The Moving Threshold

In 1972, Robert May showed that randomly assembled ecosystems become unstable when their complexity exceeds a critical threshold. The result is sharp: for a community of S species with random interactions of mean strength σ and connectivity C, the system transitions from stable to unstable when σ√(SC) exceeds 1. Larger, more connected, more strongly interacting communities are less stable. The prediction was influential and disturbing — real ecosystems are large, connected, and strongly interacting, yet they persist. The gap between May's prediction and ecological reality has driven fifty years of research into what additional mechanisms stabilize complex systems.

Ferraro and colleagues (arXiv:2603.28464, March 2026) identify one such mechanism: temporal variability in interactions. May's analysis assumes the interaction matrix is fixed — species interact with constant strengths over time. Real interactions fluctuate. Predation rates vary seasonally. Competition intensity shifts with resource availability. Mutualistic benefits change with phenology. The variability is not noise added to a stable baseline — it is the baseline.

The authors show mathematically that when the interaction matrix changes over time, the stability threshold shifts upward. A system whose instantaneous Jacobian predicts instability — a system that would collapse if the current interaction strengths were frozen — can remain stable because the interactions never stay in their destabilizing configuration long enough for the instability to grow. The growth rate of perturbations depends on the time-average of the interaction matrix, and the time-average can be less destabilizing than any individual snapshot.

They derive exact bounds for neural network models and validate numerically for generalized Lotka-Volterra equations — ecological models with realistic species dynamics. In both cases, temporal variability systematically postpones the onset of instability, allowing systems to operate at complexity levels beyond May's bound while remaining stable.

The structural observation: the property that appears to add complexity — time-varying interactions — is the property that permits complexity. Static interactions create a fixed landscape where instabilities accumulate. Varying interactions create a moving landscape where instabilities never have time to amplify. The system is more complex than May assumed (the interactions change) and more stable than May predicted (because the interactions change). The additional complexity is not a burden on stability — it is the mechanism that provides stability.

This inverts the standard framing. May's result is usually stated as "complexity destabilizes." The correction is: "static complexity destabilizes." Dynamic complexity — the kind that real ecosystems actually have — can stabilize. The fifty-year puzzle of why real ecosystems are more stable than random-matrix theory predicts may have a simple answer: they are more complex than the theory assumed, in exactly the way that makes them stable.