exclusion was assessed by t tests, treating each year as
a separate comparison. The survival on control trees
versus outside cages on experimental trees was not
significantly different (apart from fall of 1992), so we
treated these replicates as a single category, “exposed.”
Our main result was not affected by this data pooling,
because in 1993, the first year after the peak, survival
inside cages was significantly higher than survival both
outside cages on experimental trees and on control
trees (P ? 0.001 for both comparisons). The compari-
manner, except the data were first log-transformed to
stabilize the variance.
19. The difference in survival inside and outside of the
cages was not statistically significant in 1994, the
second decline year. This was probably because of
lower intraspecific competition outside of the cag-
es due to lower egg density there (17). There was
a significant negative effect of egg density on
survival inside exclusion cages (linear regression:
F1,51? 9.38, P ? 0.0035). When we corrected
survival rates by taking intraspecific effects into
account using this relation, we found a significant
difference in survival, suggesting that the preda-
tion impact on survival was still detectable 2 years
after the peak.
20. C. W. Berisford, in The Southern Pine Beetle. USDA-
Forest Service Technical Bulletin 1631, R. C. Thatcher,
J. L. Searcy, J. E. Coster, G. D. Hertel, Eds. (U.S.
Department of Agriculture, Washington, DC, 1980),
21. J. D. Reeve, Oecologia 112, 48 (1997).
22. J. D. Reeve, J. A. Simpson, J. S. Fryar, J. Entomol. Sci.
31, 123 (1996).
23. W. W. Murdoch, R. N. Nisbet, S. P. Blythe, W. S. C.
Gurney, J. D. Reeve, Am. Nat. 129, 263 (1987).
24. H. C. J. Godfray and M. P. Hassell, J. Anim. Ecol. 58,
25. S. L. Pimm, The Balance of Nature? (Univ. of Chicago
Press, Chicago, IL, 1991).
26. W. S. C. Gurney and R. M. Nisbet, Ecological Dynam-
ics (Oxford Univ. Press, New York, 1998).
27. C. J. Krebs et al., Science 269, 1112 (1995).
28. P. J. Hudson, A. P. Dobson, D. Newborn, ibid. 282,
29. T. L. Payne, in (20), pp. 7–30.
30. J. D. Reeve, D. J. Rhodes, P. Turchin, Ecol. Entomol. 23,
31. This research was supported by USDA-Forest Service
RWU-4501 and by NSF grant DEB 9509237. We
thank D. Rhodes and the staff of Kisatchie National
Forest for assistance in the field. J. Elkinton, J. Hayes,
J. Cronin, A. Berryman, and C. Godfray provided help-
ful comments and suggestions on an early version of
15 April 1999; accepted 21 June 1999
Common Dynamic Structure of
Canada Lynx Populations Within
Three Climatic Regions
Nils Chr. Stenseth,1,2* Kung-Sik Chan,3Howell Tong,4,5
Rudy Boonstra,1,6Stan Boutin,7Charles J. Krebs,1,8Eric Post,2
Mark O’Donoghue,8,9Nigel G. Yoccoz,1,10
Mads C. Forchhammer,11,12James W. Hurrell13
Across the boreal forest of Canada, lynx populations undergo regular density
cycles. Analysis of 21 time series from 1821 onward demonstrated structural
similarity in these cycles within large regions of Canada. The observed popu-
lation dynamics are consistent with a regional structure caused by climatic
features, resulting in a grouping of lynx population dynamics into three types
(corresponding to three climatic-based geographic regions): Pacific-maritime,
Continental, and Atlantic-maritime. A possible link with the North Atlantic
Oscillation is suggested.
Periodic population fluctuations of the Can-
ada lynx (Lynx canadensis) have greatly in-
fluenced both ecological theory and statistical
time series modeling [(1, 2); see (3) for a
summary]. Recent analyses have focused on
the extent of synchrony in population fluctu-
ations, assessing the importance of external
abiotic factors (such as weather) and internal
biotic factors (such as dispersal among pop-
ulations) in causing spatial patterns (4). Such
empirical and theoretical approaches have,
however, assumed that the populations were
structurally similar [that is, the density-de-
pendent relationships are identical among
populations (5)]. This assumption has never
been thoroughly evaluated. To do so requires
determining whether the lynx populations
display the same phase- and density-depen-
dent structure (3) and then searching for sim-
ilar underlying causes of the observed dy-
namics. Using new statistical methods devel-
oped for this purpose (6), we ask to what
extent the time series on the Canada lynx
(Fig. 1) compiled by the Hudson Bay Com-
pany for the period 1821 to 1939 (7) and the
corresponding more modern time series com-
piled by Statistics Canada for the period 1921
to present (8), taken together, are structurally
similar. Specifically, we ask whether the
phase- and density-dependent structure of
changes in lynx abundance cluster into
groups defined according to ecological-based
features (9) or according to climatic-based
features (10, 11).
