1. Modelling ecological effects of climate fluctuations through the statistical modelling of long-term time series data Nils Christian Stenseth Centre for Ecological and Evolutionary Synthesis (CEES) Department of Biology University of Oslo, Norway …based on work together with several collaborators 2 nd International Conference on Mathemathical Biology - Alcalá Sept 2003

2. Focus on climate and ecology

3. Ecological effects on ecological dynamics: density-dependence versus density-independence CLIMATEVARIABILITY

4. Outline 1. 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

5. Reading the fingerprint of density dependence and density independence (such as climate) from biological time series t-2t-1tt+1 X t X t+1 time X t X t+1 = X t ·R(X t )  x t+1 = a 0 + (1 + a 1 )·x t +  t+1 (i) Density dependence only Statistical density dependence (DD) (ii) Density dependence and climate, non-interactive (additive) effects X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + (1 + a 1 )·x t + g(Clim t ) +  t+1 Clim t Additive effect of climate X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + [1 + a 1 (Clim t )]·x t +  t+1 (iii) Density dependence and climate, interactive effects Clim t Climate affecting strength of DD

6. The North Atlantic Oscillation (NAO) the difference in athmospheric pressure between the Azores and Iceland Iceland the Azores

7. The North Atlantic Oscillation (NAO) negative and positive phases NAO index 1860-2000 high NAO low NAO

8. Modelling the effect(s) of climate fluctuations (and harvesting) on population dynamics …some theoretical background

9. Single-species dynamics low b high b

10. Single-species dynamics

11. How to incorporate climatic variability in population dynamic models: - additively… …or non-additively X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + [1 + a 1 (Clim t )]·x t +  t+1 (iii) Density dependence and climate, interactive effects Clim t Climate affecting strength of DD (ii) Density dependence and climate, non-interactive (additive) effects X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + (1 + a 1 )·x t + g(Clim t ) +  t+1 Clim t Additive effect of climate

12. Single-species dynamics with climate effect (here: NAO) N t+1 = N t R 1+(aN t ) b(NAO) Non-additive effect of climate Non-linear intrinsic and extrinsic processes

13. Single-species dynamics: possible effects of changing climate N t+1 = N t R 1+(aN t ) b(NAO) b(NAO)

14. An example: the soay sheep off the coast of Scotland - one single species

15. Soay sheep at Hirta, St Kilda

16. Soay sheep: dynamics depend on NAO N t = N t-1 (R 0 /  1+(N t-1 /K) b  t a 0 + a 1 (x t-1 - k) +  1,t if x t-1  k a 0 + a 2 (x t-1 - k) +  2,t if x t-1 > k x t = X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + [1 + a 1 (Clim t )]·x t +  t+1 (iii) Density dependence and climate, interactive effects Clim t Climate affecting strength of DD

17. Soay sheep: dynamics depend on NAO Using a FCTAR-model

18. Soay sheep: dynamics depend on NAO High NAO Low NAO N t+1 = N t R 1+(aN t ) b(NAO)

19. One species  to two species

20. Sætre et al., 1999 Stenseth et al., Science 2000 Changing competetive relationships dn 1 dt = k 1 – n 1 –  12 n 2 k1k1 r1n1r1n1 dn 2 dt = k 2 (NAO) – n 2 –  21 n 1 k 2 (NAO) r2n2r2n2 n 1 =log(N 1 ), n 2 =log(N 2 )

21. Pied Flycatcher Collared flycatcher Collared, high NAO Collared, low NAO Pied Sætre et al., 1999 Stenseth et al., Science 2000 Changing competetive relationships

22. Grey-sided vole in Hokkaido Seasonal forcing and ecological dynamics (back to within-population dynamics)

23. Hokkaido voles Cold and warm currents determine differential seasonal patterns Stenseth et al., PRSB, 2002

24. Seasonal forcing – an example of ”regime shift” – a bifurcation Stenseth et al., Res. Pop. Ecol. 1998 x t = b 0 + b 1 x t-1 + b 2 x t-2 N t = N t-1 exp[(a w0 –a w1 x t-1 –a w2 x t-2 )(1-  )] ·exp(a s0 –a s1 x t-1 –a s2 x t-2 )  ]

25. Hokkaido voles: observations only the fall data AR2 models Stenseth et al., PRSB, 2002

26. Hokkaido voles: observations South North Stenseth et al., PRSB, 2002 x t =  1 x t-1 +  2 x t-2 +  t

27. Hokkaido voles: can we predict the observed patterns? Stenseth et al., PRSB, 2002

28. Hokkaido voles: predictions Stenseth et al., PRSB, 2002 x t =  1 x t-1 +  2 x t-2 +  t N t = N t-1 R summer R winter R summer = C 1 exp[(–a s1 [log(C 2 ) + (1 – a w1 + a w1  ) x t-1 –a w2 (1 –  )x t-2 ] – a s2 x t-2 )  ] R winter = C 2 exp[(–a w1 x t-1 – a w2 x t-2 )(1 –  )]  1 = 1 – a w1 + (– a s1 + a s1 a w1 + a w1 )  – a s1 a w2    2 = – a w2 + (a s1 a w1 – a s2 + a w2 )  – a s1 a w1  

29. Hokkaido voles Stenseth et al., PRSB, 2002 X t X t+1 = X t ·R(X t, Clim t )  x t+1 = a 0 + [1 + a 1 (Clim t )]·x t +  t+1 (iii) Density dependence and climate, interactive effects Clim t Climate affecting strength of DD x t+1 = a 0 + [1 + a 1 (Clim t )]·x t + [1 + a 2 (Clim t )]·x t-1 +  t+1

30. Hokkaido voles: more detailed data both spring and fall data Stenseth et al., PNAS, in review

31. Hokkaido voles: observations Stenseth et al., PNAS, in review

32. Hokkaido voles: predictions Stenseth et al., PNAS, in review Melt-off highly variable in the mountains

33. Stenseth et al., Res. Pop. Ecol. 1998 Seasonal forcing is an example of ”regime shift” – a bifurcation

34. Season length determines the population dynamics changing from non-cyclic to cyclic i.e., a bifurcation

35. Season length is determined by the climate i.e., the dynamic bifurcation is casued by climatically driven seasonal forcing

36. 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

37. 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