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The Hénon equation with a critical exponent under the Neumann boundary condition
Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)
Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, Algeria |
$ x'(t)=-f(x(t))+\int_{0}^{\tau}h(a)g(x(t-a))da.$ |
References:
[1] |
L. Berezansky, E. Braverman and L. Idels,
Nicholson's blowflies differential equations revisited: Main results and open problems, Applied Math. Modelling, 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[2] |
L. Berezansky, E. Braverman and L. Idels,
Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Compt., 219 (2013), 6268-6283.
doi: 10.1016/j.amc.2012.12.043. |
[3] |
E. Braverman and D. Kinzebulatov,
Nicholson's blowflies equation with distributed delay, Can. Appl. Math. Q, 14 (2006), 107-128.
|
[4] |
E. Braverman and S. Zhukovskiy,
Absolute and delay-dependent stability of equations with a distributed delay, Discrete and Continuous Dynam. Systems, 32 (2012), 2041-2061.
doi: 10.3934/dcds.2012.32.2041. |
[5] |
H. A. El-Morshedy,
Global attractivity in a population model with nonlinear death rate and distributed delays, J. Math. Anal. Appl., 410 (2014), 642-658.
doi: 10.1016/j.jmaa.2013.08.060. |
[6] |
C. Foley and M. C. Mackey,
Dynamics hematological disease, J. Math. Biol., 58 (2009), 285-322.
doi: 10.1007/s00285-008-0165-3. |
[7] |
K. Gopalsamy,
Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.
doi: 10.1007/978-94-015-7920-9. |
[8] |
W. S. C Gurney, S. P. Blythe and R. M. Nisbet,
Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
|
[9] |
I. Gyori and S. Trofimchuk,
Global attractivity in $x'(t) = -δ x(t)+pf(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210.
|
[10] |
J. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol 25, Americal Mathetical Society, Providence, RI, 1988. |
[11] |
J. Hale and S. M. Verduyn Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[12] |
C. Huang, Z. Yang, T. Yi and X. Zou,
On the bassin of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differ. Equations, 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[13] |
A. Ivanov and M. Mammadov,
Global asymptotic stability in a class of nonlinear differential delay equations, Discrete and Continuous Dynam. Systems, 1 (2011), 727-736.
|
[14] |
T. Krisztin and H. O. Walther,
Unique periodic orbits for delayed positive feedback and the global attractor, J. Differ. Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[15] |
Y. Kuang,
Delay Differential Equations, with Application in Population Dynamics, Academic Press, INC. 1993. |
[16] |
B. Lani-Wayda,
Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999), 901-945.
doi: 10.1090/S0002-9947-99-02351-X. |
[17] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk,
A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.
doi: 10.1090/S0033-569X-05-00951-3. |
[18] |
E. Liz and G. Rost,
On the global attractor of delay differential equations with unimodal feedback, Discrete and continuous dynam. systems, 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[19] |
E. Liz, V. Tkachenko and S. Tromichuk,
A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[20] |
E. Liz, V. Tkachenko and S. Trofimchuk,
Mackey-Glass type delay differential equations near the boundary of absolute stability, J. Math. Anal. Appl., 275 (2002), 747-760.
doi: 10.1016/S0022-247X(02)00416-X. |
[21] |
M. C. Mackey,
Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.
|
[22] |
M. C. Mackey and L. Glass,
Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[23] |
M. C. Mackey and R. Rudnicki,
Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), 89-109.
doi: 10.1007/BF00160175. |
[24] |
J. Mallet-Paret and R. Nussbaum,
Global continuation and asymptotic behavior for periodic solutions of a differential delay equation, Ann. Mat. Pura. Appl., 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[25] |
J. Mallet-Paret and R. Nussbaum,
A differential-delay equation arising in optics and physiology, SIAM. J. Math. Anal., 20 (1989), 249-292.
