Fig. 2. Geometrical approach for relating the collinearity of functions to mismatches in their observations of perturbations.

A Perturbations can be viewed as displacement vectors in community state space. Here a hypothetical community of a bacterial species and a fungal species is independently impacted by two perturbations, represented by the black arrows from the initial state of the community (I) to the points A and B. B Measures of ecosystem function can be represented as positive directions in this state-space. C Perturbed states are plotted in a space where the initial state of the community is at the origin and each axis describes the response of each species to a perturbation. Here the displacement vectors associated with the independent perturbations A and B from (A) are projected onto the directions representing total biomass and plastic decomposition. For A, both functions observe negative responses. However, for B there is a mismatch in the observations of the functions: total biomass responds positively while plastic decomposition responds negatively. D For two functions, the zones of mismatches in their observations can be found by drawing lines perpendicular to the functions that go through the origin. Aggregate properties will observe different responses for perturbations that fall between these lines (i.e. in the red zones). The angle between the two functions determines the size of the zones of mismatches. The three pairs of functions include (i) a broad and a narrow function, (ii) two broad functions, and (iii) two narrow functions. E Over many in silico perturbation experiments, the proportion of mismatches between functions can be predicted by the angle between them in radians (θ) divided by the number π. Two broad functions (ii) would have high collinearity and a low proportion of mismatches, one broad and one narrow function, (i) would have moderate collinearity and a moderate proportion of mismatches, while two narrow functions, and (iii) would have low collinearity and a high proportion of mismatches (unless they were performed by the same species). The inset shows that the proportion of mismatches between a function and total biomass is a very good predictor of the broadness of that function.