Mortality trajectories for tropical trees in variable environments

1. OET 2008 La Selva Biological Station February 2, 2008 Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population Research Center, Durham, NC Shripad Tuljapurkar Stanford University, Stanford, CA

2. OET 2008 La Selva Biological Station February 2, 2008 A time to grow and a time to die Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population Research Center, Durham, NC Shripad Tuljapurkar Stanford University, Stanford, CA

3. Mortality rate: patterns and biological processes? Evolutionary theory predicts: Mortality, the risk of dying in the near future given that you have survived until now, should increase with age ?

4. Definitions • lx Survivorship to age x number of individuals surviving to age x divided by number born in a single cohort • μx Mortality rate at age x risk of dying soon given survival up to age x

5. Calculations • μx =- log (lx +1 / lx ) • in other words: the negative of the slope of the survivorship curve (when graphed on a log scale) log (Survivorship) Age

6. Mortality rate: patterns and biological processes? Evolutionary theory predicts: Mortality, the risk of dying in the near future given that you have survived until now, should increase with age ?

7. Mortality rate: patterns and biological processes? ? Plateau

8. Mortality rate: patterns and biological processes? ? Negative senescence

9. Gompertz (1825) 1. Age-independent and constant across ages 2. Age-dependent and worsening with age

10. Gompertz (1825)

11. A third possibility 1. Age-independent and constant across ages 2. Age-dependent and worsening with age A third possibility 3. Age-independent but not constant across ages Death could depend upon something else and that something else could change across ages.

12. Relevant features of organisms withEmpirically-basedstage structured demography • Cohorts begin life in particular stage • Ontogenetic stage/size/reproductive status are known to predict survival and growth in the near future • Survival rate does not determine order of stages

13. Age-from-stage theory • Markov chains, absorbing states • An individual passes through various stages before being absorbed, e.g. dying • What is the probability it will be in certain stage at age x (time t), given initial stage? • The answer can be found by extracting information from stage-based population projection matrices Cochran and Ellner 1992, Caswell 2001 Tuljapurkar and Horvitz 2006, Horvitz and Tuljapurkar in press

14. Some plant mortality patterns Silvertown et al. 2001 fitted Weibull models for these but...

15. Proportion in each stage Horvitz and Tuljapurkar in press, Am Nat

16. Mortality plateau in variable environments Megamatrix μm= - logλm Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q(Tuljapurkar & Horvitz 2006) c22

17. Mortality plateau in variable environments Megamatrix μm= - logλm Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q(Tuljapurkar & Horvitz 2006) matrix of transitions (no reproduction) for env 1 probability of changing from env 2 to env 1 c22

18. Conclusions and results • Age-from-stage methods combined with IPM’s increase library of mortality trajectories • Pioneer, canopy and emergent tropical trees solve the light challenge differently • Single time step growth and survival peak at intermediate sizes • Mortality trajectories asymmetrically “bath tub”-shaped • Life expectancies ranged from 35 to > 500 yrs • Small plants may reach canopy sooner than large ones ! • Empirically-based stage structured demographic processes : a third perspective on death

19. Application to ten tropical trees in a Markovian environment • Pioneer, canopy and emergent species • Diameter • (+/- 0.3 mm) • CI index • Every yr for 17 yrs • 3382 individuals • 1000 mortality events • (Clark and Clark 2006, Ecological Archives)

20. The La Selva Biological Station (Organization for Tropical Studies) • in Costa Rica’s Caribbean lowlands (10o26'N, 84o00'W; 37-150 m elev.;1510 ha) • tropical wet forest • mean annual rainfall • 3.9 m (> 4 yards) Slide from D. and D. Clark

21. Cecropia obtusifolia, Cecropiaceae “Guarumo” Subcanopy Pioneer Max diam = 37 cm

22. Pentaclethra macroloba Fabaceae Canopy Max diam = 88 cm

23. Balizia elegans Fabaceae (Mimosoidae) Emergent Max diam = 150 cm

24. Lecythis ampla Lecythidaceae “Monkey Pot” Emergent Max diam = 161 cm

25. Dipteryx panamensis (Fabaceae:Papilionidae) Emergent tree ( light colored) Max diam = 187 cm

26. Species arranged from smallest to largest Look at the raw data: Linear relationship on a log scale Decrease in variance with size

