Long-term Averages of the Stochastic Logistic Map

TL;DR

This paper investigates how stochastic variations in the logistic map's parameters affect long-term population behavior, showing that noise can either benefit or harm the population depending on the parameter range.

Contribution

It provides a formal analysis of the relationship between stochastic and deterministic logistic maps, including proofs and a conjecture based on numerical evidence.

Findings

Noise can increase the mean population in some cases.

Noise can decrease the mean population in other cases.

Numerical evidence supports the conjecture about long-term behavior.

Abstract

The logistic map is a nonlinear difference equation well studied in the literature, used to model self-limiting growth in certain populations. It is known that, under certain regularity conditions, the stochastic logistic map, where the parameter is varied according to a specified distribution, has a unique invariant distribution. In these cases we can compare the long-term behavior of the stochastic system with that of the deterministic system evaluated at the average parameter value. Here we examine the relationship between the mean of the stochastic logistic equation and the mean of orbits of the deterministic logistic equation at the expected value of the parameter. We formally prove that, in some cases, the addition of noise is beneficial to the populations, in the sense that it increases the mean, while for other ranges of parameters it is detrimental. A conjecture based on numerical evidence is presented at the end.