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Effective Perron-Frobenius eigenvalue for a correlated random map.

Roman R Pool¹, Manuel O Cáceres

¹Centro Atómico Bariloche, Instituto Balseiro, CONICET, 8400 Bariloche, Argentina.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|January 15, 2011

Summary

This summary is machine-generated.

We analyzed random linear maps and found that their long-term behavior is predictable using a mean-value vector state. This approach accurately characterizes population growth rates in age-structured models with random vital parameters.

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Area of Science:

Dynamical systems
Mathematical biology
Statistical physics

Background:

Random linear maps are fundamental in modeling complex systems.
Understanding the long-term dynamics of systems with disorder is crucial.
Age-structured population models are essential for ecological and demographic studies.

Purpose of the Study:

To investigate the evolution of random positive linear maps under various types of disorder.
To develop a theoretical framework for describing the long-term statistics of these maps.
To apply the developed approach to age-structured population dynamics.

Main Methods:

Analytic perturbation theory
Direct numerical simulations
Green's function diagrammatic technique with noncommutative cumulants

Main Results:

The long-term statistics of random linear maps are accurately described by the mean-value vector state.
An effective Perron-Frobenius eigenvalue characterizes the growth rate, dependent on correlation types.
The approach successfully models population growth rates in age-structured models with random vital parameters, revealing nontrivial cross-correlation effects.

Conclusions:

The mean-value vector state provides a robust method for analyzing random linear maps.
The effective growth rate in age-structured populations is sensitive to correlations in vital parameters.
The study offers a powerful analytical tool for systems with random disorder and correlations.