Evaluation of mortality trajectories in evolutionary biodemography.
ABSTRACT An important task in evolutionary biodemography is to determine the schedule of survival and reproduction as the outcome of natural selection acting on life histories. We do this by using a model in which the state of the organism is characterized by mass and accumulated damage, both of which are affected by activity and which affect the rate of mortality. Focusing on growth during the juvenile period, we determine the level of activity that maximizes reproductive value. Given this, we are able to project forward and determine the trajectory of mortality for an individual following the optimal life history, given the physiological and reproductive parameters. We show that there are two main classes of juvenile mortality trajectories: U-shaped (such as recently reported for prereproductive humans) and steadily declining and we are able to connect the shape of the mortality trajectory with the physiological and reproductive parameters characterizing the life history. Our work shows the importance of state in models of evolutionary biodemography and the power of modern computational methods to illuminate biological process.
- SourceAvailable from: Michael B Bonsall[show abstract] [hide abstract]
ABSTRACT: If mortality rate is viewed as the outcome of processes of behavior, growth and reproduction, then it should be possible to predict mortality rate as a result of those processes. We provide two examples of how this may be done. In the first, we use the method of linear chains to treat mortality that is the result of multiple physiological processes, some of which may have delays. In the second, we assume that mortality is the result of damage associated with growth and metabolism. Both approaches lead to a rich diversity of predicted mortality trajectories. Although many of these look Gompertzian at young ages, the behavior at older ages depends upon the details of the physiological models.Theoretical Population Biology 07/2004; 65(4):353-9. · 1.24 Impact Factor
- Population and Development Review 10/2003; 29 Supp(3):270-291. · 2.22 Impact Factor
- [show abstract] [hide abstract]
ABSTRACT: In so far as it is associated with declining fertility and increasing mortality, senescence is directly detrimental to reproductive success. Natural selection should therefore act in the direction of postponing or eliminating senescence from the life history. The widespread occurrence of senescence is explained by observing that (i) the force of natural selection is generally weaker at late ages than at early ages, and (ii) the acquisition of greater longevity usually involves some cost. Two convergent theories are the 'antagonistic pleiotropy' theory, based in population genetics, and the 'disposable soma' theory, based in physiological ecology. The antagonistic pleiotropy theory proposes that certain alleles that are favoured because of beneficial early effects also have deleterious later effects. The disposable soma theory suggests that because of the competing demands of reproduction less effort is invested in the maintenance of somatic tissues than is necessary for indefinite survival.Philosophical Transactions of The Royal Society B Biological Sciences 05/1991; 332(1262):15-24. · 6.23 Impact Factor
Evaluation of mortality trajectories
in evolutionary biodemography
Stephan B. Munch†and Marc Mangel‡§
†Marine Sciences Research Center, Stony Brook University, Stony Brook, NY 11794-5000; and‡Department of Applied Mathematics and Statistics, University
of California, Santa Cruz, CA 95063
Edited by Ronald D. Lee, University of California, Berkeley, CA, and approved September 6, 2006 (received for review March 2, 2006)
An important task in evolutionary biodemography is to determine
the schedule of survival and reproduction as the outcome of
natural selection acting on life histories. We do this by using a
model in which the state of the organism is characterized by mass
and accumulated damage, both of which are affected by activity
and which affect the rate of mortality. Focusing on growth during
the juvenile period, we determine the level of activity that maxi-
mizes reproductive value. Given this, we are able to project
forward and determine the trajectory of mortality for an individual
following the optimal life history, given the physiological and
reproductive parameters. We show that there are two main classes
of juvenile mortality trajectories: U-shaped (such as recently re-
ported for prereproductive humans) and steadily declining and we
are able to connect the shape of the mortality trajectory with the
physiological and reproductive parameters characterizing the life
history. Our work shows the importance of state in models of
evolutionary biodemography and the power of modern computa-
tional methods to illuminate biological process.
free-radical theory ? disposable soma ? life history theory ?
patterns, understand pattern and process, and predict the con-
sequences of change on those patterns. Evolutionary biodemog-
raphy asks about the origins of such schedules, in the context of
evolution of life histories by natural selection. Evolutionary
biodemography seeks to merge demography with evolutionary
thinking (2–6). The result, for example, will be to use the
comparative method to explore similarities and differences of
patterns across species and to understand the patterns and
mechanisms of vital statistics as the result of evolution by natural
(and sometimes artificial) selection. Raymond Pearl, one of the
founders of quantitative population biology, understood the
importance of doing this but lacked the mathematical tools to do
so. For example, with John Miner (7) he wrote ‘‘For it appears
clear that there is no one universal ‘law’ of mortality. . . different
species may differ in the age distribution of their dying just as
characteristically as they differ in their morphology’’ and that
‘‘But what is wanted is a measure of the individual’s total
activities of all sorts, over its whole life; and also a numerical
expression that will serve as a measure of net integrated effec-
tiveness of all of the environmental forces that have acted upon
the individual throughout its life’’. With the development of
state-dependent life history theory (8–10), the tools now exist.
