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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). 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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). 
[12] 
T. Erneux, "Applied Delay Differential Equations,", Surveys and Tutorials in the Applied Mathematical Sciences, 3 (2009). 
[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. 
[14] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993). 
[15] 
R. Khasminskii, "Stochastic Stability of Differential Equations,", With contributions by G. N. Milstein and M. B. Nevelson, 66 (2012). doi: 10.1007/9783642232800. 
[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. 
[17] 
T. Insperger and G. Stépán, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011). doi: 10.1007/9781461403357. 
[18] 
V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations,", Mathematics and its Applications, 463 (1999). 
[19] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). 
[20] 
H. Lütkepohl, "Handbook of Matrices,", John Wiley & Sons, (1996). 
[21] 
X. Mao, "Stochastic Differential Equations And Applications,", Second edition, (2008). 
[22] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Pitman, (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. 
[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. 
[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. 
[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. 
[27] 
B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations,, Advances in Difference Equations, 2008 (7184). 
[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. 
[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. 
[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. 
[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. 
[32] 
L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect,, in, 136 (1978), 120. 
[33] 
L. Shaikhet, Stability of predatorprey model with aftereffect by stochastic perturbations,, Stability and Control: Theory and Application, 1 (1998), 3. 
[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. 
[35] 
L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations,", Springer, (2011). 
[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. 
show all references
References:
[1] 
L. Arnold, "Stochastic Differential Equations: Theory and Applications,", WileyInterscience [John Wiley & Sons], (1974). 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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. 
[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). 
[12] 
T. Erneux, "Applied Delay Differential Equations,", Surveys and Tutorials in the Applied Mathematical Sciences, 3 (2009). 
[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. 
[14] 
J. K. Hale and S. M. Verduyn Lunel, "Introduction to Functional Differential Equations,", Applied Mathematical Sciences, 99 (1993). 
[15] 
R. Khasminskii, "Stochastic Stability of Differential Equations,", With contributions by G. N. Milstein and M. B. Nevelson, 66 (2012). doi: 10.1007/9783642232800. 
[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. 
[17] 
T. Insperger and G. Stépán, "Semidiscretization for Timedelay Systems. Stability and Engineering Applications,", Applied Mathematical Sciences, 178 (2011). doi: 10.1007/9781461403357. 
[18] 
V. B. Kolmanovskii and A. Myshkis, "Introduction to the Theory and Applications of Functional Differential Equations,", Mathematics and its Applications, 463 (1999). 
[19] 
Y. Kuang, "Delay Differential Equations with Applications in Population Dynamics,", Mathematics in Science and Engineering, 191 (1993). 
[20] 
H. Lütkepohl, "Handbook of Matrices,", John Wiley & Sons, (1996). 
[21] 
X. Mao, "Stochastic Differential Equations And Applications,", Second edition, (2008). 
[22] 
S. E. A. Mohammed, "Stochastic Functional Differential Equations,", Pitman, (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. 
[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. 
[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. 
[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. 
[27] 
B. Paternoster and L. Shaikhet, Stability of equilibrium points of fractional difference equations with stochastic perturbations,, Advances in Difference Equations, 2008 (7184). 
[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. 
[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. 
[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. 
[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. 
[32] 
L. Shaikhet, Equations for determining the moments of solutions of linear stochastic differential equations with aftereffect,, in, 136 (1978), 120. 
[33] 
L. Shaikhet, Stability of predatorprey model with aftereffect by stochastic perturbations,, Stability and Control: Theory and Application, 1 (1998), 3. 
[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. 
[35] 
L. Shaikhet, "Lyapunov Functionals and Stability of Stochastic Difference Equations,", Springer, (2011). 
[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. 
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