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Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition
Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, ALGERIA |
In this paper, we investigate a class of non-monotone reaction-diffusion equations with distributed delay and a homogenous Neumann boundary condition. The main concern is the global attractivity of the unique positive steady state. To achieve this, we use an argument based on sub and super-solutions combined with the fluctuation method. We also give a condition under which the exponential stability of the positive steady state is reached. As particular examples, we apply our results to the diffusive Nicholson blowfly equation and the diffusive Mackey-Glass equation with distributed delay. We obtain some new results on exponential stability of the positive steady state for these models.
References:
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N. Bessonov, G. Bocharov, T. M. Touaoula, S. Trofimchuk and V. Volpert,
Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.
doi: 10.3934/dcdsb.2019085. |
[2] |
E. Braverman and S. Zhukovskiy,
Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.
doi: 10.3934/dcds.2012.32.2041. |
[3] |
L. Berezansky, E. Braverman and L. Idels,
Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[4] |
L. Berezansky, E. Braverman and L. Idels,
Mackey-Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.
doi: 10.1016/j.amc.2012.12.043. |
[5] |
K. Deng and Y. Wu,
On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.
doi: 10.1016/j.aml.2015.06.013. |
[6] |
W. E. Fitzgibbon,
Semilinear functional differential equations in Banach space, J. Differ. Equ., 29 (1978), 1-14.
doi: 10.1016/0022-0396(78)90037-2. |
[7] |
I. Gyori and S. Trofimchuk,
Global attractivity in $x'(t) = -\delta x(t)+pf(x(t-h))$, Dyn. Syst. Appl., 8 (1999), 197-210.
|
[8] |
J. Hale and SM. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[9] |
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. Equ., 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[10] |
T. Krisztin and H. O. Walther,
Unique periodic orbits for delayed positive feedback and the global attractor, J. Differ. Equ., 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[11] |
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. |
[12] |
R. 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. |
[13] |
R. Martin and H. L. Smith,
Reaction-diffusion systems with time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.
doi: 10.1515/crll.1991.413.1. |
[14] |
E. Liz and A. Ruis-Herrera,
Delayed population models with Allee effects and exploitation, Math. Biosci. Eng., 12 (2015), 83-97.
doi: 10.3934/mbe.2015.12.83. |
[15] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk,
A global stability criterion for a family of delayed population models, Q. Appl. Math., 63 (2005), 56-70.
doi: 10.1090/S0033-569X-05-00951-3. |
[16] |
E. Liz and G. Rost,
On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
J. Mallet-Paret and G. R. Sell,
The Poincar$\acute{e}$-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[21] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[22] |
G. Rost and J. Wu,
Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. A - Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[23] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Providence, RI, 1995. |
[24] |
H. R. Thieme and X-Q. Zhao,
A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[25] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, Vol. 118, American Mathematical Society, Providence, RI, 2011. |
[26] |
T. M. Touaoula,
Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.
doi: 10.3934/dcds.2018191. |
[27] |
T. M. Touaoula, M. N. Frioui, N. Bessonov, V. Volpert, Dynamics of solutions of a reaction-diffusion equation with delayed inhibition, to appear in Discrete Contin. Dyn. Syst. Ser. S. |
[28] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.
![]() ![]() |
[29] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, Vol. 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[30] |
D. Xu and X. Q. Zhao,
A nonlocal reaction-diffusion population model with stage structure, Canadian Appl. Math. Quart., 11 (2003), 303-320.
|
[31] |
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. |
[32] |
T. Yi and X. Zou,
Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differ. Equ., 245 (2008), 3376-3388.
doi: 10.1016/j.jde.2008.03.007. |
[33] |
T. Yi and X. Zou,
Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Newmann condition, Proc. R. Soc. A - Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
[34] |
T. Yi and X. Zou,
Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differ. Equ., 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[35] |
T. Yi and X. Zou,
On Dirichlet problem for a class of delayed reaction-diffusion equations with Spatial Non-locality, J. Dyn. Differ. Equ., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[36] |
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. |
[37] |
Y. Yuan and X. Q. Zhao,
Global stability for non monotone delay equations (with application to a model of blood cell production), J. Differ. Equ., 252 (2012), 2189-2209.
doi: 10.1016/j.jde.2011.08.026. |
[38] |
X. Q Zhao,
Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Canadian Appl. Math. Quart., 17 (2009), 271-281.
|
show all references
References:
[1] |
N. Bessonov, G. Bocharov, T. M. Touaoula, S. Trofimchuk and V. Volpert,
Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst. Ser. B, 24 (2019), 2073-2091.
doi: 10.3934/dcdsb.2019085. |
[2] |
E. Braverman and S. Zhukovskiy,
Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst., 32 (2012), 2041-2061.
doi: 10.3934/dcds.2012.32.2041. |
[3] |
L. Berezansky, E. Braverman and L. Idels,
Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Model., 34 (2010), 1405-1417.
doi: 10.1016/j.apm.2009.08.027. |
[4] |
L. Berezansky, E. Braverman and L. Idels,
Mackey-Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control, Appl. Math. Comput., 219 (2013), 6268-6283.
