Mortality trajectories for tropical trees in variable

environments

Carol C. HorvitzUniversity of Miami, Coral Gables, FL

C. Jessica E. MetcalfDuke Population Research Center, Durham, NC

Shripad Tuljapurkar

Stanford University, Stanford, CA

OET 2008La Selva Biological Station

February 2, 2008

A time to grow and a time to die

Carol C. HorvitzUniversity of Miami, Coral Gables, FL

C. Jessica E. MetcalfDuke Population Research Center, Durham, NC

Shripad Tuljapurkar

Stanford University, Stanford, CA

OET 2008La Selva Biological Station

February 2, 2008

Mortality rate: patterns and biological processes?

?

Senescence

80 85 90 95 100 105 110 115

Age

Mo

rtal

ity

Evolutionary Evolutionary theory theory predicts: predicts:

Mortality,Mortality,

the risk of the risk of dying in the dying in the near future near future given that given that you have you have survived until survived until now, now,

should should increase with increase with ageage

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 soongiven survival up to age x

Calculations

μx = - log ( lx +1 / lx )

• in other words: the negative

of the slope of the survivorship curve

(when graphed on a log scale)

Age

log (

Surv

ivors

hip

)

Mortality rate: patterns and biological processes?

?

Senescence

80 85 90 95 100 105 110 115

Age

Mo

rtal

ity

Evolutionary Evolutionary theory theory predicts: predicts:

Mortality,Mortality,

the risk of the risk of dying in the dying in the near future near future given that given that you have you have survived until survived until now, now,

should should increase with increase with ageage

Mortality rate: patterns and biological processes?

?

Senescence

80 85 90 95 100 105 110 115

Age

Mo

rtal

ity

PlateaPlateauu

Mortality rate: patterns and biological processes?

?

Senescence

80 85 90 95 100 105 110 115

Age

Mo

rtal

ity

NegativeNegativesenescencsenescencee

Gompertz (1825)

1. Age-independent and constant across ages2. Age-dependent and worsening with age

Gompertz (1825)

A third possibility

1. Age-independent and constant across ages2. Age-dependent and worsening with age

A third possibility3. 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 2001Tuljapurkar and Horvitz 2006, Horvitz and Tuljapurkar in press

Some plant mortality patterns

Silvertown et al. 2001 fitted Weibull models for these but...

Horvitz and Tuljapurkar in press, Am Nat

Pro

port

ion in

each

sta

ge

Mortality plateau in variable environments

Megamatrix

μm= - log λm

Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by

the initial environment’s Q (Tuljapurkar & Horvitz 2006)

c22

Mortality plateau in variable environments

Megamatrix

μm= - log λm

Before the plateau things are a little messier, powers of the megamatrix pre-multiplied by

the initial environment’s Q (Tuljapurkar & Horvitz 2006)

c22

matrix of matrix of transitions (no transitions (no reproduction)reproduction)

for env 1for env 1probability of changing probability of changing

from env 2 to env 1from 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 Pioneer, canopy and emergent and emergent speciesspecies•Diameter Diameter (+/- 0.3 mm)(+/- 0.3 mm)•CI indexCI index•Every yr for 17 yrsEvery yr for 17 yrs•3382 individuals3382 individuals•1000 mortality 1000 mortality eventsevents

(Clark and Clark (Clark and Clark 2006, Ecological 2006, Ecological Archives)Archives)

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 rainfall3.9 m (> 4 yards)

Slide from D. and D. Clark

Cecropia obtusifolia, Cecropia obtusifolia, CecropiaceaeCecropiaceae““Guarumo”Guarumo”

SubcanopySubcanopyPioneerPioneer

Max diam = 37 cmMax diam = 37 cm

Pentaclethra macrolobaPentaclethra macrolobaFabaceae Fabaceae CanopyCanopyMax diam = 88 cmMax diam = 88 cm

Balizia elegansBalizia elegansFabaceae Fabaceae (Mimosoidae)(Mimosoidae)

EmergentEmergent

Max diam = 150 cmMax diam = 150 cm

Lecythis amplaLecythis amplaLecythidaceaeLecythidaceae““Monkey Pot”Monkey Pot”EmergentEmergentMax diam = 161 cmMax diam = 161 cm

Dipteryx panamensisDipteryx panamensis((Fabaceae:Papilionidae)Fabaceae:Papilionidae) Emergent tree ( light Emergent tree ( light colored)colored)Max diam = 187 cmMax 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.Metcalf, Horvitz and Tuljapurkar, in prep.

