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

A striking result of our data analysis is the large and fairly sudden increase in the amplitude of the oscillations in the animal populations, around 1810. This could be seen directly in the raw fur-counts plotted against time (not shown, but please see the appendices), and it explains why the first half of the century-long records is tightly grouped along the negative PC-1 axis (Figure 1a). We know that the hunting pressure on the fur-yielding mammals increased during the time interval under study (1752–1849), mainly because fur clothing became more fashionable and the market for it increased.

A very simple predator-prey model of Lotka–Volterra type, in which the prey population is harvested, leads to an increase in the oscillation amplitude of this population when the harvesting parameter increases; our model is described in the Methods section and the results are shown in Figure 4. References [10,11] also discussed how harvesting may destabilize population dynamics, using a somewhat different, discrete-time model [10] and a single predator population [11].

Figure 4 Changes in the solution behavior of a predator–prey model subject to environmental pressures and given by system (2). (a) Phase plane showing the limit cycles exhibited by the populations of prey and predator species for different values of the environmental pressure parameter e. Note that the amplitude of the oscillations increases as e decreases, i.e. when environmental effects are less favorable. This may be the case when the prey population is submitted to intense hunting. (b) Amplitude of the limit cycle as a function of the parameter e.

The meaning of the mode that we refer to as "the 160–170-year trend" is the following. Both SSA [8,9,12] and MTM [3] permit the reliable extraction of trends that are more robust than the usual linear ones, which are obtained by least-square fitting. Such a nonlinear trend may take at times the shape of an incomplete sine function. If the curve in question is close to or exceeds about one-half the period of a sine function, both Monte Carlo SSA and MTM can determine the period of the sine function of closest fit. In our case, this period equals 160–170 years for the leading PCs of all four data sets we consider, as well as for the NHT time series, taken separately. We cannot, therefore, attach a true statistical significance to this period, but believe that longer data sets might support its presence in both NH temperatures and North American mammal populations.

This interpretation would suggest that long-term variations of the animal populations are linked to long-term variations of temperature and that the high-latitude animals we study have benefited from the temperature increase associated with the NH recovery from the "Little Ice Age" [9]. Note that the presence of the 160–170-year trend in the ENSO and NAO indices, too, reflects large-scale climatic interactions and has, obviously, nothing to do with the fur-counts we concentrate on here.

The 2.5-year period is also common to the spectral analysis of the NHT index and of the fur-counts. The role of temperature in population dynamics has been documented for many species (e.g., [13]); it may be due to the sensitivity of reproduction, survival, or intra- and inter-specific interactions to temperature.

The well-known ENSO periods of 4 and 2 years do not arise unambiguously from the spectral analysis of the leading PCs of the fur data alone (Table 2). Still, PC analysis of the {(fur-counts) + ENSO} data set clearly underlines the contribution of ENSO to the variance of the fur-counts (Figure 1b).

To clarify the reason for this apparent discrepancy, we carried out the spectral analysis of each of the climatic indices and fur-count records by itself (Table 3). Indeed, the ENSO periods are seldom highly significant in the individual population records; only in the muskrat time series is the 2-year peak significant at the 95% level. Furthermore, a well known 10-year period [14,15] does appear at the 99% level in our lynx record and at the 95% level in the marten population, but it does not show up in the leading PCs of Table 2.

These two discrepancies between the spectral results for individual time series and for the leading PCs of the whole population arise because of the nonlinearity present in combining PC analysis and spectral analysis. Each of these analyses separately involves a linear operator; for a finite record, finitely sampled, this operator takes the form of a matrix. The combination of the two analyses, however, is not a linear operator, i.e., a matrix product, acting between the individual records and the spectra of the PCs (or the PCs of the spectra). The PC analysis renders more significant the collective impact of ENSO on all mammalian populations combined, while it renders less significant the 10-year mode of variability that seems to be restricted to the lynx and the marten populations.

The second component of the {(fur-counts) + ENSO} data set is highly correlated with ENSO, as shown by the ordering of the different species along the second axis of Figure 1b. Although the role of ENSO in the dynamics of North American mammals had not been documented so far, its impact on Canadian climate is well-known by now [16,17].

