1. A new integrated measure of selection: when both demography and selection vary over time Carol Horvitz 1, Tim Coulson 2, Shripad Tuljapurkar 3, Douglas Schemske 4 1 University of Miami, Coral Gables, FL 2 Imperial College, Silwood Park, London, UK 3 Stanford University, Stanford, CA 4 Michigan State University, East Lansing, MI Theoretical Ecology Department Lund University, Lund, Sweden, 29 April 2008

3. Institute for Theoretical and Mathematical Ecology University of Miami Coral Gables, FL USA Mathematics Steve Cantrell Chris Cosner Shigui Ruan Biology Don De Angelis Carol Horvitz Matthew Potts Marine Science Jerry Ault Don Olson

4. floral tube length and birth date How can we integrate variable selection across years? *for structured populations and overlapping generations

5. Preview: Integrated selection on Calathea floral tube length El niño driven -0.071 Stasis -0.098 Tree-fall -0.156 Dry season severity -0.103 Environmental driver Selection _____________________________________________

6. Preview: Integrated selection on red deer birth date NAO driven -0.247 26-yr cycle -0.289 IID and equal -0.287 Quality correlated -0.239 Environmental driver Selection _____________________________________________

7. Preview: Integrated selection Environment-specific elasticity X Environment-specific selection gradient summed across all relevant life history and environmental paths Horvitz, Coulson, Tuljapurkar, Schemske (in prep)

8. a small tropical Mexican herb and a large Scottish mammal Fitness components and stochastic growth rate Selection gradients vary Demographic transitions vary Environmental states are dynamic Environmental driver matters

9. a small tropical Mexican herb and a large Scottish mammal Floral tube length (pollinator related) 3 yrs of selection gradients Fruit production Demographic projection matrices for 4 yrs Local, regional and global environmental dynamics Birth date (seasonal advantage) 26 yrs of selection gradients Recruitment and survival for two classes Demographic projection matrices for 26 yrs Local, regional and global environmental dynamics

10. Phenotypic selection theory Relative fitness regressed against quantitative trait value The slope of the regression = selection gradient for the trait (Lande and Arnold 1983 Evolution) fitness something quantitative

11. Candidate parameters for measuring fitness Fitness components Reproduction (stage-specific) Survival (stage-specific) Growth (stage-specific) Population growth rate

12. Candidate parameters for measuring fitness Fitness components Reproduction (stage-environment-specific) Survival (stage-environment-specific) Growth (stage-environment-specific)  Stochastic growth rate

13. Schemske and Horvitz 1989 Evolution ** Fitness component vs floral tube length (relative, mean-standardized) Standardized selection Years => Environments mature fruits

14. Years => Environments Fitness component vs birth date (relative, mean-standardized) Coulson et al. 2003 Evolution

15. Demographic transitions and fitness in a constant world N(t+1) = A N(t) A is a population projection matrix Transitions and contributions between stages, a ij = fitness components λ = population growth rate

16. N(t+1) = X(t) N(t) X(t) is a random variable A 1, A 2, A 3 …A K, K environments Transitions and contributions in each environment, a ijβ λ S = stochastic growth rate Tuljapurkar 1982, 1990 Demographic transitions and fitness in a variable world

17. In a variable world : sequences, frequencies and new sensitivities Environmental dynamics sequences along sample paths an expected long run stationary distribution λ s is sensitive to perturbations of means, variances, and transitions in particular environmental states E δ, E μ, E β and others… Tuljapurkar et al. 2003 Am Nat Horvitz et al. 2005 Ecology

18. Environmental dynamics: Scaling up using climate data Calathea : “Dry season ” driver Red Deer: “NAO” driver

19. Sample years in context of historical record Monthly rainfall during the dry season only

20. Sample years in context of historical record Monthly rainfall during the dry season only Annual Deviations from Mean NAO 1864-2006 Year, starting with 1864