Here we respond to the challenge of Wachter (11), who noted
that the evolutionary theories of aging generally fail to be able
to predict the characteristics of mortality trajectories. To do this,
we apply a recent development in state-dependent life history
theory (12) that accounts for activity, the generation of cellular
damage through metabolism and reinforcement of damage,
repair of damage, and the mortality consequences of damage.
This approach may be thought of as a fusion of the free-radical
theory of aging (13) and the disposable soma theory (14) in an
optimal life history context. Other efforts along these lines have
emography is, in part, the study of the implications of a
schedule of survival and mortality. The goal is to describe
(15, 16) or constructed models specific to a given system (17–19).
In framing the aging problem as one driven by energy acquisition
as well as allocation, our model is analogous to that recently
developed by Yearsley et al. (20). However, our approach differs
from these prior approaches in three important respects. First,
the mortality trajectories we find are not a priori governed by a
particular functional form but rather are an emergent property
of the life history optimization. Thus, the range of trajectories
that are possible is not constrained to lie within some traditional
family of functions. Second, we focus on the development of
mortality trajectories before the onset of maturation. Clearly,
this departure from the traditional approach leaves open the
question of what happens after maturation. Nevertheless, we
obtain worthwhile insights into the shape of prereproductive
mortality trajectories. Third, rather than focusing on parameters
attributable to a particular species, we attempt to enumerate all
parameter space. We used 3,000 parameter sets to characterize
the mortality trajectories that this framework is capable of
producing. After visual inspection of all 3,000 optimal mortality
trajectories, four main classes of shapes emerged (Fig. 1). We
tabulated the frequency with which each trajectory shape oc-
curred and note the fitness and final size associated with each
Author contributions: S.B.M. and M.M. designed research, performed research, analyzed
data, and wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS direct submission.
§To whom correspondence should be addressed. E-mail: firstname.lastname@example.org.
© 2006 by The National Academy of Sciences of the USA
framework are steadily declining or broadly U-shaped.
The four classes of mortality trajectories produced by our modeling
October 31, 2006 ?
vol. 103 ?
trajectory. We categorized the kinds of mortality trajectories
that result from the life history and interpret our results in that
context. (More details are given in Materials and Methods).
As in ref. 12, rather than fit the model to a particular data set,
we sampled the parameter space (Table 1) and examined the
resulting mortality trajectories. In addition to finding that a
variety of trajectories are possible (also see ref. 21), ranging from
Gompertz like behavior to declines of mortality rate, we are able
to decompose the mortality trajectory into size dependent and
damage dependent components (Fig. 2). Because we compute
the fitness associated with a set of parameter values, we are also
able to compute both raw and fitness-normalized frequencies of
each class of mortality trajectory (Fig. 3). Finally, although the
relationship between the physiological parameters and the mor-
tality trajectory they produce is quite complicated, a discrimi-
interactions explains 90% of the variability in parameter space
among trajectory types and provides a simple means to visualize
these relationships (Fig. 4).
Discussion and Conclusion
Our approach to the computation of mortality trajectories in
evolutionary biodemography is based on phenotypic modeling;
in this sense it is a version of the disposable soma theory (22,23).
However, we are able to reproduce the U-shaped trajectories
that are commonly observed in natural, human and engineering
systems. In our model, in all cases the early decline of mortality
as the organism grows and the increased mortality later in life is
due to the accumulation of damage. Thus, our phenotypic theory
developed for prereproductive individuals, can predict the com-
mon U-shape of mortality trajectories, something that purely
genetic theories are currently unable to do (and noted by
Hamilton in his original work; ref. 24). Although type I and II
recent analyses for humans (25) revealed a U-shaped trajectory
before maturation providing striking empirical support for the
type III and IV curves.
