(or, chaos meets data in population ecology)

Many natural populations fluctuate in abundance, in ways that appear random to the eye and to some standard time-series analyses. This paper reviews the methods, and results, of recent attempts to determine if the fluctuations are chaotic rather than random. Early "indirect" methods (fit a model to the data, and see if the model is chaotic) found no evidence of chaos, but the results are sensitive to the choice of model and parameter-estimation methods. "Direct" methods (based on reconstructing trajectories in time-delay coordinates) appear to reveal "strange" (i.e., fractal) attractors that are a hallmark of chaotic dynamics. Reconstruction is model-free, but offers no way of objectively evaluating one's visual impression of the attractor. Fractal dimension calculations, using a new maximum-likelihood method for short time-series (Ellner 1988), can be used to test for fractal structure (non­ integer dimension) if there's enough data. For some measles incidence data, dimension calculations suggest an attractor with dimension between 2 and 3, supporting the visual impression of a strange attractor underlying the fluctuations.