Higher-order interactions in random Lotka-Volterra communities

We use generating functionals to derive a dynamic mean-field description for generalised Lotka-Volterra systems with higher-order quenched random interactions. We use the resulting single effective species process to determine the stability diagram in the space of parameters specifying the statistics of interactions, and to calculate the properties of the surviving community in the stable phase. We find that the behaviour as a function of the model parameters is often similar to the pairwise model. For example, the presence of more exploitative interactions increases stability. However we also find differences. For instance, we confirm in more general settings an observation made previously in model with third-order interactions that more competition between species can increase linear stability, and the diversity in the community, an effect not seen in the pairwise model. The phase diagram of the model with higher-order interactions is more complex than that of the model with pairwise interactions. We identify a new mathematical condition for a sudden onset of diverging abundances.