Scaling of the number of stable and uninvadable equilibria with species richness

Determine whether, in ecosystems that exhibit local multistability, the number of feasible, stable, and uninvadable equilibria scales exponentially with the number of species or instead grows sub‑exponentially, in order to clarify the landscape complexity of large ecological communities and its implications for tipping behavior and resilience.

Background

The lectures discuss how ecosystems can display multiple stable states, with abrupt transitions (tipping points) driven by small perturbations. In high-dimensional ecological models and disordered systems, the number and structure of equilibria determine the dynamical landscape (e.g., spin-glass versus structural-glass analogies). Establishing how the count of stable and uninvadable equilibria scales with the number of species is crucial for understanding whether large ecosystems have exponentially many attractors (implying extensive barriers and complex dynamics) or a smaller, sub-exponential set.

This question directly connects to the diversity-stability debate and has practical ramifications for predicting regime shifts in ecosystems. It also informs the design of simulations and laboratory experiments aimed at observing multistability and invasion resistance at scale.

References

Whether the number of such stable and uninvadable states grows exponentially with the number of species -- a behavior that could be observed in simulations or tailored laboratory experiments -- or on a much smaller scale remains an open question.

— Les Houches lectures on Theoretical Ecology: High-dimensional models and extreme events (2503.02792 - Altieri, 4 Mar 2025) in Introduction (Section 1)