  Logarithmic growth-rates are fundamental observables for describing
ecological systems and the characterization of their distributions with
analytical techniques can greatly improve their comprehension. Here a neutral
model based on a stochastic differential equation with demographic noise, which
presents a closed form for these distributions, is used to describe the
population dynamics of microbiota. Results show that this model can
successfully reproduce the log-growth rate distribution of the considered
abundance time-series. More significantly, it predicts its temporal dependence,
by reproducing its kurtosis evolution when the time lag $\tau$ is increased.
Furthermore, its typical shape for large $\tau$ is assessed, verifying that the
distribution variance does not diverge with $\tau$. The simulated processes
generated by the calibrated stochastic equation and the analysis of each
time-series, taken one by one, provided additional support for our approach.
Alternatively, we tried to describe our dataset by using a logistic model with
an environmental stochastic term. Analytical and numerical results show that
this model is not suited for describing the leptokurtic log-growth rates
distribution found in our data. These results effectively support a neutral
model with demographic stochasticity for describing the growth-rate dynamics
and the stationary abundance distribution of the considered microbiota. This
suggests that there are no significant parametric demographic differences among
the species, which can be statistically characterized by the same vital rates.
