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 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 OET 2008 La Selva Biological Station February 2, 2008

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

Definitions l x Survivorship to age x number of individuals surviving to age x divided by number born in a single cohort μ x Mortality rate at age x risk of dying soon given survival up to age x

Calculations μ x = - log ( l x +1 / l x ) in other words: the negative of the slope of the survivorship curve (when graphed on a log scale) Age log (Survivorship)

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

Mortality rate: patterns and biological processes? ? Plateau

? Negativesenescence

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

Gompertz (1825)

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.

Relevant features of organisms with Empirically-based stage 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

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

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

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

Mortality plateau in variable environments Megamatrix μ m = - log λ m Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q (Tuljapurkar & Horvitz 2006) c 22

Mortality plateau in variable environments Megamatrix μ m = - log λ m Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by the initial environment’s Q (Tuljapurkar & Horvitz 2006) c 22 matrix of transitions (no reproduction) for env 1 for env 1 probability of changing from env 2 to env 1

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

Application to ten tropical trees in a Markovian environment Pioneer, canopy and emergent speciesPioneer, canopy and emergent species DiameterDiameter (+/- 0.3 mm) (+/- 0.3 mm) CI indexCI index Every yr for 17 yrsEvery yr for 17 yrs 3382 individuals3382 individuals 1000 mortality events1000 mortality events (Clark and Clark 2006, Ecological Archives)

The La Selva Biological Station (Organization for Tropical Studies) in Costa Rica’s Caribbean lowlands (10 o 26'N, 84 o 00'W; m elev.;1510 ha) tropical wet forest mean annual rainfall 3.9 m (> 4 yards) Slide from D. and D. Clark

Cecropia obtusifolia, Cecropiaceae“Guarumo”SubcanopyPioneer Max diam = 37 cm

Pentaclethra macroloba FabaceaeCanopy Max diam = 88 cm

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

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

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

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

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”

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

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

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

Integral projection model 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) Numerical estimation: Construct matrix We used one 300 x 300 matrix for each Light environment 300 size categories Ellner and Rees 2006

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

Model development/parameterization

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

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

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

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!

Variance in growth highest for small plants Cecropia spp

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

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

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

Thanks! Deborah Clark, David Clark Deborah Clark, David Clark

Age from stage methods follow

A is population projection matrix F is reproduction death is an absorbing state

Q = A – F S = 1- death = column sum of Q

Q’s and S’s in a variable environment At each age, A(x) is one of {A 1, A 2, A 3 …A k } and Q(x) is one of {Q 1, Q 2, Q 3 …Q k } and S(x) is one of {S 1, S 2, S 3 …S K } Stage-specific one-period survival

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 Q 1, Q 2,…,Q K Cohort dynamics with stage structure, variable environment

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

one period survival of cohort at age x = stage-specific survivals weighted by stage structure l(x+1)/l(x) = l(x+1)/l(x) = Z is a random variable that takes on values S 1, S 2,…,S K Mortality rate at age x μ(x) = - log [ l(x+1)/l(x) ] Mortality from weighted average of one-period survivals

Mortality directly from survivorship Survivorship to age x, l(x), is given by the sum of column 1* of Powers of Q (constant environment) Random matrix product of Q(x)’s (variable environment) Age-specific mortality, the risk of dying soon after reaching age x, given that you have survived to age x, is calculated as, μ(x) = - log [ l(x+1)/l(x)] ____________________________________ *assuming individuals are born in stage 1

N, “the Fundamental Matrix” and Life Expectancy Constant: N = I + Q 1 + Q 2 + Q 3 + …+Q X which converges to (I-Q) -1 Life expectancy: column sums of N e.g., for stage 1, column 1 Variable: Variable: N = I + Q(1) + Q(2)Q(1) + Q(3)Q(2)Q(1) + …etc N = I + Q(1) + Q(2)Q(1) + Q(3)Q(2)Q(1) + …etc which is NOT so simple; described for several cases in Tuljapurkar and Horvitz 2006 which is NOT so simple; described for several cases in Tuljapurkar and Horvitz 2006 Life expectancy: column sums of N Life expectancy: column sums of N e.g., for stage 1, column 1 e.g., for stage 1, column 1