vol. 172, no. 2
the american naturalist

august 2008
Stage Dynamics, Period Survival, and Mortality Plateaus
Carol C. Horvitz1,* and Shripad Tuljapurkar2,†
1. Department of Biology, University of Miami, Coral Gables,
Florida 33124;
2. Biological Sciences, Stanford University, Stanford, California 94305
Received April 15, 2007; Accepted February 12, 2008;
Electronically published July 9, 2008
Online enhancements: appendixes.
abstract: Mortality plateaus at advanced ages have been found in
many species, but their biological causes remain unclear. Here, we
exploit age-from-stage methods for organisms with stage-structured
demography to study cohort dynamics, obtaining age patterns of
mortality by weighting one-period stage-specific survivals by expected age-specific stage structure. Cohort dynamics behave as a
killed Markov process. Using as examples two African grasses, one
pine tree, a temperate forest perennial herb, and a subtropical shrub
in a hurricane-driven forest, we illustrate diverse patterns that may
emerge. Age-specific mortality always reaches a plateau at advanced
ages, but the plateau may be reached rapidly or slowly, and the
trajectory may follow positive or negative senescence along the way.
In variable environments, birth state influences mortality at early
but not late ages, although its effect on the level of survivorship
persists. A new parameter mq summarizes the risk of mortality averaged over the entire lifetime in a variable environment. Recent
aging models for humans that employ nonobservable abstract states
of “vitality” are also known to produce diverse trajectories and similar
asymptotic behavior. We discuss connections, contrasts, and implications of our results to these models for the study of aging.
Keywords: cohort dynamics, killed Markov processes, age from stage,
long-run stochastic mortality rate.
Does mortality rate, the instantaneous risk of death, always
increase with age? Gompertz (1825), analyzing human
mortality tables, developed a famous model in which mortality accelerates with age. Hamilton (1966), studying the
evolution of senescence, argued that mortality should in* E-mail: [email protected].
†
E-mail: [email protected].
Am. Nat. 2008. Vol. 172, pp. 203–215. 䉷 2008 by The University of Chicago.
0003-0147/2008/17202-42549$15.00. All rights reserved.
DOI: 10.1086/589453
crease with age because the force of selection against deleterious mutations declines with age. More recent empirical (e.g., Finch 1998; Pletcher and Curtsinger 1998; Vaupel
et al. 1998; Roach and Gampe 2004) and theoretical studies
(e.g., Yashin et al. 2000; Vaupel et al. 2004; Baudisch 2005)
are concerned with causal mechanisms that produce
changes in the rate of mortality with age and have been
particularly interested in the appearance of a “mortality
plateau,” the leveling of mortality rate, at late ages. Plateaus
have been found in many species, including humans (Carey 2003; Horiuchi 2003), and many models have been
proposed that can produce plateaus; however, data that
clearly link a particular biological mechanism with an observable plateau are relatively few.
For many plants and animals, demography is stage
structured rather than age structured. Stage (e.g., size and/
or developmental stage) rather than age predicts survival
and reproduction each year as well as the probability of
moving to other stages; thus, stage has been used to structure populations and to model population dynamics (Caswell 2001). The assumption here is that all individuals
within a stage are subject to the same demographic rates.
There is no hidden age-within-stage structure; rates are
independent of time spent in the stage and of age at arrival
in the stage. We refer to these as populations with empirically based stage structure. Even though age is not a
determinant of demographic rates in these populations,
individuals do have ages and age-specific stage profiles.
Age-from-stage methods developed by Cochran and Ellner
(1992) and Caswell (2001, 2006) for constant environments and by Tuljapurkar and Horvitz (2006) for variable
environments make it possible to add species with stagebased demography to the library of age-specific mortality
trajectories (e.g., Silvertown et al. 2001), thus broadening
the comparative study of life-history evolution and the
evolution of senescence. This article explores how a cohort’s stage structure and its one-period survival change
with age, resulting in an observable age trajectory of
mortality.
Using as examples empirically based stage-structured
plant data for constant environments (Bierzychudek 1982,
1999; Platt et al. 1988; O’Connor 1993; Silvertown et al.
