vol. 158, no. 5 the american naturalist november 2001
A Dynamic Model of Size-Dependent Reproductive Effort
in a Sequential Hermaphrodite: A Counterexample
to Williams?s Conjecture
Lock Rogers* and Robert Craig Sargent?
Center for Ecology, Evolution, and Behavior, School of Biological
Sciences, University of Kentucky, Lexington, Kentucky 40506-0225
Submitted October 12, 2000; Accepted May 25, 2001
abstract: In 1966, G. C. Williams showed that for iteroparous
organisms, the level of reproductive effort that maximizes fitness is
that which balances the marginal gains through current reproduction
against the marginal losses to expected future reproduction. When,
over an organism?s lifetime, the value of future reproduction declines
relative to the value of current reproduction, the level of effort al-
located to current reproduction should always increase with increas-
ing age. Conversely, when the value of future reproduction increases
relative to the value of current reproduction, the level of effort al-
located to current reproduction should decrease or remain at zero.
While this latter pattern occurs commonly in species that exhibit a
delayed age at first reproduction, it may also occur following an
initial period of reproduction in some sex-changing organisms that
experience a dramatic increase in reproductive potential as they grow
larger. Indeed, this schedule of reproductive effort is predicted by
models of ?early? sex change; however, these models may arrive at
this result incidentally because they consider only two reproductive
states: on and off. In order to examine the schedule of reproductive
effort in greater detail in a system where the potential reproductive
rate increases sharply, we adapt the logic and methods of time-
dependent dynamic-programming models to develop a size-
dependent model of reproductive effort for an example species that
experiences a dramatic increase in reproductive potential at large
sizes: the bluehead wrasse, Thalassoma bifasciatum. Our model shows
that the optimal level of reproductive effort will decline with in-
creasing size or age when increases to the residual reproductive value
outpace the increases to current reproductive potential. This result
confirms the logic of Williams?s analysis of optimal life histories,
while offering a realistic counterexample to his conjecture of ever-
increasing allocation to current reproduction.
* Corresponding author. Present address: Department of Biology, Lewis and
Clark College, Portland, Oregon 97219; e-mail: lrogers@lclark.edu.
? E-mail: csargent@pop.uky.edu.
Am. Nat. 2001. Vol. 158, pp. 543?552. 
 2001 by The University of Chicago.
0003-0147/2001/15805-0008$03.00. All rights reserved.
Keywords: reproductive effort, reproductive value, life history,
dynamic programming, sex change, Thalassoma bifasciatum.
The goal of life-history theory is to explain the evolution
of the schedules of reproduction and survival over an or-
ganism?s lifetime (Fisher [1930] 1958; Williams 1966a,
1966b; Gadgil and Bossert 1970; Schaffer 1974, 1979, 1983;
Pianka and Parker 1975). The importance of these sched-
ules in an evolutionary context is clear because these two
fitness components jointly determine an organism?s ex-
pected lifetime reproductive success (R 0), which in sta-
tionary populations is equivalent to fitness (Fisher [1930]
1958; Schaffer 1979; Roff 1992). Related to R 0 is Fisher?s
concept of reproductive value. Reproductive value(Vx) is
the expected contribution of an individual of age x to the
growth of the population (Fisher [1930] 1958). At age 0,
R 0 is equivalent to V0 in stationary populations (Stearns
1992), and the schedules of survival and fecundity that
maximize reproductive value at each age x also maximize
R 0 and consequently fitness (Fisher [1930] 1958; Williams
1966b; Schaffer 1979).
