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.
Publié le : 1994-08-14
Classification:
Evolutionary theory,
random dynamical system,
products of random matrices,
Perron-Frobenius theory,
Markov chain in a random environment,
thermodynamic formalism,
Gibbs measures,
variational principle,
equilibrium states,
28D99,
58F11,
92D15,
60J10,
54H20,
92D25
@article{1177004975,
author = {Arnold, Ludwig and Gundlach, Volker Matthias and Demetrius, Lloyd},
title = {Evolutionary Formalism for Products of Positive Random Matrices},
journal = {Ann. Appl. Probab.},
volume = {4},
number = {4},
year = {1994},
pages = { 859-901},
language = {en},
url = {http://dml.mathdoc.fr/item/1177004975}
}
Arnold, Ludwig; Gundlach, Volker Matthias; Demetrius, Lloyd. Evolutionary Formalism for Products of Positive Random Matrices. Ann. Appl. Probab., Tome 4 (1994) no. 4, pp. 859-901. http://gdmltest.u-ga.fr/item/1177004975/