# American Institute of Mathematical Sciences

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)

 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.
Citation: Tarik Mohammed Touaoula. Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models). Discrete & Continuous Dynamical Systems - A, 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. Google Scholar [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. Google Scholar [3] E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with distributed delay, Can. Appl. Math. Q, 14 (2006), 107-128. Google Scholar [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. Google Scholar [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. Google Scholar [6] C. Foley and M. C. Mackey, Dynamics hematological disease, J. Math. Biol., 58 (2009), 285-322. doi: 10.1007/s00285-008-0165-3. Google Scholar [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. Google Scholar [8] W. S. C Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. Google Scholar [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. Google Scholar [10] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol 25, Americal Mathetical Society, Providence, RI, 1988. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] Y. Kuang, Delay Differential Equations, with Application in Population Dynamics, Academic Press, INC. 1993. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. Google Scholar [22] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics V. 118, AMS, 2011. Google Scholar [32] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003. Google Scholar [33] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar

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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. Google Scholar [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. Google Scholar [3] E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with distributed delay, Can. Appl. Math. Q, 14 (2006), 107-128. Google Scholar [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. Google Scholar [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. Google Scholar [6] C. Foley and M. C. Mackey, Dynamics hematological disease, J. Math. Biol., 58 (2009), 285-322. doi: 10.1007/s00285-008-0165-3. Google Scholar [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. Google Scholar [8] W. S. C Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. Google Scholar [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. Google Scholar [10] J. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol 25, Americal Mathetical Society, Providence, RI, 1988. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [15] Y. Kuang, Delay Differential Equations, with Application in Population Dynamics, Academic Press, INC. 1993. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [21] M. C. Mackey, Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 51 (1978), 941-956. Google Scholar [22] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [31] H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, Graduate Studies in Mathematics V. 118, AMS, 2011. Google Scholar [32] H. R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton 2003. Google Scholar [33] D. Xu and X.-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q., 11 (2003), 303-319. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar
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