## Time series analysis counting and characterizing lags

In other, related cases, the emphasis shifts to deriving the optimal statistical model because the number of lags in that model may provide clues as to how abundance is being determined. It may do so because Takens' theorem (see Section 5.8.5) indicates that a system that can be represented with three lags, for example, comprises three functional interacting elements, whereas two lags imply just two elements, and so on.

One example of this approach hares and lynx ... (another is described in Section 12.7.1)

is the study of Stenseth et al. (1997) of the hare-lynx system in Canada to which we have already referred briefly in Section 10.2.5. We noted there that the optimal model for the hare time series suggested three lags, whereas that for the lynx suggested two. The density dependences for these lags are illustrated in Figure 14.10a. For the hares, direct density dependence was weakly negative (remember that the slope shown is 1 + Pj) and density dependence with a delay of 1 year was negligible, but there was significant density dependence with a delay of 2 years. For the lynx, direct density dependence was effectively absent, but there was strong density dependence with a delay of 1 year.

support for the specialist predation hypothesis

Figure 14.10 (a) Functions for the autoregressive equations (see Equation 14.1) for the snowshoe hare, above (three 'dimensions': direct density dependence and delays of 1 and 2 years), and the lynx, below (two dimensions: direct density dependence and a delay of 1 year). In each case, therefore, the slope indicates the estimated parameters, 1 + Pj, P2 and P3, respectively, reflecting the intensity of density dependence. The 95% confidence intervals are also shown.

This, combined with a detailed knowledge of the whole community of which the hare and lynx are part

(Figure 14.10b, c), provided justification for Stenseth et al. (1997) to go on to construct a three-equation model for the hares and a two-equation model for the lynx. Specifically, the model for the lynx comprised just the lynx and the hares, since the hares are by far the lynx's most important prey (Figure 14.10b). Whereas the model for the hares comprised the hares themselves, 'vegetation' (since hares feed relatively indiscriminately on a wide range of vegetation), and 'predators' (since a wide range of predators feed on the hares and even prey on one another in the absence of hares, adding a strong element of self-regulation within the predator guild as a whole) (Figure 14.10c).

Lastly, then, and again without going into technical details, Stenseth et al. were able to recaste the two- and three-equation models of the lynx and hare into the general, time-lag form of Equation 14.1. In so doing, they were also able to recaste the P values in the time-lag equations as appropriate combinations of the interaction strengths between and within the hares, the lynx, and so on. Encouragingly, they found that these combinations were entirely consistent with the slopes (i.e. the P values) in Figure 14.10a. Thus, the elements that appeared to determine hare and lynx abundance were first counted (three and two, respectively) and then characterized. What we have here, therefore, is a powerful hybrid of a statistical (time series) analysis of densities and a mechanistic approach (incorporating into mathematical models knowledge of the specific interactions impinging on the species concerned).

Finally, note that related methods of time series analysis have been used in the search for chaos in ecological systems, as described in Section 5.8.5. The motivations in the two cases, of course, are somewhat different. The search for chaos, none the less, is, in a sense, an attempt to identify as 'regulated' populations that appear, at first glance, to be anything but.