The second component of the {(fur-counts) + NAO} data set is correlated to NAO (Figure 2), even though the relation is somewhat less pronounced than in the ENSO case. Consequently NAO effects help explain part of the differences in the variability observed between the different animal species. The presence of a 40–44-year period in PC-2 for furs alone, as well as in the {(fur-counts) + NAO} and {(fur-counts) + NHT} data sets, may be related to a similar period being present in variations of the North Atlantic Ocean's thermohaline circulation [18,19]. Post et al. [20] underscored the correlation between NAO and several parameters that describe animal behavior and their interactions. For example, they explain how NAO may influence wolf-pack sizes and the risk of predation exerted on moose. Stenseth et al. [21] discuss more specifically how NAO may influence the dynamics of lynx populations in three distinct climatic regions of Canada. Fur-counts of both lynx and wolf are among our eight longest data sets.

The exact mechanism by which the NAO and ENSO have an impact on the group of mammal populations we studied remains to be determined. We know that these two climatic indices are both linked to diverse features of North American climate, such as seasonal temperature means, liquid precipitation, freezing and snowfall. All these climatic variables may influence the individual fitness of the animal species used in the present work.

Having discussed the influence of external factors on the population dynamics of North American mammals, it is time to turn to the effect of intrinsic factors. In each PC analysis we carried out here, climatic indices are correlated to the second axis (Figures 1b and 2), while the leading component, which captures at least 54% of the variance (Table 1), is representative of the animal populations themselves: the correlation between the fur-counts of each species and PC-1 of the furs-only data set ranges between 0.63 (for the wolf) and 0.94 (for the bear). The spectral analysis of PC-1 displays, in all four data sets, two dominant components that are significant at the 99% level: a 160–170-year trend (significant at this level only in the MTM analysis) and a 3-year oscillation (highly significant in both the MTM and SSA analyses).

The 160–170-year trend is probably linked to long-term variations in temperature, while the 3-year period does not appear as a significant peak in any of the climatic indices. As explained in the caption of Table 2, our spectral resolution distinguishes clearly between this 3-year period and the climate-related ones of 2.5 and 3.5 years. The 3-year period in the fur-counts must therefore be linked either to the intrinsic population dynamics of the mammal species under consideration or to an external factor that acts on all the populations but has entirely escaped our attention. It may also arise from density-dependent ecological interactions, whether intra- or interspecific, and their interplay with seasonality. More detailed spectral analyses of individual species (not shown) have indicated that this 3-year period is strongest in those North-American mammals that are linked by predation, especially beaver, mink, muskrat and wolf. This seems to confirm the role of interspecific mechanisms in giving rise to the 3-year peak in our data.

Seldal et al. [22] related the 3-year period observed for lemmings populations to plant palatability and herbivory tolerance, while Turchin et al. [23] also based their explanation of the 3-year lemming cycles on two inter-specific interactions: predation and herbivory. The present analysis of North American fur-count data extends the range of interspecific dynamics that leads to a 3-year cycle to species other than lemmings. Indeed, we find this dominant period for data sets that include as many as eleven mammal species.

The results of the present study may also be linked to the issues of synchroneity among variations in spatially separated populations. The Moran [24] effect refers to a population that is subject to a density-independent process, while being fragmented into subpopulations within which density dependence is important. Synchroneity between the subpopulations may be due to the density-independent process (Moran effect) or to dispersal of individuals between the subpopulations. Several studies have described such synchronization, based either on natural data or on models [15,24-27]. Our results clearly show that climatic phenomena, which are clearly independent of mammalian populations, do have an influence on these populations, and that synchroneity between the variations of the latter may then be expected. Since we do not possess geographical details on the mammalian populations concerned, it is not possible at present to discuss the relative role of dispersal vs the Moran effect in the population variations we find. Nonetheless, we feel that our methodological framework, in general, and particularly the advanced spectral methods used, in particular, may be of great help in future synchroneity studies.

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