Figure 1. Geometrical properties of feasibility.

The panels show the size and shape of the feasibility domain for three interaction matrices, each defining the interactions between three populations. If r corresponds to a feasible equilibrium, so does cr for any positive c; one can therefore study the feasibility domain on the surface of a sphere²⁵ (Supplementary Note 3). The grey sphere represents the S=three-dimensional space of growth rates, while the coloured part corresponds to the combination of growth rates leading to stable coexistence. The area (or volume for higher-dimensional systems) of the coloured part is measured by Ξ. Larger values of Ξ correspond to a higher fraction of growth rate combinations leading to coexistence: the red interaction matrix (panel a) is therefore more robust against perturbations of r than the green one (b). The size of this region (that is, the value of Ξ) does not capture all the properties relevant for coexistence. The red (a) and blue (c) systems have the same Ξ, but the two regions—despite having the same area—have very different shapes, summarized in d, where we show the length of each side for the red and blue systems. In the red system (a), the three sides have about the same length, and thus moving from the centre in any direction will have about the same effect. In the blue system (c), however, one side is much shorter than the other two, implying that even small perturbations falling along this direction may drive the system outside the feasibility domain. One of our main results is that, roughly speaking, if the red system corresponds to the random case, then the green one to food webs (having the same heterogeneity in side lengths as the random case but with a smaller Ξ overall), and the blue one to empirical mutualistic networks (Ξ rougly the same as in the random case but with the heterogeneity in side lengths much greater).