The available time series (Fig. 1A) cover
two ecosystems (referred to below as ecolog-
ical regions): the northern, open boreal forest
(Fig. 1B) and the southern, closed boreal
forest. In western Canada, the mountainous
topography adds complexity. Additionally,
the series cover three climatic regions defined
by the spatial influences of the North Atlantic
Oscillation (NAO) [Fig. 1C; see (12)], which
may contribute to spatial differences in tro-
phic interactions (13).
Previously, we fitted a piecewise linear
autoregressive model (14) to each of the
series (3). A general hare-lynx model (3, 15)
may be expressed as an equivalent model in
delay coordinates of the lynx (the species for
which we have data). Here we check whether
all the time series, or some subsets of these,
display the same underlying phase- and den-
sity-dependent structure. For this purpose we
use a piecewise linear model (14, 15):
?s,1,0? ?s,1,1ys,t?1? ?s,1,2ys,t?2? ?s,1,t
?s,2,0? ?s,2,1ys,t?1? ?s,2,2ys,t?2? ?s,2,t
where ys,tis the log-transformed abundance of
lynx at site s and for year t [that is, ys,t?
log(Ys,t) where Ys,tis the abundance of lynx at
site s and in year t, and where s ? 1, 2, . . . ,
represent the sites corresponding to the individ-
ual time series; see Fig. 1A]; ?s,i,jare the sta-
tistical parameters that determine the phase-
? 1 and 2 correspond to the lower and the
upper regimes of the model; j ? 0, 1, 2 corre-
spond to the constant term, the first lag, and the
second lag, respectively) at site s; εs,i,tis nor-
the log-transformed density d years earlier.
1Center for Advanced Study, The Norwegian Acade-
my of Science and Letters, Drammensveien 78,
N-0271 Oslo, Norway.2Division of Zoology, Depart-
ment of Biology, University of Oslo, P.O. Box 1050
Blindern, N-0316 Oslo, Norway.3Department of Sta-
tistics and Actuarial Science, University of Iowa, Iowa
City, IA 52242, USA.4Department of Statistics, Lon-
don School of Economics, London WC2A 2AE, UK.
5Department of Statistics, University of Hong Kong,
Hong Kong.6Division of Life Sciences, University of
Toronto at Scarborough, Scarborough, Ontario M1C
University of Alberta, Edmonton, Alberta T6G 2E9,
Canada.8Department of Zoology, University of British
Columbia, Vancouver, British Columbia V6T 1Z4,
Canada.9Department of Renewable Resources, Fish
and Wildlife Branch, P.O. Box 310, Mayo, Yukon Y0B
1M0, Canada.10Department of Arctic Ecology, Nor-
wegian Institute for Nature Research, Polar Environ-
mental Centre, N-9296 Tromsø, Norway.11Depart-
ment of Zoology, University of Cambridge, Downing
Street, Cambridge CB2 3EJ, UK.
Landscape Ecology, National Environmental Research
Institute, Kalø, Grenåvej 12, DK-8410 Rønde, Den-
mark.13National Center for Atmospheric Research,
Climate Analysis Section, P. O. Box 3000, Boulder, CO
7Department of Biological Sciences,
*To whom correspondence should be addressed. E-
R E P O R T S
www.sciencemag.orgSCIENCE VOL 28513 AUGUST 1999
advantages (16). The upper (respective lower)
regime has been found to correspond to the
decrease (respective increase) phase (3).
Because of varying carrying capacities
and trapping efforts across sites, we expect
series to have different means and standard
deviations. Therefore, Chan et al. (6) consid-
ered the hypothesis of common structure that
all series enjoy the same dynamics up to their
means and standard deviations. The hypoth-
esis of common structure is equivalent to
two hypotheses: the hypothesis of common
slopes ?s,i,j? ?1,i,j, and the hypothesis of
common ratio of intercepts [(the intercept
in the upper regime)/(the intercept in the
lower regime)] takes the same value at the
threshold across the different sites. Chan et
al. (6) derived some test statistics for
checking these hypotheses.