2001) and for variable environments (Pascarella and Hor-
204 The American Naturalist
vitz 1998; Tuljapurkar et al. 2003), we illustrate distinct
patterns that may emerge. The mortality plateau may be
reached rapidly or slowly and may follow positive or negative senescence along the way. In variable environments,
birth habitat additionally influences the age trajectory of
mortality, and there are two kinds of mortality plateaus,
one related to average over cohorts at each age and the
other related to the time average for a single cohort. Biologically, the latter represents risk of mortality averaged
over an entire lifetime. The concept and computation of
this long-run-time average of mortality is presented here
for the first time.
Age-from-stage models are based on the theory of
discrete-time finite Markov chains. Rates of production of
new individuals are separated from probabilities of growth
and survival of existing individuals. Using the latter probabilities and adding death as an absorbing state yields a
Markov chain. Each individual starting in a particular stage
passes through various stages before being absorbed, that
is, dying. Markov chain theory gives us the probability
that an individual will be in a certain stage at a certain
age (time), given its initial stage. Age-specific survivorship
curves are obtained from the powers of a matrix in constant environments (Cochran and Ellner 1992; Caswell
2001, 2006) and from random matrix products in variable
environments (Tuljapurkar and Horvitz 2006). It has gone
largely unappreciated by those studying empirically based
stage-structured population dynamics that this setup always produces an old-age mortality plateau (but see Tuljapurkar and Horvitz 2006), although models from other
fields that share essential mathematical structure with these
models are well known to produce this type of asymptotic
behavior (Yashin et al. 2000; McNamara et al. 2001; Steinsaltz and Evans 2004).
Cohort dynamics based on age-from-stage models belong to a general class of stochastic processes known as
killed Markov processes, in which the probability mass of
the system (here representing living individuals) decreases
over time. These processes are not usually irreducible or
primitive, but when there is a set of states that is accessible
from all other states, conditional on survival to entry into
this set and depending on initial state, the process eventually becomes concentrated on this set (Steinsaltz and
Evans 2004). (State 2 is considered to be accessible from
state 1 if, given that the process starts at state 1, it has a
nonzero probability of eventually arriving at state 2.) The
system resides for a long time in this persistent set of states
before eventual absorption (i.e., death). The probability
distribution of being in each state of this set is called the
quasi-stationary distribution. It has an associated killing
rate (Darroch and Seneta 1965; Seneta 1981; see app. B
in the online edition of the American Naturalist for mathematical notes). In our application, the state variable is an
empirically based stage (not age), and the killing rate will
yield the mortality plateau. These general results apply to
continuous stages as well as discrete stages (Steinsaltz and
Evans 2004) and to those cases with rates that depend on
stage duration that can be described by Markovian models
by expanding the set of stages.
These are not new mathematical results (Darroch and
Seneta 1965), and it is not the first time this kind of
mathematics has been brought to bear on aging. Analogous results for related models of human mortality include
work by Gavrilov and Gavrilova (1991), Yashin et al.
(2000), Aalen and Gjessing (2001), Weitz and Fraser
(2001), and Steinsaltz and Evans (2004). Also, the same
kind of process was studied by McNamara et al. (2001),
to address problems in optimal animal behavior for scenarios involving survivorship. What is new in this article
is (1) an emphasis on using projection to examine transient
dynamics of stage structure as well as the onset of asymptotic
dynamics for a cohort with empirically based stage structure,
(2) presentation of the stochastic mortality rate mq to summarize the risk of mortality averaged over an entire lifetime
in a variable environment, and (3) drawing connections
between the analysis of age-specific mortality trajectories
from stage-based data for empirically stage-structured populations and other recent aging models.
Transient Dynamics of Populations
and Cohorts: Projection
Consider an empirically stage-structured population as defined above with S distinct life-history stages censused at
discrete times t, t ⫹ 1, and so on. At time t, the number
ni(t) of individuals in each stage i are listed in the vector
n(t). The population dynamics are given by n(t ⫹ 1) p
A(t)n(t), where A(t) is an S # S population projection
matrix and Aij(t) gives the per capita rate at which individuals in stage j at time t contribute to or become individuals in stage i at time t ⫹ 1. (Note: for all transition
matrices, we follow the column-to-row ( j r i) parameterization that is the rule in the field of population dynamics.) Transition probabilities and reproductive rates
are estimated from field data on marked individuals belonging to observable stages. The use of these rates in agefrom-stage methods assumes that all individuals in a stage
are subject to the same demographic rates.
