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March  2013, 18(2): 453-465. doi: 10.3934/dcdsb.2013.18.453

Asymptotic behaviour of random tridiagonal Markov chains in biological applications

Peter E. Kloeden 1, and  Victor Kozyakin 2,

1. 

Institut für Mathematik, Goethe Universität, D-60054 Frankfurt am Main
2. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127994 GSP-4

Received  February 2011 Revised  May 2012 Published  November 2012

Discrete-time discrete-state random Markov chains with a tridiagonal generator are shown to have a random attractor consisting of singleton subsets, essentially a random path, in the simplex of probability vectors. The proof uses the Hilbert projection metric and the fact that the linear cocycle generated by the Markov chain is a uniformly contractive mapping of the positive cone into itself. The proof does not involve probabilistic properties of the sample path $\omega$ and is thus equally valid in the nonautonomous deterministic context of Markov chains with, say, periodically varying transitions probabilities, in which case the attractor is a periodic path.
Keywords: linear cocycles, positive cones, Hilbert metric, random attractors., Random Markov chain, uniformly contracting cocycles.
Mathematics Subject Classification: Primary: 15B48, 15B52, 37H10; Secondary: 15B51, 60J10, 92C9.
Citation: Peter E. Kloeden, Victor Kozyakin. Asymptotic behaviour of random tridiagonal Markov chains in biological applications. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 453-465. doi: 10.3934/dcdsb.2013.18.453
References:
[1]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," CRC Press, Boca Raton, FL, second edn., 2011.

[2]

L. Arnold, "Random Synamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[3]

E. Asarin, P. Diamond, I. Fomenko et al., Chaotic phenomena in desynchronized systems and stability analysis, Comput. Math. Appl., 25 (1993), 81-87, doi:10.1016/0898-1221(93)90214-G, URL http://www.sciencedirect.com/science/article/pii/089812219390214G. doi: 10.1016/0898-1221(93)90214-G.

[4]

J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009, reprint of the 1990 edition [MR1048347].

[5]

M. F. Barnsley, A. Vince and D. C. Wilson, Real projective iterated function systems, ArXiv.org e-Print archive, arXiv:1003.3473.

[6]

P. J. Bushell, Hilbert's metric and positive contraction mappings in a Banach space, Arch. Rational Mech. Anal., 52 (1973), 330-338.

[7]

D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.

[8]

I. Chueshov, "Monotone Random Systems Theory and Applications," 1779 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.

[9]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. doi:10.1080/1468936042000207792, URL http://www.tandfonline.com/doi/abs/10.1080/1468936042000207792. doi: 10.1080/1468936042000207792.

[10]

H. Cohn, Products of stochastic matrices and applications, Internat. J. Math. Math. Sci., 12 (1989), 209-233. doi:10.1155/S0161171289000268, URL http://www.hindawi.com/journals/ijmms/1989/656040/abs/. doi: 10.1155/S0161171289000268.

[11]

D. J. Hartfiel, "Markov Set-Chains," 1695 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998.

[12]

D. J. Hartfiel, "Nonhomogeneous Matrix Products," World Scientific Publishing Co. Inc., River Edge, NJ, 2002.

[13]

A. E. Hutzenthaler, "Mathematical Models for Cell-Cell Coomunication on Different Time Scales," Ph.D. thesis, Zentrum Mathematik, Technische Universität München, 2009, URL http://deposit.ddb.de/cgi-bin/dokserv?idn=100332925x&dok_var=d1&dok_ext=pdf&filename=100332925x.pdf.

[14]

P. Imkeller and P. Kloeden, On the computation of invariant measures in random dynamical systems, Stoch. Dyn., 3 (2003), 247-265. doi:10.1142/S0219493703000711, URL http://www.worldscinet.com/sd/03/0302/S0219493703000711.html. doi: 10.1142/S0219493703000711.

[15]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," 176 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011.

[16]

M. A. Krasnosel$'$skij, J. A. Lifshits and A. V. Sobolev, "Positive Linear Systems," 5 of Sigma Series in Applied Mathematics, Heldermann Verlag, Berlin, 1989, The method of positive operators, Translated from the Russian by Jürgen Appell.

[17]

A. Leizarowitz, On infinite products of stochastic matrices, Linear Algebra Appl., 168 (1992), 189-219. http://dx.doi.org/10.1016/0024-3795(92)90294-K doi:10.1016/0024-3795(92)90294-K, URL http://www.sciencedirect.com/science/article/pii/002437959290294K. doi: 10.1016/0024-3795(92)90294-K.

[18]

M. Neumann and H. Schneider, The convergence of general products of matrices and the weak ergodicity of Markov chains, Linear Algebra Appl., 287 (1999), 307-314, http://dx.doi.org/10.1016/S0024-3795(98)10196-9 doi:10.1016/S0024-3795(98)10196-9, URL http://www.sciencedirect.com/science/article/pii/S0024379598101969, special issue celebrating the 60th birthday of Ludwig Elsner. doi: 10.1016/S0024-3795(98)10196-9.

[19]

B. Noble and J. W. Daniel, "Applied Linear Algebra," Prentice-Hall Inc., Englewood Cliffs, N. J., second edn., 1977.

[20]

B. S. Thomson, J. B. Bruckner and A. M. Bruckner, "Elementary Real Analysis," www.classicalrealanalysis.com, second edn., 2008.

