Modelling ecological effects Modelling ecological effects of climate fluctuations of climate fluctuations through the statistical through the statistical

modelling of long-term time modelling of long-term time series dataseries dataNils Christian Stenseth

Centre for Ecological and Evolutionary Synthesis (CEES)Department of Biology

University of Oslo, Norway

…based on work together with several collaborators

2nd International Conference on Mathemathical Biology - Alcalá Sept 2003

Focus on climate and ecology

Ecological effects on ecological dynamics: density-dependence

versus density-independence

CLIMATECLIMATEVARIABILITYVARIABILITY

Outline1. Some few conceptual introductory remarks

2. Large-scale climate indices (e.g., the North Atlantic Oscillation (NAO), El Nino)

3. Modelling ecological effects of climate fluctuations (e.g., linear/non-linear, additive/non-additive)

4. Population ecology: The dynamics of the Soay sheep off Scotland: non-linear, non-additive climate effects

5. Two species – Community ecology: Climatic influence on competitive relationships among species

6. Population ecology: Voles in Hokkaido, Japan

7. Conclusion

Reading the fingerprint of density dependence and density independence (such as climate) from biological time series

t-2 t-1 t t+1

Xt Xt+1 time

Xt Xt+1 = Xt·R(Xt) xt+1 = a0 + (1 + a1)·xt + t+1

(i) Density dependence only

Statistical density dependence (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

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

The North Atlantic Oscillation (NAO)the difference in athmospheric 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

Modelling the effect(s) of climate fluctuations (and harvesting) on population

dynamics

…some theoretical background

Single-species dynamics

bt

tt aN

RNN

)(11

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10

low b

high b

btaN

R

)(1

tN

Single-species dynamics

bt

tt aN

RNN

)(11

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

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

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

0

5 0 0

1 0 0 0

1 5 0 0

2 0 0 0

2 5 0 0

1 9 5 5 1 9 6 5 1 9 7 5 1 9 8 5 1 9 9 5

Y e a r

Nu

mb

er o

f in

div

idu

als

-6

-4

-2

0

2

4

6

1 9 5 5 1 9 6 5 1 9 7 5 1 9 8 5 1 9 9 5

NA

O

Soay sheep: dynamics depend on NAO

Nt = Nt-1(R0/1+(Nt-1/K)bt

a0 + a1(xt-1 - k) + 1,t if xt-1 k

a0 + a2(xt-1 - k) + 2,t if xt-1 > k xt =

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

Soay sheep: dynamics depend on NAO

Using a FCTAR-model

Soay sheep: dynamics depend on NAO

High NAO

Low NAONt+1 = Nt R

1+(aNt )b(NAO)

One species to two species

Sætre et al., 1999Stenseth et al., Science 2000

Changing competetive relationships

dn1

dt=

k1 – n1 –12n2

k1r1n1

dn2

dt=

k2 (NAO) – n2 –21n1

k2(NAO)r2n2

n1 =log(N1 ), n2 =log(N2 )

Pied Flycatcher

Col

lare

d fl

ycat

cher

Collared, high NAO

Collared, low NAO

Pied

Sætre et al., 1999Stenseth et al., Science 2000

Changing competetive relationships

Grey-sided vole in Hokkaido

Seasonal forcing and ecological dynamics (back to within-population dynamics)

Hokkaido voles

Cold and warm currents determine differential seasonal patternsStenseth et al., PRSB, 2002

Seasonal forcing – an example of ”regime shift” – a bifurcation

Stenseth et al., Res. Pop. Ecol. 1998

xt = b0 + b1xt-1 + b2xt-2

Nt = Nt-1exp[(aw0–aw1xt-1–aw2xt-2)(1-)] ·exp(as0–as1xt-1–as2xt-2)]

Hokkaido voles: observations only the fall data

AR2 models

Stenseth et al., PRSB, 2002

Hokkaido voles: observations

SouthNorth

Stenseth et al., PRSB, 2002

xt = 1xt-1 + 2xt-2 + t

Hokkaido voles: can we predict the observed patterns?

Stenseth et al., PRSB, 2002

Hokkaido voles: predictions

Stenseth et al., PRSB, 2002

xt = 1xt-1 + 2xt-2 + t

Nt = Nt-1 Rsummer Rwinter

Rsummer = C1exp[(–as1 [log(C2) + (1 – aw1 + aw1) xt-1–aw2 (1 – )xt-2] – as2 xt-2)]Rwinter = C2exp[(–aw1xt-1 – aw2xt-2 )(1 – )]

1 = 1 – aw1 + (– as1 + as1aw1 + aw1)– as1aw2

2 = – aw2 + (as1aw1 – as2 + aw2)– as1aw1

Hokkaido voles

Stenseth et al., PRSB, 2002

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

xt+1 = a0 + [1 + a1(Climt)]·xt + [1 + a2(Climt)]·xt-1 + t+1

Hokkaido voles: more detailed databoth spring and fall data

Stenseth et al., PNAS, in review

Hokkaido voles: observations

Stenseth et al., PNAS, in review

Hokkaido voles: predictions

Stenseth et al., PNAS, in review

Melt-off highly variable in the mountains

Stenseth et al., Res. Pop. Ecol. 1998

Seasonal forcing is an example of ”regime shift” – a bifurcation

Season length determines the population dynamics

changing from non-cyclic to cyclici.e.,

a bifurcation

Season length is determined by the climate

i.e.,the dynamic bifurcation is casued

by climatically driven seasonal forcing

Conclusions

1. Indices (North Atlantic Oscillation and the like) are found to be good climate proxies useful for understanding how climatic fluctuations have affected ecological pattern and processes in the past.

2. Climatic variation affect ecological dynamics (e.g., Soay sheep) through behavioral changes having dynamic effects

3. Climatic variation affect ecological dynamics (e.g., Hokkaido voles) through the length of the seasons having dynamic effects

Methodological coda

1. Understanding what the response of ecological systems to environmental change has been in the past will help us be prepared for what might happen in the future.

2. For this, monitoring data is essential – and the statistical modeling thereof is important.

3. Mathematical modeling is important to understand the dynamic consequences of possible climate change