The distribution of distances to the edge of species coexistence

In Lotka-Volterra community models, given a set of biotic interactions, recent approaches have analysed the probability of finding a set of species intrinsic growth rates (representing intraspecific demographic features) that will allow coexistence. Several metrics have been used to quantify the fragility of coexistence in the face of variations in those intrinsic growth rates (representing environmental perturbations), thus probing a notion of distance to the edge of coexistence of the community. Here, for any set of interacting species, we derive an analytical expression for the whole distribution of distances to the edge of their coexistence. Remarkably, this distribution is entirely driven by (at most) two characteristic distances that can be directly computed from the matrix of species interactions. We illustrate on data from experimental plant communities that our results offer new ways to study the contextual role of species in maintaining coexistence, and allow us to quantify the extent to which intraspecific features and biotic interactions combine favorably (making coexistence more robust than expected), or unfavourably (making coexistence less robust than expected). Our work synthesizes different study of coexistence and proposes new, easily calculable metrics to enrich research on community persistence in the face of environmental disturbances.