[21]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling," World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.

[22]

J. Wolfowitz, Products of indecomposable, aperiodic, stochastic matrices, Proc. Amer. Math. Soc., 14 (1963), 733-737.

show all references

References:
[1]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," CRC Press, Boca Raton, FL, second edn., 2011.

[2]

L. Arnold, "Random Synamical Systems," Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998.

[3]

E. Asarin, P. Diamond, I. Fomenko et al., Chaotic phenomena in desynchronized systems and stability analysis, Comput. Math. Appl., 25 (1993), 81-87, doi:10.1016/0898-1221(93)90214-G, URL http://www.sciencedirect.com/science/article/pii/089812219390214G. doi: 10.1016/0898-1221(93)90214-G.

[4]

J.-P. Aubin and H. Frankowska, "Set-valued Analysis," Modern Birkhäuser Classics, Birkhäuser Boston Inc., Boston, MA, 2009, reprint of the 1990 edition [MR1048347].

[5]

M. F. Barnsley, A. Vince and D. C. Wilson, Real projective iterated function systems, ArXiv.org e-Print archive, arXiv:1003.3473.

[6]

P. J. Bushell, Hilbert's metric and positive contraction mappings in a Banach space, Arch. Rational Mech. Anal., 52 (1973), 330-338.

[7]

D. N. Cheban, P. E. Kloeden and B. Schmalfuß, The relationship between pullback, forward and global attractors of nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.

[8]

I. Chueshov, "Monotone Random Systems Theory and Applications," 1779 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2002.

[9]

I. Chueshov and M. Scheutzow, On the structure of attractors and invariant measures for a class of monotone random systems, Dyn. Syst., 19 (2004), 127-144. doi:10.1080/1468936042000207792, URL http://www.tandfonline.com/doi/abs/10.1080/1468936042000207792. doi: 10.1080/1468936042000207792.

[10]

H. Cohn, Products of stochastic matrices and applications, Internat. J. Math. Math. Sci., 12 (1989), 209-233. doi:10.1155/S0161171289000268, URL http://www.hindawi.com/journals/ijmms/1989/656040/abs/. doi: 10.1155/S0161171289000268.

[11]

D. J. Hartfiel, "Markov Set-Chains," 1695 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1998.

[12]

D. J. Hartfiel, "Nonhomogeneous Matrix Products," World Scientific Publishing Co. Inc., River Edge, NJ, 2002.

[13]

A. E. Hutzenthaler, "Mathematical Models for Cell-Cell Coomunication on Different Time Scales," Ph.D. thesis, Zentrum Mathematik, Technische Universität München, 2009, URL http://deposit.ddb.de/cgi-bin/dokserv?idn=100332925x&dok_var=d1&dok_ext=pdf&filename=100332925x.pdf.

[14]

P. Imkeller and P. Kloeden, On the computation of invariant measures in random dynamical systems, Stoch. Dyn., 3 (2003), 247-265. doi:10.1142/S0219493703000711, URL http://www.worldscinet.com/sd/03/0302/S0219493703000711.html. doi: 10.1142/S0219493703000711.

[15]

P. E. Kloeden and M. Rasmussen, "Nonautonomous Dynamical Systems," 176 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2011.

[16]

M. A. Krasnosel$'$skij, J. A. Lifshits and A. V. Sobolev, "Positive Linear Systems," 5 of Sigma Series in Applied Mathematics, Heldermann Verlag, Berlin, 1989, The method of positive operators, Translated from the Russian by Jürgen Appell.

[17]

A. Leizarowitz, On infinite products of stochastic matrices, Linear Algebra Appl., 168 (1992), 189-219. http://dx.doi.org/10.1016/0024-3795(92)90294-K doi:10.1016/0024-3795(92)90294-K, URL http://www.sciencedirect.com/science/article/pii/002437959290294K. doi: 10.1016/0024-3795(92)90294-K.

[18]

M. Neumann and H. Schneider, The convergence of general products of matrices and the weak ergodicity of Markov chains, Linear Algebra Appl., 287 (1999), 307-314, http://dx.doi.org/10.1016/S0024-3795(98)10196-9 doi:10.1016/S0024-3795(98)10196-9, URL http://www.sciencedirect.com/science/article/pii/S0024379598101969, special issue celebrating the 60th birthday of Ludwig Elsner. doi: 10.1016/S0024-3795(98)10196-9.

[19]

B. Noble and J. W. Daniel, "Applied Linear Algebra," Prentice-Hall Inc., Englewood Cliffs, N. J., second edn., 1977.

[20]

B. S. Thomson, J. B. Bruckner and A. M. Bruckner, "Elementary Real Analysis," www.classicalrealanalysis.com, second edn., 2008.

[21]

D. Wodarz and N. Komarova, "Computational Biology of Cancer: Lecture Notes and Mathematical Modeling," World Scientific Publishing Co. Pte. Ltd., Singapore, 2005.

[22]

J. Wolfowitz, Products of indecomposable, aperiodic, stochastic matrices, Proc. Amer. Math. Soc., 14 (1963), 733-737.

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