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

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

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

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.
Citation: Elena Braverman, Sergey Zhukovskiy. Absolute and delay-dependent stability of equations with a distributed delay. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1016/j.camwa.2005.09.001.  Google Scholar

[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), 103.   Google Scholar

[3]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Math. Comput. Modelling, 48 (2008), 287.  doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[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.  doi: 10.1016/j.apm.2009.08.027.  Google Scholar

[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 (2007).   Google Scholar

[6]

E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay,, Can. Appl. Math. Quart., 14 (2006), 107.   Google Scholar

[7]

W. A. Coppel, The solution of equations by iteration,, Proc. Cambridge Philos. Soc., 51 (1955), 41.  doi: 10.1017/S030500410002990X.  Google Scholar

[8]

C. Corduneanu, "Functional Equations with Causal Operators,", Stability and Control: Theory, 16 (2002).   Google Scholar

[9]

K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis,, Ukrainian Math. J., 50 (1998), 3.  doi: 10.1007/BF02514684.  Google Scholar

[10]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17.  doi: 10.1038/287017a0.  Google Scholar

[11]

I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications,", Oxford Mathematical Monographs, (1991).   Google Scholar

[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.   Google Scholar

[13]

K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations,, Arch. Rat. Mech. Anal., 65 (1977), 87.  doi: 10.1007/BF00289359.  Google Scholar

[14]

A. F. Ivanov, On global stability in a nonlinear discrete model,, Nonlinear Anal., 23 (1994), 1383.  doi: 10.1016/0362-546X(94)90133-3.  Google Scholar

[15]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, in, 1 (1992), 164.   Google Scholar

[16]

G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models,, J. Dynam. Diff. Eq., 4 (1992), 161.   Google Scholar

[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.  doi: 10.1023/A:1009091930589.  Google Scholar

[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 (1999).   Google Scholar

[19]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[20]

I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics,, Math. Comput. Modelling, 35 (2002), 295.   Google Scholar

[21]

M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies,, Appl. Anal., 43 (1992), 109.   Google Scholar

[22]

B. Lani-Wayda, Erratic solutions of simple delay equations,, Trans. Am. Math. Soc., 351 (1999), 901.  doi: 10.1090/S0002-9947-99-02351-X.  Google Scholar

[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.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[24]

E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations,, Differential Integral Equations, 15 (2002), 875.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2009.24.1215.  Google Scholar

[27]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596.  doi: 10.1137/S0036141001399222.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00416-X.  Google Scholar

[29]

J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations,, Chaos, 3 (1993), 167.  doi: 10.1063/1.165982.  Google Scholar

[30]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar

[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.  doi: 10.1007/BF01790539.  Google Scholar

[32]

J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology,, SIAM J. Math. Anal., 20 (1989), 249.  doi: 10.1137/0520019.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[34]

A. J. Nicholson, An outline of the dynamics of animal populations,, Austral. J. Zool., 2 (1954), 9.  doi: 10.1071/ZO9540009.  Google Scholar

[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.   Google Scholar

[36]

S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time,, Appl. Math. Comput., 136 (2003), 241.  doi: 10.1016/S0096-3003(02)00035-8.  Google Scholar

[37]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

show all references

References:
[1]

L. Berezansky and E. Braverman, Mackey-Glass equation with variable coefficients,, Comput. Math. Appl., 51 (2006), 1.  doi: 10.1016/j.camwa.2005.09.001.  Google Scholar

[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), 103.   Google Scholar

[3]

L. Berezansky and E. Braverman, Linearized oscillation theory for a nonlinear equation with a distributed delay,, Math. Comput. Modelling, 48 (2008), 287.  doi: 10.1016/j.mcm.2007.10.003.  Google Scholar

[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.  doi: 10.1016/j.apm.2009.08.027.  Google Scholar

[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 (2007).   Google Scholar

[6]

E. Braverman and D. Kinzebulatov, Nicholson's blowflies equation with a distributed delay,, Can. Appl. Math. Quart., 14 (2006), 107.   Google Scholar

[7]

W. A. Coppel, The solution of equations by iteration,, Proc. Cambridge Philos. Soc., 51 (1955), 41.  doi: 10.1017/S030500410002990X.  Google Scholar

[8]