Next, we split birthrates from transitions of existing
individuals to obtain A(t) p Q(t) ⫹ F(t), where F(t) contains all rates involving reproduction or fission. The term
Q(t) is an S # S matrix that projects future fates of any
individuals alive at time t; Qij(t) are transition probabilities, and the sum of the jth column of Q(t) is the oneperiod probability of survival sj(t) for stage j. We assume
sj(t) is !1. This assumption is not necessary, but it is suf-
Stage Dynamics and Mortality
ficient to guarantee eventual death of the cohort. The t is
the row vector of one-period survivals for all stages.
For a constant environment, the Qij are elements of a
fixed transition matrix Q. For a variable environment, at
each time step, the environmental state determines stage
transition rates; the matrix Q(t) takes on values Q1,
Q 2 , … , QK, and the one-period survivals s(t) take on values s1, s 2 , … , sK, where K is the number of environmental
states. We assume that environmental transitions follow a
Markov chain with a transition matrix P p (Pab), for environments a, b p 1, … , K (transitions are b r a). Each
cohort experiences some random sequence of environmental states (sample path q) as it ages.
Start at t p 0 with a cohort of newborn (age p 0) individuals all starting in stage 1. Since we are interested in
the proportion of the initial cohort rather than the absolute
numbers at each age, we normalize the initial cohort to
sum to 1, setting n 1(0) p 1, ni(0) p 0, i 1 1. As this cohort
ages, some individuals die while survivors may spread out
among stages. For t ≥ 0, we computed the sequence
n(t ⫹ 1) p Q(t)n(t) to track the proportion of the initial
cohort in each stage at each time. Noting that time here
equals age, survivorship to age x at time t is l(x) p
冘i ni(x), and l(x) decreases with age. The matrix chosen to
project the cohort from age x to age x ⫹ 1 is denoted Q(x).
The stage structure is the proportion in each stage class
at age x conditional on survival to age x, denoted by
u(x) p
n(x)
.
l(x)
(1)
Note that u(x) includes the effects of history, as it results
from the cumulative dynamics prescribed by the sequence
of matrices between birth and age x ⫺ 1 given by the recursion equation. Then the proportion of those alive at
age x that survive to age x ⫹ 1 is
l(x ⫹ 1)
p
l(x)
冘
si(x)ui(x),
i
p! s(x), u(x) 1 .
(2)
Thus, the one-period survival of the entire cohort at
age x is a weighted average that can be computed as the
scalar product of two vectors: the one-period stage-specific
survival and the stage structure. The age-specific mortality
rate m(x) is the rate of decrease in survivorship given by
Asymptotic Dynamics of Cohorts: Analytical Results
For constant environments, the matrix Q (as described
above and given that there exists a set of stages accessible
from the initial stage) has positive real dominant eigenvalue l ! 1, and at old ages, surviving individuals eventually will be distributed among stages according to the
elements of quasi-stationary distribution u̇, the right eigenvector of the matrix Q associated with the dominant
eigenvalue. The quasi-stationary distribution yields mortality at the plateau: m∗ p ⫺ log (! s, u˙ 1) . Thus, as age x
increases, survivorship must change exponentially:
log l(x) ∼ x(log l).
[
]
l(x ⫹ 1)
p log l(x) ⫺ log l(x ⫹ 1).
l(x)
(3)
(4)
This equation reveals a key property of populations with
stage-structured demographic transition probabilities; that
is, at old ages the age-specific mortality rate m(x) must approach an age-independent plateau value m∗ p ⫺(log l).
