Analysis of community interactions using linear transfer function models

Abstract

Interactions among populations simulated with several ecosystem models and the interactions among populations in a sealed aquatic microcosm were analyzed using linear transfer function models. A transfer function is composed of the sum of three components: a deterministic transfer function ( F), a stochastic input function ( G), and a white-noise error time series ( e). The model has the form: y(t) = F{x(t), y ̂ (t−1)} + G{n(t), r ̂ (t−1)} + e(t) , where y( t) is an output time series, x( t) is an input time series, ^ y( t−1) is the estimate of y( t−1) based on F alone, n( t) is an unobserved white-noise time series, and r ̂ (t−1) is the estimate of the residual series r(t−1) = y(t−1) − y ̂ (t−1) based on G alone. The deterministic transfer function accounts for the pattern in the output time series that is attributable to the interaction between the input and output variables. The stochastic input function accounts for the remaining pattern in the output time series. The deterministic portion of these simple linear models could recover most of the pattern in complex modeled systems ( r-square generally greater than 0.9). Deterministic interactions were not as readily detected in the microcosm although a significant portion of the pattern could often be recovered with the stochastic input function. The ability to detect relationships among variables in modeled systems but not in the living system suggests that may extant ecological models may be poorly posed. Temporal correlation between variables is a necessary (but not sufficient) condition for a mechanistic model based on the interaction between these variables to be successful. The transfer function estimation procedure, therefore, can be a useful tool for identifying potentially important interactions in poorly understood systems before more mechanistic models are developed.