Feasibility as a moving target: Fluctuating species interactions lead to universal power law in equilibrium abundances

Abstract

Theoretical ecology has traditionally equated persistence with the stability of a fixed equilibrium point. Here we argue that the primary threat to ecosystem persistence need not be the loss of stability, but instead the escape of the stable equilibrium to a negative orthant. In a realistic setting, fluctuations in interactions do not merely disturb abundances about an equilibrium but can displace the equilibrium point itself. We theoretically and empirically analyze such displacements of the equilibrium point in a complex community. Theoretically, we find that light-tailed fluctuations in species interactions, no matter how small, lead to a heavy-tailed power law $P(y)=1/y^α$ for the equilibrium abundance $y$ of a species. Remarkably, the exponent $α=2$ is a universal value independent of interaction structure, community size, and species. Empirically, our analysis of 34 species reveals a power law signal for most, with a median exponent $α\sim2.56$. Next, we derive a formula for the critical noise, $σ_c$, beyond which the community experiences feasibility loss ``with near certainty''. We find that $σ_c(N)\sim N^{-1}$, implying that larger communities are significantly more fragile to noise induced feasibility loss. Lastly, we define and calculate biologically measurable analytical metrics for both global and species-specific feasibility escape rates, and implement these metrics in dynamic simulations of 98 real world mutualistic and food web networks, to successfully predict their fragility.

Figures (6)

• Figure 1: Probability distribution of equilibrium abundances for different noise levels. We plot the exact distribution $h(\vec{y})$ for ${\bf A}$={{-1.4, 0.1} , {0.5, -1.4} } perturbed by $10\%$ (left), $100\%$ (center) and $1000\%$ (right) noise. As the noise increases, more probability mass moves to negative quadrants, signifying increased risk. The unperturbed equilibrium is marked by $\bm{\circ}$.
• Figure 2: The bulk and tail behavior of $h(\vec{y})$. As we move along the main diagonal $|y|=y_1=y_2$ the distribution turns from a Gaussian to a power law $1/|y|^{N+1}$. The power law scaling is the same in every direction. The parameters used in the figure are same as Fig.\ref{['fig:contour_plots']} with $\sigma=8\%$.
• Figure 3: Evidence of heavy tailed abundance distributions in aquatic communities (A) Complementary Cumulative Distribution Functions (CCDF) for the tails of abundance on a log-log scale. Empirical data are shown as gold circles; the solid blue line represents the best fit power law distribution ($P(X \geq x) \propto x^{-(\alpha-1)}$). In the insets of each panel abundance time series of the species/functional groups are given where units are fresh weight concentration (mg/L) for Mesocosm dataset and dry mass density for Lake dataset. carpenter_synthesis_2018. (B) Stacked histogram showing the distribution of power law exponents ($\alpha$) for all species identified as having plausible power law tails ($p \geq 0.1$). The data combines results from a 33 year field study of a reference lake (dark blue) carpenter_synthesis_2018 and a 7 year constant condition laboratory mesocosm (light blue) beninca_chaos_2008. (C) Assessment of the power law hypothesis. Bars indicate the $p$-value from a Kolmogorov-Smirnov goodness of fit test using 10,000 semiparametric bootstrap simulations; values $p \geq 0.1$ indicate that the power law model cannot be rejected. Labels denote the estimated exponent ($\alpha$) for each species. An asterisk (*) indicates a statistically significant preference for the power law over an exponential distribution (Likelihood Ratio Test, $\mathcal{R} > 0, p < 0.1$). In all analyses, no statistically significant preference for the log-normal distribution over the power law was found.
• Figure 4: A mutualistic seed dispersal network observed in Spain guitian1983relacionesWebOfLifeDataset and the risk predictions for each species. The top panel illustrates the structure of a representative mutualistic network, composed of 12 disperser species (black nodes) and 7 plant species (white nodes). As is characteristic of this community, interactions are defined due to competition (negative, red edges) within each group and interspecific mutualism (positive, blue edges) between the two groups. The bottom panel directly compares our analytical predictions with simulation results. It shows the normalized escape rate for each species, calculated from Eq.\ref{['eq:escape_rates']}, alongside the corresponding feasibility loss ratios from stochastic simulations. The results demonstrate a strong correspondence, with the species predicted to have the highest escape rate also accounting for approximately one third of all simulated feasibility loss events. This predictive power, which also captures both the ranked importance and the relative contribution of each species to community fragility is a good example of predictive power of derived escape rate formula Eqn. \ref{['eq:escape_rates']}.
• Figure 5: Feasibility loss rate and mean time to loss for empirical networks alongside analytical predictions. Each panel compares an analytical prediction (x-axis) against results from stochastic Monte Carlo simulations (y-axis), with the solid black line representing the line of perfect agreement. (a, b) The relationship between the analytical escape rate ratio (normalized Eqn. \ref{['eq:escape_rates']}) and the simulated corresponding loss probability. Each point corresponds to a single species within an ecosystem. Larger escape rate values indicate that the ecosystem is more likely to lose feasibility through that species, signifying its higher fragility. Panel (a) shows results for mutualistic networks, and (b) shows results for food webs. (c, d) The relationship between analytical and simulated mean time to feasibility loss. Each point represents an entire ecosystem, calculated as the inverse of the total escape rate. A larger persistence time indicates greater ecosystem robustness against fluctuations. Panel c) shows results for mutualistic networks, and d) shows results for food webs. For all panels, the color of each point corresponds to the size of the ecosystem (S), as indicated by the color bars. Marker shapes describe how the network is parameterized: for mutualistic networks, they distinguish between weak and strong mutualism and low and high variance (CV). For food webs, markers distinguish between a low and high ratio of negative to positive interaction, $\mu_{ratio}$ and low and high variance (CV). The model shows substantial agreement for food webs (Concordance Correlation Coefficient, CCC = 0.96 for both metrics), while showing moderate agreement for the mutualistic networks (CCC = 0.72 for persistence time, CCC = 0.90 for escape rates). For the panels a) and b) we excluded the species with escape rate or corresponding loss ratio values below $0.05$. For each of the parametrized networks we ensured the starting system is globally stable and feasible. For panels a) and d) weak mutualism corresponds to average positive interaction value $0.2 \times \mu_{max}$ where $\mu_{max}$ is maximum allowed average of positive interactions to ensure global stability. Similarly for strong mutualism this value is $0.6 \times \mu_{max}$. For all the panels, low variance and high variance correspond to $CV=0.5$ and $CV=2.0$, respectively. Also, negative in-group interactions representing competition are taken as $0.25$ for all the panels.