For variable environments, the picture is a little more
complicated because there are two distinct kinds of longterm expectations of mortality. One is the average mortality of an assemblage of cohorts at each age, and the
other is the long-run time-averaged mortality of a single
cohort. This distinction is analogous to different kinds of
averages for asymptotic population growth in variable environments (Tuljapurkar et al. 2003). For Markovian environments, Tuljapurkar and Horvitz (2006) presented the
expected old-age mortality plateau for an assemblage of
cohorts. It is obtained by first finding the dominant eigenvalue l M of a megamatrix M (necessary conditions for
its existence are discussed in app. B) that summarizes all
possible one-period transitions among life-history stages
and environmental states, with dimension given by the
product of the number of stages and the number of environmental states SK # SK . For the average described by
the megamatrix, there is an asymptotic habitat state by
stage quasi-stationary distribution given by the right eigenvector of the megamatrix associated with its dominant
eigenvalue (when there is one). Survivorship of an average cohort changes exponentially at old age at the rate
given by the old-age plateau in age-specific mortality
m∗ p ⫺(log l M).
The long-run-time-averaged mortality rate of a single
cohort, the stochastic mortality risk, is presented here for
the first time and is given by
log l q p lim
xr⬁
m(x) p ⫺ log
205
() [
]
1
l(xFa, q)
log
,
x
l(0)
(5)
where survivorship to age x, l(xFa, q), is calculated conditional on environmental state at birth a and the sequence
206 The American Naturalist
of environments experienced since birth, the sample path
q, and
mq p ⫺(log l q ).
(6)
We obtain mq from numerical simulations such as those
performed to calculate the stochastic growth rate of a population (Caswell 2001; Tuljapurkar et al. 2003), except here
we do not discard early time steps associated with transient
dynamics, because we are interested in including potential
effects of birth state. This parameter measures the risk of
mortality averaged over an entire lifetime. While it is calculated along a particular very long sample path, there is
convergence in its value among sample paths. A stationary
distribution of environmental states and of life-history
stages independent of initial conditions is obtained over
the long run.
In variable environments, as in constant environments,
during the transient phases, age-specific mortality can be
lower, higher, or close to the eventual old-age plateau. Of
course for finite populations, the size of the initial cohort,
the speed with which the asymptote is approached, and
the old-age level of survivorship interact in determining
whether any individuals remain in the cohort at the age
when the plateau would be realized.
Examples
We use four plant species that exemplify distinct patterns
of age-specific mortality in constant environments. Example 5 illustrates the concepts and calculations for a variable environment. Broad patterns are presented here, and
details are in appendix A in the online edition of the
American Naturalist. Stage transition matrices are visually
depicted in figure A1A–A1D for the constant environment
examples and figure A2A–A2G for the variable environment example. In the constant environment examples 1–
4, the stage structure of a cohort as it ages is tracked in
figure 1A–1D, and the one-period survivals of each stage
are in figure 1E–1H. Age-specific mortality (fig. 1I–1L)
exhibits joint effects of stage structure at each age and oneperiod stage-specific survival rates. Transient dynamics reveal changes in stage structure between birth and late ages.
Asymptotic behavior is observed once the steady state distribution of stages is reached, often not until old age.
During ages when transient dynamics apply, mortality can
be lower, higher, or close to the eventual old-age plateau
(fig. 1I–1L).
Example 1. Pinus palustris (Pinaceae); longleaf pine.
Found, and historically dominant, in the forested coastal
plains of the southeastern United States (Platt et al. 1988).
Stage structure gradually becomes dominated by large trees
(stages 7 and 8, respectively, 60–70 and ≥ 70 cm diameter
at breast height; see app. A; fig. 1A) with high survival
(fig. 1 E), but the highest survival is for trees of intermediate size. Age-specific mortality decreases substantially
from birth until age about 50 years and then increases
slowly toward a plateau at old ages (fig. 1I). The oldest
individual at the study site was about 450 years old, an
age at which mortality is approaching its plateau and
l(x) p 0.0014. The age pattern is faintly reminiscent of
human mortality, with a rate of senescence (the slope of
m[x] in fig. 1I ) that is negative at young ages and then
positive at old ages.
Example 2. Digitaria eriantha (Poaceae). An African savanna grass that rarely reproduces by seed but frequently
reproduces by aboveground vegetative offshoots (O’Connor
1993). This species displays a much simpler pattern of
mortality than longleaf pine, increasing (positive senescence) from birth until it plateaus at age about 10 years
(fig. 1J) when l(x) p 0.3092. The stage distribution then
becomes dominated by plants of intermediate tuft circumference (7–18 cm; fig. 1B), which have the lowest survival
probability and the highest mortality (fig. 1F).
