August  2013, 18(6): 1521-1531. doi: 10.3934/dcdsb.2013.18.1521

A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations

1. 

Johannes Kepler University, Institute for Stochastics, Altenbergerstraße 69, 4040 Linz, Austria, Austria

Received  December 2011 Revised  April 2012 Published  March 2013

The stability of equilibrium solutions of a deterministic linear system of delay differential equations can be investigated by studying the characteristic equation. For stochastic delay differential equations stability analysis is usually based on Lyapunov functional or Razumikhin type results, or Linear Matrix Inequality techniques. In [7] the authors proposed a technique based on the vectorisation of matrices and the Kronecker product to transform the mean-square stability problem of a system of linear stochastic differential equations into a stability problem for a system of deterministic linear differential equations. In this paper we extend this method to the case of stochastic delay differential equations, providing sufficient and necessary conditions for the stability of the equilibrium. We apply our results to a neuron model perturbed by multiplicative noise. We study the stochastic stability properties of the equilibrium of this system and then compare them with the same equilibrium in the deterministic case. Finally the theoretical results are illustrated by numerical simulations.
Citation: Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[2]

M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913.  doi: 10.1088/0951-7715/18/2/022.  Google Scholar

[3]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations,, Mathematics and Computers in Simulation, 45 (1998), 269.  doi: 10.1016/S0378-4754(97)00106-7.  Google Scholar

[4]

J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delay-differential model of delayed visual feedback effects on postural dynamics,, Philosophical Transactions of the Royal Society A, 368 (2010), 423.   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 Dynamics in Nature and Society, 2007 (9295).  doi: 10.1155/2007/92959.  Google Scholar

[6]

D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation,, Proc. Natl. Acad. Sci. USA, 102 (2005), 14593.   Google Scholar

[7]

E. Buckwar and T. Sickenberger, A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems,, Applied Numerical Mathematics, 62 (2012), 842.  doi: 10.1016/j.apnum.2012.03.002.  Google Scholar

[8]

M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,, Mathematical Biosciences, 175 (2002), 117.  doi: 10.1016/S0025-5564(01)00089-X.  Google Scholar

[9]

M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays,, Mathematical Biosciences, 210 (2007), 395.  doi: 10.1016/j.mbs.2007.05.009.  Google Scholar

[10]

G. Decoa, V. Jirsa, A. R. McIntosh, O. Sporns and R. Kötter, Key role of coupling, delay, and noise in resting brain fluctuations,, Proceedings of the National Academy of Sciences, 106 (2009), 10302.   Google Scholar

[11]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations,, Techreport TW-330, (2001).   Google Scholar

[12]

T. Erneux, "Applied Delay Differential Equations,", Surveys and Tutorials in the Applied Mathematical Sciences, 3 (2009).   Google Scholar

[13]

K. Green and T. Wagenknecht, Pseudospectra and delay differential equations,, Journal of Computational and Applied Mathematics, 196 (2006), 567.  doi: 10.1016/j.cam.2005.10.011.  Google Scholar

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[15]

R. Khasminskii, "Stochastic Stability of Differential Equations,", With contributions by G. N. Milstein and M. B. Nevelson, 66 (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[16]

C. Hauptmann, O. Popovych and P. A.Tass, Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks,, Stochastics and Dynamics, 5 (2005), 307.  doi: 10.1142/S0219493705001420.  Google Scholar

[17]

T. Insperger and G. Stépán, "Semi-discretization for Time-delay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011).  doi: 10.1007/978-1-4614-0335-7.  Google Scholar

[18]

V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations,", Mathematics and its Applications, 463 (1999).   Google Scholar

[19]

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

[20]

H. Lütkepohl, "Handbook of Matrices,", John Wiley & Sons, (1996).   Google Scholar

[21]

X. Mao, "Stochastic Differential Equations And Applications,", Second edition, (2008).   Google Scholar

[22]

S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Pitman, (1984).   Google Scholar

[23]