21. Hypothetical environmental drivers markov chain models El niño Stasis Dry season Tree-falls

22. Hypothetical environmental drivers markov chain models Quality correlatedNAO IID and equal 26-yr cycle

23. Hypothetical environmental drivers markov chain models Quality correlatedNAO IID and equal 26-yr cycle

24. …2 3 4 3 3 2 2 2 4 4 3 1 3 2 2 3… El Niño …1 3 2 1 1 1 1 3 1 3 1 3 2 1 1 1… Dry season …3 3 3 3 3 3 3 1 4 4 2 2 3 3 3 3… Tree-falls …1 1 2 2 2 2 2 2 3 3 3 3 3 3 3 3… Stasis Sequences for hypothetical environmental drivers

25. … 17 11 11 17 6 6 11 6 6 11 4 11 6 22 20 … NAO … 7 15 22 10 22 3 13 3 22 21 13 2 20 … Quality correlated …2 18 19 6 2 4 1 25 3 17 4 20 10 9 4 … IID and equal …11 12 13 14 15 16 17 18 19 20 21 22 23 … 26-yr cycle

26. Environment-specific elasticity of λ s for each driver (to seed production) El niño Stasis Dry season Tree-falls Years => Environments Environment-specific Elasticity Stage class

27. Environment-specific elasticity of λ s with NAO driver Survival Recruitment Environment-specific Elasticity Age class Years => Environments NAO

28. Survival Recruitment Environment-specific Elasticity Age class Years => Environments Quality correlated Environment-specific elasticity of λ s for different driver…

29. Integrated elasticity stage-specific elasticity, e ij = change in λ due to a change in one element of the matrix X selection gradient = change in one element of the matrix due to a change in the trait value (van Tienderen 2000 Ecology, Coulson et al. 2003 Evolution)

30. Integrated stochastic elasticity Integrated selection Environment-specific elasticity, e ijβ = change in λ S due to a change in one matrix element in one state of the environment X selection gradient = change in one matrix element in one state of the environment due to a change in the trait value Horvitz, Coulson, Tuljapurkar, Schemske (in prep)

31. Calathea Each matrix is 8x8 4 environments (let’s look at 1) Selection gradient only on top row All reproductive stages have same value

32. 1 example (there are 4 per driver) “dry season” driver, envt 83-84 × = (elementwise multiplication)

33. Red deer Each matrix is 20 x 20 26 environments environment-specific elasticity males : zero females : recruitment and survival each age females are in 11 x 11 matrix, top left

34. 1 example (there are 26 per driver) “NAO” driver, envt 5 × = (elementwise multiplication)

35. Integrated selection by environmental state and TOTAL El niño Total= -0.071 Stasis Total -0.098 Tree- falls Total= -0.156 Dry season Total= -0.103 Years => Environments

36. Integrated selection by environmental state and TOTAL 26-yr cycle Total = -0.289 IID and equal Total= -0.287 Quality correlated Total = -0.239 NAO Total = -0.247 Years => Environments

37. Note: These are ALL negative Integrated selection by transition rate

38. Integrated selection by stage and type

39. Conclusions New parameter Integrates selection across the life cycle and across changing environments Uses λ s and its sensitivities (by environment) The force of selection on a trait depends upon environmental dynamics Historical climate data combined with a few years of demographic observations: plausible long run patterns

40. Thanks Per Lundberg 2006-08 NSF OPUS 1982-84, 1984-88 NSF 1982National Geographic NERC Royal Society Biotechnology and Biological Research Council Field assistants, students and colleagues

41. Extras for questions…

42. Sample years in context of historical record Monthly rainfall during the dry season only

43. Difference from the long-term mean, Years = Environments

44. Sample years in context of historical record Monthly rainfall during the dry season only Annual Deviations from Mean NAO 1864-2006 Year, starting with 1864

45. Standardized annual deviations (observed/SD) of NAO Historical data 1864-2006

46. Flowers with trigger

47. Tongue length Floral tube length

48. zNext slides exemplify differences due to sequence once frequency is accounted for…

49. Elasticity of λ s to stage-specific reproduction for each environmental state, normalized for frequency Stasis Dry season Tree-falls Years => Environments El niño

50. Elasticity of λ s to the first 3 age-specific survivals for each environmental state, normalized for frequency 26 yr Cycle Quality correlated Iid and equal Years => Environments NAO

51. × = Example: “dry season” driver, envt 84-85 (elementwise multiplication)

52. Years => Environments