Figures and data in The dominant–egalitarian transition in species-rich communities

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5 figures, 1 table and 1 additional file

Figures

Figure 1 with 1 supplement

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Probability distribution of species abundance.

Red circles: results of a simulation of the process described in Equation 1 with parameters $S = 200$ , $μ = 0.6$ , $K = 2 \cdot 10^{6}$ , ${\tilde{σ}}_{e} = 0.1$ and $τ = 8$ . The values of the extracted parameters for Equation 4 are $c = 0.0034$ and ${\tilde{σ}}_{e}^{2} = 0.0131$ . The solid line in blue is the expression of Equation 4 with parameters $α = 91.2$ and $β = 1.56$ .

Figure 1—figure supplement 1

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Species abundances (left), log-abundance (middle), and the egality parameter $S_{1 / 2} / S$ (right) for the model considered in the main text of van Nes et al., 2024, Parameters are $S = 100$ , ${\tilde{σ}}_{e} = 0.005$ and $K = 100$ .

The dynamics of Equation SA2 was integrated. The log-abundance of all species performs a random walk (middle), hence, as time goes by, more and more species get stuck at negligible abundances (this is the diffusive trapping, or stickiness of the rare state, left), and only a single species dominates the community and the egality parameter decreases to $1 / S$ .

Figure 2

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The egaility parameter as a function of the number of species.

Left panel (A): the egality parameter $S_{1 / 2} / S$ is plotted against $1 / S$ , the total number of species, for ${\tilde{σ}}_{e} = 0.015$ (blue) 0.02 (red) 0.04 (orange) and 0.08 (green). For all these simulations $μ = 0.6$ , $τ = 8$ and $K = 10^{4}$ . The values of the parameter $β$ , as extracted from the simulation, are plotted (again, vs. $1 / S$ ) in the inset. As expected, for $β > 2$ (blue) the egality parameter $S_{1 / 2} / S$ extrapolates, at $S \to \infty$ , to a finite value, meaning that the fraction of high-abundance species is finite and the community is egalitarian. This feature is also reflected in the right panel (B), where at large $S$ , $S_{1 / 2} \sim 0.04 S$ for ${\tilde{σ}}_{e} = 0.02$ (red) and $S_{1 / 2} \sim 0.08 S$ for ${\tilde{σ}}_{e} = 0.015$ (blue). On the other hand, when ${\tilde{σ}}_{e} = 0.08$ (green), the value of $β$ approaches 1.8 and is definitely smaller than two. In addition, the egality parameter extrapolates to zero at large $S$ , meaning that the community is dominated by a small number of species. The results shown in the right panel suggest $S_{1 / 2} \sim S^{0.3}$ . The case ${\tilde{σ}}_{e} = 0.04$ (orange) represents a near marginal case, for which $β$ extrapolates to values slightly smaller than 2, and it would appear that $S_{1 / 2} / S$ is tending toward zero. Indeed, the right panel shows a power law with exponent smaller than unity and decreasing with increasing ${\tilde{σ}}_{e}$ .

Figure 3

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Species abundance distributions, as obtained for a few diverse communities.

The gut microbial community (OTUs of subject A from David et al., 2014) fits our formula Equation 4 quite well. Immigration is extremely weak, but the sampling power is strong enough to reveal some of the effects of the decrease in the number of species at low densities due to immigration. The corresponding distribution for the oceanic prokaryote population Eguíluz et al., 2019 is very close to a pure power law, possibly because the sampling is not deep enough (see text). In both cases, $β < 2$ , indicating that the community is dominated by only a few species. The distributions for the global bird population (Callaghan et al., 2021) and tropical trees in the Amazon Basin (Cooper et al., 2024) show a clear transition between two power-law behaviors. As explained in the text, for these macroorganisms the effect of demographic stochasticity must be taken into account. When this is done (see ‘SI D’), we obtain an excellent fit for the results using the corrected formula. Equation 6. For birds, we find $β \approx 2$ , placing them on the margin between egalitarian and dominance communities. For tropical trees in the Amazon Basin, $β$ is definitely larger than two, indicating that the community is indeed egalitarian. Note the similarities between the values of $S_{1 / 2} / S$ in the empirical results and the numerical experiments in Figure 2. For more information, see ‘SI E’.

Figure 4 with 1 supplement

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From stochasticity-driven to chaotic dynamics.

$S_{1 / 2}$ , as obtained from simulations of Equation 7, is plotted against $ϵ$ , for $N = 200$ , $K = 2 \cdot 10^{6}$ and $μ = 0.6$ . For ${\tilde{σ}}_{e} = 0.01$ (blue circles), the system is in the egalitarian phase and the main effect of the transition to chaos is an effective enhancement of the strength of stochasticity that makes the system less egalitarian, hence $S_{1 / 2}$ decreases. The inverse effect is observed for $σ_{e} = 0.08$ (red squares). For large values of $ϵ$ , the effect of $σ_{e}$ becomes negligible and the dependence of $S_{1 / 2}$ on $σ_{e}$ weakens significantly. The data was obtained by averaging over at least 20 different realizations of the interaction matrix $α_{i, j}$ .

