Predicting global community properties from uncertain estimates of interaction strengths
The community matrix measures the direct effect of each species on every other in ecological communities. Community matrices summarize a large amount of information about a system: they can be used to determine whether the system is stable (returns to equilibrium after small perturbations of the population abundances), reactive (perturbations initially amplify before damping out), and they may be used to determine the response of any individual species to environmental perturbations. However, several studies have shown that small errors in estimating the entries of the community matrix translate into large errors in predicting those species responses. Unfortunately, it is usually impossible to measure the entries of the community matrix with high precision due to the inherent noisiness of ecological data. Therefore, here we ask if there are properties of complex communities one can still predict using only a crude, order-of-magnitude estimate of the community matrix entries.
Using empirical data, numerically generated community matrices, and matrices generated by the Allometric Trophic Network model, we show that the eigenvalue distribution of a large community matrix is not strongly affected by having only an order-of-magnitude knowledge of its entries. We demonstrate mathematically that inaccuracies do not influence our ability to predict the stability and reactivity properties of the system as a whole, though they do hamper our ability to predict individual species responses. Even for matrices that are very close to a bifurcation point, the approximate matrices accurately capture their stability properties in 91% of all cases. Drawing on the theory of random matrices and pseudospectra, we show when and why our crude approximations are expected to yield an accurate description of communities. Our results indicate that even rough estimates of interaction strengths can be useful for assessing global properties of large systems.