L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation,, HFSP Journal, 3 (2009), 55.   Google Scholar

[24]

B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virus-tumor-immune system interaction: Deterministic and stochastic analysis,, Stochastic Analysis and Applications, 27 (2009), 409.  doi: 10.1080/07362990802679067.  Google Scholar

[25]

M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations,, Physical Review E (3), 52 (1995), 3366.  doi: 10.1103/PhysRevE.52.3366.  Google Scholar

[26]

S. Pal, S. Chatterjee, K. pada Das and J. Chattopadhyay, Role of competition in phytoplankton population for the occurrence and control of plankton bloom in the presence of environmental fluctuations,, Ecological Modelling, 220 (2009), 96.   Google Scholar

[27]

B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations,, Advances in Difference Equations, 2008 (7184).   Google Scholar

[28]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations,, Random Operators and Stochastic Equations, 18 (2010), 267.  doi: 10.1515/ROSE.2010.015.  Google Scholar

[29]

R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach,, Mathematical Biosciences, 196 (2005), 65.  doi: 10.1016/j.mbs.2005.04.001.  Google Scholar

[30]

R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology,, Journal of Mathematical Biology, 57 (2008), 613.  doi: 10.1007/s00285-008-0178-y.  Google Scholar

[31]

E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Time-delayed feedback in neurosystems,, Philosophical Transactions of the Royal Society A: Mathematical, 367 (2009), 1079.  doi: 10.1098/rsta.2008.0258.  Google Scholar

[32]

L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect,, in, 136 (1978), 120.   Google Scholar

[33]

L. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations,, Stability and Control: Theory and Application, 1 (1998), 3.   Google Scholar

[34]

L. Shaikhet, Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations,, Dynamic Systems and Applications, 17 (2008), 235.   Google Scholar

[35]

L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations,", Springer, (2011).   Google Scholar

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,", Texts in Applied Mathematics, 57 (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

show all references

References:
[1]

L. Arnold, "Stochastic Differential Equations: Theory and Applications,", Wiley-Interscience [John Wiley & Sons], (1974).   Google Scholar

[2]

M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913.  doi: 10.1088/0951-7715/18/2/022.  Google Scholar

[3]

E. Beretta, V. Kolmanovskii and L. Shaikhet, Stability of epidemic model with time delays influenced by stochastic perturbations,, Mathematics and Computers in Simulation, 45 (1998), 269.  doi: 10.1016/S0378-4754(97)00106-7.  Google Scholar

[4]

J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delay-differential model of delayed visual feedback effects on postural dynamics,, Philosophical Transactions of the Royal Society A, 368 (2010), 423.   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 Dynamics in Nature and Society, 2007 (9295).  doi: 10.1155/2007/92959.  Google Scholar

[6]

D. Bratsun, D. Volfson, L. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation,, Proc. Natl. Acad. Sci. USA, 102 (2005), 14593.   Google Scholar

[7]

E. Buckwar and T. Sickenberger, A structural analysis of asymptotic mean-square stability for multi-dimensional linear stochastic differential systems,, Applied Numerical Mathematics, 62 (2012), 842.  doi: 10.1016/j.apnum.2012.03.002.  Google Scholar

[8]

M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment,, Mathematical Biosciences, 175 (2002), 117.  doi: 10.1016/S0025-5564(01)00089-X.  Google Scholar

[9]

M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays,, Mathematical Biosciences, 210 (2007), 395.  doi: 10.1016/j.mbs.2007.05.009.  Google Scholar

[10]

G. Decoa, V. Jirsa, A. R. McIntosh, O. Sporns and R. Kötter, Key role of coupling, delay, and noise in resting brain fluctuations,, Proceedings of the National Academy of Sciences, 106 (2009), 10302.   Google Scholar

[11]

K. Engelborghs, T. Luzyanina and G. Samaey, DDE-BIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations,, Techreport TW-330, (2001).   Google Scholar

[12]

