Explaining the colour of power spectra in chaotic ecological models

Abstract

Power spectrum analysis is often used to determine whether population time series are dominated by particular frequencies. Results for chaotic time series are often reported in terms of the colour of the spectra whereby red spectra indicate a dominance of low frequency (long-term) fluctuations, white spectra indicate that all frequencies are equally dominant and blue spectra indicate a dominance of high frequency (short-term) fluctuations. Several studies have employed such analysis and much discussion has been provoked by an apparent conflict between the fact that the time series of natural populations produce reddened power spectra whereas chaotic, single species ecological models can produce blue, white or red spectra. Here, we resolve the question of which parameter values give rise to particular colour spectra by analysing simple models in terms of ‘universal’ parameters allowing direct comparisons between models to be drawn. This suggests that some models are not capable of producing reddened spectra, which would question their usefulness in describing ecological systems. The population behaviour associated with each colour spectrum is described and compared with models that incorporate simple modifications to represent delayed density dependence, spatial structure and environmental effects.