We also compared model fits between var-
ious possible groupings of the time series into
autoregression) models (Eq. 1) may be con-
strained to have some coefficients identical
across series within a given subgroup. Different
groupings can be compared in terms of their
respective AICs (Akaike information criteria)
(14, 17). For groupings involving different se-
ries to be comparable, each grouping will be
defined for all series. A model with minimal
AIC strikes a good balance between parsimony
and goodness of fit to the data.
Time series coming from the same loca-
tions exhibit the same dynamic structure, in-
dicating a common underlying dynamic mod-
el (18). The time series come from forested
biomes across Canada and thus from areas
with greatly different plant species composi-
tion and habitat structure. Focusing on the
vegetation, we may classify the time series
into two ecological-based groups (Fig. 1B):
the northern forest tundra, which consists of
shrub tundra and low-density trees (19), and
the true boreal forest, which is a mixture of
conifer and deciduous trees (19). [A western
ecological-based group with heterogeneous
topography and habitat, as well as climate
(20) may also be identified.] However, treat-
ing the old and modern series separately, the
ecological-based grouping represents no im-
provement over the baseline of no grouping
[Table 1; see also (21, 22)].
As an alternative to this ecological-based
grouping, the Canada lynx series may be
grouped according to three major climate-based
features: the Pacific-maritime region, the Con-
tinental region, and the Atlantic-maritime re-
gion (Fig. 1C). This grouping clearly provides a
better description of the data [Table 1; see also
(22)]. The similarity is particularly strong for
of climate-based properties contributing to the
structuring of the lynx dynamics.
Over much of central and western Cana-
da, surface climate is most strongly influ-
enced by the atmospheric circulation up-
stream over the North Pacific and in particu-
lar by a natural mode of large-scale atmo-
spheric variability known as the Pacific–
North American (PNA) teleconnection pattern
(23). However, the influence of the PNA on
Canadian surface temperature is spatially ho-
mogeneous. In contrast, the influence of the
NAO on surface winter temperatures varies
considerably from coast to coast (Fig. 1C)
and shows spatial variation corresponding
well to the best grouping of the lynx series
(Table 1). Hence, it is the winter atmospheric
circulation, for which the NAO may serve as
a proxy, that probably contributes to making
the nonlinear structure of the hare-lynx dy-
namics similar within each of the three Ca-
nadian groups. Although it is generally
known that climate profoundly influences re-
gional variation in vegetation [for example,
see (9)], our results suggest that the spatio-
temporal patterns of climatic variation also
influence the trophic interaction between the
lynx and its main prey, the snowshoe hare,
differently across these regions.
Because the NAO may have a delayed
effect on the lynx dynamics, we have to
choose between using lag-0 or lag-1 NAO as
the covariate. Several statistical techniques
are available, including Cox’s test of separate
families of hypotheses, AIC or its many vari-
ants, and others (17). Even though the effect
of the NAO on lynx abundance is not strong,
the lynx series fall along an east-west gradi-
ent progressing from negative to positive and
Hudson Bay Company
Fig. 1. Time series data studied. (A) Map of Canada with demarcations of the studied time series [red
indicates the Hudson Bay Company time series (7) and black indicates the recent series (8)]. See (3) for
definitions of names of the individual time series used. (B) Ecological regions of Canada (24). (C)
Climatic regions of Canada (10). The NAO refers to a meridional oscillation in surface pressures with
centers of action near Iceland and over the subtropical Atlantic. When surface pressures are lower than
normal near Iceland and higher than normal over the subtropical Atlantic (the positive phase of the
NAO), enhanced northerly flow over eastern Canada cools surface temperatures and enhanced
southerly flow from the Gulf of Mexico into much of central Canada produces warm surface anomalies.
Over the Pacific-maritime region, there is no significant NAO signature.
R E P O R T S
13 AUGUST 1999VOL 285 SCIENCEwww.sciencemag.org
finally to no effect of the NAO. The previ- Download full-text
ously observed phase-dependent nature of the
density-dependent structure (3) remains even
after the NAO is included as a covariate. As
a result, this study is consistent with earlier
results but adds the geographic component to
the structure of the lynx time series.
