Time-series modeling in ecology: a synoptic overview Nils Chr. Stenseth Centre for Ecological and Evolutionary Synthesis

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

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)


Charles Sutherland Elton (March 29, 1900 - May 1, 1991)


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


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


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


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)



Lynx time series

Stenseth et al., Proc. Natl. Acad. Sci. 1998

1820-1940

1920-1994


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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ttttt

ttttt

XaXabXX

XaXabXX

Xt = b + (I+A)Xt–1 + t

a second order delay equation in the variable we have data on (typically the lynx)

…


log-transformed time series normalized to mean zero


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Fur returns are good proxies for actual abundance



Linearity or non-linearity? What do the data “say”?


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



Phase-dependence


Functional responsePhase dependency

Rochester, Alberta Kluane Lake, Yukon


Let us ask the lynx (or the data on the lynx)...

Is there any spatial

structuring of these time-

series data?


What is the spatial structuring force(s)?

Stenseth et al., Science 1999


Canada divided by climatic regions



The North Atlantic Oscillation (NAO)the difference in atmospheric pressure

between the Azores and Iceland

Iceland

the Azores


The North Atlantic Oscillation (NAO)negative and positive phases

NAO index 1860-2000

high NAO

low NAO


A package of weather- Climate indices


Climatic zonation



This grouping was a result of statistical modeling



What is the underlying causes of the geographic structuring?



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)


… 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



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


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


Cycles & Non-Cycles: a synoptic account (after Stenseth 1999, Oikos)


The Fennoscandian gradient

Bjørnstad et al. PRSB, 1996.Stenseth et al. PRSB, 1996.


A continental European gradient

Tkadlec & Stenseth PRSB, 1996.


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.


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


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.


Modelling the effect(s) of climate fluctuations on population dynamics

…some theoretical background


Single-species dynamics

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high b

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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


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


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


Single-species dynamics: possible effects of changing climate

Nt+1 = Nt R

1+(aNt )b(NAO)

b(NAO)


An example: the soay sheep off the coast of

Scotland- one single species



Soay sheep at Hirta, St Kilda

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1955 1965 1975 1985 1995

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The effect of climatic fluctuation on population dynamics


ResultsSoay sheep: dynamics depend on NAO

Using a FCTAR non-linear and non-additive model

Stenseth et al. (2004)


High NAO

Low NAONt+1 = Nt R

1+(aNt )b(NAO)

Soay sheep: dynamics depend on 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.


Long-term ecological time series, ecology and statistical modeling


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


… … 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 …


Valuable data in HokkaidoValuable data in Hokkaido


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)


Open access web-basedOpen access web-based data bases data bases


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


We need to save such data filesWe need to save such data files

This we must avoid!This we must avoid!

F. Finse, Norway: Norwegian lemming, L. lemmus

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1945 1955 1965 1975 1985 1995

Documents

Time-series modeling in ecology: a synoptic overview Nils Chr. Stenseth Centre for Ecological and Evolutionary Synthesis