Emergent feasibility in random ecological systems with higher-order interactions

A recurring lesson from random ecological models is that coexistence is hard to come by: in the Generalized Lotka-Volterra (GLV) model with pairwise interactions, the probability that randomly sampled parameters admit a positive (feasible) equilibrium – a necessary condition for coexistence – is exactly 1 / 2 ⁿ in n species, vanishing rapidly with diversity. This rarity is often read as evidence that coexistence demands specific ecological mechanisms. Real interactions, however, are rarely strictly pairwise: any nonlinear dependence of one species’ growth rate on another’s abundance, Taylor-expanded, generates higher-order interactions (HOIs) of increasing degree. Treating the interaction order d as a knob that indexes this nonlinearity, we map the random GLV with HOIs onto the Kostlan-Shub-Smale class of random polynomial systems and approximate the probability of feasibility ( P _f ) analytically. We find a phase transition at d = 4: below this threshold, P _f decays with diversity as in the pairwise case; above it, the exponential proliferation of equilibria outpaces the probability that any given equilibrium is feasible, and the probability of feasibility increases with n , approaching one. The transition appears to be universal across symmetric coefficient distributions, but vanishes when sign symmetry of the parameter distribution is broken. This work uncovers a route by which feasibility emerges from nonlinearity alone, with no fine-tuning of parameters and no appeal to specific ecological mechanisms.