Time-series modeling in ecology:
a synoptic overview
Nils Chr. Stenseth
Centre for Ecological and
Evolutionary Synthesis
Outline
1. The British Ecologist, Charles Elton – the first
ecologist to appreciate the importance of long-term
monitoring ecological data.
2. The Canadian lynx.
3. Vole, mice and lemmings.
4. The Soay sheep off the coast of Scotland.
5. Statistical modeling of long-term monitoring data.
6. A French-Norwegian data bank for ecological data
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Charles Sutherland Elton (March 29, 1900 - May 1, 1991)
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Elton: a zoologist – and the founding father of
(modern) ecology
• Lemmings – the Norwegian lemming – and long-term data on abundance of
lemmings – played a key role in his intellectual development
• Julian Huxley invited him as a field assistant to Spitsbergen/Svalbard in 1921
– the first of several expeditions
“… I did go, and the experience had a profound influence
upon my ideas in ecology …”
• While returning from Spitzbergen in September 1923:
“I bought a book in a Tromsø
shop that changed my whole
life. It was bought with one of
the three pounds I had left in
my pocket – Robert Collett’s
‘Norges Pattedyr’ (=Norwegian
Mammals) .. it was the part
about lemmings that enthralled
me”.
Spitzbergen
Tromsø
Oxford
Matematisk Institutt, Oslo (05.04.05)
Lemming and vole cycles
lemmings
In the Bible: “..swarms of grasshoppers
sweeping over the country ..”
14th century
A frustratingly distinct pattern with an
ennoyingly elusive explanation
Begon, 1998
From Olaus Magnus (1555) A Description of the Northern Peoples
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But much story telling and myths around the
lemmings and the lemming/vole cycles
- raining from the sky
- returning to Atlantis: a debate between Crotch and Collett in
the pages of Nature in 1876
- Walt Disney in Barrow [Biology today (1971)]
- Donald Duck in the Norwegian fjords.
The Norwegian fiords are well known
Donald Duck is well known
But few know that he has watched lemmings
running down the from the Norwegian mountains into the
Norwegian fjords
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Elton contributed to make ecology quantitative at
the Bureau of Animal Population
A definition of ecology: “Ecology is the scientific
endeavor aiming at explaining the distribution
and abundance – and their changes thereof – of
species in space through time by studying the
environment of individuals in natural populations”
(after CJ Krebs)
That is, a quantitative definition of
ecology
“George” (PH) Leslie: The Leslie population
matrix and Capture-Mark-Recapture modelling
(e.g., Caswell 2003)
Matematisk Institutt, Oslo (05.04.05)
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Lynx time series
1920-1994
1820-1940
Stenseth et al.,
Proc. Natl. Acad. Sci. 1998
Matematisk Institutt, Oslo (05.04.05)
Snowshoe hare and lynx are
highly interconnected
– but can we (through a second
order autoregressive model)
considerer only one of the
species
– and believe that we’ve gotten
a “full” understanding of the
dynamic interaction in the
system?
Matematisk Institutt, Oslo (05.04.05)
X t1  X t11  b1  a11 X t11  a12 X t21   t1
X X
2
t
2
t 1
 b2  a21 X
1
t 1
 a22 X
2
t 1

2
t
Xt = b + (I+A)Xt–1 + t
…
a second order delay equation in the variable we
have data on (typically the lynx)
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log-transformed time series normalized
to mean zero
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 a11 a12   .28  .48 .08 .12 
A





a
a
.
26

.
24
.
05
.
07

 