Example 3. Setaria incrassata (Poaceae). An African savanna grass that reproduces both by seed and belowground
vegetative offshoots (O’Connor 1993). Mortality in this
species provides an interesting contrast to Digitaria. For
Setaria, it (fig. 1K) rises rapidly with age (positive senescence) to a peak at age about 25 years and then falls
(negative senescence) slowly toward a final plateau near
age 145, when l(x) p 2.8 # 10⫺27. During intermediate
ages, the stage distribution is dominated by relatively small
plants (tuft circumference 7–12 cm; fig. 1 C) that have the
lowest survival rate (fig. 1G), while the terminal stage distribution is dominated by large plants (with tuft circumference 136 cm; fig. 1C) that also have relatively low survival (fig. 1G). Plants with intermediately large tuft
circumference (21–36 cm) have the highest survival (fig.
1G), and their relative abundance rises and falls across the
ages (fig. 1C).
Example 4. Arisaema triphyllum (Araceae); jack-in-thepulpit. A perennial herb of North American temperate
deciduous forests (Bierzychudek 1982, 1999). Jack-in-thepulpit differs from all above examples by displaying decreasing age-specific mortality (negative senescence) at all
ages until the mortality plateau is reached at about age 45
(fig. 1L) when l(x) p 0.0006. The proportion of individuals in each successively larger size class increases as the
cohort ages (fig. 1D), as survival generally increases (while
mortality generally decreases) with size class (fig. 1H). All
six vegetative (i.e., nonseed) size classes are present in the
terminal set of stages, the most abundant being the largest
(leaf area 1340 cm2).
Example 5. Ardisia escallonioides (Myrsinaceae); marlberry. A subtropical understory shrub in southern Florida
Stage Dynamics and Mortality
207
Figure 1: A–D, Stage distribution at each age for Pinus, Digitaria, Setaria, and Arisaema. E–H, One-period stage survival sj for each species. I–L,
Age-specific mortality mx and the mortality plateau ⫺ log (l) (dotted lines) for each species.
hurricane-disturbed hardwood forests (Pascarella and
Horvitz 1998; Tuljapurkar et al. 2003). The environmental
state is measured by openness of the forest canopy. States
1–7 represent a gradient of canopy openness, from the
lightest sky (65% open) to the darkest sky (5% open; see
app. A; Pascarella and Horvitz 1998.) Each environmental
state has its own set of stage transition rates (fig. A2A–
A2G). A cohort experiences a sequence of environments,
a sample path generated by the matrix of environmental
dynamics (fig. A2H). We examined 200 paths and their
resulting mortality trajectories. Here, we illustrate transient
dynamics for one such path, that is, for a single cohort
(fig. 2A) and its corresponding age-specific stage structures
(fig. 2B) and mortality rates (fig. 2C). The eight stages are
seeds, three size classes of nonreproductives, and four size
classes of reproductives (fig. 2; app. A; Pascarella and
Horvitz 1998). Asymptotic dynamics of environments,
stages, and mortalities, obtained by long-run simulation,
are also illustrated (fig. 2D–2G).
Some features are common to all paths, while others
differ. For example, common to all is a repeating pattern
where a number of years of darkness (environment 7) is
followed by a sudden opening up of the canopy (environments 1–3) and its gradual closing (increase in habitat
state) over a several-year period (fig. 2A, 3D). Another
feature common to all paths is that newborns have the
highest mortality and that old plants have lowest mortality
(fig. 2C). Thus, there is an overall trend of negative se-
208 The American Naturalist
Figure 2: Environmental state, stage structure, and mortality rate at each age for a single cohort for Ardisia in the variable environment. A–C,
Transient dynamics from birth to age 250. D–F, Asymptotic dynamics. G, Stochastic mortality mq.
nescence, because survival of seeds (stage 1) is lowest and
survival of largest plants (stage 8) is highest in all environmental states (fig. 3). In all paths, there are also spikes
of intermediate mortality, but the ages at which spikes
occur differ. What differs among paths is how stage transitions interact with environmental transitions.