““A time to grow and a time to die: IPMs for ten A time to grow and a time to die: IPMs for ten tropical trees in a Markovian environment” 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

ExceptPIONEERS

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)

Numericalestimation:

Construct matrix

We used one300 x 300matrix 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 --Crown Illumination Index: Darkest = 1 -->> Lightest = 5, 6 Lightest = 5, 6

Model development/parameterization Regression of survival on size, by lightRegression of survival on size, by light Regression of size(t+1) on size(t), by Regression of size(t+1) on size(t), by

lightlight Integral projection model (IPM), by lightIntegral projection model (IPM), by light Markov chain of light dynamicsMarkov 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 …

Highest juvenile

Lowest intermediate age Plateau way below juvenile level

Light matters

EXCEPTIONSPioneers, 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 cmForest inventory threshold

30 cmDiameter when canopy height is attained

Max Diameterobserved

Size, mmSize, mm

First passage timeto 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. Thanks, D. and D. Clark!!!! Clark!!!!

National Institute on Aging, NIH, National Institute on Aging, NIH, P01 AG022500-01P01 AG022500-01

Duke Population Research CenterDuke Population Research Center John C. Gifford Arboretum at the John C. Gifford Arboretum at the

University of MiamiUniversity of Miami Jim Carey, Jim Vaupel Jim Carey, Jim Vaupel

And also to Benjamin GompertzAnd also to Benjamin Gompertz[that we may not quickly][that we may not quickly] “… “…lose [our] remaining power to oppose destruction…”lose [our] remaining power to oppose destruction…”

Thanks! Thanks!

Deborah Clark, David ClarkDeborah Clark, David Clark

Age from stage methods follow

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

Stage at time t+1

Stage at time t

seed seedling juvenile reproductive

seed 0.1 0 0 12

seedling 0.2 0.1 0 0

juvenile 0 0.3 0.1 0

reproductive

0 0.1 0.2 0.4

dead 0.7 0.5 0.7 0.6

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

Stage at time t+1

Stage at time t

seed seedling juvenile reproductive

seed 0.1 0 0 0

seedling 0.2 0.1 0 0

juvenile 0 0.3 0.1 0

reproductive

0 0.1 0.2 0.4

S 0.3 0.5 0.3 0.4

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

Individuals are born into stage 1

N(0)N(0) = [1, 0, … ,0]’ = [1, 0, … ,0]’

As the cohort ages, its dynamics are given by

NN(x+1) = (x+1) = X (t) X (t) NN (x), (x), X is a random variable that takes on values

QQ11, Q, Q22,…,Q,…,QKK

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)l(x) = = ΣΣ N(x)N(x) survivorship of cohort survivorship of cohort

UU(x) = (x) = NN(x)/l(x) (x)/l(x) stage structure of cohort stage structure of cohort

Cohort dynamics with stage structure

one period survival of cohort at age one period survival of cohort at age xx = = stage-specific survivals weighted by stage structurestage-specific survivals weighted by stage structure

l(x+1)/l(x) = < Z (t), U(x) >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 xμμ(x) = - (x) = - log [log [ l(x+1)/l(x) 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 + Q1 + Q2 + Q3 + …+QX

which converges to (I-Q) -1

Life expectancy: column sums of N e.g., for stage 1, column 1

Variable: Variable: NN = = II + Q(1) + Q(1) + Q(2)Q(1) + Q(2)Q(1) + Q(3)Q(2)Q(1) + Q(3)Q(2)Q(1) + …etc+ …etc which is NOT so simple; described for several which is NOT so simple; described for several

cases in Tuljapurkar and Horvitz 2006cases in Tuljapurkar and Horvitz 2006 Life expectancy: column sums of NLife expectancy: column sums of N e.g., for stage 1, column 1e.g., for stage 1, column 1