# American Institute of Mathematical Sciences

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 and 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], New York-London-Sydney, 1974. [2] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022. [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), 269-277. doi: 10.1016/S0378-4754(97)00106-7. [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), 423-438. [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, Article ID 92959, 25 pp. doi: 10.1155/2007/92959. [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), 14593-14598. [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), 842-859. doi: 10.1016/j.apnum.2012.03.002. [8] M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117-131. doi: 10.1016/S0025-5564(01)00089-X. [9] M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395-414. doi: 10.1016/j.mbs.2007.05.009. [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), 10302-10307. [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, Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001. [12] T. Erneux, "Applied Delay Differential Equations," Surveys and Tutorials in the Applied Mathematical Sciences, 3, Springer, New York, 2009. [13] K. Green and T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, 196 (2006), 567-578. doi: 10.1016/j.cam.2005.10.011. [14] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [15] R. Khasminskii, "Stochastic Stability of Differential Equations," With contributions by G. N. Milstein and M. B. Nevelson, Completely revised and enlarged second edition, Stochastic Modelling and Applied Probability, 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [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), 307-319. doi: 10.1142/S0219493705001420. [17] T. Insperger and G. Stépán, "Semi-discretization for Time-delay Systems. Stability and Engineering Applications," Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7. [18] V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations," Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 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] H. Lütkepohl, "Handbook of Matrices," John Wiley & Sons, Ltd., Chichester, 1996. [21] X. Mao, "Stochastic Differential Equations And Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. [22] S. E. A. Mohammed, "Stochastic Functional Differential Equations," Pitman, Boston, 1984. [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), 55-66. [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), 409-429. doi: 10.1080/07362990802679067. [25] M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations, Physical Review E (3), 52 (1995), 3366-3376. doi: 10.1103/PhysRevE.52.3366. [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), 96-110. [27] B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations, Advances in Difference Equations, 2008, Article ID 718408, 21 pp. [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), 267-284. doi: 10.1515/ROSE.2010.015. [29] R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach, Mathematical Biosciences, 196 (2005), 65-81. doi: 10.1016/j.mbs.2005.04.001. [30] R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology, Journal of Mathematical Biology, 57 (2008), 613-648. doi: 10.1007/s00285-008-0178-y. [31] E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Time-delayed feedback in neurosystems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (2009), 1079-1096. doi: 10.1098/rsta.2008.0258. [32] L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect, in "Theory of Stochastic Processes," No. 6 (Russian), 136, "Naukova Dumka," Kiev, (1978), 120-123. [33] L. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Application, 1 (1998), 3-13. [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), 235-253. [35] L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations," Springer, New York, 2011. [36] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.

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##### References:
 [1] L. Arnold, "Stochastic Differential Equations: Theory and Applications," Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1974. [2] M. Bandyopadhyay and J. Chattopadhyay, Ratio-dependent predator-prey model: Effect of environmental fluctuation and stability, Nonlinearity, 18 (2005), 913-936. doi: 10.1088/0951-7715/18/2/022. [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), 269-277. doi: 10.1016/S0378-4754(97)00106-7. [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), 423-438. [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, Article ID 92959, 25 pp. doi: 10.1155/2007/92959. [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), 14593-14598. [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), 842-859. doi: 10.1016/j.apnum.2012.03.002. [8] M. Carletti, On the stability properties of a stochastic model for phage-bacteria interaction in open marine environment, Mathematical Biosciences, 175 (2002), 117-131. doi: 10.1016/S0025-5564(01)00089-X. [9] M. Carletti, Mean square stability of a stochastic model for bacteriophage infection with time delays, Mathematical Biosciences, 210 (2007), 395-414. doi: 10.1016/j.mbs.2007.05.009. [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), 10302-10307. [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, Department of Computer Science, K. U. Leuven, Leuven, Belgium, 2001. [12] T. Erneux, "Applied Delay Differential Equations," Surveys and Tutorials in the Applied Mathematical Sciences, 3, Springer, New York, 2009. [13] K. Green and T. Wagenknecht, Pseudospectra and delay differential equations, Journal of Computational and Applied Mathematics, 196 (2006), 567-578. doi: 10.1016/j.cam.2005.10.011. [14] J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences, 99, Springer-Verlag, New York, 1993. [15] R. Khasminskii, "Stochastic Stability of Differential Equations," With contributions by G. N. Milstein and M. B. Nevelson, Completely revised and enlarged second edition, Stochastic Modelling and Applied Probability, 66, Springer, Heidelberg, 2012. doi: 10.1007/978-3-642-23280-0. [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), 307-319. doi: 10.1142/S0219493705001420. [17] T. Insperger and G. Stépán, "Semi-discretization for Time-delay Systems. Stability and Engineering Applications," Applied Mathematical Sciences, 178, Springer, New York, 2011. doi: 10.1007/978-1-4614-0335-7. [18] V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations," Mathematics and its Applications, 463, Kluwer Academic Publishers, Dordrecht, 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] H. Lütkepohl, "Handbook of Matrices," John Wiley & Sons, Ltd., Chichester, 1996. [21] X. Mao, "Stochastic Differential Equations And Applications," Second edition, Horwood Publishing Limited, Chichester, 2008. [22] S. E. A. Mohammed, "Stochastic Functional Differential Equations," Pitman, Boston, 1984. [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), 55-66. [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), 409-429. doi: 10.1080/07362990802679067. [25] M. C. Mackey and I. G. Nechaeva, Solution moment stability in stochastic differential delay equations, Physical Review E (3), 52 (1995), 3366-3376. doi: 10.1103/PhysRevE.52.3366. [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), 96-110. [27] B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations, Advances in Difference Equations, 2008, Article ID 718408, 21 pp. [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), 267-284. doi: 10.1515/ROSE.2010.015. [29] R. R. Sarkar and S. Banerjee, Cancer self remission and tumor stability - a stochastic approach, Mathematical Biosciences, 196 (2005), 65-81. doi: 10.1016/j.mbs.2005.04.001. [30] R. Schlicht and G. Winkler, A delay stochastic process with applications in molecular biology, Journal of Mathematical Biology, 57 (2008), 613-648. doi: 10.1007/s00285-008-0178-y. [31] E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Time-delayed feedback in neurosystems, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 367 (2009), 1079-1096. doi: 10.1098/rsta.2008.0258. [32] L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect, in "Theory of Stochastic Processes," No. 6 (Russian), 136, "Naukova Dumka," Kiev, (1978), 120-123. [33] L. Shaikhet, Stability of predator-prey model with aftereffect by stochastic perturbations, Stability and Control: Theory and Application, 1 (1998), 3-13. [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), 235-253. [35] L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations," Springer, New York, 2011. [36] H. Smith, "An Introduction to Delay Differential Equations with Applications to the Life Sciences," Texts in Applied Mathematics, 57, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.
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