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Evolutionary Formalism for Products of Positive Random Matrices

Ludwig Arnold, Volker Matthias Gundlach and Lloyd Demetrius
The Annals of Applied Probability
Vol. 4, No. 3 (Aug., 1994), pp. 859-901
Stable URL: http://www.jstor.org/stable/2245067
Page Count: 43
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Evolutionary Formalism for Products of Positive Random Matrices
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Abstract

We present a formalism to investigate directionality principles in evolution theory for populations, the dynamics of which can be described by a positive matrix cocycle (product of random positive matrices). For the latter, we establish a random version of the Perron-Frobenius theory which extends all known results and enables us to characterize the equilibrium state of a corresponding abstract symbolic dynamical system by an extremal principle. We develop a thermodynamic formalism for random dynamical systems, and in this framework prove that the top Lyapunov exponent is an analytic function of the generator of the cocycle. On this basis a fluctuation theory for products of positive random matrices can be developed which leads to an inequality in dynamical entropy that can be interpreted as a directionality principle for the mutation and selection process in evolutionary dynamics.

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