There is no simple relationship between the values of the
physiological parameters and the mortality trajectories they
generate. For any given parameter, we find trajectories from
each class throughout the sampled range. However, within this
which there is a smooth transition from declining to increasing
mortality (Fig. 4), which is dominated by interactions among
parameters. The loadings on this axis (Table 2) allow us to
interpret the conditions under which each mortality trajectory
may occur. For example, trajectories in class 1, for which
mortality continues to decline are characterized by high rates of
consumption and activity, relatively high damage accumulation,
and very low dependence of mortality and reproductive value
upon damage. This class of trajectory represents ?50% of the
simulated parameter values and is commensurately common
trajectories in which mortality is increasing with age are char-
acterized by high sensitivity of fitness to damage. Although we
focus on a juvenile period, these results suggest that increasing
late-life mortality trajectories are common in nature because
accumulated damage incurs nonnegligible fitness costs. We also
note that the rise in mortality rates with increasing age cannot
dependent components. The solid line indicates the total mortality as repre-
sented in Fig. 1. The dotted line indicates the size-dependent mortality,
whereas the dashed line indicates damage-dependent mortality.
Decomposition of the mortality trajectory into size- and damage-
frequencies of each trajectory type, open bars indicate the geometric mean
fitness-weighted frequencies of each.
Frequencies of each mortality trajectory. Filled bars indicate the raw
order parameter interactions associated with each type of mortality trajec-
tory. Trajectory types are indicated by the different marker types and line up
along the horizontal axis in order. That is, trajectory types I–IV are indicated
by the gray circles, black circles, white squares, and gray squares, respectively.
Linear discriminant function analysis of the parameters and first-
Table 1. Summary of the parameter space searched
ParameterDescription Range searched
Maximum consumption rate
Damage reinforcement rate
Energetic efficiency of repair
Maximum repair rate
Half saturation for repair
Value exponent for size
Value exponent for damage
Munch and Mangel PNAS ?
October 31, 2006 ?
vol. 103 ?
no. 44 ?
be simply due to the use of a fixed end of the juvenile period
because the rise in mortality rates does not occur in the majority
of parameter sets. Furthermore, other models (16–18) using the
same general approach (but for very different specific problems)
the rise in mortality.
In conclusion, we have shown that phenotypic life history
optimization predicts most of the major classes of mortality
trajectories that are observed in biodemography and that when
mortality trajectories are U-shaped, they are predicted to begin
to rise before the onset of reproductive activity. Our theory
shows how these trajectories can be explained in the context of
the parameters characterizing the life history. Our approach
illustrates the power of modern computation to illuminate
biology by methods beyond merely fitting models to data (cf. 26)
Materials and Methods
The analysis we present is based on a model developed to predict
the evolution of compensatory growth (12). We consider a life
history governed by two state variables, size (X) and damage (D).
Activity (a), parameterized as the multiples of basal metabolism
spent on foraging, is the control variable through which indi-
viduals regulate growth, the accumulation of damage, and
predation risk. Growth in size (dX?dt) is due to the difference
between energy intake, C, and energy spent either on metabo-
lism, R, or the repair of damage, U, each of which may be
functions of size, damage, and activity. That is, we model growth
and the accumulation of damage (dD?dt) as
dt? C?X, a? ? R?X, a? ? U?X, D?
dt? ?R?R?X, a? ? vU?X, D?? ? ?DD.
The specific functional forms for the intake and loss rates
Intake: C?X, a? ? ?
a ? ?X3/4
Losses: R?X, a? ? ?1 ? a?X
Repair U?X, D? ? ?X
The specific functional forms for C and R are analogous to the
model of West et al. (27) modified to allow for variable rates of
activity. A recent review (28) of the bioenergetics literature
supports the mass scalings used here, at least for vertebrates. The
dependence of R on activity is true given the definition of a and
the dependence of C on activity arises from asserting that
encounter rates depend linearly on the energy invested in
searching, whereas handling time remains fixed in a Holling type
II foraging model. Our choice for the repair function asserts that
when damage is low, the energy invested in repair will be near
zero and that as damage increases investment in repair will
increase until the maximum allocation to repair (?) is reached.
