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Invariance and monotonicity for stochastic delay differential equations
A note on the analysis of asymptotic meansquare stability properties for systems of linear stochastic delay differential equations
1.  Johannes Kepler University, Institute for Stochastics, Altenbergerstraße 69, 4040 Linz, Austria, Austria 
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications,", WileyInterscience [John Wiley & Sons], (1974). Google Scholar 
[2] 
M. Bandyopadhyay and J. Chattopadhyay, Ratiodependent predatorprey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913. doi: 10.1088/09517715/18/2/022. 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/S03784754(97)001067. Google Scholar 
[4] 
J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delaydifferential 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, Delayinduced 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 meansquare stability for multidimensional 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 phagebacteria interaction in open marine environment,, Mathematical Biosciences, 175 (2002), 117. doi: 10.1016/S00255564(01)00089X. 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, DDEBIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations,, Techreport TW330, (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: 10.1007/9783642232800. 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, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011). doi: 10.1007/9781461403357. 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 virustumorimmune 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/s002850080178y. Google Scholar 
[31] 
E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Timedelayed 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 predatorprey 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: 10.1007/9781441976468. Google Scholar 
show all references
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications,", WileyInterscience [John Wiley & Sons], (1974). Google Scholar 
[2] 
M. Bandyopadhyay and J. Chattopadhyay, Ratiodependent predatorprey model: Effect of environmental fluctuation and stability,, Nonlinearity, 18 (2005), 913. doi: 10.1088/09517715/18/2/022. 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/S03784754(97)001067. Google Scholar 
[4] 
J. Boulet, R. Balasubramaniam, A. Daffertshofer and A. Longtin, Stochastic two delaydifferential 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, Delayinduced 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 meansquare stability for multidimensional 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 phagebacteria interaction in open marine environment,, Mathematical Biosciences, 175 (2002), 117. doi: 10.1016/S00255564(01)00089X. 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, DDEBIFTOOL v. 2.00: A Matlab package for bifurcation analysis of delay differential equations,, Techreport TW330, (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: 10.1007/9783642232800. 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, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011). doi: 10.1007/9781461403357. 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 virustumorimmune 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/s002850080178y. Google Scholar 
[31] 
E. Schöll, G. Hiller, P. Hövel and M. Dahlem, Timedelayed 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 predatorprey 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: 10.1007/9781441976468. Google Scholar 
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