doi: 10.1137/0520019. |
[26] |
J. Mallet-Paret and G. R. Sell,
The poincar?Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[27] |
G. Rost and J. Wu,
Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[28] |
H. L. Smith,
Monotone Dynamical Systems: An introduction to the theory of Competitive and Cooperative Systems, Math, Surveys Monogr, vol 41, Amer. Math. Soc. 1995. |
[29] |
H. L. Smith,
An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[30] |
H. L. Smith and H. R. Thieme,
Monotone semiflows in scalar non quasi-monotone functional differential equations, J. Math. Anal. Appl., 150 (1990), 289-306.
doi: 10.1016/0022-247X(90)90105-O. |
[31] |
H. L. Smith and H. R. Thieme,
Dynamical Systems and Population Persistence, Graduate Studies in Mathematics V. 118, AMS, 2011. |
[32] |
H. R. Thieme,
Mathematics in Population Biology, Princeton University Press, Princeton 2003. |
[33] |
D. Xu and X.-Q. Zhao,
A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.
|
[34] |
T. Yi, Y. Chen and J. Wu,
Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.
doi: 10.1007/s00033-012-0224-x. |
[35] |
T. Yi and X. Zou,
Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Newmann condition, Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
[36] |
T. Yi and X. Zou,
Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differ. Equations, 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[37] |
T. Yi and X. Zou,
On Dirichlet Problem for a Class of Delayed Reaction-Diffusion Equations with Spatial Non-locality, J. Dyn. Diff. Equat., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[38] |
Y. Yuan and J. Belair,
Stability and Hopf bifurcation analysis for functional differential equation with distributed delay, SIAM, J. Appl. Dyn. Syst., 10 (2011), 551-581.
doi: 10.1137/100794493. |
[39] |
Y. Yuan and X. Q. Zhao,
Global stability for non monotone delay equations (with application to a model of blood cell production), J. Differ. Equations, 252 (2012), 2189-2209.
doi: 10.1016/j.jde.2011.08.026. |
show all references
References:
[1] |
L. Berezansky, E. Braverman and L. Idels,
Nicholson's blowflies differential equations revisited: Main results and open problems, Applied Math. Modelling, 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[2] |
L. Berezansky, E. Braverman and L. Idels,
Mackey-Glass model of hematopoiesis with non-monotone feedback: Stability, oscillation and control, Appl. Math. Compt., 219 (2013), 6268-6283.
doi: 10.1016/j.amc.2012.12.043. |
[3] |
E. Braverman and D. Kinzebulatov,
Nicholson's blowflies equation with distributed delay, Can. Appl. Math. Q, 14 (2006), 107-128.
|
[4] |
E. Braverman and S. Zhukovskiy,
Absolute and delay-dependent stability of equations with a distributed delay, Discrete and Continuous Dynam. Systems, 32 (2012), 2041-2061.
doi: 10.3934/dcds.2012.32.2041. |
[5] |
H. A. El-Morshedy,
Global attractivity in a population model with nonlinear death rate and distributed delays, J. Math. Anal. Appl., 410 (2014), 642-658.
doi: 10.1016/j.jmaa.2013.08.060. |
[6] |
C. Foley and M. C. Mackey,
Dynamics hematological disease, J. Math. Biol., 58 (2009), 285-322.
doi: 10.1007/s00285-008-0165-3. |
[7] |
K. Gopalsamy,
Stability and Oscillation in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, Dordrecht, Boston, London, 1992.
doi: 10.1007/978-94-015-7920-9. |
[8] |
W. S. C Gurney, S. P. Blythe and R. M. Nisbet,
Nicholson's blowflies revisited, Nature, 287 (1980), 17-21.
|
[9] |
I. Gyori and S. Trofimchuk,
Global attractivity in $x'(t) = -δ x(t)+pf(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210.
|
[10] |
J. Hale,
Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol 25, Americal Mathetical Society, Providence, RI, 1988. |
[11] |
J. Hale and S. M. Verduyn Lunel,
Introduction to Functional Differential Equations, Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[12] |
C. Huang, Z. Yang, T. Yi and X. Zou,
On the bassin of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, J. Differ. Equations, 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[13] |
A. Ivanov and M. Mammadov,
Global asymptotic stability in a class of nonlinear differential delay equations, Discrete and Continuous Dynam. Systems, 1 (2011), 727-736.
