Can higher-order interactions resolve the species coexistence paradox?

Most ecological models are based on the assumption that species interact in pairs. Emerging in diverse communities, however, are higher-order interactions, in which two or more species jointly impact the growth of a third species. A pitfall of the pairwise approach is that it misses the higher-order interactions potentially responsible for maintaining diversity in nature. Here, we explore how well higher-order interactions enable coexistence when pairwise interactions are insufficient to do so. Specifically, we explore the stability properties of systems where higher-order interactions guarantee that a specified set of abundances are a feasible equilibrium of the dynamics. Even these higher-order interactions do not necessarily produce robust coexistence. We find that facilitative higher-order interactions that counter pairwise competitive interactions are less likely to generate stable equilibria than competitive interactions countering pairwise facilitation. In both cases the system is more likely to be stable when the pairwise interactions are weak, and when the equilibrium abundances are near those produced by the pairwise interactions. Last, we show that correlations between the pairwise and higher-order interactions permit robust coexistence in diverse systems. Our work reveals the challenges in generating stable coexistence through higher-order interactions, but also uncovers higher-order interaction patterns that can enable diversity.