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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 and 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.

[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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