|
[14] |
T. Krisztin and H. O. Walther,
Unique periodic orbits for delayed positive feedback and the global attractor, J. Differ. Equations, 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[15] |
Y. Kuang,
Delay Differential Equations, with Application in Population Dynamics, Academic Press, INC. 1993. |
[16] |
B. Lani-Wayda,
Erratic solutions of simple delay equations, Trans. Amer. Math. Soc., 351 (1999), 901-945.
doi: 10.1090/S0002-9947-99-02351-X. |
[17] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk,
A global stability criterion for a family of delayed population models, Quart. Appl. Math., 63 (2005), 56-70.
doi: 10.1090/S0033-569X-05-00951-3. |
[18] |
E. Liz and G. Rost,
On the global attractor of delay differential equations with unimodal feedback, Discrete and continuous dynam. systems, 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[19] |
E. Liz, V. Tkachenko and S. Tromichuk,
A global stability criterion for scalar functional differential equations, SIAM. J. Math. Anal., 35 (2003), 596-622.
doi: 10.1137/S0036141001399222. |
[20] |
E. Liz, V. Tkachenko and S. Trofimchuk,
Mackey-Glass type delay differential equations near the boundary of absolute stability, J. Math. Anal. Appl., 275 (2002), 747-760.
doi: 10.1016/S0022-247X(02)00416-X. |
[21] |
M. C. Mackey,
Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.
|
[22] |
M. C. Mackey and L. Glass,
Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289.
doi: 10.1126/science.267326. |
[23] |
M. C. Mackey and R. Rudnicki,
Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), 89-109.
doi: 10.1007/BF00160175. |
[24] |
J. Mallet-Paret and R. Nussbaum,
Global continuation and asymptotic behavior for periodic solutions of a differential delay equation, Ann. Mat. Pura. Appl., 145 (1986), 33-128.
doi: 10.1007/BF01790539. |
[25] |
J. Mallet-Paret and R. Nussbaum,
A differential-delay equation arising in optics and physiology, SIAM. J. Math. Anal., 20 (1989), 249-292.
doi: 10.1137/0520019. |
[26] |
J. Mallet-Paret and G. R. Sell,
The poincar?Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equations, 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[27] |
G. Rost and J. Wu,
Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[28] |
H. L. Smith,
Monotone Dynamical Systems: An introduction to the theory of Competitive and Cooperative Systems, Math, Surveys Monogr, vol 41, Amer. Math. Soc. 1995. |
[29] |
H. L. Smith,
An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer, 2011.
doi: 10.1007/978-1-4419-7646-8. |
[30] |
H. L. Smith and H. R. Thieme,
Monotone semiflows in scalar non quasi-monotone functional differential equations, J. Math. Anal. Appl., 150 (1990), 289-306.
doi: 10.1016/0022-247X(90)90105-O. |
[31] |
H. L. Smith and H. R. Thieme,
Dynamical Systems and Population Persistence, Graduate Studies in Mathematics V. 118, AMS, 2011. |
[32] |
H. R. Thieme,
Mathematics in Population Biology, Princeton University Press, Princeton 2003. |
[33] |
D. Xu and X.-Q. Zhao,
A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319.
|
[34] |
T. Yi, Y. Chen and J. Wu,
Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys., 63 (2012), 793-812.
doi: 10.1007/s00033-012-0224-x. |
[35] |
T. Yi and X. Zou,
Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Newmann condition, Proc. R. Soc. London. Ser. A Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
[36] |
T. Yi and X. Zou,
Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differ. Equations, 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[37] |
T. Yi and X. Zou,
On Dirichlet Problem for a Class of Delayed Reaction-Diffusion Equations with Spatial Non-locality, J. Dyn. Diff. Equat., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[38] |
Y. Yuan and J. Belair,
Stability and Hopf bifurcation analysis for functional differential equation with distributed delay, SIAM, J. Appl. Dyn. Syst., 10 (2011), 551-581.
doi: 10.1137/100794493. |
[39] |
Y. Yuan and X. Q. Zhao,
Global stability for non monotone delay equations (with application to a model of blood cell production), J. Differ. Equations, 252 (2012), 2189-2209.
doi: 10.1016/j.jde.2011.08.026. |
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