In 80% of the cases surveyed, the allocation to repair was ?0.5
and ?0.9 in only 9%. In preliminary model testing, reducing the
exponent from 2 to 1 did not appreciably change the qualitative
Mortality in this life history model has size- and damage-
dependent components. We model the size-dependence of mor-
tality as approximately inversely proportional to length; here a
power function of mass with exponent ?1?3, with the assump-
tion that size-dependent mortality arises through predation and
that activity increases exposure to predators proportionally. This
scaling is consistent with several reviews of the size dependence
of mortality. For simplicity and based on limited empirical
evidence (29), damage-dependent mortality is proportional to
the accumulated damage. Thus the rate of mortality is
M?X, D, a? ? ?1 ? ?a?X?1/3? ?DD,
where ?a is the rate at which predation risk increases with
activity. Note that although Eq. 7 is an equation for mortality, it
does not define the shape of the mortality trajectory because
neither X(t) nor D(t) are known at the outset. Rather, X(t), D(t),
and M(t) are defined by the optimal life history and may a priori
take on almost any shape. The model as presented is already
nondimensionalized to eliminate redundant parameters. Specif-
ically, we have eliminated proportionality constants for meta-
bolic losses and the size dependence of mortality by rescaling
time and size.
We model fitness (F) using lifetime reproductive output
where ?(x, d) is reproductive output per unit time at size x and
accumulated damage d. We focus on the evolution of mortality
associated with prereproductive growth and activity. Assuming
that a switch from growth to reproduction occurs at age T, F may
be decomposed as follows
F ? ??XT, DT?e??0
Conditioning F on surviving to age T, we have residual repro-
V ? ??XT, DT??
Note that the integral here depends only on mortality subse-
quent to the growth interval and is clearly a function of size and
Table 2. Loadings for the 30 most important parameter
combinations determining the shape of the mortality trajectories
on axis 1
on axis 2
on axis 1
on axis 2
? ? ?
? ? ?1
? ? ?
? ? ?D
? ? ?R
? ? ?2
? ? ?d
? ? ?d
? ? ?a
? ? ?
? ? ?2
? ? ?
? ? ?2
? ? ?1
Parameter combinations are arranged in descending order of importance.
Although there are another 36 parameter combinations, these have negligi-
ble loadings on the primary axis, which accounts for ?80% of the variance.
www.pnas.org?cgi?doi?10.1073?pnas.0601735103Munch and Mangel
accumulated damage through their effects on fecundity and
mortality after the growth interval. Evaluation of this expression
would require dynamical assumptions beyond those already
presented. To simplify, we use a product of two power functions
to approximate the size and damage dependence of residual
reproductive value, i.e.,
V?X, D? ? X?X?1 ? D???D,
where the exponents for size ?Xand damage ?Dare treated as
parameters and allowed to range widely.
Working backwards through the growth interval, we define a
fitness function F(x, d, t) by
F?x, d, t? ? max
?V?X?T?, D?T??X?t? ? x, D?t? ? d?,
so that F(x, d, t) is the maximum fitness at the end of the growth
interval [t, T] taken over activity levels at each instant of time
throughout the interval. When t ? T, fitness is given by the
residual reproductive value, That is, we have (from Eq. 7) F(x,
d, T) ? V(x, d) ? x?X(1 ? d)??D.
Fo previous times, F(x, d, t) satisfies an equation of dynamic
F?x, d, t?
??1 ? M?x, d, a?dt?F?x ? dX, d ? dD, t ? dt??,
which is solved backwards in time, from t ? T ? dt to t ? 0.
Essentially, Eq. 9 works backwards through time calculating
present reproductive value by discounting the future reproduc-
tive value with the probability of surviving the time interval. The
boundaries of the size ? damage grid were chosen such that they
had no effect on the optimal trajectories for the starting sizes we
used. In some of our initial runs, the upper bound for damage
was low enough to influence the outcome and mortality plateaus
were observed. At t ? 0, F(x, d, 0) is the maximum value of F
(lifetime reproductive output) attainable given initial size and
damage. At each time and state, we generate the optimal level
of activity a*(x, d, t), which can then be used to predict growth
trajectories by application of Eqs. 1 and 2. For all simulations, we
set T ? 5 and in general we used dt ? 0.05, but setting dt ? 0.001
did not change the results.
All of the physiological and life history parameters in the
determined experimentally (the parameters governing damage,
though, may be more difficult to quantify). However, the range
of plausible parameter combinations is too broad to investigate
completely. So, to analyze the model, we adopted a Monte Carlo
approach. We defined ranges for each parameter such that a
reasonable amount of growth could occur in the interval [0,T]
and that there was some nonzero probability of surviving to the
end of the growth interval. We acknowledge that this approach
leaves open the question of what happens to mortality trajecto-
ries beyond the ranges from which we sampled. However,
preliminary sampling outside the ranges reported here did not
produce any new trajectory types. For each parameter, this range
was divided into up to 30 distinct values (see ref. 12, Table 1)
resulting in ?1010possible parameter combinations. Because
this space is too large for complete enumeration, we randomly
chose 10,000 parameter sets from these possible combinations
and evaluated the optimal life history for each. Parameter sets
from this range that produced implausible results (e.g., that the
optimal growth trajectory was to never grow) were discarded,
leaving ?3,000 viable parameter sets. The vast majority of
nonviable parameter sets resulted in damage accumulation rates
that were so costly as to eliminate the feasibility of growth.