C. Corduneanu, "Functional Equations with Causal Operators,", Stability and Control: Theory, 16 (2002).   Google Scholar

[9]

K. Gopalsamy, N. Bantsur and S. Trofimchuk, A note on global attractivity in models of hematopoiesis,, Ukrainian Math. J., 50 (1998), 3.  doi: 10.1007/BF02514684.  Google Scholar

[10]

W. S. C. Gurney, S. P. Blythe and R. M. Nisbet, Nicholson's blowflies revisited,, Nature, 287 (1980), 17.  doi: 10.1038/287017a0.  Google Scholar

[11]

I. Győri and G. Ladas, "Oscillation Theory of Delay Differential Equations. With Applications,", Oxford Mathematical Monographs, (1991).   Google Scholar

[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.   Google Scholar

[13]

K. P. Hadeler and J. Tomiuk, Periodic solutions of difference-differential equations,, Arch. Rat. Mech. Anal., 65 (1977), 87.  doi: 10.1007/BF00289359.  Google Scholar

[14]

A. F. Ivanov, On global stability in a nonlinear discrete model,, Nonlinear Anal., 23 (1994), 1383.  doi: 10.1016/0362-546X(94)90133-3.  Google Scholar

[15]

A. F. Ivanov and A. N. Sharkovsky, Oscillations in singularly perturbed delay equations,, in, 1 (1992), 164.   Google Scholar

[16]

G. Karakostas, Ch. Philos and Y. Sficas, Stable steady state of some population models,, J. Dynam. Diff. Eq., 4 (1992), 161.   Google Scholar

[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.  doi: 10.1023/A:1009091930589.  Google Scholar

[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 (1999).   Google Scholar

[19]

Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993).   Google Scholar

[20]

I. Kubiaczyk and S. H. Saker, Oscillation and stability in nonlinear delay differential equations of population dynamics,, Math. Comput. Modelling, 35 (2002), 295.   Google Scholar

[21]

M. R. S. Kulenović, G. Ladas and Y. Sficas, Global attractivity in Nicholson's blowflies,, Appl. Anal., 43 (1992), 109.   Google Scholar

[22]

B. Lani-Wayda, Erratic solutions of simple delay equations,, Trans. Am. Math. Soc., 351 (1999), 901.  doi: 10.1090/S0002-9947-99-02351-X.  Google Scholar

[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.  doi: 10.3934/dcdsb.2007.7.191.  Google Scholar

[24]

E. Liz, C. Martínez and S. Trofimchuk, Attractivity properties of infinite delay Mackey-Glass type equations,, Differential Integral Equations, 15 (2002), 875.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.3934/dcds.2009.24.1215.  Google Scholar

[27]

E. Liz, V. Tkachenko and S. Trofimchuk, A global stability criterion for scalar functional differential equations,, SIAM J. Math. Anal., 35 (2003), 596.  doi: 10.1137/S0036141001399222.  Google Scholar

[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.  doi: 10.1016/S0022-247X(02)00416-X.  Google Scholar

[29]

J. Losson, M. C. Mackey and A. Longtin, Solution multistability in first-order nonlinear differential delay equations,, Chaos, 3 (1993), 167.  doi: 10.1063/1.165982.  Google Scholar

[30]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287.  doi: 10.1126/science.267326.  Google Scholar

[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.  doi: 10.1007/BF01790539.  Google Scholar

[32]

J. Mallet-Paret and R. Nussbaum, A differential-delay equation arising in optics and physiology,, SIAM J. Math. Anal., 20 (1989), 249.  doi: 10.1137/0520019.  Google Scholar

[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.  doi: 10.1006/jdeq.1996.0037.  Google Scholar

[34]

A. J. Nicholson, An outline of the dynamics of animal populations,, Austral. J. Zool., 2 (1954), 9.  doi: 10.1071/ZO9540009.  Google Scholar

[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.   Google Scholar

[36]

S. H. Saker, Oscillation and global attractivity in hematopoiesis model with delay time,, Appl. Math. Comput., 136 (2003), 241.  doi: 10.1016/S0096-3003(02)00035-8.  Google Scholar

[37]

D. Singer, Stable orbits and bifurcation of maps of the interval,, SIAM J. Appl. Math., 35 (1978), 260.  doi: 10.1137/0135020.  Google Scholar

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