Determining natural scales of ecological systems

A key issue in ecology is to identify the appropriate scale(s) at which to observe trends in ecosystem behavior. The characteristic length scale (CLS) is a natural scale of a system at which the underlying deterministic dynamics are most clearly observed. Any approach to estimating CLSs of a natural system must be able to accommodate complex nonlinear dynamics and must have realistic requirements for data. Here, we compare the robustness of two methods to estimate CLSs of dynamical systems, both of which use attractor reconstruction to account for the complex oscillatory dynamics of ecological systems. We apply these techniques to estimate CLSs of spatial multispecies systems of varying complexity, and show that both methods are robust for the simplest system, but as model complexity increases, the Pascual and Levin metric is more robust than that of Keeling et al. Both methods demonstrate some sensitivity to the choice of species used in the analysis, with closely connected species producing more similar CLSs than loosely connected species. In this context, connectivity is determined both by the topology of the interaction network and spatial organization in the system. Notably, systems showing complex spatial self-organization can yield multiple CLSs, with larger length scales indicating the emergent dynamics of interactions between patches. While the prediction r to the power of 2 metric of Pascual and Levin is suitable to estimate CLSs of complex systems, their method is not suitable to apply to most real ecosystems because of the requirement of long time series for attractor reconstruction. We offer two alternatives, both based on prediction r to the power of 2, but where repetition in space is largely (the "short time series" method) or wholly (the "sliding window" method) substituted for repetition in time in attractor reconstruction. Both methods, and in particular the short time series based on only three or four sequential observations of a system, are robust in detecting the primary length scale of complex systems. We conclude that the modified techniques are suitable for application to natural systems. Thus they offer, for the first time, an opportunity to estimate natural scales of real ecosystems, providing objectivity in important decisions about scaling in ecology