To analyze the influence of the different physiological and life
history parameters on the resultant mortality trajectories, we
used linear and quadratic discriminant functions (30) on the
original parameter set and an augmented set that included all
pairwise products. Using leave-one-out cross-validation as a
model selection criterion, we found that a linear model in the
augmented parameter space was optimal and had a cross-
validation classification success of 76%.
All of the analyses reported here were carried out by using
code written by SM in Matlab v. 6.5 (Mathworks, Natick, MA).
This code is available upon request.
We thank Ron Lee for inspiring talks about these questions. S.B.M. was
a postdoc supported by the Center for Stock Assessment Research, a
partnership between the Southwest Fisheries Science Center Santa Cruz
Laboratory and the University of California, Santa Cruz. M.M. was
partially supported by National Science Foundation Grant DMS 031054.
1. Gompertz B (1825) Philos Trans R Soc London 115:513–583.
2. Gavrilov LA, Gavrilova NS (1991) The Biology of Life Span: A Quantitative
Approach (Harwood, London).
3. Wachter KW, Finch CE, eds (1997) Between Zeus and the Salmon: The
Biomdemography of Longevity (Natl Acad Press, Washington, DC).
4. Carey JR (2001) Annu Rev Ent 46:79–110.
5. Carey JR, Judge DS (2001) Population 13:9–40.
6. Carey JR (2003) Longevity: The Biology and Demography of Life Span (Prince-
ton Univ Press, Princeton).
7. Pearl R, Miner JR (1935) Q Rev Biol 10:60–79.
8. Mangel M, Clark CW (1988) Dynamic Modeling in Behavioral Ecology (Prince-
ton Univ Press, Princeton).
9. Houston AI, McNamara JM (1999) Models of Adaptive Behavior (Cambridge
Univ Press, Cambridge, UK).
10. Clark CW, Mangel M (2000) Dynamic State Variable Modeling in Ecology:
Methods and Applications (Oxford Univ Press, Oxford).
11. Wachter KW (2003) Popul Dev Rev 29(Suppl):270–291.
12. Mangel M, Munch SB (2005) Am Nat 166:E155–E176.
13. Harman D (1956) J Gerontol 2:298–300.
14. Kirkwood TBL, Rose MR (1991) Philos Trans R Soc London B 332:15–24.
15. Chu CYC, Lee RD (2006) Theor Popul Biol 69:193–201.
16. Cichon M, Kozlowski J (2000) Evol Ecol Res 2:857–870.
17. Mangel M (2001) J Theor Biol 213:559–571.
18. Mangel M (2003) Popul Dev Rev 29(Suppl):57–70.
19. Novoseltsev VN, Arking R, Novoseltseva JA, Yashin AI (2002) Evolution
(Lawrence, Kans) 56:1136–1149.
20. Yearsley JM, Kyriazakis A, Gordon IJ, Johnston SL, Speakman JR, Tolkamp
BJ, Illius AW (2005) J Theor Biol 235:305–317.
21. Mangel M, Bonsall MB (2004) Theor Popul Biol 65:353–359.
22. Kirkwood TBL, Holliday R (1979) Proc R Soc London B 205:531–546.
23. Drenos F, Kirkwood TBL (2005) Mech Age Dev 126:99–103.
24. Hamilton,WD (1966) J Theor Biol 12:12–45.
25. Milne EMG (2006) Mech Ag Dev 127:290–297.
26. Lander AD (2004) PLoS Biol 2:0712–0714.
27. West GB, Brown JH, Enquist BJ (2001) Nature 413:628–631.
28. Essington TE, Kitchell JF, Walters CJ (2001) Can J Fish Aquat Sci 58:2129–
29. Collins AR, Gedik CM, Olmedilla B, Southon S, Beillizzi M (1998) FASEB J
30. Hastie T, Tibshirani R, Friedman J (2001) The Elements of Statistical Learning:
Data Mining, Inference, and Prediction (Springer, Berlin), pp 84–111.
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