We can now reach a comprehensive synthe-
sis of the time series of the Canadian lynx—
namely, the lynx cycle is a direct result of
trophic interactions varying structurally in three
different regions of Canada, a grouping that is
associated with the large-scale climatic effects
known to be associated with the NAO. We
argue that the extensive similarity during the
decrease phase is to a large extent a result of
region-specific winter conditions and suggest
that these may be linked to the state of NAO.
We do not yet know how these winter climatic
events influence the lynx cycle, but we suggest
that lynx hunting efficiency needs to be mea-
sured in the three climatic regions.
References and Notes
Lemmings (Clarendon, Oxford, 1942); see also (3).
2. See P. A. P. Moran [Aust. J. Zool. 1, 163 (1953)], who
fitted a linear autoregressive model of order two, which
exhibits quasi-periodicity, but was well aware of its
inadequacy. In particular, he pointed out the inhomoge-
neity of the fitted residuals, which violated the assump-
tion of a common and constant variance for the white
noise term in the fitted model. As an interesting histor-
ical point, it should be noted that Moran learned about
the lynx data when he visited Charles Elton and Dennis
Chitty in the Bureau of Animal Population at Oxford
after World War II.
3. N. C. Stenseth et al. Proc. Natl. Acad. Sci. U.S.A. 95,
4. E. Ranta, V. Kaitala, P. Lundberg, Science 278, 1621
(1997); E. Ranta, V. Kaitala, J. Lindstro ¨m, Ecography
20, 454 (1997).
5. The focus on one particular lynx series has to some
extent distracted both ecologists and statisticians up
to the present from the fact that similar time series
exist for the entire continent [but see T. Royama,
Analytical Population Dynamics (Chapman and Hall,
6. K. S. Chan, H. Tong, N. C. Stenseth, unpublished data.
7. C. S. Elton and M. Nicholson, J. Anim. Ecol. 11, 215
8. Statistics Canada; Dominion Bureau of Statistics
1965; Statistics Canada 1983–1995.
9. H. Walters, Vegetation of the Earth and Ecological Sys-
R. G. Bailey, Ecoregions: The Ecosystem Geography of the
Oceans and Continents (Springer-Verlag, Berlin, 1998).
10. J. W. Hurrell and H. Van Loon, Clim. Change 36, 301
11. In principle there are more than 221classifications of
the 21 lynx series. A “black-box” approach means
doing an exhaustive search for the optimum combi-
nation in some sense—an approach troubled by mul-
tiple comparison in statistical modeling. Instead, we
focus on two sets of combinations, one based on eco-
logical features and the other on climatic condition.
12. J. W. Hurrell, Geophys. Res. Lett. 23, 665 (1996).
13. E. Post and N. C. Stenseth, J. Anim. Ecol. 67, 537 (1998);
M. C. Forchhammer, N. C. Stenseth, E. Post, R. Langvatn,
Proc. R. Soc. London Ser. B 265, 341 (1998); E. Post and
N. C. Stenseth, Ecology 80, 1322 (1999).
14. H. Tong, Non-Linear Time Series: A Dynamical System
Approach (Clarendon Press, Oxford, 1990); H. Tong,
Threshold Models in Non-Linear Time Series Analysis
(Springer-Verlag, Berlin, 1983). The model given by
Eq. (1) is a SETAR(2;2,2) model. For a nontechnical
presentation of the SETAR models, see Stenseth et al.
[N. C. Stenseth, K.-S. Chan, E. Framstad, H. Tong,
Proc. R. Soc. London Ser. B 265, 1957 (1998)].
15. N. C. Stenseth, W. Falck, O. N. Bjørnstad, C. J. Krebs,
Proc. Natl. Acad. Sci. U.S.A. 94, 5147 (1997).
16. N. C. Stenseth and K.-S. Chan, Nature 395, 620
17. D. R. Cox and D. V. Hinkley, Theoretical Statistics
(Chapman & Hall, London, 1974); H. Akaike, in Sec-
ond International Symposium on Information Theory,
B. N. Petrov and F. Csaki, Eds. (Akademiai Kiado,
Budapest, 1973); H. Akaike, IEEE Trans. Autom. Cont.
AC-19, 716 (1974).
