Introducing Mathematical Modeling in an Undergraduate Ecology Laboratory
Time-series modeling in ecology: a synoptic overview
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Transcript of Time-series modeling in ecology: a synoptic overview
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Time-series modeling in ecology: Time-series modeling in ecology: a synoptic overviewa synoptic overview
Nils Chr. StensethNils Chr. Stenseth
Centre for Ecological and Centre for Ecological and Evolutionary SynthesisEvolutionary Synthesis
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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
Matematisk Institutt, Oslo (05.04.05)
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Matematisk Institutt, Oslo (05.04.05)
Charles Sutherland Elton (March 29, 1900 - May 1, 1991)
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Matematisk Institutt, Oslo (05.04.05)
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”.
Oxford
Spitzbergen
Tromsø
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Matematisk Institutt, Oslo (05.04.05)
Lemming and vole cycles
In the Bible: “..swarms of grasshoppers sweeping over the country ..”
A frustratingly distinct pattern with an ennoyingly elusive explanation
Begon, 1998
lemmings
14th century
From Olaus Magnus (1555) A Description of the Northern Peoples
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Matematisk Institutt, Oslo (05.04.05)
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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Matematisk Institutt, Oslo (05.04.05)
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)
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Matematisk Institutt, Oslo (05.04.05)
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Matematisk Institutt, Oslo (05.04.05)
Lynx time series
Stenseth et al., Proc. Natl. Acad. Sci. 1998
1820-1940
1920-1994
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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?
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22122
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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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Matematisk Institutt, Oslo (05.04.05)
log-transformed time series normalized to mean zero
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Matematisk Institutt, Oslo (05.04.05)
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Matematisk Institutt, Oslo (05.04.05)
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Matematisk Institutt, Oslo (05.04.05)
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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Matematisk Institutt, Oslo (05.04.05)
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]
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
is equivalent to
yt-2
2,2 y
t-2
yt-2
1,2 y
t-2
LowerUpperPhase dependency: threshold model
non-linear
Stenseth et al., Proc. Natl. Acad. Sci. 1998
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Matematisk Institutt, Oslo (05.04.05)
Phase-dependence
Stenseth et al., Proc. Natl. Acad. Sci. 1998
Functional responsePhase dependency
Rochester, Alberta Kluane Lake, Yukon
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Matematisk Institutt, Oslo (05.04.05)
Let us ask the lynx (or the data on the lynx)...
Is there any spatial
structuring of these time-
series data?
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Matematisk Institutt, Oslo (05.04.05)
What is the spatial structuring force(s)?
Stenseth et al., Science 1999
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Matematisk Institutt, Oslo (05.04.05)
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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Matematisk Institutt, Oslo (05.04.05)
The North Atlantic Oscillation (NAO)negative and positive phases
NAO index 1860-2000
high NAO
low NAO
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Matematisk Institutt, Oslo (05.04.05)
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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Matematisk Institutt, Oslo (05.04.05)
What is the underlying causes of the geographic structuring?
Stenseth et al., Science 1999
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Matematisk Institutt, Oslo (05.04.05)
Snow is a key factor for the trophic interaction between hare and lynx
‘X’ = locations (stations) that exhibit statistical significance at the 5% level
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Stenseth et al., Proc. Natl. Acad. Sci. (2004)
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… 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
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Matematisk Institutt, Oslo (05.04.05)
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A synoptic account of the legacy of Elton’s work on the cycle problem – particularly on voles, mice and lemmings
Population studies on voles, mice and lemmings
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Matematisk Institutt, Oslo (05.04.05)
A way to summarize small rodent dynamics:
Direct annual density dependence (a1)
Del
ayed
an
nu
al d
ensi
ty d
epen
den
ce (
a 2)
2.0
3.0 4.0 5.0 6.0
Proper multiannual cycles
2-year ’cycles’
Stable
xt = a1xt-1 + a2xt-2 + t
Population dynamics: cycles and non-cycles
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Matematisk Institutt, Oslo (05.04.05)
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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Matematisk Institutt, Oslo (05.04.05)
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 Hokkaidoand seasonal forcing
Stenseth et al. Res Pop Ecol, 1998.Stenseth et al. PRSB, 1999.
