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September  2018, 38(9): 4391-4419. doi: 10.3934/dcds.2018191

Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models)

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  August 2017 Revised  April 2018 Published  June 2018

Global asymptotic and exponential stability of equilibria for the following class of functional differential equations with distributed delay is investigated
$ x'(t)=-f(x(t))+\int_{0}^{\tau}h(a)g(x(t-a))da.$
We make our analysis by introducing a new approach, combining a Lyapunov functional and monotone semiflow theory. The relevance of our results is illustrated by studying the well-known integro-differential Nicholson's blowflies and Mackey-Glass equations, where some delay independent stability conditions are provided. Furthermore, new results related to exponential stability region of the positive equilibrium for these both models are established.
Keywords: Monotone semi-flow, global asymptotic and exponential stability, fluctuation method.
Mathematics Subject Classification: Primary: 34K20, 37L15; Secondary: 92B05.
Citation: Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4391-4419. doi: 10.3934/dcds.2018191
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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