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

Averaging in random systems of nonnegative matrices

Janusz Mierczyński 1,

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.
Keywords: nonnegative matrix, averaging, Lyapunov exponent, Random matrix system, metapopulation growth rate..
Mathematics Subject Classification: Primary: 37H15, 15B48; Secondary: 92D2.
Citation: Janusz Mierczyński. Averaging in random systems of nonnegative matrices. Conference Publications, 2015, 2015 (special) : 835-840. doi: 10.3934/proc.2015.0835
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998.

[2]

L. Arnold, V. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., 4(3) (1994), 859-901.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, revised reprint of the 1979 original, Classics Appl. Math., 9, SIAM, Philadelphia, PA, 1994.

[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(11) (2015), 2052-2081.

[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(6) (2001), 1669-1679.

[6]

J. Mierczyński, Estimates for principal Lyapunov exponents: A survey, Nonauton. Dyn. Syst., 1(1) (2014), 137-162,

[7]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.

[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(2) (2013), 438-458.

[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), 1907-1914.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monogr. Math., Springer, Berlin, 1998.

[2]

L. Arnold, V. M. Gundlach and L. Demetrius, Evolutionary formalism for products of positive random matrices, Ann. Appl. Probab., 4(3) (1994), 859-901.

[3]

A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, revised reprint of the 1979 original, Classics Appl. Math., 9, SIAM, Philadelphia, PA, 1994.

[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(11) (2015), 2052-2081.

[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(6) (2001), 1669-1679.

[6]

J. Mierczyński, Estimates for principal Lyapunov exponents: A survey, Nonauton. Dyn. Syst., 1(1) (2014), 137-162,

[7]

J. Mierczyński and W. Shen, Spectral Theory for Random and Nonautonomous Parabolic Equations and Applications, Chapman Hall/CRC Monogr. Surv. Pure Appl. Math., Chapman & Hall/CRC, Boca Raton, FL, 2008.

[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(2) (2013), 438-458.

[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), 1907-1914.

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