
[27] |
M. A. Kaashoek and S. M. Verduyn Lunel,
Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334 (1992), 479-517.
doi: 10.1090/S0002-9947-1992-1155350-0. |
[28] |
F. Kappel, Linear autonomous functional differential equations, Delay Differential Equations and Applications, 41–139, NATO Sci. Ser. Ⅱ Math. Phys. Chem., 205, Springer, Dordrecht, 2006.
doi: 10.1007/1-4020-3647-7_3. |
[29] |
H. Kellermann and M. Hieber,
Integrated semigroups, J. Funct. Anal., 84 (1989), 160-180.
doi: 10.1016/0022-1236(89)90116-X. |
[30] |
Z. Liu, P. Magal and S. Ruan,
Projectors on the generalized eigenspaces for functional differential equations using integrated semigroups, Journal of Differential Equations, 244 (2008), 1784-1809.
doi: 10.1016/j.jde.2008.01.007. |
[31] |
Z. Liu, P. Magal and S. Ruan,
Hopf bifurcation for non-densely defined Cauchy problems, Z. Angew. Math. Phys., 62 (2011), 191-222.
doi: 10.1007/s00033-010-0088-x. |
[32] |
Z. Liu, P. Magal and S. 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. |
[33] |
P. Magal,
Compact attractors for time periodic age-structured population models, Electr. J. Differential Equations, 2001 (2001), 1-35.
|
[34] |
P. Magal and S. Ruan,
On integrated semigroups and age structured models in Lp spaces, Differential and Integral Equations, 20 (2007), 197-139.
|
[35] |
P. Magal and S. Ruan, Center manifolds for semilinear equations with non-dense domain and applications to hopf bifurcation in age structured models, Memoirs of the American Mathematical Society, 202 (2009), vi+71 pp. |
[36] |
P. Magal and S. Ruan,
On semilinear cauchy problems with non-dense domain, Advances in Differential Equations, 14 (2009), 1041-1084.
|
[37] |
P. Magal and S. Ruan, Theory and Applications of Abstract Semilinear Cauchy Problems, Applied Mathematical Sciences, 201, Springer International Publishing, 2018. |
[38] |
P. Magal and X.-Q. Zhao,
Global attractors in uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275.
doi: 10.1137/S0036141003439173. |
[39] |
R. H. Martin and H. L. Smith,
Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.
doi: 10.2307/2001590. |
[40] |
H. Matsunaga, S. Murakami, Y. Nagabuchi and N. Van Minh,
Center manifold theorem and stability for integral equations with infinite delay, Funkcialaj Ekvacioj, 58 (2015), 87-134.
doi: 10.1619/fesi.58.87. |
[41] |
C. C. McCluskey,
Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 6 (2009), 603-610.
doi: 10.3934/mbe.2009.6.603. |
[42] |
G. Rost and J. Wu,
SEIR epidemiological model with varying infectivity and infinite delay, Math. Biosci. Eng., 5 (2008), 389-402.
doi: 10.3934/mbe.2008.5.389. |
[43] |
S. Ruan and G. S. K. Wolkowicz,
Bifurcation analysis of a chemostat model with a distributed delay, Journal of Mathematical Analysis and Applications, 204 (1996), 786-812.
doi: 10.1006/jmaa.1996.0468. |
[44] |
W. R. Ruess,
Flow invariance for nonlinear partial differential delay equations, Trans. Amer. Math. Soc., 361 (2009), 4367-4403.
doi: 10.1090/S0002-9947-09-04833-8. |
[45] |
G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer, New York, 2002. |
[46] |
H. R. Thieme,
Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations, 3 (1990), 1035-1066.
|
[47] |
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. |
[48] |
H. R. Thieme, Quasi-compact semigroups via bounded perturbation, Advances in Mathematical Population Dynamics–Molecules, Cells and Man (Houston, TX, 1995), 691–711, Ser. Math. Biol. Med., 6, World Sci. Publishing, River Edge, NJ, 1997. |
[49] |
C. C. Travis and G. F. Webb,
Existence and stability for partial functional differential equations, Trans. Amer. Math. Soc., 200 (1974), 395-418.
doi: 10.1090/S0002-9947-1974-0382808-3. |
[50] |
C. C. Travis and G. F. Webb,
Existence, stability, and compactness in the $\alpha-$norm for partial functional differential equations, Trans. Amer. Math. Soc., 240 (1978), 129-143.
doi: 10.2307/1998809. |
[51] |
H.-O. Walther,
Differential equations with locally bounded delay, Journal of Differential Equations, 252 (2012), 3001-3039.
doi: 10.1016/j.jde.2011.11.004. |
[52] |
G. F. Webb,
Functional differential equations and nonlinear semigroups in $L^p$-spaces, J. Differential Equations, 20 (1976), 71-89.
doi: 10.1016/0022-0396(76)90097-8. |
[53] |
G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985. |
[54] |
G. F. Webb,
An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303 (1987), 751-763.
doi: 10.1090/S0002-9947-1987-0902796-7. |
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