• 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
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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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Matematisk Institutt, Oslo (05.04.05)
Single-species dynamics
bt
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Matematisk Institutt, Oslo (05.04.05)
Single-species dynamics
bt
tt aN
RNN
)(11
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Matematisk Institutt, Oslo (05.04.05)
Single-species dynamics
How to incorporate climatic variability in population dynamic models:- additively…
…or non-additively
Xt Xt+1 = Xt·R(Xt, Climt) xt+1 = a0 + [1 + a1(Climt)]·xt + t+1
(iii) Density dependence and climate, interactive effects
Climt
Climate affecting strength of DD
(ii) Density dependence and climate, non-interactive (additive) effects
Xt Xt+1 = Xt·R(Xt, Climt) xt+1 = a0 + (1 + a1)·xt + g(Climt) + t+1
Climt
Additive effect of climate
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Matematisk Institutt, Oslo (05.04.05)
Mathematical and statistical modeling
Nt = Nt-1(R0/1+(Nt-1/K)bt Maynard-Smith – Slatkin model
a0 + a1(xt-1 - k) + 1,t if xt-1 k
a0 + a2(xt-1 - k) + 2,t if xt-1 > k xt =
Statistical model
Xt Xt+1 = Xt·R(Xt, Climt) xt+1 = a0 + [1 + a1(Climt)]·xt + t+1
(iii) Density dependence and climate, interactive effects
Climt
Climate affecting strength of DD
…generalized statistical model
t
1,t
2,t
b
a1
a2
Much statistical work needs to be done – and is been done
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Matematisk Institutt, Oslo (05.04.05)
Single-species dynamics with climate effect (here: NAO)
Nt+1 = Nt R
1+(aNt )b(NAO)
• Non-additive effect of climate
• Non-linear intrinsic and extrinsic processes
exp(κ)
Using a piecewise linear model (FCTAR) for estimating parameters and functions
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Matematisk Institutt, Oslo (05.04.05)
Single-species dynamics: possible effects of changing climate
Nt+1 = Nt R
1+(aNt )b(NAO)
b(NAO)
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Matematisk Institutt, Oslo (05.04.05)
An example: the soay sheep off the coast of
Scotland- one single species
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Matematisk Institutt, Oslo (05.04.05)
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Matematisk Institutt, Oslo (05.04.05)
Soay sheep at Hirta, St Kilda
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The effect of climatic fluctuation on population dynamics
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Matematisk Institutt, Oslo (05.04.05)
ResultsSoay sheep: dynamics depend on NAO
Using a FCTAR non-linear and non-additive model
Stenseth et al. (2004)
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High NAO
Low NAONt+1 = Nt R
1+(aNt )b(NAO)
Soay sheep: dynamics depend on NAO
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Matematisk Institutt, Oslo (05.04.05)
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.
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Matematisk Institutt, Oslo (05.04.05)
Long-term ecological time series, ecology and statistical modeling
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Matematisk Institutt, Oslo (05.04.05)
Elton and Elton and the Oxford Bureauthe 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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Matematisk Institutt, Oslo (05.04.05)
… … long-term ecological data need to store long-term ecological data need to store in a bank so that others can use them …in a bank so that others can use them …
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Matematisk Institutt, Oslo (05.04.05)
Valuable data in HokkaidoValuable data in Hokkaido
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Matematisk Institutt, Oslo (05.04.05)
Banking and maintaining long-term Banking and maintaining long-term data of two kinds: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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Matematisk Institutt, Oslo (05.04.05)
Open access web-basedOpen access web-based data bases data bases
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Matematisk Institutt, Oslo (05.04.05)
Such a data-bank should be organized Such a data-bank should be organized according the principles of our traditional according the principles of our traditional museums museums
… … but should take fully advantage of mordern but should take fully advantage of mordern computer technologycomputer technology
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We need to save such data filesWe need to save such data files
This we must avoid!This we must avoid!
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F. Finse, Norway: Norwegian lemming, L. lemmus
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