Growth-rate distributions at stationarity

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.

Figures (3)

• Figure 1: A decision tree illustrating our classification framework.
• Figure 2: On the left: Columns correspond to the abundance distribution (I), the LGR distribution at $\tau=1$ (II), and the LGR distribution at $\tau>>1$ (III) for typical time-series with abundance distribution best fitted by a Gamma (a), a Lognormal (b) and an Inverse (c) distribution. The solid lines represent the fits: Gamma (black), Lognormal (green), and Inverse (red). In column II the solid black lines represent the fits with $P^*(g,\tau)$, in column III the fits with $P_\infty(g)$. Yellow lines are adjusted using a Normal distribution. On the right: column IV shows typical examples of the different behaviours of the evolution of $Var_P=Var[P(g,\tau)]$ as a function of $\tau$. For more details, see SuppMat.
• Figure 3: On the left: In this scatter plot blue points represent the $\hat{\alpha}_{\tau=1}$ versus $\hat{\alpha}_{\tau>>1}$. Green points stand for $\hat{\alpha}_{\tau=1}$ versus $\hat{\alpha}_{Ab}$. The Pearson correlation is equal to 0.73 and 0.76, respectively. On the right: Red points stand for $\hat{\alpha}_{\tau>>1}$ versus $\hat{\alpha}_{Ab}$. The Pearson correlation is equal to 0.98. The solid lines are $x=y$.