Re-examination of evidence for low-dimensional chaos in the Canadian Lynx data

Abstract

The time series of the annual number ofCanadian lynx caught by the Hudson Bay company be-tween 1821 and 1935 exhibits pseudo-cyclic behaviour andhas long been considered as an archetypal example of irreg-ularly fluctuating population dynamics. Recently proposedglobal polynomial models of this data have been found toexhibt chaotic dynamics and were therefore presented asdirect evidence of chaos in a real ecosystem. In this pa-per we re-examine that evidence by constructing global ra-dial basis models subject to information theoretic parame-ter constraints. We find that the models exhibit very goodagreement with the data and are able to accurately repro-duce the qualitative long term dynamical behaviour. Themodels also often exhibit “almost” chaotic dynamics, ei-ther: (a) very long period periodicity, (b) a periodic or-bit embedded in a dissipative mixing region, or (c) verylong time transient irregular aperiodic dynamics with anasymptotically periodic orbit. In each case the dynamicsexhibit a very rich range of behaviour and can also pro-vide a qualitatively accurate deterministic model of the ap-parently chaotic dynamics when subjected to a delay re-construction. We conclude that, while the data and thesemodels are consistent with the hypothesis of chaos in a realecosystem, the data may also be adequately explained byperiodic “almost chaotic” behaviour.