18. Using the tests described in (22) and adopting a 5%
significance level, we cannot reject the hypothesis of
common slopes and the hypothesis of the common
ratio of intercepts for the pairs (L1 and L2) and (L4
and L5). For L6 and L7, we cannot reject the hypoth-
esis of common slopes in the upper regimes and that
the lag-2 coefficients may be the same for the two
series. However, L6 and L7 appear to have a different
lag-1 coefficient in the lower regime. Thus, the evi-
dence of common dynamics for L6 and L7 is some-
what weak. For details, see (6).
19. B. Wiken, D. Gauthier, I. Marshall, K. Lawton, H.
Hirvonen, A Perspective on Canada’s Ecosystems,
CCEA Occasional Paper No. 14, Ottawa (1996).
20. J. S. Rowe, Forest Regions of Canada (publ. 1300,
Environment Canada, Canadian Forestry Service, Ot-
21. We have investigated the constraint structure further
(Table 1): assuming only a common structure in the
lag-1 and lag-2 in the upper regime provides worse fit.
This suggests that the additional constraints [found by
testing (6)] are significant. Relocating L3 to the Pacific-
maritime group yields an AIC slightly larger than
?1819.38. This implies some fuzziness in the boundary
of the three zones. One can classify L3 into the Pacific-
maritime zone or the Continental zone. The case of
Yukon seems rather clear cut: including it in the Pacific
zone substantially increases the AIC and hence provides
a poorer description.
22. Supplementary information is available at www.
23. J. M. Wallace and D. S. Gutzler, Mon. Weather Rev.
109, 784 (1981).
24. Ecosystem Stratification Working Group, A National
Ecological Framework for Canada (Agriculture and Agri-
Food Canada/Environment Canada, Ottawa, 1995).
25. This work was initiated while several of the authors
were at the Centre for Advanced Study at the Norwe-
gian Academy for Sciences and Letters, Oslo. N.C.S.
acknowledges generous support from the Norwegian
Science Council and the University of Oslo. K-S.C. ac-
knowledges generous support from the University of
Iowa. H.T. acknowledges the Engineering and Physical
Sciences Research Council (UK) and the University of
Hong Kong for support. N.C.S. and H.T. acknowledge
support from UK Biotechnology and Biological Sciences
Research Council/Engineering and Physical Sciences Re-
search Council (UK) grant 96/MMI09785. R.B., S.B., and
C.J.K. acknowledge support from the Natural Sciences
and Engineering Research Council of Canada. E.P. ac-
knowledges the National Science Foundation (grant
DBI-98-04178) for support. M.C.F. acknowledges the
Danish National Research Council for support. We
thank A. Kenney for preparing the figure and three
helping us to clarify our argument.
17 December 1998; accepted 21 June 1999
Table 1. Common structure in the underlying dynamics. Groupings with the smallest AIC value represent
the best ones. (Only the AIC differences between the particular grouping and the baseline comparison are
given [see (22), where the absolute AIC values are also given]. In some of the subgroups, the SETAR
models are constrained to have some coefficients identical across series within the subgroup (see
“Identical constraint structure” column). These constraints are suggested by the new statistical tests
developed by Chan et al. (6). For different groupings involving different series to be comparable, each
grouping is defined for all series in the panel; each series not in a grouping implicitly forms a singleton
group. For definition of names of the series (L1, L2, . . . , L22), see (3).
No groupings(Individual series without any constraints)
(L1), (L2), . . . (L14), (L15), . . . (L22)
All Hudson Bay Co. series (L1, L2, . . ., L14)
All Statistics Canada series (L15, L16, . . ., L22)
All series together (L1, L2, L3, . . ., L22)
Old and modern as
Old and modern
Western (L1, L2)
Northern (L3, L10)
Southern (L4, L5, L6, L7, L9)
(L18, L19, L20)
Eastern (L11, L12, L14)
(maritime vs. continental)
Pacific-maritime (L1, L2)
Continental (L3, L4, L5, L6, L7, L9, L10)
(L16, L17, L18, L19, L20)
Atlantic-maritime‡ (L11, L12, L14)
*AIC differences given in parentheses correspond to assuming only constraints on lag-1 and lag-2 in the upper regime.
†1, ?s,2,1and ?s,2,2each being common for different series (different s); 2, ?s,i,jfor each i and each j being common for
different series (different s); 3, ?s,1,2, ?s,2,1, and ?s,2,2each being common for different series (different s).
and modern series in the Eastern (Atlantic-maritime) seem to share the same lag-1 and lag-2 coefficients in the decrease
phase, as imposing these constraints further decreases the AIC by 3.39 for both groupings.
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