Chaos in ecology

As noted by Hao Bai-Lin in the preface to his admirable collection [29,30] of in uential papers on nonlinear dynamics, the discovery [48,55,56] of chaos in ecological di€erence equations, as much as anything else, fertilized a owering of interest in this subject some twenty-®ve years ago. Perhaps not surprisingly, it was in the physical, as opposed to the biological, sciences that ``chaos theory'', as it is often (and inaccurately!) referred to, really took hold. In ecology itself, the ubiquity of chaos and other non-linear phenomena in both discrete and continuous models was subsequently con®rmed 1 [1,3,4,9,27,34,46,57,70±72]. At the same time, convincing evidence for chaos in natural systems proved harder to come by [17,33,64]. For example, in the case of pre-vaccination epidemics of measles in large, ®rst world cities, what was once judged [63] to be one of the more likely examples of real-world ecological chaos, is now the subject of divergent opinion [16,18]. In as much as ecological systems have all the necessary ingredients: palpable non-linearity, multiple state variables, etc., for chaotic dynamics, the lack of evidence for chaos in nature may strike some as surprising. The conventional view [64] is that ecological data sets are too short and too corrupted by observational error and process noise to allow for accurate characterization of the underlying dynamics. This has prompted a search for better methods of extracting the deterministic signal from noisy time series [17,19,20,58,80±82]. The inherent diculty of this task is underscored by two facts: in the ®rst place, as emphasized by Auerbach and others [7,47], chaotic motion can be viewed as a choreography involving non-stable cycles, with the lead dancer, the evolving orbit, successively favoring di€erent partners. It follows that chaotic time series will often have strong periodic components rendering it dicult to distinguish them from null models [17] consisting of periodic motions in the presence of noise. 2 An additional problem results from the fact that the time evolution of non-linear systems is often re ective not only of the attractors to which trajectories tend in the limit of large time, but also of the presence of non-stable sets elsewhere in the phase space. 3 For example, in the case of the Lorenz equations, Yorke and Yorke [87] have pointed out some time ago that there exist parameter values for www.elsevier.nl/locate/chaos Chaos, Solitons and Fractals 12 (2001) 197±203