Averaging in random systems of nonnegative matrices

TL;DR

This paper proves that the top Lyapunov exponent of a certain class of random nonnegative matrix systems is bounded below by that of the averaged system, providing insights into stability properties of such systems.

Contribution

It establishes a lower bound for the top Lyapunov exponent in systems combining nonnegative matrices with random diagonal matrices, a novel theoretical result.

Findings

The top Lyapunov exponent of the system is bounded below by that of the averaged system.

The result contrasts with biological metapopulation models, indicating different stability behavior.

Provides a mathematical foundation for analyzing stability in random matrix systems.

Abstract

It is proved that for 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.