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 $P(y)=1/y^\alpha$ for the equilibrium abundance $y$ of a species. Remarkably, the exponent $\alpha=2$ 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 $\alpha \sim2.56$. Next, we derive a formula for the critical noise, $\sigma_c$, beyond which the community experiences feasibility loss ``with near certainty''. We find that $\sigma_c(N)\sim N^{-1}$, 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.