22 
 21
  11   .48   .08 

      
  22   .15   .04 
  .36  0.10
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Fur returns are good proxies for actual
abundance
Stenseth et al., Proc. Natl. Acad. Sci. 1998
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Linearity or non-linearity?
What do the data “say”?
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Predator-prey model with
phase-dependence
Hares:
Ht+1= Ht exp[ai,0 - ai,1xt - ai,2yt]
Predators:
Pt+1= Pt exp[bi,0 - bi,1yt - bi,2xt]
is equivalent to
yt = (ai,0bi,2 + ai,1bi,0) + (2 - ai,1 - bi,1)yt-1
+ (ai,1 + bi,1 - ai,1bi,1 - ai,2bi,2 - 1)yt-2 + t
non-linear
Phase dependency:
Lower
b1,2 yt-2
Upper
b2,2 yt-2
threshold model
Stenseth et al.,
Proc. Natl. Acad. Sci. 1998
Matematisk Institutt, Oslo (05.04.05)
yt-2
yt-2
Phase-dependence
Rochester, Alberta
Kluane Lake, Yukon
Phase dependency
Functional response
Stenseth et al., Proc. Natl. Acad. Sci. 1998
Matematisk Institutt, Oslo (05.04.05)
Let us ask the lynx
(or the data on the lynx)...
Is there any
spatial
structuring of
these timeseries data?
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What is the spatial structuring
force(s)?
Stenseth et al.,
Science 1999
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Canada divided by climatic regions
Stenseth et al.,
Science 1999
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The North Atlantic Oscillation (NAO)
the difference in atmospheric pressure
between the Azores and Iceland
Iceland
the Azores
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The North Atlantic Oscillation (NAO)
negative and positive phases
low NAO
high NAO
NAO index 1860-2000
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A package of weather
- Climate indices
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Climatic zonation
Stenseth et al.,
Science 1999
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This grouping was a result of statistical modeling
Stenseth et al.,
Science 1999
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What is the underlying causes of the geographic
structuring?
Stenseth et al.,
Science 1999
Matematisk Institutt, Oslo (05.04.05)
Difference in frequency of winter warm spells
between opposite polarity of the NAO
Snow is a key factor for the trophic interaction
between hare and lynx
‘X’ = locations (stations) that exhibit statistical significance at the 5% level
Stenseth et al., Proc. Natl. Acad. Sci. (2004)
Matematisk Institutt, Oslo (05.04.05)
… the snow condition may be a
key factor in structuring the
dynamic interaction between the
hare and the lynx
Source: Rudolfo's Usenet Animal Pictures Gallery
Matematisk Institutt, Oslo (05.04.05)
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Population studies on voles, mice and
lemmings
A synoptic account of the legacy of Elton’s work on the cycle
problem – particularly on voles, mice and lemmings
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Population dynamics:
cycles and non-cycles
A way to summarize small rodent dynamics:
Delayed annual density dependence (a2)
xt = a1xt-1 + a2xt-2 + t
2-year
’cycles’
Stable
2.0
3.0
4.0
5.0 6.0
Proper multiannual cycles
Direct annual density dependence (a1)
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Cycles & Non-Cycles: a synoptic account (after Stenseth 1999, Oikos)
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The Fennoscandian gradient
Bjørnstad et al. PRSB, 1996.
Stenseth et al. PRSB, 1996.
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A continental European gradient
Tkadlec & Stenseth PRSB, 1996.
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Grey-sided voles in Hokkaido
Stenseth et al. PRSB, 1996; Stenseth et al. Res Pop Ecol, 1998.
Stenseth & Saitoh Pop Ecol, 1998.
Stenseth et al. PRSB, 2002; Stenseth et al. PNAS 2003.
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Grey-sided voles in Hokkaido
and seasonal forcing
• the density dependent
structure differ between
seasons
• the variation in density
dependences among sites is –
it seems – fully accounted for
by the length of the seasons
• long winters tend to generate
cycles
Stenseth et al. Res Pop Ecol, 1998.
Stenseth et al. PRSB, 1999.
Matematisk Institutt, Oslo (05.04.05)
Vole, Mice and Lemmings:
some conclusions
1. Populations within a given species might be both
cyclic and non-cyclic.
2. Typically there are geographic gradients in the
periodic structure.
3. Statistical work lead us to understand that the
relative length of the seasons might determine
whether cycles or non-cycles occur.