Although these schedules may be viewed as separate
characters, in reality they are bound together by the fact
that organisms have limited resources. This forces the in-
vestments toward reproduction and survival to be traded
off against one another (Bell 1980; Reznick 1983; Warner
1984a; Partridge and Harvey 1985; Schluter et al. 1991;
Roff 1992; Stearns 1992), and a single schedule of allo-
cation to current versus future reproduction can therefore
be considered. The optimal solution to this trade-off was
examined by Williams (1966a, 1966b), who modified the
concept of reproductive value by splitting it into two
elements: one representing the contribution to fitness
through current reproduction and another representing
the contribution to fitness through expected reproduction
in the future. This latter element is termed the ?residual
reproductive value? (RRV). Williams?s (1966a, 1966b)
analysis shows that the allocation to current reproduction
(E) maximizing Vx is that which balances the marginal
544 The American Naturalist
Figure 1: Modified from Pianka and Parker (1975). Plot shows the trade-offs between the contributions to reproductive value (or fitness) through
current and future expected reproduction over the lifetime of an iteroparous organism. The dots and the heavy line on the curved surface of the
three-dimensional solid trace the optimal life history balancing the allocation to current reproduction (E opt) versus future reproduction ( ).1
 Eopt
The drop lines and the shaded area on the floor of the surface trace the realized reproductive rate that is a function of the potential (maximal)
reproductive rate and the allocation of effort toward reproduction. Note that as the residual reproductive value (RRVs) declines with increasing age/
size, the proportional allocation of effort toward current reproduction is predicted to increase. This is Williams?s (1966a) conjecture.
fitness benefits of increased current reproduction against
the marginal fitness costs of decreased expected future
reproduction resulting from increases in E.
Over a lifetime, an organism?s reproductive value may
change in a predictable manner when increasing age or
increasing size affect current reproductive potential, future
reproductive expectations, or both. As a result, the optimal
investment to current versus future reproduction will
change in a similarly predictable way. The classic example
of this is Williams?s (1966a) conjecture that for iteroparous
organisms, the optimal level of reproductive effort (E opt)
is always expected to increase with age. This prediction is
founded on the assumption that as an individual ages, its
RRV declines. Such a decline may result from senescence
(i.e., an age-specific reduction in either survival or repro-
ductive capacity) but could also result from changes in the
environment (e.g., the onset of winter) that similarly affect
future fitness but are independent of an organism?s con-
dition. In either case, as the value of current reproduction
increases relative to RRV, the optimal level of effort al-
located to current reproduction increases (see fig. 1 for a
graphical presentation taken from Pianka and Parker
1975).
Although theoretical and empirical studies have gen-
erally supported Williams?s (1966a) prediction (see Roff
1992 and Stearns 1992 for reviews), three counterexamples
in which the optimal level of reproductive effort is ex-
pected to decrease as age increases have been described
(Fagen 1972; Charlesworth and Leo?n 1976; Roff 1992).
Here, we briefly review and discuss these before focusing
on a fourth counterexample. Fagen (1972) showed that
E opt can decline with age when the functions describing
the age-specific patterns of growth, reproduction, or sur-
vival are not monotonic. Specifically, the example de-
scribed by Fagen (1972) outlines a four-age (0, 1, 2, and
3) life history in which reproduction is possible at ages 1,
2, and 3 and growth is possible at ages 0, 2, and 3. In this
case, because the maximum growth increment from age
Reproductive Effort under Sex Change 545
1 to age 2 is 0, the function describing the growth rate is
not monotonic but rather concave. The resulting optimal
allocation schedule is a relatively high investment to cur-
rent reproduction at age 1 because any negative effect of
high levels of E on fitness is eliminated by the fact that
individual size at future ages is independent of E at this
age; E opt then falls at age 2 because size, and therefore
future fecundity, is a function of E at this age. Finally, Eopt
increases to the maximum at age 3 because there is no
future reproduction for which to withhold some invest-
ment (i.e., ).RRV p 0age 3
In Roff?s (1992) counterexample, the probability of sur-
vival to the next age is a declining step function of E, and
growth is determinate. Let E crit equal some critical level of
reproductive effort. Levels of allocation in excess of E crit
result in zero survival, while levels less than E crit result in
a high probability of survival. Under these circumstances,
the optimal solution at each point in the life history is for
E to be the maximal value that still results in high sur-
vivorship ( ). If E crit decreases as a function ofE ! Eopt crit
age, the E opt will likewise decrease over age. Although this
model assumes that survival is a nonlinear step function
of reproductive effort, a sigmoidal approximation of this
step function would have the same effect and yet would
formally be a monotone function. Thus, in theory, E opt
can decrease with age in life histories where survival is
either a monotone or a nonmonotone function of age.
