  Identifying directed interactions between species from time series of their
population densities has many uses in ecology. This key statistical task is
equivalent to causal time series inference, which connects to the Granger
causality (GC) concept: $x$ causes $y$ if $x$ improves the prediction of $y$ in
a dynamic model. However, the entangled nature of nonlinear ecological systems
has led to question the appropriateness of Granger causality, especially in its
classical linear Multivariate AutoRegressive (MAR) model form. Convergent-cross
mapping (CCM), a nonparametric method developed for deterministic dynamical
systems, has been suggested as an alternative. Here, we show that linear GC and
CCM are able to uncover interactions with surprisingly similar performance, for
predator-prey cycles, 2-species deterministic (chaotic) or stochastic
competition, as well as 10- and 20-species interaction networks. There is no
correspondence between the degree of nonlinearity of the dynamics and which
method performs best. Our results therefore imply that Granger causality, even
in its linear MAR($p$) formulation, is a valid method for inferring
interactions in nonlinear ecological networks; using GC or CCM (or both) can
instead be decided based on the aims and specifics of the analysis.
