Absolute and delay-dependent stability of equations with a distributed delay

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June 2012, 32(6): 2041-2061. doi: 10.3934/dcds.2012.32.2041

Absolute and delay-dependent stability of equations with a distributed delay

Elena Braverman ^1, and Sergey Zhukovskiy ^2,

1.	Department of Mathematics and Statistics, University of Calgary, 2500 University Drive NW, Calgary, AB, T2N 1N4
2.	Department of Mathematics, Moscow PF University, Miklukho-Maklaya str. 6, Moscow 117198, Russian Federation

Received July 2010 Revised December 2011 Published February 2012

Abstract
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We study delay-independent stability in nonlinear models with a distributed delay which have a positive equilibrium. Models with a unique positive equilibrium frequently occur in population dynamics and other applications. In particular, we construct a relevant difference equation such that its stability implies stability of the equation with a distributed delay and a finite memory. This result is, generally speaking, incorrect for systems with infinite memory. If the relevant difference equation is unstable, we describe the general delay-independent lower and upper solution bounds and also demonstrate that the equation with a distributed delay is stable for small enough delays.

Keywords: Nicholson's blowflies equation, Mackey-Glass equation., permanent solutions, global attractivity, Equations with a distributed delay.

Mathematics Subject Classification: Primary: 34K20; Secondary: 92D25, 34K60, 34K2.

Citation: Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete and Continuous Dynamical Systems, 2012, 32 (6) : 2041-2061. doi: 10.3934/dcds.2012.32.2041

References:

[1]	L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16. doi: 10.1016/j.camwa.2005.09.001.
[2]	L. Berezansky and E. Braverman, On existence and attractivity of periodic solutions for the hematopoiesis equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13B (2006), suppl., 103-116.
[3]	L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Math. Comput. Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.
[4]	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.
[5]	N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007, Art. ID 92959, 25 pp.
[6]	E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay, Can. Appl. Math. Quart., 14 (2006), 107-128.
[7]	W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43. doi: 10.1017/S030500410002990X.
[8]	C. Corduneanu, "Functional Equations with Causal Operators," Stability and Control: Theory, Methods and Applications, 16, Taylor & Francis, London, 2002.
[9]	K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12. doi: 10.1007/BF02514684.
[10]	W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.
[11]	I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.
[12]	I. Győri and S. Trofimchuk, Global attractivity in $x^{\'}(t)= -\delta x(t) +p f(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210.
[13]	K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95. doi: 10.1007/BF00289359.
[14]	A. F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal., 23 (1994), 1383-1389. doi: 10.1016/0362-546X(94)90133-3.
[15]	A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in "Dynamics Reported: Expositions in Dynamical Systems," Dynam. Rep. Expositions Dynam. Systems (New Series), 1, Springer, Berlin, (1992), 164-224.
[16]	G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models, J. Dynam. Diff. Eq., 4 (1992), 161-190.
[17]	T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57. doi: 10.1023/A:1009091930589.
[18]	T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Inst. Monogr., 11, AMS, Providence, RI, 1999.
[19]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[20]	I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301.
[21]	M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies, Appl. Anal., 43 (1992), 109-124.
[22]	B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Am. Math. Soc., 351 (1999), 901-945. doi: 10.1090/S0002-9947-99-02351-X.
[23]	E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199. doi: 10.3934/dcdsb.2007.7.191.
[24]	E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential Integral Equations, 15 (2002), 875-896.
[25]	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.
[26]	E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete and Continuous Dynamical Systems, 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.
[27]	E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.
[28]	E. Liz, E. Trofimchuk 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.
[29]	J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations, Chaos, 3 (1993), 167-176. doi: 10.1063/1.165982.
[30]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[31]	J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.
[32]	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.
[33]	J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.
[34]	A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2 (1954), 9-65. doi: 10.1071/ZO9540009.
[35]	G. Röst 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.
[36]	S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250. doi: 10.1016/S0096-3003(02)00035-8.
[37]	D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.

show all references

References:

