2015, 2015(special): 835-840. doi: 10.3934/proc.2015.0835

Averaging in random systems of nonnegative matrices

1. 

Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wybrzeże Wyspiańskiego 27, PL-50-370 Wrocław, Poland

Received  August 2014 Revised  April 2015 Published  November 2015

It is proved that the top Lyapunov exponent of a random matrix system of the form $\{A D(\omega)\}$, where $A$ is a nonnegative matrix and $D(\omega)$ is a diagonal matrix with positive diagonal entries, is bounded from below by the top Lyapunov exponent of the averaged system. This is in contrast to what one should expect of systems describing biological metapopulations.
Citation: Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835
References:
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L. Arnold, Random Dynamical Systems,, Springer Monogr. Math., (1998).   Google Scholar

[2]

L. Arnold, V. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices,, Ann. Appl. Probab., 4 (1994), 859.   Google Scholar

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, revised reprint of the 1979 original, 9 (1979).   Google Scholar

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G. Froyland, C. González-Tokman and A. Quas, Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles,, Comm. Pure Appl. Math., 68 (2015), 2052.   Google Scholar

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J. Mierczyński, Estimates for principal Lyapunov exponents: A survey,, Nonauton. Dyn. Syst., 1 (2014), 137.   Google Scholar

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J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., (2008).   Google Scholar

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S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence,, Proc. Roy. Soc. Ser. B Biol. Sci., 277 (2010), 1907.   Google Scholar

show all references

References:
[1]

L. Arnold, Random Dynamical Systems,, Springer Monogr. Math., (1998).   Google Scholar

[2]

L. Arnold, V. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices,, Ann. Appl. Probab., 4 (1994), 859.   Google Scholar

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, revised reprint of the 1979 original, 9 (1979).   Google Scholar

[4]

G. Froyland, C. González-Tokman and A. Quas, Stochastic stability of Lyapunov exponents and Oseledets splittings for semi-invertible matrix cocycles,, Comm. Pure Appl. Math., 68 (2015), 2052.   Google Scholar

[5]

V. Hutson, W. Shen and G. T. Vickers, Estimates for the principal spectrum point for certain time-dependent parabolic operators,, Proc. Amer. Math. Soc., 129 (2001), 1669.   Google Scholar

[6]

J. Mierczyński, Estimates for principal Lyapunov exponents: A survey,, Nonauton. Dyn. Syst., 1 (2014), 137.   Google Scholar

[7]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications,, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., (2008).   Google Scholar

[8]

J. Mierczyński and W. Shen, Principal Lyapunov exponents and principal Floquet spaces of positive random dynamical systems. II. Finite-dimensional case,, J. Math. Anal. Appl., 404 (2013), 438.   Google Scholar

[9]

S. J. Schreiber, Interactive effects of temporal correlations, spatial heterogeneity and dispersal on population persistence,, Proc. Roy. Soc. Ser. B Biol. Sci., 277 (2010), 1907.   Google Scholar

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