Lechon-Alonso, P.; Miller, Z. R.; Liaghat, A.; Breiding, P.; Pascual, M.; Allesina, S.

Whether species-rich communities erode gradually or collapse abruptly under environmental change is a central question in ecology. Classical pairwise theory predicts that coexistence is always lost gradually, through smooth declines to extinction, yet real ecological interactions are often strongly state-dependent -- shaped by nonlinearities that fixed pairwise coefficients cannot capture. Here we show that higher-order (nonlinear) interactions make abrupt, irreversible loss of coexistence a typical route to community collapse: across diverse random communities, the equilibrium supporting coexistence disappears suddenly at a fold bifurcation. Using polynomial homotopy continuation to track equilibria as environmental conditions change, we find that folds progressively dominate the boundary of the coexistence domain as nonlinearity strengthens, replacing the gradual extinctions of pairwise theory. Furthermore, the sign structure of higher-order interactions controls both the onset of tipping-points and whether biodiversity buffers or amplifies collapse. Because higher-order and nonlinear interactions are intimately linked, tipping points also arise generically in pairwise models with strong nonlinearity. Applying our continuation framework to a canonical model of plant-pollinator collapse, we formally resolve its bifurcation structure as fold-mediated, and we show that fold bifurcations are typical across published multispecies models spanning mutualistic, competitive, and consumer-resource interactions. These results challenge the expectation that monitoring abundances suffices to anticipate collapse, and unify structural-stability theory, which delineates the safe operating space for coexistence, with critical transition theory, which characterizes the nature of its boundaries.