Linking ecological patterns to environmental forcing via nonlinear time series models

The identification of key environmental forcings responsible for population patterns is a pervasive ecological problem and an important application of time series analysis. A common approach, implemented with methods such as cross-correlation and cross-spectral analysis, relies on matching scales of variability. This approach concludes that a population pattern is caused by a physical factor if their variances share a dominant period. In a nonlinear system, however, forcing at one temporal period can produce a response with variability at one or more different periods. Thus, scale-matching methods will be most successful at establishing cause-effect relationships in linear systems, or close to equilibria, where nonlinear systems are well approximated by linear ones. Here, we propose an alternative approach that does not assume linearity and relies on time series models that are both nonlinear and nonparametric. We specifically apply these models to determine the correct but unknown frequency of a periodic forcing. The time series are generated by simulation of a predator-prey model. Under periodic forcing, this type of model is known to be capable of different dynamic regimes, including chaos and quasi-periodicity, in which the power spectra of population numbers exhibit variance at frequencies other than that of the forcing. We show that nonlinear time series models, built with feedforward neural networks, are able to distinguish the correct forcing period in the predator-prey simulations. These results hold under two common limitations of ecological data: the presence of dynamical and measurement noise, and the availability of time series data for only one variable. We discuss future applications of the approach to more general environmental forcings, other than periodic.