For instance, the stage that varies most among environments in survival is juveniles (stage 3), with lowest
Figure 2: (Continued)
210 The American Naturalist
Figure 3: One-period stage-specific survival in each state of the environment sa for Ardisia in the variable environment.
survival in environment 3 (fig. 3). If a transition to environment 3 occurs when there are many juveniles, high
mortality results. This spike is seen at age 20 of our example cohort (fig. 2), but it is not seen in most cohorts.
The example cohort was born into the darkest state of the
environment and remained in that state until age 20. Starting out as seeds with high mortality, its stage structure
shifts soon to juveniles, who dominate until the environment changes to state 3, when a mortality spike occurs
and the stage structure dramatically shifts. For the next
31 years, juveniles share the distribution fairly equally with
small, medium, and large reproductives. Gradually, extralarge reproductives come to dominate the stage structure
as the canopy closes, and then the lowest mortality is observed (fig. 2C).
Survivorship l(x) of the example cohort to ages 50, 150,
and 250 years is 0.0058, 0.0011, and 0.0003, respectively.
An initial cohort would have to number nearly 4,000 individuals for any to still be alive by age 250. In the lightest
(rarest) state of the environment, reproductives produce
3,000 seeds each (Horvitz et al. 2005), and cohorts of this
size would be expected. But in the darkest (most frequent)
state, reproductives produce 25 seeds each, and cohorts
are rather small.
Using closed-form equations (Tuljapurkar and Horvitz
2006), we obtained expected survivorship and mortality
for an assemblage of cohorts born into each of the seven
states of the environment. As with individual cohort simulations, analytical results indicate negative senescence.
Analytical results show that birth state a significantly affects mortality ma(x) (fig. 4A) in the first 30 years of life,
but by age about 50 years, the mortality plateau m∗ p
0.0135 common to all birth states has been reached. Survivorship la(x) (fig. 4B) differs considerably by birth state
at young ages, but at old ages, it has only slightly different
levels. The coefficient of variation of survivorship increases
with age (fig. 4C; more details and mathematical notes in
fig. A2 and app. B).
The long-run stochastic mortality rate (figs. 2G, 5) is
normally distributed among sample paths, with mean Ⳳ
standard error mq p 0.01497 Ⳳ 0.00002 for sample size
N p 200 sample paths. Note the extremely low standard
Figure 4: A, Age-specific mortality for each birth environment ma(x) for Ardisia and the mortality plateau ⫺ log (l) (dotted line) common to all
birth environments. B, Survivorship la(x). C, Coefficient of variation of survivorship (standard deviation/mean) of la(x).
212 The American Naturalist
Figure 5: Stochastic mortality mq for 200 sample paths (each computed to 40,000 time steps, not implying that plants survive to 40,000 years old
but rather illustrating the computational method).
error. Birth state does not have a significant effect on mq.
The value of mq in this particular case was rather close to
the plateau of the megamatrix analysis for the average
cohort, but that is not generally expected.
Discussion
Stage, Age, and Plant Senescence
Our analysis of empirically based stage-structured models
extends previous work on the age-pattern of mortality and
senescence in plants. Silvertown et al. (2001) used the
methods of Cochran and Ellner (1992) to obtain age trajectories of mortality from stage-based models of demography in constant environments for 65 plant species, including those we discuss. They fit parameters that were
proposed to measure extrinsic versus intrinsic mortality
according to a Weibull model (Ricklefs 1998) for these
trajectories. They distinguished three age patterns of plant
mortality: type I (e.g., Digitaria) showed an asymptotic
increase in mortality with age, type II (e.g., Arisaema)
showed an asymptotic decrease in mortality with age, and
type III (e.g., Pinus) showed a minimum mortality occurring between the youngest and oldest observed ages.
Our work reveals additional age patterns of mortality. For
example, Setaria had maximum mortality occurring between young and old ages. We might call this type IV. On
the other hand, our results indicate that any pattern is
possible, because transitions among stages are not necessarily ordered across ages, and survival does not necessarily follow a monotonic pattern across stages.
Silvertown et al. (2001) reported plateaulike patterns
from visual inspection for some plant species but not all.
