Laboratoire d'Ecologie, UMR 7625, Ecole Normale Supérieure, 46 rue d'Ulm, F-75230 Paris cedex 05, France

Département Terre-Atmosphère-Océan, Ecole Normale Supérieure, 24 rue Lhomond, F-75231, Paris cedex 05, France, and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, CA 90095-1567, USA

Abstract

Background

Extensive work has been done to identify and explain multi-year cycles in animal populations. Several attempts have been made to relate these to climatic cycles. We use advanced time series analysis methods to attribute cyclicities in several North-American mammal species to abiotic vs. biotic factors.

Results

We study eleven century-long time series of

Conclusions

Our results show that all three climatic indices influence the animal-population dynamics: they explain a substantial part of the variance in the fur-counts and share characteristic periods with the fur-count data set. In addition to the climate-related periods, the fur-count time series also contain a significant 3-year period that is, in all likelihood, caused by biological interactions.

Background

The dynamics of animal populations are driven by both biotic and abiotic factors. Following the seminal work of Volterra

The present work aims at separating the influence of biotic and climatic factors in the dynamics of eleven North-American mammal populations. The animal species we study are bear, beaver, fisher, fox, lynx, marten, mink, muskrat, otter, wolf and wolverine. The variations in these populations are determined by using the Hudson Bay Company's database of annual fur-counts

The application of both principal component (PC) analysis and spectral analysis helps separate the different factors that influence the dynamics of the animal community under study. Using a fairly large set of long population records makes the application of PC analysis necessary: it allows us to distinguish between climatic factors that affect all the populations and those that do not. Advanced spectral methods permit us, on the other hand, to detect subtle but systematic variations in one or more of the mammalian populations under study. The results of this combined data analysis approach allow us to conclude that both climatic and intrinsic factors affect this community and to quantify, at least approximately, their relative role.

Results

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

Principal component (PC) analysis of the data set composed of the eight longest fur-counts and the Niño-3 sea surface temperatures (SSTs)

Principal component (PC) analysis of the data set composed of the eight longest fur-counts and the Niño-3 sea surface temperatures (SSTs). EOF-^{th }largest eigenvalue of the covariance matrix of the data set. Each time series, whether fur count or climatic index, is centered and normalized (

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

Correlation circle corresponding to the PC analysis of the data set that includes the same eight fur-counts as in Figure

Correlation circle corresponding to the PC analysis of the data set that includes the same eight fur-counts as in Figure

Eigenvalues of the principal component (PC) analysis for the four data sets, given as percent of the total variance (rounded off to the nearest whole percentage point). Note that the first component captures the lion's share of the variance, and that the second component is also quite sizable.

Data set

Egv 1

Egv 2

Egv 3

Egv4

Egv5

Egv6

Egv7

Egv8

Egv9

Furs

61%

13%

9%

6%

5%

4%

1%

1%

--

Furs + NAO

55%

12%

10%

7%

5%

5%

3%

2%

1%

Furs + ENSO

54%

14%

9%

7%

5%

5%

3%

2%

1%

Furs + temp

56%

12%

11%

7%

5%

4%

3%

2%

1%

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

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

Figure

Projection of the {(fur-counts) + ENSO} data set on axes 1 and 2 of the PC analysis; see legend inside the figure

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

Spectral analysis of the two leading components of the four data sets; the periods are given in years. The first column for each PC gives the dominant periodicities, the second one gives the less pronounced peaks; for economy of presentation, the 160–170-year nonlinear trend [5] is also included in the first column for PC-1. The analysis reported in this Table was performed using the median-filter MTM [3,5] with the bandwidth parameter

**PC-1**

**PC-2**

Main periods

Secondary periods

Main periods

Secondary periods

Furs

170 **3**

**2.5**

Furs+NAO

**3**

**3.5**

**40 **

Furs +ENSO

160 **3**

**2.5 **2

**30**

Furs+temp

170 **3**

**2.5**

44 2.5

3.5

The results of spectrally analyzing PC-1 and PC-2 in all four data sets are shown in Table

In order to verify the interpretation of the results in Table

Spectral analysis of the individual climatic indices and fur-count records. As in Table

Time series

Main periods

Secondary periods

NAO

3, 3.5

ENSO

4

2

NHT

170, 2.5

2

Bear

170, 3, 2.5

Beaver

170, 3, 2.5

Fisher

170, 3, 2.5

Fox

170

3, 2.5

Lynx

170, 10, 2.5

3

Marten

170, 3

10, 2.5

Mink

170, 3, 2.5

Muskrat

170, 3

2

Otter

170, 3, 2.5

Wolf

170, 2.5

Wolverene

170, 3, 2.5

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

Discussion

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

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

Changes in the solution behavior of a predator–prey model subject to environmental pressures and given by system (2)

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

The meaning of the mode that we refer to as "the 160–170-year trend" is the following. Both SSA

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"

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 (

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

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

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,

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

The second component of the {(fur-counts) + NAO} data set is correlated to NAO (Figure

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

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

Seldal

The results of the present study may also be linked to the issues of synchroneity among variations in spatially separated populations. The Moran

Conclusions

Our two-step methodology led us to distinguish between intrinsic and external factors in the dynamics of over ten North American mammal populations. PC analysis shows that internal dynamics is most important but also captures the role of ENSO, NAO and NH temperatures in the animal population dynamics. The striking change in the amplitude of the oscillations present in our fur-count data is probably linked to an increase of hunting pressure over the century-long interval of study.

