Howell Tong conference The importance of TAR- modeling for understanding the structure of ecological dynamics: the hare-lynx population cycle as an example

Howell Tong conference

The importance of TAR-The importance of TAR-modeling for understanding modeling for understanding the structure of ecological the structure of ecological dynamics: the hare-lynx dynamics: the hare-lynx population cycle as an population cycle as an

exampleexample

Nils Chr. Stenseth and othersHowell Tong Conference

Nils Chr. StensethNils Chr. StensethCentre of Ecological and Evolutionary Synthesis (Centre of Ecological and Evolutionary Synthesis (CEESCEES))

Institute of Biology, University of Oslo, NorwayInstitute of Biology, University of Oslo, Norway


Maps and pictures from Big Cats Online (dialspace.dial.pipex.com/agarman/bco)

The Lynx species

Canadian lynx (Lynx canadensis)

Iberian lynx (Lynx pardinus)

Bobcat (Lynx rufus)

Eurasian lynx (Lynx lynx)


... after which it has become a standard text-book example ...

Charles Elton (1924, 1942)started it all


Raven & Johnson 1996: Biology


Krebs 2001: Ecology


Futuyma 1998: Evolutionary Biology


Edelstein-Keshet 1988: Mathematical Models

in Biology


… and, of course, Howell Tong’s time series book at OUP …


The lynx’s food-web …

… data from south of the Hudson Bay in Canada. Stenseth et al., Proc. Natl. Acad. Sci. 1997


… and the snowshoe hare food-web



What do the experimental data tell us?

... and what do observational (time-series) data tell us?

… and do they tell the same story?


The experiment by Krebs et al. (Kluane lake 1987-94)

Treatment I Treatment II Treatment III

Reducing predators Adding food Reducing predatorsand adding food

2x increase of hares 2x increase of hares 10x increase of hares

Evidence for three trophic level effects

Snowshoe hare populations: Squeezed from below and above

Statistical model

ht+1 = 0 + 1 ht + 2 ht-1 + 3 ht -2 + ... i.e., an order 3 process

Krebs et al., Science 1995; Stenseth, Science 1995

pt+1 = 0 + 1 pt + 2 pt-1 + ... i.e., an order 2 process


As the lynx sees its food web …



… and as the snowshoe hare sees its food-web


Predators

Food species


That is, the hare and the lynx see their food web slightly differently

Stenseth et al., Proc. Natl. Acad. Sci. 1997

• The hare sees all its enemies (and doesn’t care who kills it)

• The hare also sees all its food species (and doesn’t really care who provides its dinner)

• The hare is also somewhat self-regulated

Hence, an order three process.

• The lynx gets its dinner from a great variety of prey species – but primarily from the hare

• The lynx is also somewhat self-regulated

Hence, an order two process


Robert May – Lord May of Oxford – has worked on the same topic:May, R.M. 1973. Time-delays versus stability in population dynamics with two and three trophic levels. Ecology 54: 315-325.

Bob May says (in an e-mail to me on July 6, 2004): “Send him my very best wishes for the success of the meeting, and my most sincere apologies for not being able to be there with you”


Lynx time series


1820-1940

1920-1994


Fur returns are good proxies for actual abundance



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


The non-linearity is due to phase-dependent relations

between the hare and the lynx

Functional responseThis phase-dependency

may furthermore be due to fluctuating

weather conditions

… indeed, snow-condition enters as a significant covariate producing a similar functional response


Understanding the phase dependency was a direct result of statistical modelling

Stenseth et al., PNAS 1998


… 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


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



Climatic zonation



This grouping was a result of statistical modelling



Are the genetics structured similar to the ecology?

... we obtained samples …

Rueness et al., Nature (2003)


The sampling scheme

We aimed at as good coverage of the entire Canada as possible ...



That is: (1) the Pacific region is genetically different from the Continental region, and

(2) the Continental region is different from the Atlantic region

Genetics of lynx



Remember:Snow condition

affects the success of hare and lynx

That genetic differentiation between the Pacific region and the Continental region is ”easy”: the Rockies

But why the genetic differentiation between the Continental region and the Atlantic region?


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. (in press)


Understanding the link between the grouping of the lynx dynamics, climate and snow condition – and how that operates through the direct hare-lynx food-web link –

was a direct result of statistical modelling

Stenseth et al., PNAS 2004a,b


All having a similar ecological structuring as the lynx (Yao et al., 2000. PRSB.)

Genetics of mink and muskrat

Lynx Mink Muskrat

Pictures/maps from:www.washington.edu/burkemuseum/

If the peculiarities of the lynx life history is important, similar genetic differentiation south of the Hudson Bay is not expected for mink and muskrat


Working on the dynamics of the Canadian lynx has led me into stimulating collaborative work with Ecologists:

and Geneticists

Statisticians:

-Charles Krebs, Wilhelm Falck, Ottar Bjørnstad, Mark O’Donoghue, Stan Boutin, Rudy Boonstra, Hildegunn Viljugrein,Nigel Yoccoz and Eric Post,

-Kung-Sik Chan, Howell Tong andOle Chr. Lingjærde

- Kjetill Jakobsen, Hans Ellegren, Eli Rueness and Dorothee Ehrich.

Documents

Howell Tong conference The importance of TAR- modeling for understanding the structure of ecological dynamics: the hare-lynx population cycle as an example