“…As such, our approach can only describe the dynamics near the equilibrium produced by the higher-order interactions. The unstable equilibria we identified may sometimes lead to coexistence, rather than species exclusions, either through other equilibria (Section 11 of the Supplementary Information) or more complex dynamics (Hastings & Powell, 1991;Huisman & Weissing, 1999, 2001Lever et al, 2020;McCann & Yodzis, 1994;Rogers et al, 2022Rogers et al, , 2023. Conversely, even when the target abundance equilibrium is stable, it may not be globally stable (Section 11 of the Supplementary Information).…”