Some Combinatorial Aspects of Discrete Non-linear Population Dynamics

TL;DR

This paper explores the application of Carleman linearization to analyze discrete non-linear population models, including those with immigration and the logistic map, providing new insights into invariant densities.

Contribution

It extends Carleman linearization techniques to population models with immigration and offers new results on the invariant density of the logistic map.

Findings

Carleman linearization can be applied to models with immigration

New results on the invariant density of the logistic map

Method simplifies analysis of non-linear population dynamics

Abstract

Motivated by issues arising in population dynamics, we consider the problem of iterating a given analytic function a number of times. We use the celebrated technique known as Carleman linearization that turns (for a certain class of functions) this problem into simply taking the power of a real number. We expand this method, showing in particular that it can be used for population models with immigration, and we also apply it to the famous logistic map. We also are able to give a number of results for the invariant density of this map, some being related to the Carleman linearization.