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![Page 1: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/1.jpg)
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
![Page 2: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/2.jpg)
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
![Page 3: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/3.jpg)
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
![Page 4: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/4.jpg)
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
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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
)
![Page 6: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/6.jpg)
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
![Page 7: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/7.jpg)
Mortality rate: patterns and biological processes?
?
Senescence
80 85 90 95 100 105 110 115
Age
Mo
rtal
ity
PlateaPlateauu
![Page 8: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/8.jpg)
Mortality rate: patterns and biological processes?
?
Senescence
80 85 90 95 100 105 110 115
Age
Mo
rtal
ity
NegativeNegativesenescencsenescencee
![Page 9: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/9.jpg)
Gompertz (1825)
1. Age-independent and constant across ages2. Age-dependent and worsening with age
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Gompertz (1825)
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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.
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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
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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
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Some plant mortality patterns
Silvertown et al. 2001 fitted Weibull models for these but...
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Horvitz and Tuljapurkar in press, Am Nat
Pro
port
ion in
each
sta
ge
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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
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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
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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
![Page 19: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/19.jpg)
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)
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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
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Cecropia obtusifolia, Cecropia obtusifolia, CecropiaceaeCecropiaceae““Guarumo”Guarumo”
SubcanopySubcanopyPioneerPioneer
Max diam = 37 cmMax diam = 37 cm
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Pentaclethra macrolobaPentaclethra macrolobaFabaceae Fabaceae CanopyCanopyMax diam = 88 cmMax diam = 88 cm
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Balizia elegansBalizia elegansFabaceae Fabaceae (Mimosoidae)(Mimosoidae)
EmergentEmergent
Max diam = 150 cmMax diam = 150 cm
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Lecythis amplaLecythis amplaLecythidaceaeLecythidaceae““Monkey Pot”Monkey Pot”EmergentEmergentMax diam = 161 cmMax diam = 161 cm
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Dipteryx panamensisDipteryx panamensis((Fabaceae:Papilionidae)Fabaceae:Papilionidae) Emergent tree ( light Emergent tree ( light colored)colored)Max diam = 187 cmMax diam = 187 cm
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Species arranged from smallest to largest
Look at the raw data:
Linear relationship on a log scale
Decrease in variance with size
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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”
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Growth as given by parameters of regression
Growth increment peaks at intermediate sizes
Interaction of size with initial light is complicated
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Survival as given by parameters of logistic regression
Survival peaks at fairly small sizes
Survival lower in the dark
ExceptPIONEERS
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Tropical trees
Growth and survival vary with size and depend upon light
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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
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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
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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 …
![Page 34: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/34.jpg)
Highest juvenile
Lowest intermediate age Plateau way below juvenile level
Light matters
EXCEPTIONSPioneers, Pentaclethra
Age, yrs
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Rapid rise at small size
Peak ~ 5 cm
Initial diam (mm)
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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
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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!
![Page 38: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/38.jpg)
Variance in growth highest for small plants
Cecropia spp
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Rapid growth associated with lower life expectancy
Some species not expected to make it to canopy
Initial light matters
Initial Light
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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
![Page 41: Mortality trajectories for tropical trees in variable environments Carol C. Horvitz University of Miami, Coral Gables, FL C. Jessica E. Metcalf Duke Population.](https://reader038.fdocuments.in/reader038/viewer/2022110212/56649f315503460f94c4c9cd/html5/thumbnails/41.jpg)
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…”
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Thanks! Thanks!
Deborah Clark, David ClarkDeborah Clark, David Clark
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Age from stage methods follow
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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
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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
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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
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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
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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
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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
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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
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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