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

Tarik Mohammed Touaoula

doi:10.3934/dcds.2018191

Article Contents

2018, Volume 38, Issue 9: 4391-4419. Doi: 10.3934/dcds.2018191

This issue Previous Article The Hénon equation with a critical exponent under the Neumann boundary condition Next Article The maximal entropy measure of Fatou boundaries

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: September 2018

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Abstract

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.

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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.