27. Model development/parameterization • Regression of size(t+1) on size(t), by light • Regression of survival on size, by light • Integral projection model (IPM), by light • Markov chain of light dynamics • Megamatrix for age-from-stage analysis: transitions by light (5-6 categories) and size (300 size categories) Metcalf, Horvitz and Tuljapurkar, in prep. “A time to grow and a time to die: IPMs for ten tropical trees in a Markovian environment”

28. Growth as given by parameters of regression Growth increment peaks at intermediate sizes Interaction of size with initial light is complicated

29. Survival as given by parameters of logistic regression Survival peaks at fairly small sizes Survival lower in the dark Except PIONEERS

30. Tropical trees Growth and survival vary with size and depend upon light

31. Integral projection model Numerical estimation: Construct matrix We used one 300 x 300 matrix for each Light environment 300 size categories Integrates over size x at time t and projects to size y at time t+1, according to growth and survival functions, g(y, x) and s(x) Ellner and Rees 2006

32. Light environment dynamics: transitions in CI index by individual trees of each species Crown Illumination Index: Darkest = 1 --> Lightest = 5, 6

33. Model development/parameterization • Regression of survival on size, by light • Regression of size(t+1) on size(t), by light • Integral projection model (IPM), by light • Markov chain of light dynamics • Megamatrix for age-from-stage analysis: transitions by light (5-6 categories) and size (300 size categories) • Track expected transitions among stages and light environments for cohorts born into each light environment …

34. Highest juvenile Lowest intermediate age Plateau way below juvenile level Light matters EXCEPTIONS Pioneers, Pentaclethra Age, yrs

35. Rapid rise at small size • Peak ~ 5 cm Initial diam (mm)

36. First passage times (yrs) quicker when initial environment is lighter Size, mm 10 cm Forest inventory threshold 30 cm Diameter when canopy height is attained Max Diameter observed

37. First passage time to reach canopy vs initial size has a hump! Small plants may get there faster than somewhat larger plants Stage is different than age!

38. Cecropia spp Variance in growth highest for small plants

39. Initial Light Rapid growth associated with lower life expectancy Some species not expected to make it to canopy Initial light matters

40. Conclusions and results • Age-from-stage methods combined with IPM’s increase library of mortality trajectories • Pioneer, canopy and emergent tropical trees solve the light challenge differently • Single time step growth and survival peak at intermediate sizes • Mortality trajectories asymmetrically “bath tub”-shaped • Life expectancies ranged from 35 to > 500 yrs • Small plants may reach canopy sooner than large ones ! • Empirically-based stage structured demographic processes : a third perspective on death

41. Thanks, D. and D. Clark!!!! • National Institute on Aging, NIH, P01 AG022500-01 • Duke Population Research Center • John C. Gifford Arboretum at the University of Miami • Jim Carey, Jim Vaupel And also to Benjamin Gompertz [that we may not quickly] “…lose [our] remaining power to oppose destruction…”

42. Thanks! • Deborah Clark, David Clark

44. Age from stage methods follow

45. A is population projection matrixF is reproductiondeath is an absorbing state

46. Q =A –FS = 1- death = column sum of Q

47. Q’s and S’s in a variable environment At each age, A(x) is one of {A1, A2, A3…Ak} and Q(x) is one of {Q1, Q2, Q3…Qk} and S(x) is one of {S1, S2, S3…SK} Stage-specific one-period survival

48. Cohort dynamicswith stage structure, variable environment Individuals are born into stage 1 N(0) = [1, 0, … ,0]’ As the cohort ages, its dynamics are given by N(x+1) = X (t) N (x), X is a random variable that takes on values Q1, Q2,…,QK

49. Cohort dynamicswith stage structure As the cohort ages, it spreads out into different stages and at each age x, we track l(x) = ΣN(x) survivorship of cohort U(x) = N(x)/l(x) stage structure of cohort

50. Mortality from weighted average of one-period survivals one period survival of cohort at age x = stage-specific survivals weighted by stage structure l(x+1)/l(x) = < Z (t), U(x) > Z is a random variable that takes on values S1, S2,…,SK Mortality rate at age x μ(x) = - log [ l(x+1)/l(x) ]