T. Erneux, "Applied Delay Differential Equations,", Surveys and Tutorials in the Applied Mathematical Sciences, 3 (2009).   Google Scholar

[13]

K. Green and T. Wagenknecht, Pseudospectra and delay differential equations,, Journal of Computational and Applied Mathematics, 196 (2006), 567.  doi: 10.1016/j.cam.2005.10.011.  Google Scholar

[14]

J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993).   Google Scholar

[15]

R. Khasminskii, "Stochastic Stability of Differential Equations,", With contributions by G. N. Milstein and M. B. Nevelson, 66 (2012).  doi: 10.1007/978-3-642-23280-0.  Google Scholar

[16]

C. Hauptmann, O. Popovych and P. A.Tass, Multisite coordinated delayed feedback for an effective desynchronization of neuronal networks,, Stochastics and Dynamics, 5 (2005), 307.  doi: 10.1142/S0219493705001420.  Google Scholar

[17]

T. Insperger and G. Stépán, "Semi-discretization for Time-delay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011).  doi: 10.1007/978-1-4614-0335-7.  Google Scholar

[18]

V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations,", Mathematics and its Applications, 463 (1999).   Google Scholar

[19]

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

[20]

H. Lütkepohl, "Handbook of Matrices,", John Wiley & Sons, (1996).   Google Scholar

[21]

X. Mao, "Stochastic Differential Equations And Applications,", Second edition, (2008).   Google Scholar

[22]

S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Pitman, (1984).   Google Scholar

[23]

L. G. Morelli, S. Ares, L. Herrgen, C. Schröter, F. Jülicher and A. C. Oates, Delayed coupling theory of vertebrate segmentation,, HFSP Journal, 3 (2009), 55.   Google Scholar

[24]

B. Mukhopadhyay and R. Bhattacharyya, A nonlinear mathematical model of virus-tumor-immune system interaction: Deterministic and stochastic analysis,, Stochastic Analysis and Applications, 27 (2009), 409.  doi: 10.1080/07362990802679067.  Google Scholar

[25]

M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations,, Physical Review E (3), 52 (1995), 3366.  doi: 10.1103/PhysRevE.52.3366.  Google Scholar

[26]

S. Pal, S. Chatterjee, K. pada Das and J. Chattopadhyay, Role of competition in phytoplankton population for the occurrence and control of plankton bloom in the presence of environmental fluctuations,, Ecological Modelling, 220 (2009), 96.   Google Scholar

[27]

B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations,, Advances in Difference Equations, 2008 (7184).   Google Scholar

[28]

M.-K. von Renesse and M. Scheutzow, Existence and uniqueness of solutions of stochastic functional differential equations,, Random Operators and Stochastic Equations, 18 (2010), 267.  doi: 10.1515/ROSE.2010.015.  Google Scholar

[29]

R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach,, Mathematical Biosciences, 196 (2005), 65.  doi: 10.1016/j.mbs.2005.04.001.  Google Scholar

[30]

R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology,, Journal of Mathematical Biology, 57 (2008), 613.  doi: 10.1007/s00285-008-0178-y.  Google Scholar

[31]

E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Time-delayed feedback in neurosystems,, Philosophical Transactions of the Royal Society A: Mathematical, 367 (2009), 1079.  doi: 10.1098/rsta.2008.0258.  Google Scholar

[32]

L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect,, in, 136 (1978), 120.   Google Scholar

[33]

L. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations,, Stability and Control: Theory and Application, 1 (1998), 3.   Google Scholar

[34]

L. Shaikhet, Stability of a positive point of equilibrium of one nonlinear system with aftereffect and stochastic perturbations,, Dynamic Systems and Applications, 17 (2008), 235.   Google Scholar

[35]

L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations,", Springer, (2011).   Google Scholar

[36]

H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences,", Texts in Applied Mathematics, 57 (2011).  doi: 10.1007/978-1-4419-7646-8.  Google Scholar

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