A time to grow and a time to die: a new way to analyze the dynamics of size, light, age and death of tropical trees C. Jessica E. Metcalf Duke Population Research Center, Durham, NC Carol C. Horvitz University of Miami, Coral Gables, FL Shripad Tuljapurkar Stanford University, Stanford, CA Evolution 2008 June 24, 2008

Tropical wet forest trees Time to death (and other times…) Why it is important Tree population and life history parameters Forest turnover rates Carbon dynamics Why it is hard to know Long-lived and mostly no annual rings Size = Age (High variability in growth rate)

Light in tropical wet forests Light is rare at the bottom of the forest abundant at the top of the forest Two ways for trees to get it Use gaps Grow tall Two light-related niche dimensions Juvenile recruitment Ultimate vertical position of adults Can you live long enough/grow fast enough to get to the light?

Photo courtesy of D.A. & D.B. Clark Time to death? Can you live long enough to get to the light?

Things to include in a model Light dependency of demography Dynamics of light environment over the life times of individuals Variability of growth across size Conditionality of growth on survival

Take the ingredients: Integral Projection Model (Ellner and Rees 2006, 2007) Age from stage methods (Tuljapurkar and Horvitz 2006, Horvitz and Tuljapurkar 2008) Data on individuals’ size, survival, environment at single time steps (Clark and Clark 2006)

Combine and shake well… Stochastic integral projection cohort dynamics model Metcalf, Horvitz and Tuljapurkar, in prep.

Use it: Time to reach & to reside in different sizes and environments before dying Time of death and other key events Metcalf, Horvitz and Tuljapurkar, in prep.

cohort 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) Statistical (regression) models g(y,x) s(x) Numerical estimation of integral: a high dimensional matrix Ellner and Rees 2006, 2007

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 2008

In variable environments, at the heart of the analysis is a Megamatrix e.g., survivorship to each age is given by 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

The Data: every yr for 17 yrs Pioneer, canopy and emergent speciesPioneer, canopy and emergent species DiameterDiameter (+/- 0.3 mm) (+/- 0.3 mm) Light: CI indexLight: CI index 3382 individuals3382 individuals 1000 mortality events1000 mortality events (Clark and Clark 2006, Ecological Archives) Photo courtesy of D.A. & D.B. Clark

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 courtesy of D.A and D.B Clark

Results and conclusions Regression analyses reveal Mean single time step growth and survival peak at intermediate sizes higher in lighter environments Variance in single time step growth is highest for juveniles, especially pioneers Analysis of stochastic integral projection cohort model reveals Age-specific mortality trajectories asymmetrically “bath tub”-shaped Life expectancies range from 35 to > 500 yrs Small individuals may reach canopy sooner than larger ones ! Trade-off between life expectancy and growth rate among species Initial light environment influences expectation of surviving to reach the canopy for some species Improvements over previous modeling approaches for trop trees stochasticity of growth environmental dynamics over the life course of individuals consistent with radiocarbon dated individuals!!!!

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

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

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

Main steps in analysis  Statistical regression analyses  Integral projection cohort model (IPM) for each light environment  Markov chain of light dynamics Numerical estimation of the stochastic IPM megamatrix: 5-6 light categories X 300 size categories Use the stochastic IPM age patterns of mortality life expectancy expected first passage times to key events

Growth increment peaks at intermediate sizes Interaction of size with initial light is complicated Size at time t CI = 1 Dark CI = 2 CI = 3 CI = 4 CI = 5 CI = 6 Light Growth increment based on regression of size(t+1) on size(t)

Survival peaks at fairly small sizes Survival lower in the dark Pioneers different CI <=3 Dark CI > 3 Light Annual survival based on logistic regression of alive(t+1) vs size(t)

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

Main steps in analysis Statistical regression analyses Integral projection cohort model (IPM) for each light environment Markov chain of light dynamics Numerical estimation of the stochastic IPM megamatrix: 5-6 light categories X 300 size categories  Use the stochastic IPM  age patterns of mortality  life expectancy  expected first passage times to key events

High juvenile mortality Minimum: age of “escape” from juvenile mortality Old age plateau (way below juvenile level) Light matters Pioneers different Age, yrs CI = 1 Dark CI = 2 CI = 3 CI = 4 CI = 5 CI = 6 Light Age- specific mortality patterns by birth environment

These are means Rapid rise at small size Peak ~ 5 cm Light matters Pioneers different Remaining time to death by current size and current environment CI = 1 Dark CI = 2 CI = 3 CI = 4 CI = 5 CI = 6 Light

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! CI = 1 Dark CI = 2 CI = 3 CI = 4 CI = 5 CI = 6 Light Higher variance in growth of small guys leads to an interesting result

All species: first passage times (yrs) to key events (from 10 mm) Forest inventory threshold Size, mm Size, mm Max size Canopy is attained Max Diameter observed Expected time to reach the canopy

Rapid growth associated with lower life expectancy Some species not expected to make it to canopy Initial light matters more for some spp. Initial Light Can you live long enough to reach the canopy? > or < the dotted line?

Results and conclusions Regression analyses reveal Mean single time step growth and survival peak at intermediate sizes higher in lighter environments Variance in single time step growth is highest for juveniles, especially pioneers Analysis of stochastic integral projection cohort model reveals Age-specific mortality trajectories asymmetrically “bath tub”-shaped Life expectancies range from 35 to > 500 yrs Small individuals may reach canopy sooner than larger ones ! Trade-off between life expectancy and growth rate among species Initial light environment influences expectation of surviving to reach the canopy for most species Improvements over previous modeling approaches for trop trees stochasticity of growth environmental dynamics over the life course of individuals consistent with radiocarbon dated individuals!!!!

CI = 1 Dark CI = 2 CI = 3 CI = 4 CI = 5 CI = 6 Light Our results consistent with the radiocarbon-aged* individuals At the same site! *Fichtler et al. 2003

Thanks, D.A. and D.B. 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, University of Miami John C. Gifford Arboretum, University of Miami Institute for Theoretical and Mathematical Ecology, University of Miami