[1]	L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients, Comput. Math. Appl., 51 (2006), 1-16. doi: 10.1016/j.camwa.2005.09.001.
[2]	L. Berezansky and E. Braverman, On existence and attractivity of periodic solutions for the hematopoiesis equation, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13B (2006), suppl., 103-116.
[3]	L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay, Math. Comput. Modelling, 48 (2008), 287-304. doi: 10.1016/j.mcm.2007.10.003.
[4]	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.
[5]	N. Bradul and L. Shaikhet, Stability of the positive point of equilibrium of Nicholson's blowflies equation with stochastic perturbations: Numerical analysis, Discrete Dyn. Nat. Soc., 2007, Art. ID 92959, 25 pp.
[6]	E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay, Can. Appl. Math. Quart., 14 (2006), 107-128.
[7]	W. A. Coppel, The solution of equations by iteration, Proc. Cambridge Philos. Soc., 51 (1955), 41-43. doi: 10.1017/S030500410002990X.
[8]	C. Corduneanu, "Functional Equations with Causal Operators," Stability and Control: Theory, Methods and Applications, 16, Taylor & Francis, London, 2002.
[9]	K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis, Ukrainian Math. J., 50 (1998), 3-12. doi: 10.1007/BF02514684.
[10]	W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited, Nature, 287 (1980), 17-21. doi: 10.1038/287017a0.
[11]	I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications," Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1991.
[12]	I. Győri and S. Trofimchuk, Global attractivity in $x^{\'}(t)= -\delta x(t) +p f(x(t-h))$, Dynam. Syst. Appl., 8 (1999), 197-210.
[13]	K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations, Arch. Rat. Mech. Anal., 65 (1977), 87-95. doi: 10.1007/BF00289359.
[14]	A. F. Ivanov, On global stability in a nonlinear discrete model, Nonlinear Anal., 23 (1994), 1383-1389. doi: 10.1016/0362-546X(94)90133-3.
[15]	A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations, in "Dynamics Reported: Expositions in Dynamical Systems," Dynam. Rep. Expositions Dynam. Systems (New Series), 1, Springer, Berlin, (1992), 164-224.
[16]	G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models, J. Dynam. Diff. Eq., 4 (1992), 161-190.
[17]	T. Krisztin and H.-O. Walther, Unique periodic orbits for delayed positive feedback and the global attractor, J. Dynam. Differential Equations, 13 (2001), 1-57. doi: 10.1023/A:1009091930589.
[18]	T. Krisztin, H.-O. Walther and J. Wu, "Shape, Smoothness and Invariant Stratification of an Attracting Set for Delayed Monotone Positive Feedback," Fields Inst. Monogr., 11, AMS, Providence, RI, 1999.
[19]	Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics," Mathematics in Science and Engineering, 191, Academic Press, Inc., Boston, MA, 1993.
[20]	I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics, Math. Comput. Modelling, 35 (2002), 295-301.
[21]	M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies, Appl. Anal., 43 (1992), 109-124.
[22]	B. Lani-Wayda, Erratic solutions of simple delay equations, Trans. Am. Math. Soc., 351 (1999), 901-945. doi: 10.1090/S0002-9947-99-02351-X.
[23]	E. Liz, Local stability implies global stability in some one-dimensional discrete single-species models, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 191-199. doi: 10.3934/dcdsb.2007.7.191.
[24]	E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations, Differential Integral Equations, 15 (2002), 875-896.
[25]	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.
[26]	E. Liz and G. Röst, On the global attractor of delay differential equations with unimodal feedback, Discrete and Continuous Dynamical Systems, 24 (2009), 1215-1224. doi: 10.3934/dcds.2009.24.1215.
[27]	E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations, SIAM J. Math. Anal., 35 (2003), 596-622. doi: 10.1137/S0036141001399222.
[28]	E. Liz, E. Trofimchuk 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.
[29]	J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations, Chaos, 3 (1993), 167-176. doi: 10.1063/1.165982.
[30]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[31]	J. Mallet-Paret and R. Nussbaum, Global continuation and asymptotic behavior for periodic solutions of a differential-delay equation, Ann. Mat. Pura Appl. (4), 145 (1986), 33-128. doi: 10.1007/BF01790539.
[32]	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.
[33]	J. Mallet-Paret and G. R. Sell, The Poincaré-Bendixson theorem for monotone cyclic feedback systems with delay, J. Differential Equations, 125 (1996), 441-489. doi: 10.1006/jdeq.1996.0037.
[34]	A. J. Nicholson, An outline of the dynamics of animal populations, Austral. J. Zool., 2 (1954), 9-65. doi: 10.1071/ZO9540009.
[35]	G. Röst 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.
[36]	S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time, Appl. Math. Comput., 136 (2003), 241-250. doi: 10.1016/S0096-3003(02)00035-8.
[37]	D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math., 35 (1978), 260-267. doi: 10.1137/0135020.

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