From individuals to population densities: Searching for the intermediate scale of nontrivial determinism

The degree of stochasticity or determinism in the dynamics of ecological systems varies with sampling scale. We propose the application of a determinism test from nonlinear data analysis to describe this variation and to identify a characteristic length scale at which to average spatiotemporal systems. Specifically, we investigate the spatial scale at which to aggregate individuals into densities in a system that combines demographic noise with local density-dependent interactions. The proposed approach is applied to the dynamics of a spatial and individual-based predator-prey model. The selected spatial scale is smaller than the one obtained by a previously proposed method whose similarities and differences we discuss. Two models, the simplest candidates for approximating the dynamics of densities at the selected scale, are examined: a predator-prey system of differential equations that ignores the local nature of the dynamics, and an extension of it that adds demographic noise. These approximations perform poorly, failing to capture broad statistical features of the predator and prey fluctuations. These findings indicate that spatial factors are nonnegligible at the selected intermediate scale of aggregation. Thus, predator-prey systems and other oscillatory ecological systems may display a dynamic regime at an intermediate scale of aggregation in which local interactions are still important. We discuss the type of model needed to approximate population densities in this dynamic regime.