Emergent self-inhibition governs the landscape of stable states in complex ecosystems

TL;DR

Describing and explaining the landscape of stable states in the random Generalized Lotka-Volterra (GLV) model, in which multistability is widespread, reveals emergent self-inhibition as a fundamental organizing principle for the attractor landscape of complex ecosystems.

Abstract

Species-rich ecosystems often exhibit multiple stable states with distinct species compositions. Yet, the factors determining the likelihood of each state's occurrence remain poorly understood. Here, we characterize and explain the landscape of stable states in the random Generalized Lotka-Volterra (GLV) model, in which multistability is widespread. We find that the same pool of species with random initial abundances can result in different stable states, whose likelihoods typically differ by orders of magnitude. A state's likelihood increases sharply with its total biomass, or inverse self-inhibition. We develop a simplified model to predict and explain this behavior, by coarse-graining ecological interactions so that each stable state behaves as a unit. In this setting, we can accurately predict the entire landscape of stable states using only two macroscopic properties: the biomass of each state and species diversity. Our theory also provides insight into the biomass-likelihood relationship: High-biomass states have low self-inhibition and thus grow faster, outcompete others, and become much more likely. These results reveal emergent self-inhibition as a fundamental organizing principle for the attractor landscape of complex ecosystems - and provide a path to predict ecosystem outcomes without knowing microscopic interactions.

Figures (21)

• Figure 1: Multistability and biomass--likelihood relationship in the random GLV model with strong interactions. (a) Phase diagram showing the typical number of stable states for each interaction matrix $A_{ij}$ with $S=100$ species given its mean $\mu$ and standard deviation $\sigma$. Multistability is widespread (red region); black line shows the analytically-derived multistability boundary (Appendix \ref{['main:app:multistabilityboundary']}) separating the multistable and unique stable state (white) phase. (b) For a fixed $A$, varying initial conditions reveals different stable states. Shown are example dynamics (each species is a different color). Each state's likelihood is estimated as the fraction of simulations that end in the state, while biomass is computed as the total abundance of surviving species. (c) The likelihood $p$ of a stable state increases sharply with biomass $B$, following a roughly hyperbolic relationship: $\log(p) \propto (B^\ast - B)^{-1}$ (solid line). Data show results from $10^6$ simulations with random initial conditions using a single $A$ (inset) with $\mu=0.5$, $\sigma=0.3$.
• Figure 2: Coarse-graining states reveals emergent self-inhibition as a predictor of biomass and likelihood. (a) Species that interact weakly with each other form uninvadable stable states. The interactions among coexisting species are separated from the original interaction distribution, with a reduced mean. (b) Species-level interaction matrix. (c) We can conceptualize this as an effective matrix where different states compete with each other and one eventually emerges as the winner. (d) In a monodominant matrix where different stable states contain only a single species, both biomass and likelihood can be computed analytically from the self-inhibition (Appendix \ref{['main:app:monodominantCalc']}). (e) In a block matrix where each different state corresponds to a block of $L$ species, we can treat each block as an effective species with self-inhibition inversely proportional to its biomass. (f) Predictions combining this self-inhibition and block size accurately predict simulated likelihoods. Shown are block sizes $L=\{4,6\}$.
• Figure 3: Biomass predicts state likelihoods in ecosystems with disordered interactions. (a) Using the emergent self-inhibition of each state as the inverse of its biomass, we can predict the overall biomass--likelihood relationship. (b) The predicted likelihoods of each state, without any detailed knowledge of interactions, match for most states. For states with dramatically low likelihoods, our theory slightly overestimates likelihoods due to our mean-field approximation (gray region) (Appendix \ref{['main:app:likelihoodCalcRandMatrix']}).
• Figure S1: Flowchart of our algorithm to identify feasible, stable and uninvadable states for a given GLV interaction matrix in our model. Briefly, we choose a random initial condition, numerically integrate the dynamics, and detect surviving species using an self-consistently determined extinction threshold. We then finally ensure the resulting state is feasible and stable both to perturbations in surviving species abundances and to invasions of extinct species.
• Figure S2: Biomass--likelihood distributions for asymmetric interaction matrices. (a) Correlation $\rho=0.9$, (b) correlation $\rho=0.8$. In both cases a positive correlation between likelihood and biomass is observed, although the strength decreases as $\rho$ becomes smaller. Curves indicate fits to the log-hyperbolic relation $\log\left(p\right) \propto (B^\ast - B)^{-1}$.