Growth-rate distributions at stationarity

TL;DR

This paper introduces new analytical tools to describe stationary growth-rate distributions, revealing that deviations from normality are due to statistical factors rather than pathological behavior, and provides models for macroecological patterns.

Contribution

It offers a general framework for understanding growth-rate distributions, emphasizing the role of statistical considerations over traditional assumptions of normality, and introduces a pragmatic workflow for model selection.

Findings

Growth-rate distributions can be described by a generalized logistic distribution.

Deviations from normality are explained by statistical considerations, not pathology.

Large time lags lead to time-independent, finite variance growth-rate distributions.

Abstract

We propose new analytical tools for describing growth-rate distributions generated by stationary time-series. Our analysis shows how deviations from normality are not pathological behaviour, as suggested by some traditional views, but instead can be accounted for by clean and general statistical considerations. In contrast, strict normality is the effect of specific modelling choices. Systems characterized by stationary Gamma or heavy-tailed abundance distributions produce log-growth-rate distributions well described by a generalized logistic distribution, which can describe tent-shaped or nearly normal datasets and serves as a useful null model for these observables. These results prove that, for large enough time lags, in practice, growth-rate distributions cease to be time-dependent and exhibit finite variance. Based on this analysis, we identify some key stylized macroecological patterns and specific stochastic differential equations capable of reproducing them. A pragmatic workflow for heuristic selection between these models is then introduced. This approach is particularly useful for systems with limited data-tracking quality, where applying sophisticated inference methods is challenging.