Fig. 3. Universal behaviour of the robustness of ecosystem service supply as a function of network fragility.

a For 125 random ecosystem services of the form shown in Fig. 2, we uniformly sampled 5 values for species richness S between 10 and 200, 20 for functional traits N between 10 and 100 and 10 for connectance p between 0 and 0.5 and for each considered 600 extinction scenarios leading to a loss of service, drawn uniformly over the set of all possible extinction sequences. We see that the median robustness of ecosystem service supply (that is, R_0.5(E)) in our simplified Boolean model behaves as a simple decreasing linear function of median network fragility (f_0.5). b For three values of number of traits (that is, N = 4, 16 and 128) and random values of number of species S and connectance p, we used the analytical formula in Eq. (1) to draw the contours of the bulk of the robustness distribution (that is, 10th–90th percentiles) as a function of f_0.5, the value of fragility associated with median robustness. We see that, at fixed N, the percentiles can be expressed as functions of f_0.5, with the largest spread at intermediate values of network fragility. At fixed fragility, increasing the number of traits N reduces the width of the distribution.