They did not provide a quantitative measure of the level
of the plateau nor recognize its existence in all cases. Our
work shows that plateaus always occur for these empirically based stage-structured models even at late ages. Cohort size will interact with convergence rate to determine
whether a given finite population will experience the plateau. Also, our work reveals that age patterns of mortality
Stage Dynamics and Mortality
arise directly from the dynamics of observable stage structure and stage-specific survival rather than from a combination of contributions from hypothetical and unobserved “intrinsic” and “extrinsic” factors represented by
the parameters of the Weibull model. Of course, neither
Silvertown et al.’s analyses nor ours can estimate rates of
death at advanced ages that are not due to stage. Such
estimates require different data that are not available for
these species.
Roach and Gampe (2004) examined the role of age versus size as a determinant of mortality in a 4-year experiment on the perennial plant Plantago. They found no
evidence of an increase in mortality at late ages; maximum
mortality was at an intermediate age (similar to our type
IV above). Further, they found that size was an important
predictor of demographic rates and concluded that “increasing size after reproductive maturity may allow this
plant species to escape from demographic senescence”
(Roach and Gampe 2004, p. 60). These results support the
view that stage and not age is an important determinant
of mortality for plants.
Convergence: The Plateau Is out There
Our results show that the mortality plateau is a general feature of organisms with empirically based stagestructured demography. It is always out there at old ages.
Whether it is relevant to a particular finite population
depends on the rate at which mortality converges to the
plateau, the value of cohort survivorship when the plateau
is reached, and initial cohort size. Mathematically, the rate
of convergence to the mortality plateau depends on subdominant eigenvalues and associated eigenvectors of stage
transition matrices and also on environmental transition
rates. In our examples, the most rapid convergence is !10
years for the African grass Digitaria, and the slowest is
11,000 years for the long-lived temperate longleaf pine
Pinus. For the subtropical forest shrub Ardisia, cohorts
born when the environment is in state 7 (darkest) reach
the plateau about three decades later than those born in
state 1 (lightest). Before convergence, transient dynamics
govern mortality patterns; mortality at early ages can be
above or below the plateau and can rise and fall over the
course of the life cycle before reaching the old-age plateau.
If convergence is slow or the initial cohort is small, the
mortality plateau may not be attained and only transient
mortality patterns may be realized.
Senescence in Humans and Other Species
Models of senescence that hypothesize abstract state variables are also known to generate diverse trajectories of
age-specific mortality (reviewed by Yashin et al. [2000]
213
and by Aalen and Gjessing [2001] and synthesized by
Steinsaltz and Evans [2004]). These include Markov process models of discrete or continuous state variables, with
transitions among states happening in discrete or continuous time. For the most part, state variables are assumed
to be abstract unobservable traits called “vitality,” “senescent state,” “frailty,” and so on. Observable traits are sometimes included as covariates but rarely as the state variable
itself. Individuals in a population may belong to different
fixed or dynamic states. In general, these models assume
a tendency toward increasing death rate. In some models,
only one state can transition to death, but the tendency
is to move toward it (e.g., fig. 3 in Aalen and Gjessing
2001). In others, many states can make the transition to
death, with the intensity of death increasing in order
among states as does the intensity of moving to ever-worse
states (Le Bras 1976; Gavrilov and Gavrilova 1991; Yashin
et al. 2000, sect. 5). In other models, death occurs when
a certain threshold is crossed, and although a diffuion term
allows for some movement closer to and farther from
death, the drift coefficient that measures the transition
toward death is assumed to be stronger (Weitz and Fraser
2001). Mathematical results emphasizing first passage time
to death, hazard rates, and approach to quasi-stationary
distributions of state that are analogous to those that we
present here show that all such models yield an eventual
constant killing rate (synthesized by Steinsaltz and Evans
[2004]). Further, they show that the shape of the trajectory on its way to the plateau depends on how far an
initial state of the population is from death and from
the eventual quasi-stationary distribution, that is, the tension created by the relative strength of the forces between
rates that take the system closer to and further away from
death and closer to and further away from the eventual
quasi-stationary distribution. The size of the initial population is also recognized (e.g., Weitz and Fraser 2001) as
being important in determining whether an asymptotic
plateau is realized by a finite population.