doi: 10.1016/j.amc.2012.12.043. |
[5] |
K. Deng and Y. Wu,
On the diffusive Nicholson's blowflies equation with distributed delay, Appl. Math. Lett., 50 (2015), 126-132.
doi: 10.1016/j.aml.2015.06.013. |
[6] |
W. E. Fitzgibbon,
Semilinear functional differential equations in Banach space, J. Differ. Equ., 29 (1978), 1-14.
doi: 10.1016/0022-0396(78)90037-2. |
[7] |
I. Gyori and S. Trofimchuk,
Global attractivity in $x'(t) = -\delta x(t)+pf(x(t-h))$, Dyn. Syst. Appl., 8 (1999), 197-210.
|
[8] |
J. Hale and SM. Verduyn Lunel, Introduction to Functional Differential Equations, Applied Mathematical Sciences, Vol. 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[9] |
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. Equ., 256 (2014), 2101-2114.
doi: 10.1016/j.jde.2013.12.015. |
[10] |
T. Krisztin and H. O. Walther,
Unique periodic orbits for delayed positive feedback and the global attractor, J. Differ. Equ., 13 (2001), 1-57.
doi: 10.1023/A:1009091930589. |
[11] |
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. |
[12] |
R. 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. |
[13] |
R. Martin and H. L. Smith,
Reaction-diffusion systems with time delay: Monotonicity, invariance, comparison and convergence, J. Reine Angew. Math., 413 (1991), 1-35.
doi: 10.1515/crll.1991.413.1. |
[14] |
E. Liz and A. Ruis-Herrera,
Delayed population models with Allee effects and exploitation, Math. Biosci. Eng., 12 (2015), 83-97.
doi: 10.3934/mbe.2015.12.83. |
[15] |
E. Liz, M. Pinto, V. Tkachenko and S. Tromichuk,
A global stability criterion for a family of delayed population models, Q. Appl. Math., 63 (2005), 56-70.
doi: 10.1090/S0033-569X-05-00951-3. |
[16] |
E. Liz and G. Rost,
On the global attractor of delay differential equations with unimodal feedback, Discrete Contin. Dyn. Syst., 24 (2009), 1215-1224.
doi: 10.3934/dcds.2009.24.1215. |
[17] |
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. |
[18] |
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. |
[19] |
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. |
[20] |
J. Mallet-Paret and G. R. Sell,
The Poincar$\acute{e}$-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differ. Equ., 125 (1996), 441-489.
doi: 10.1006/jdeq.1996.0037. |
[21] |
C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.
![]() ![]() |
[22] |
G. Rost and J. Wu,
Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc. A - Math. Phys. Eng. Sci., 463 (2007), 2655-2669.
doi: 10.1098/rspa.2007.1890. |
[23] |
H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Vol. 41, American Mathematical Society, Providence, RI, 1995. |
[24] |
H. R. Thieme and X-Q. Zhao,
A non-local delayed and diffusive predator-prey model, Nonlinear Anal. Real World Appl., 2 (2001), 145-160.
doi: 10.1016/S0362-546X(00)00112-7. |
[25] |
H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics, Vol. 118, American Mathematical Society, Providence, RI, 2011. |
[26] |
T. M. Touaoula,
Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst., 38 (2018), 4391-4419.
doi: 10.3934/dcds.2018191. |
[27] |
T. M. Touaoula, M. N. Frioui, N. Bessonov, V. Volpert, Dynamics of solutions of a reaction-diffusion equation with delayed inhibition, to appear in Discrete Contin. Dyn. Syst. Ser. S. |
[28] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.
![]() ![]() |
[29] |
J. Wu, Theory and Applications of Partial Functional Differential Equations, Applied Mathematical Sciences, Vol. 119, Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4612-4050-1. |
[30] |
D. Xu and X. Q. Zhao,
A nonlocal reaction-diffusion population model with stage structure, Canadian Appl. Math. Quart., 11 (2003), 303-320.
|
[31] |
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. |
[32] |
T. Yi and X. Zou,
Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: A non-monotone case, J. Differ. Equ., 245 (2008), 3376-3388.
doi: 10.1016/j.jde.2008.03.007. |
[33] |
T. Yi and X. Zou,
Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Newmann condition, Proc. R. Soc. A - Math. Phys. Eng. Sci., 466 (2010), 2955-2973.
doi: 10.1098/rspa.2009.0650. |
[34] |
T. Yi and X. Zou,
Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differ. Equ., 251 (2011), 2598-2611.
doi: 10.1016/j.jde.2011.04.027. |
[35] |
T. Yi and X. Zou,
On Dirichlet problem for a class of delayed reaction-diffusion equations with Spatial Non-locality, J. Dyn. Differ. Equ., 25 (2013), 959-979.
doi: 10.1007/s10884-013-9324-3. |
[36] |
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. |
[37] |
Y. Yuan and X. Q. Zhao,
Global stability for non monotone delay equations (with application to a model of blood cell production), J. Differ. Equ., 252 (2012), 2189-2209.
doi: 10.1016/j.jde.2011.08.026. |
[38] |
X. Q Zhao,
Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Canadian Appl. Math. Quart., 17 (2009), 271-281.
|
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