- Intrinsic and climatic factors in North-American animal population dynamics

We analyzed four different data sets: one contains the fur-counts alone, the other three contain in addition one climatic index each (see Methods section). Using PC analysis, the individual years that constitute each data set may be separated into two groups. In the plane spanned by the first two PCs (Figure 1a), the first half of the data set that contains the eight longest fur-counts and the ENSO index is concentrated in a small area, whereas the other years are much more dispersed. This separation holds for all the data sets studied (not shown). It reflects the fact that the amplitude of the variations of the fur-counts is fairly low for the first half of the data, and much higher thereafter.

Figure 1 Principal component (PC) analysis of the data set composed of the eight longest fur-counts and the Niño-3 sea surface temperatures (SSTs). EOF-k is the eigenvector corresponding to the kth largest eigenvalue of the covariance matrix of the data set. Each time series, whether fur count or climatic index, is centered and normalized (i.e., it has zero mean and unit variance), and the EOFs so obtained have length one. (a) Distribution of the annual values of the two leading PCs with respect to time; points are grouped by quarter-century intervals (see legend inside the figure). Note that the first half of the record (1752–1800) lies entirely within a small area in the left half-plane, and that the amplitude of the variations from one year to the next increases significantly later in the record. The results are similar when using just the eight longest fur-count records or the combination of these with either one of the other two climatic indices (not shown). (b) Correlation circle corresponding to the same PC analysis as in panel (a). The abscissa (PC-1) captures 54% of the total variance and is highly correlated with each animal-population index. The animal populations included here are bear, beaver, fox, lynx, marten, otter, wolf and wolverine. The ordinate (PC-2) captures 14% of the variance and is very well correlated (r = 0.76) with the Niño-3 SSTs; see legend in the figure for symbols.

The variance captured by each component is given by the corresponding eigenvalue. The eigenvalues of the PC analysis for all four data sets are collected in Table 1. The meaning of the leading principal axes is given by the correlation circles. The one presented in Figure 1b corresponds to the data set of {(fur-counts) + ENSO}. The first axis is clearly correlated to all the animal populations, whereas the second axis is correlated to the ENSO index. The correlation coefficient r between PC-1 and each animal population in Figure 1b ranges between r = 0.94 (for the bears) and r = 0.63 (for the wolves). The same plot for the other three data sets (shown only, in Figure 2, for the {(fur-counts) + NAO}) indicates that the first component is always by far the largest, since it embodies at least 54% of the total variance (see Table 1); PC-1 correlates, most clearly and exclusively, with the animal populations, in all four data sets. The animal populations are thus most strongly correlated to each other.

Figure 2 Correlation circle corresponding to the PC analysis of the data set that includes the same eight fur-counts as in Figure 1, but replaces the ENSO climate index there with the NAO index here. The abscissa (PC-1) captures 55% of the total variance and is highly correlated with each one of the animal populations; same notation as in Figure 1b. The ordinate (PC-2) captures 12% of the variance and is correlated fairly well (r = 0.52) with the NAO index, but not as highly as for ENSO in Fig. 1b.

When a climatic index is added to the data set of fur-counts, it is strongly correlated, in all three cases, with the second axis. This correlation is shown for the data set of {(fur-counts) + ENSO} in Figure 1b and the {(fur-counts) + NAO} in Figure 2; it is also true for the NH temperature (NHT) index (not shown). The different animal species are thus affected to various degrees by climatic factors, but apparently less so than by biotic interactions among species.

The PC analysis also allows one to separate the signal from the noise in the data sets. As already mentioned, the first component contains the lion's share of the variance (54% to 61%) and PC-2 is quite important, too (12% to 14%; see Table 1). We studied PC-5 as well, because its variance is very close to that of PC-6 and so this pair may jointly capture a single mode of, possibly oscillatory, behavior; if this were the case, the combined mode would also represent 9%–10% of the variance. Consequently, spectral analysis was performed on all four of these components, PC-1, PC-2, PC-5 and PC-6. The results for PC-5 and PC-6, however, turned out to be less interesting and are thus omitted here.

Figure 3 displays the projection of the {(fur-counts) + ENSO} data set on the two leading components. The projection on PC-1 (Figure 3) is quite similar in the other three data sets, given that it stands essentially for the animal populations, which are the same in all four sets. Projection on the other components (shown for this particular data set and PC-2 in Figure 3), on the other hand, does depend on the climatic index chosen, or its absence (not shown for the other PCs and other data sets).

Figure 3 Projection of the {(fur-counts) + ENSO} data set on axes 1 and 2 of the PC analysis; see legend inside the figure. Each of these two projections is then analyzed, using the SSA-MTM Toolkit, for all four data sets, to give the spectral results shown in Table 2.

The results of spectrally analyzing PC-1 and PC-2 in all four data sets are shown in Table 2. We cross-checked the spectral peaks obtained by the median-filter version [3] of the multi-taper method (MTM), as listed in the Table, with those given by the Monte Carlo version [4] of Singular Spectrum Analysis (SSA; not shown). The two sets of results agree overall very well for all species. Occasionally, slight differences arise for minor peaks that are statistically significant at the 90% level in one of the analyses but not the other. The main periods are all significant at the 99% level when tested against a red-noise null hypothesis [5], whereas the significance threshold for secondary periods is 95%.

In order to verify the interpretation of the results in Table 2, we also carried out a spectral analysis of each climatic index by itself (Table 3). For the ENSO index, the main period is of 4 years and the secondary period is 2-year long, in agreement with known results ([6,7], and references therein). The NAO results are much less clear cut: two periods are emerging, 3.5 and 3 years, but their level of significance is quite low (90%). For the NHT index, the main modes of variation are a 170-year trend [8,9] and a 2.5-year period; a secondary 2-year period also arises in this signal. A 160–170-year trend is present in the ENSO and the NAO indices as well, but is less significant than in the NHT index.

The spectral analysis results for the first component are independent of the data set, because PC-1 always embodies the animal populations' behavior. The main modes are a 160–170 year trend and a 3-year periodicity; a secondary, 2.5-year peak is also highly significant (see Table 2). The main periods of PC-2 correspond to the characteristic periods of the corresponding climatic index: 3.5 years for the {(fur-counts) + NAO} data set, 4 years and 2 years for the {(fur-counts) + ENSO}, and 2.5 years for the {(fur-counts) + NHT}. Other periods appear for this second component, especially a 30–45-year peak. The pair (PC-5) + (PC-6) also contains an 8.5-year peak, as a main or secondary period (not shown), for all three data sets that do include a climatic index.

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