Figure 4—figure supplement 1

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Time traces of species abundances for $ϵ = 0$ (left panel, time-averaged neutral) and $ϵ = 0.05$ (right panel, chaotic), with $S = 200$ , $K = 10^{4} S$ , and $τ = 8$ .

Since ${\tilde{σ}}_{e} = 0.01$ , the system is in its egalitarian phase when $ϵ = 0$ , and the abundance variations span approximately two orders of magnitude. In the chaotic phase (right panel), these variations span the entire range between $K$ and $μ = 0.6$ . Another notable feature is the emergence of cliques of high-abundance species in the chaotic phase, whereas in the stochastic phase there is a clear hierarchy of dominance. As a result, the chaotic phase exhibits a characteristic ‘shoulder’ of high-abundance species, as discussed in the main text.

Figure 5

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$\ln P (n)$ vs. $\ln n$ for $ϵ = {0 (solid blue line), 0.05 (dashed red line)}$ , ${\tilde{σ}}_{e} = 0.01$ .

The distribution for $ϵ = 0$ is typical of the egalitarian phase with a large slope for large $n$ . The distribution for $ϵ = 0.05$ , however, has the slower falloff characteristic of the dominance phase. In addition, there is a shoulder for very large $n$ , which is due to the existence of multiple sets of several transient quasi-stable species, arising from the chaotic dynamics present in the absence of environmental stochasticity.

Tables

Appendix 1—table 1

Summary of fitted models and parameter estimates (with 1σ uncertainties) for the four systems analyzed.

System (panel)	Fitted equation	Fitted parameters (±1σ)
Gut microbiome (a)	Fit to Equation 4 with $α = 2 μ / σ_{e}^{2}$ , $β = 1 + 2 c / σ_{e}^{2}$ , A normalization factor	$A = 1.77 \pm 0.060$ ; $α = 5.5 \times 10^{- 6} \pm 5 \times 10^{- 7}$ ; $β = 1.31 \pm 0.01$
Plankton (b)	Fit to Equation 4 with $α = 2 μ / σ_{e}^{2}$ , $β = 1 + 2 c / σ_{e}^{2}$ , A normalization factor	$A = 6.7 \pm 0.2$ ; $α = 0.8 \pm 0.5$ ; $β = 1.7 \pm 0.03$
Birds (c)	Fit to Equation 6 with $γ = 2 μ / σ_{d}^{2}$ , $β = 1 + 2 c / σ_{e}^{2}$ , $k = σ_{d}^{2} / σ_{e}^{2}$ , A normalization factor	$A = 23 \pm 1$ ; $γ = 0.34 \pm 0.01$ ; $β = 2.05 \pm 0.05$ ; $k = 3.0 \times 10^{6} \pm 6 \times 10^{5}$
Amazon trees (d)	$A = 14 \pm 1$ ; $γ = 2 μ / σ_{d}^{2}$ , $β = 1 + 2 c / σ_{e}^{2}$ , $k = σ_{d}^{2} / σ_{e}^{2}$ , A normalization factor	$A = 14 \pm 1$ ; $γ = 0.013 \pm 0.096$ ; $β = 2.4 \pm 0.2$ ; $k = 380 \pm 200$

Additional files

MDAR checklist: https://cdn.elifesciences.org/articles/103999/elife-103999-mdarchecklist1-v2.pdf
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David A Kessler
Nadav M Shnerb

(2025)

The dominant–egalitarian transition in species-rich communities

eLife 14:e103999.

https://doi.org/10.7554/eLife.103999

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Probability distribution of species abundance.

Species abundances (left), log-abundance (middle), and the egality parameter S1/2/S (right) for the model considered in the main text of van Nes et al., 2024, Parameters are S=100, σ~e=0.005 and K=100.

The egaility parameter as a function of the number of species.

Species abundance distributions, as obtained for a few diverse communities.

From stochasticity-driven to chaotic dynamics.

Time traces of species abundances for ϵ=0 (left panel, time-averaged neutral) and ϵ=0.05 (right panel, chaotic), with S=200, K=104S, and τ=8.

ln⁡P(n) vs.ln⁡n for ϵ={0 (solid blue line),0.05 (dashed red line)}, σ~e=0.01.

Summary of fitted models and parameter estimates (with 1σ uncertainties) for the four systems analyzed.

MDAR checklist

Downloads (link to download the article as PDF)

Open citations (links to open the citations from this article in various online reference manager services)

Cite this article (links to download the citations from this article in formats compatible with various reference manager tools)

Species abundances (left), log-abundance (middle), and the egality parameter $S_{1 / 2} / S$ (right) for the model considered in the main text of van Nes et al., 2024, Parameters are $S = 100$ , ${\tilde{σ}}_{e} = 0.005$ and $K = 100$ .

Time traces of species abundances for $ϵ = 0$ (left panel, time-averaged neutral) and $ϵ = 0.05$ (right panel, chaotic), with $S = 200$ , $K = 10^{4} S$ , and $τ = 8$ .

$\ln P (n)$ vs. $\ln n$ for $ϵ = {0 (solid blue line), 0.05 (dashed red line)}$ , ${\tilde{σ}}_{e} = 0.01$ .