Figure 8.5 Stability map for a second-order autoregressive process as in equations 8.15 and 8.16. Different regions in parameter space numbered with Roman numerals correspond to dynamics illustrated in figure 8.6. From Royama (1992).

ln[Mt + 1)] = a0 + (1 + a 1)ln[^(t)] + a2ln[N(t - 1)] (8.16)

Royama (1992) has mapped the regions of parameter space with different stability properties (figures 8.5 and 8.6).

Bj0rnstad et al. (1995) used autoregressive procedures to estimate (1 + a1) and a2 from a number of vole populations and then studied geographic variation in the autoregressive coefficients. This approach holds promise for revealing the ecological correlates of predator-prey dynamics. The usual interpretation is that the first autoregressive term represents density dependence and the second and higher-order terms are a consequence of trophic-level interactions. Certainly the autoregression coefficients do not yield to such simple interpretations, but work is just beginning in this area.

Akaike's Information Criterion (AIC) has been used to optimize the dimensionality and magnitude of coefficients for autoregressive models (Bj0rnstad et al. 1995). Although such models are primarily descriptive rather than mechanistic, attempts to interpret the autoregressive coefficients have been encouraging.

Figure 8.6 Population dynamics emerging from second-order autoregressive models. Each plot is representative of the patterns coming from regions plotted with corresponding Roman numerals in figure 8.5. The horizontal axis is time for each of these plots. From Royama (1992).

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