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.
Lechon-Alonso, P., Strang, A., Breiding, P., Allesina, S.
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