To show where our models fit in this broader picture,
we discuss some of the salient features of age-from-stage
models derived from population projection matrices for
empirically stage-structured populations. The organisms
we refer to here are modeled by discrete-space discretetime Markov chain models with a set of transient states
and one absorbing state. The absorbing state, death, can
be reached from any stage class. The state variable is observed size or developmental category. The initial state is
defined by the size or developmental category into which
newborns are recruited. The initial state does not necessarily have the lowest mortality, and in general, states are
not ordered by survival rate. Organisms change readily to
states with either higher or lower survival than the current
state and may jump states or regress by more than one
214 The American Naturalist
state in a single time step. Survival rates and stage transitions depend on habitat, sometimes in complex ways.
Stage transition rates are measured directly from field data
on marked individuals.
Some features of these organisms might illuminate directions for future work on human mortality models. Recent work on human data is beginning to show how mortality depends on measured physiological variables and on
physiological transitions (Manton et al. 1994; Aalen and
Gjessing 2001; Seeman et al. 2004). Vaupel’s work on heterogeneity of frailty within human populations describes
a process that tracks the average mortality of a heterogeneous cohort at each age. In this formulation, frailty is
regarded as a continuous-stage variable z, and age-specific
mortality is given by weighting the age-specific mortality
of each frailty state m(a, z) by its frequency in the cohort
at a particular age fa(z). Vaupel integrates over frailty
stages at each age, m̄(a) p ∫ m(a, z)fa(z)dz (see Vaupel and
Canudas-Romo 2002), which is analogous to the scalar
product at each age that we describe here for discrete
stages. The distinction is in the source and dynamics of
the heterogeneity of survival rates. In Vaupel’s examples,
individuals do not make transitions among frailty states;
in our examples, they do. However, our results together
with these suggest that it is timely to estimate stage transition models of human mortality, using biological measurements to define stages and stage transition rates.
Our results for variable environments show birth conditions persistently influence the risk of death during early
life, but as cohorts approach the plateau, the effect of birth
state on risk of death fades away. Postbirth environmental
variability generates accumulating differences in survivorship between what we might call “lucky” and “unlucky”
cohorts, affecting the fraction of a cohort that survives to
old age but not the rate of change of survivorship at old
age. These properties suggest that stage-structured models
are a natural way to model the effect of changing environmental conditions on age-specific mortality. For example, current debate over the Barker effect (Barker et
al. 1989; Scrimshaw 1997) concerns the extent to which
there are correlations between conditions at birth and
mortality in later life. The cohort birth effects we have
illustrated provide one possible model for such correlations; in a stage-structured setting, one could also analyze non-Markovian models in which there are longerlived correlations between stage transition rates over
time. Stage-structured models also provide a natural way
of modeling observations on the reversibility of the effects
of environmental stresses (Vaupel et al. 2003), since
stresses can be viewed as producing short-term changes
in physiological state. Our results on the accumulation of
environmental variability provide a mechanism that generates mortality heterogeneity between individuals who ex-
perience different environments through life. Such environmentally driven heterogeneity also provides an
explanation for unusually high survivorship in cohorts that
experience particularly long runs of favorable environments. We think it likely that recorded instances of unusually long-lived plants (Lewington and Parker 2002)
may be the result of such accumulated variability.
Our discrete model of environmental variability has a
continuous analog in continuous-state models that are
subject to random perturbations (Freidlin and Wentzell
1984; Kifer 1990). Our conclusions should be generalizable
to continuous models.
Finally, we note that evolutionary models of senescence
have in the main been age structured (Hamilton 1966;
Charlesworth 1994). Stage-structured models have been
used to study senescence but in a framework of optimization rather than evolutionary genetics (Mangel and Bonsall 2004; Vaupel et al. 2004). Our results suggest there
would be much to learn from analyzing the evolution of
senescence in genetically based stage-structured models.
Acknowledgments
We thank the National Institutes of Health, the National
Institute of Aging (P01 AG022500-01), and the National
Science Foundation (DEB-0614457) for support and S.
Orzack and two anonymous reviewers for useful critiques.
This is contribution 658 of the University of Miami Program in Tropical Biology, Ecology, and Behavior.
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Associate Editor: John M. McNamara
Editor: Donald L. DeAngelis