Giant fluctuations in logistic growth

TL;DR

This paper investigates the stochastic fluctuations in the populations of two competing species undergoing logistic growth, revealing giant fluctuations near carrying capacity when their growth rates are similar, and emphasizing the importance of stochastic analysis.

Contribution

It demonstrates the occurrence of giant fluctuations in competing species with logistic growth and clarifies when deterministic models are insufficient, extending to various growth laws.

Findings

Giant fluctuations occur when species have similar growth rates.

Deterministic models are valid only with significantly different growth rates.

Results apply to a broad class of growth laws.

Abstract

We analyze the fluctuation of the number of individuals when two competing species, beginning with a few initial individuals, are submitted to a logistic growth. We show that when the total number of individuals reaches the carrying capacity, the number of each species is subject to giant fluctuations (variance $\sim$ mean 2) if the two species have similar growth rate. We show that the deterministic logistic equation can be used only when the growth rates are significantly different, otherwise such growth has to be investigated by stochastic processes tools. These results generalize to a wide class of growth law.