Our spectral analysis determines the key periods of the three climatic indices we use. They are 170 years and 2.5 years for the mean NH temperatures, and 4 years and 2 years for the Niño-3 SSTs, while the NAO has only spectral peaks that are both weaker and marginally significant. The key periods of all four data sets that comprise the animal fur-counts are 170 years and 3 years. The latter has to be attributed to biological interactions.

Methods

Data sets

Four different data sets were analyzed. The first set includes fur-counts of the eleven animal populations within a given year. The three other sets were obtained by augmenting the fur records by one climatic record in each case. The three records we use are the North Atlantic Oscillation (NAO) index, as defined in

The time series of eight animal populations (bear, beaver, fox, lynx, marten, otter, wolf and wolverine) are 98-year long, extending from 1752 to 1849; those of fisher, mink and muskrat start in 1767 and are thus only 83-year long. All the fur-counts are provided in Appendices 1 through 11. The main results reported in this paper were obtained using the eight species with 98-year long records. When adding the three species with shorter records to the previous eight, the results are very similar (not shown). All the climatic data were available throughout the 1752–1849 time interval.

The NAO index represents a suitably normalized difference in sea-level pressure between the Açores High and the Icelandic Low; it determines the strength of the westerly winds over the North Atlantic Ocean and is correlated with several climatic patterns over the adjacent land masses

The cold-season Niño-3 SSTs are a reliable index of the phase of the El Niño-Southern Oscillation (ENSO); ENSO dominates interannual climate variability in the Tropical Pacific

Principal component analysis

Our data analysis includes two steps: principal component (PC) analysis and spectral analysis. PC analysis has been extensively used in the biomedical sciences

Advanced spectral methods

Once the population-count or climatic signal was isolated, we used a battery of spectral analysis tools to determine the main periods it contains. Singular-spectrum analysis (SSA) is based on eigenvalue-eigenvector decomposition of a time series' lag-covariance matrix _{k}, ordered from largest to smallest. When two eigenvalues are nearly equal, and the corresponding pair of (odd and even) EOFs are in phase quadrature, they may capture, subject to statistical significance tests, an anharmonic (

SSA can decompose a short, noisy time series into a (variable) trend, periodic oscillations, other statistically significant components that are aperiodic, and noise. The projection of the time series onto an EOF yields the corresponding principal component (PC) of length

The multi-taper method (MTM) is designed to reduce the variance of spectral estimates by using a small set of tapers rather than the unique data taper or spectral window used by Blackman-Tukey methods

Mann & Lees _{N}/4, _{R}}, where _{N }is the Nyquist frequency, _{R }is the Rayleigh frequency, and _{N }= 0.5 cycles.year^{-1}, _{R }= 0.02 cycles.year^{-1}, and we used ^{-1}, and so all the periods listed in Table ^{-1}, which is marginally useful for separating the shortest periodicities in the Table, but the fact that the peaks coincide in both methods lends further credence to their independent existence.

The SSA-MTM Toolkit was developed by the Theoretical Climate Dynamics group at the University of California, Los Angeles (UCLA)

Heuristic population model

We propose here to link the striking and fairly sudden increase in variance of all the fur-count records to a slight variation of environmental conditions. To do so, consider the very simple model:

of the interaction of a prey population

All parameters are strictly positive, except

Non-dimensionalizing the model (1) gives:

where

The equilibrium associated with system (2) was studied using the CONTENT software _{H}: if _{h}, a single equilibrium, with finite and nonzero _{h}, this equilibrium becomes unstable and gives rise to a stable limit cycle. The larger the difference _{h}, the larger the amplitude of the oscillations (see Figure _{h }-

This very simple model, with only one prey and one predator species, shows that, as the external conditions deteriorate even slightly, the amplitude of both the predator's and the prey's population cycles increases. The large change in the oscillations' amplitude for all the population records studied here could therefore be linked to a deterioration in Canada's environmental conditions in the early 1800s. This deterioration might also be linked to increased hunting pressures.

Other results show that harvesting pressure may have a destabilizing effect. Basson & Fogarty

Acknowledgement

This work was initiated as part of a collaboration between the Ecole Normale Supérieure and UCLA. We thank both institutions for their respective support and mutual hospitality. We also thank Régis Ferrière at the ENS for helpful advice on ecological modeling, Pascal Yiou of the Laboratoire des Sciences du Climat et de l'Environnement for advice on and help with the acquisition of paleoclimatic data, and Andrew W. Robertson for constructive comments on an earlier version of the manuscript. The SSA-MTM Toolkit is available as freeware at