Heavy-tailed abundance distributions from stochastic Lotka-Volterra models

Microbial communities found in nature are composed of many rare species and few abundant ones, as reflected by their heavy-tailed abundance distributions. How a large number of species can coexist in those complex communities and why they are dominated by rare species is still not fully understood. We show how heavy-tailed distributions arise as an emergent property from large communities with many interacting species in population-level models. To do so, we rely on generalized Lotka-Volterra models for which we introduce a global maximal capacity. This maximal capacity accounts for the fact that communities are limited by available resources and space. In a parallel ad hoc approach, we obtain heavy-tailed abundance distributions from logistic models, without interactions, through specific distributions of the parameters. We expect both mechanisms, interactions between many species and specific parameter distributions, to be relevant to explain the observed heavy tails.

Figure 1

Properties of experimental data. We selected one time series, stool B, for illustrative purposes and refer to additional figures in the Supplemental Material [6] for time series listed in Figs. S1, S2, S3, and S4. (a) Experimental data have heavy-tailed distributions. The log-normal distribution (full line) fits best. The power law (dash-dotted) and exponential (dotted) distributions are rejected for most communities because the $p$ values of the KS test are lower than 0.05 (lines are colored in blue). The widths $s$ of the log-normal distribution range between 1.42 and 2.78. For data coming from a time series, we considered the first time point. (b) Coefficient of variation of longitudinal experimental data: Longitudinal experimental data of microbial communities have coefficients of variation around one. For species with a large coefficient of variation, we can reject the hypothesis that the population follows a $\mathrm{\Gamma }$ distribution. The dotted horizontal lines denote the median coefficient of variation per community. (c) The color code for the KS test shown in panels (a), (b), and (e). (d) Jensen-Shannon distance of longitudinal experimental data: Jensen-Shannon (JS) distance of longitudinal experimental data as a function of the time interval between both compositions. The median value (red line) ranges from 0.3 to 0.4. The Pearson correlation function and $p$ value for noncorrelation are given. (e) Stability of heavy-tailed distribution of experimental data: The species abundances of microbial communities show large fluctuations over time, but the width of the abundance distribution remains nearly stable over time. The gray horizontal band extends from 1 to 3. The color of the dots shows the $p$ value of the KS test to a log-normal distributions; blue values are poor fits. The log-normal fits for which the width is higher than 3 can all be rejected. (f) The time traces of species with ranks 1, 5, 10, and 50.

Figure 2

A finite maximal capacity and nonzero immigration rate allow for high diversity in the presence of interactions (a). Notice that the maximal capacity does not limit the number of species; e.g., there can be 200 species even when $N_{\max }=100$ because the species abundances can be smaller than 1 in population-level models. For decreasing maximal capacities, the abundance distribution becomes log normal (b). To assess the quality of the different fits, we used a method described in section Fit distributions (see Supplemental Material [6]). The gray area represents distributions that are not log normal, power law, or normal. The width of this log-normal distribution increases for decreasing immigration and increasing connectivity (C). Note that the color code has been chosen such that the white range corresponds to experimental values.

Figure 3

Heavy-tailed distributions for logistic equations, the influence of the maximal capacity and noise strength. (a) Abundances mostly fit a log-normal distribution best, with a tendency to normal distributions for small noise (not shown). The width of the log-normal distribution decreases in this direction (b). The species diversity is maintained except for small maximal capacities and high noise (b). The average coefficient of variation (c) and average JS distance (d) increase for decreasing maximal capacities and increasing levels of noise. Because logistic equations are used for these plots, the fixed parameters are the interaction strength $a=0$, the connectance $C=0$, and immigration rate $\lambda =0$. Note that the color code has been chosen such that the white range corresponds to experimental values.

Figure 4

The ratio of uniforms distribution in a linear scale on the left and in a log-log scale on the right. In the right plot, a log-normal distribution is drawn for comparison.

Figure 5

Log-normal parameter distributions for the logistic equation lead to log-normal abundance distributions. (a) The width of the abundance distribution $s$ does not depend on the width of the growth rate ${\sigma }_g$. (b) For large noise and large width of the growth rate, species go extinct. The parameters of the fluctuation, average coefficient of variation (c) and average JS distance (d), are negatively correlated with respect to the width of the growth rate distribution. They both increase for increasing noise. The fixed parameters of these plots are the imposed width of the abundance distribution $s=2$, the immigration $\lambda =0$, the interaction strength $a=0$, and connectivity $C=0$.