Definition of the variables and parameters of the chemostat system Eq (1) and of the growth rates Eq (2).

<p>Definition of the variables and parameters of the chemostat system <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0197462#pone.0197462.e001" target="_blank">Eq (1)</a> and of the growth rates <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0197462#pone.0197462.e013" target="_blank">Eq (2)</a>.</p

Phase plane (<i>X</i>₁, <i>X</i>₂) with nullclines for Eq (4) for increasing growth rates: <i>r</i> = [0.02, 0.02] (a), [0.027, 0.027] (b), [0.05, 0.02] (c), [0.05, 0.05](<i>d</i>).

<p>Different steady state configurations are found at the intersections of the nullclines: both species become extinct <i>E</i>, both species survive <i>L</i>₁₂, only one species survives <i>L</i>₁, <i>L</i>₂. Stable solutions are indicated by the solid circle, while unstable saddle solutions are shown by the open circle.</p

Different dynamical regimes for the mutualist-competitive system exist, depending on the values of the parameters.

<p>We define these regimes as follows: R1: Extinction. R2: Competitive exclusion, only species <i>X</i>₁ survives. R3: Competitive exclusion, only species <i>X</i>₂ survives. R4: bistability between extinction and coexistence. R5: bistability between survival of <i>X</i>₁ and coexistence. R6: bistability between survival of <i>X</i>₂ and coexistence. R7: coexistence of <i>X</i>₁ and <i>X</i>₂. (a) Influence of the growth rates <b><i>μ</i></b> in the chemostat system (b) Influence of the growth rates <b><i>μ</i></b> in the extended LV model (c) Influence of the flow rate Φ and the inflow in the chemostat for <b><i>μ</i></b> = [1600, 800]. (d) Influence of the flow rate Φ and the inflow in the extended LV model for <b><i>r</i></b> = [0.04, 0.02].</p

Zoom of Fig 3(b), using the same notation.

<p>The blue dashed lines are the linear approximations of the nullclines and need to intersect in order to have bistability. The regions (1)-(5) are discussed in the text. The blue and red areas correspond to the basins of attraction of each steady state.</p

Time simulations for different values of the growth rates <i>μ</i> illustrate the different behavioral regimes of the system.

<p>For each panel 10 simulations are shown using initial densities varied between 0 and 20. (a) Extinction of the species <i>X</i>₁ and <i>X</i>₂ for all initial densities (<b><i>μ</i></b> = [800, 800]). (b) Bistability: depending on the initial densities the species will either survive (state 1) or become extinct (state 2) (<b><i>μ</i></b> = [1600, 1600]). (c) Bistability: There are two final states possible: coexistence of the species, <i>X</i>₁ ≠ 0 and <i>X</i>₂ ≠ 0 (state 1) or extinction of species <i>X</i>₂ (state 2) (<b><i>μ</i></b> = [2400, 1200]). (d) Survival of <i>X</i>₁ and <i>X</i>₂ for all initial densities (<b><i>μ</i></b> = [2400, 2400]).</p

Schemes of the two-species system for the chemostat model (a) and the simplified model (b).

<p>Microbial species are represented by variables <i>X</i> (blue), nutrients by <i>S</i> (red), arrows represent the consumption or production of a nutrient by a species. In this system <i>X</i>₁ and <i>X</i>₂ compete for the consumption of <i>S</i>₀ and they are mutualistic due to the cross-feeding through nutrients <i>S</i>₁ and <i>S</i>₂. Self-inhibition in the simplified system is determined via the parameter <i>b</i>_<i>ii</i> for species <i>X</i>_<i>i</i> (<i>i</i> = 1, 2), <i>b</i>_<i>ij</i> (<i>i</i> ≠ <i>j</i>) quantifies the strength of mutualism and <i>c</i>_<i>i</i> the strength of competition, <i>d</i>_<i>i</i> is the death rate.</p

Additional file 14: of Signatures of ecological processes in microbial community time series

Figure S12. Variability of noise-type classification across rarefactions. The noise types of 100 taxa selected to be top abundant in one rarefaction were computed for repeated rarefactions in the stool data set of individual A [3]. (PDF 5 kb

Additional file 12: of Signatures of ecological processes in microbial community time series

Figure S10. The accuracy of network inference with LIMITS decreases more strongly when applied to the last 100 than to the first 100 time points. (a) LIMITS accuracy, i.e., mean correlation of inferred and known interaction matrix, for the first 100 time points. (b) LIMITS goodness of fit for the first 100 time points. The goodness of fit was computed as the mean correlation between original and predicted time series. (c) LIMITS accuracy for the last 100 time points. Since gLV time series are constant, no network could be inferred for them. (d) LIMITS goodness of fit for the last 100 time points. The correlation between the goodness of fit to the Ricker model and the intrinsic noise strength observed in noise-free time series is lost. The data points are colored according to the connectance in panels (a) and (c), according to interval in panel (b) and according to the intrinsic noise strength sigma in panel (d). (PDF 17 kb

Additional file 9: of Signatures of ecological processes in microbial community time series

Figure S7. Increasing the time series length improves the accuracy of the test for temporal structure. Noise types were computed for time series sub-sets from 1000 to 1010 (a) and 1000 to 1025 (b) for all data sets with more than 1000 time points. Labels for time series are colored according to the level of non-zero intrinsic noise (sigma) for Ricker, according to the death rate if larger than one for Hubbell, according to the interval if larger than one (with interval coloring taking precedence over sigma) and black otherwise. (PDF 8 kb

Additional file 6: of Signatures of ecological processes in microbial community time series

Figure S4. The noise-type classification and the neutrality test are robust for a wide parameter range in the Hubbell model, but noise types are affected by the death rate. (a) The percentage of taxa with black, brown, pink and white noise types is plotted against the death rate. There is a significant negative correlation between the percentage of brown species and the death rate (Spearman’s rho: − 0.85, p value < 0.000001) and a corresponding positive correlation of the percentage of pink species to the death rate (Spearman’s rho: 0.94, p value < 0.000001). (b) The p values of the neutrality test are plotted against the death rate. (c) The percentage of taxa with black, brown, pink, and white noise types is plotted against the number of individuals. (d) The p values of the neutrality test are plotted against the number of individuals. (d) The percentage of taxa with black, brown, pink, and white noise types is plotted against the immigration rate. (e) The p values of the neutrality test are plotted against the immigration rate. Neutrality is rejected for a p value below 0.05. The p value of 0.05 is indicated by a dashed horizontal line. Time series were generated for 100 species and 3000 time points. For the immigration rate, the percentage of noise types of taxa with non-zero abundances was plotted, since for the low immigration rates tested in this simulation, many taxa have abundances of zero. (PDF 40 kb