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Modelling the effect(s) of
climate fluctuations on
population dynamics
…some theoretical background
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Single-species dynamics
Nt R
N t 1 
b
1  (aN t )
R
1 (aN t ) b
0.25
0.2
0.15
0.1
low b
0.05
high b
0
0
2
4
6
8
10
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Nt
Single-species dynamics
Nt R
N t 1 
1  (aN t ) b
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Single-species dynamics
How to incorporate climatic variability in
population dynamic models:
- additively…
(ii) Density dependence and climate, non-interactive (additive) effects
Climt
Xt
Xt+1 = Xt·R(Xt, Climt)  xt+1 = a0 + (1 + a1)·xt + g(Climt) + t+1
Additive effect of climate
…or non-additively
(iii) Density dependence and climate, interactive effects
Xt
Climt
Xt+1 = Xt·R(Xt, Climt)  xt+1 = a0 + [1 + a1(Climt)]·xt + t+1
Climate affecting strength of DD
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Mathematical and statistical modeling
Maynard-Smith – Slatkin model
Nt = Nt-1(R0/1+(Nt-1/K)b
t
Statistical model
xt =
a0 + a1(xt-1 - k) + 1,t
if xt-1  k
a0 + a2(xt-1 - k) + 2,t
if xt-1 > k
…generalized statistical model
(iii) Density dependence and climate, interactive effects
Xt
Climt
Xt+1 = Xt·R(Xt, Climt)  xt+1 = a0 + [1 + a1(Climt)]·xt + t+1
Climate affecting strength of DD
Much statistical work needs to be done – and is been done
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Single-species dynamics with climate
effect (here: NAO)
Nt R
Nt+1 =
1+(aNt )b(NAO)
exp(κ)
• Non-additive effect of climate
• Non-linear intrinsic and extrinsic processes
Using a piecewise linear model (FCTAR) for
estimating parameters and functions
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Single-species dynamics: possible effects of changing climate
Nt R
Nt+1 =
1+(aNt )b(NAO)
b(NAO)
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An example: the soay sheep
off the coast of Scotland
- one single species
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The effect of climatic fluctuation on population dynamics
Soay sheep at Hirta, St Kilda
Number of individuals
2500
2000
1500
1000
500
0
1955
1965
1975
1985
1995
Year
6
4
NAO
2
0
-2
-4
-6
1955
1965
1975
1985
1995
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Results
Soay sheep: dynamics depend on NAO
Using a FCTAR non-linear and non-additive model
Stenseth et al. (2004)
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Soay sheep: dynamics depend on NAO
Low NAO
High NAO
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Nt R
Nt+1 =
1+(aNt )b(NAO)
Soay sheep: some
conclusions
1. There is a clear density dependent structure due to
within population interaction.
2. The strength of this density dependency is affected
by climate.
3. Hence, climate may influence the dynamics
properties of the population.
Matematisk Institutt, Oslo (05.04.05)
Long-term ecological time
series, ecology and
statistical modeling
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Elton and
the Oxford Bureau
A great naturalist who founded modern ecology and by so doing
stated the development of making ecology a quantitative field
Observational
field studies
Providing
important long
term data series…
Time series on total count
Time series on individuals
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… long-term ecological data need to store
in a bank so that others can use them …
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Valuable data in Hokkaido
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Banking and maintaining long-term
data of two kinds:
1. Open access data-bases: scientists might be reluctant to
store their data in such an open bank – and users might not
obtain the proper background information for using the
stored data properly
2. Rather, the data should be stored in what resembles old
traditional museums (with staff members, i.e., curators,
which can provide background knowledge about the stored
data)
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Open access web-based data bases
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Such a data-bank should be organized
according the principles of our traditional
museums
… but should take fully advantage of mordern
computer technology
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We need to save such data files
This we must avoid!
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40 F. Finse, Norway: Norwegian lemming, L. lemmus
35
30
25
20
15
10
5
0
1945
1955
1965
1975
1985
1995