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May  2020, 19(5): 2473-2490. doi: 10.3934/cpaa.2020108

Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition

Tarik Mohammed Touaoula

Département de Mathématiques, Faculté des Sciences, Université de Tlemcen, Laboratoire d'Analyse Non Linéaire et Mathématiques Appliquées, Tlemcen, BP 119, 13000, ALGERIA

Received  October 2018 Revised  October 2019 Published  March 2020

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.

Keywords: Reaction-diffusion equation; distributed delay, sub and super-solution, global attractivity; exponential stability.
Mathematics Subject Classification: Primary: 35R10, 35B40, 35Q92.
Citation: Tarik Mohammed Touaoula. Global dynamics for a class of reaction-diffusion equations with distributed delay and neumann condition. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2473-2490. doi: 10.3934/cpaa.2020108
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[28] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.   Google Scholar
[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.  Google Scholar

[30]

D. Xu and X. Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canadian Appl. Math. Quart., 11 (2003), 303-320.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21] C. V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[28] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003.   Google Scholar
[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.  Google Scholar

[30]

D. Xu and X. Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Canadian Appl. Math. Quart., 11 (2003), 303-320.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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