A third set of circumstances leading to an age-specific
decrease in E opt was identified by Charlesworth and Leo?n
(1976), who show that under density independence, when
the intrinsic rate of increase for the population (r) is high
and the cost of reproduction is low (i.e., survival and
growth rates do not drop sharply with increasing E), the
optimal schedule of reproductive effort can decline with
age (Charlesworth 1994). In this case, the decline in E opt
is driven by the fact that early life reproduction has a
greater contribution to fitness than late life reproduction
when r is high and is permitted by the fact that RRV is
not greatly reduced by relatively high levels of allocation
to current reproduction.
While these three counterexamples show that the op-
timal schedule of reproductive effort will not inevitably
increase with age, they are each somewhat artificial (Roff
[1992, p. 261] in reference to Fagen?s [1972] model and
his own but not that of Charlesworth and Leo?n [1976]).
Nonmonotone vital-rate functions of the form described
by Fagen (1972) are conceivable, but it is difficult to imag-
ine a realistic system that would fit his model. Fagen (1972)
does note that certain large mammal species (specifically
male African elephants, some seals, and toothed whales)
exhibit a pattern of growth that matches the predictions
of the nonmonotone model (size increases until puberty,
remains constant for some period, then increases again
following the latent period). However, it is not readily
apparent that such a latent period results from incapacity
for growth, as suggested by Fagen (1972), or from an
allocation of resources to other components of the life
history (i.e., survival and reproduction). The model sug-
gested by Charlesworth and Leo?n (1976) is more realistic;
however, its key condition, that the intrinsic rate of in-
crease is relatively high due to density independence, is
unlikely to be sustained for long in natural populations.
Thus, the Charlesworth and Leo?n (1976) model can ex-
plain why reproductive effort might decline over size or
age for some interval; however, it cannot explain why this
pattern should hold for populations at equilibrium den-
sities.
Sex Change as a Counterexample: When Gains to RRV
Outpace the Gains in Current Reproduction
Here we suggest that iteroparous organisms that experi-
ence a dramatic increase in potential reproductive success
as they grow larger (a pattern observed in some species
of sex-changing fishes) constitute an unrecognized and
biologically realistic counterexample to Williams?s (1966a)
conjecture. In such species, as some critical point in the
life history is approached, the size-specific gains in ex-
pected future reproduction can outpace the size-specific
gains to current reproductive potential, which drives the
predicted optimal level of reproductive effort down. This
case is then simply the converse of the conditions outlined
by Williams (1966a), who assumed that RRV generally
declines with age or size and simply makes the opposite
prediction (see fig. 2 for a graphical presentation of this
adapted from Pianka and Parker 1975).
This pattern of size-specific reproductive potential is
observed in some species of protogynous (i.e., female first,
male second) sex-changing fish, such as the bluehead
wrasse, Thalassoma bifasciatum (Labridae). In this species,
small, initial-phase females spawn once a day, while very
large, terminal-phase males spawn with an average of
20?30 females per day (Warner and Schultz 1992). Fur-
thermore, individuals in this species exhibit what is termed
?early? sex change. Early sex change occurs when, after
reproducing for some period in the initial sex, an indi-
vidual changes sex but immediately enters a protracted
nonreproductive period (i.e., E falls to 0) before eventually
resuming reproduction in the second sex. This observed
nonreproductive period is predicted by existing life-history
models of early sex change developed by Hoffman et al.
(1985) and Iwasa (1991). However, we suggest that these
models may arrive at this prediction inadvertently. In sup-
port of this perspective, we caution that the model of early
sex change by Hoffman et al. (1985) explicitly considers
only two reproductive states (off [ ] and on [E p 0 E p
546 The American Naturalist
Figure 2: Adapted from Pianka and Parker (1975). As in figure 1, plot shows the trade-offs between current and future expected reproduction in
an iteroparous organism. The dots and heavy line on the surface of the three-dimensional solid trace the optimal life history. Here, there is a
dramatic increase in the potential reproductive rate at the size horizon (S). In species such as the bluehead wrasse, Thalassoma bifasciatum, the size-
specific potential reproductive rate increases at sex/color-phase change to 20?30 times the average rate before this point. This increase in reproductive
potential at S (size at sex/color-phase change) generates an increase in the residual reproductive value at sizes leading up to S. When the size-specific
gains to RRVs outpace the size-specific increase in the potential reproductive rate, the optimal allocation to current reproduction will fall. This
provides a counterexample to Williams?s (1966a) conjecture while reinforcing the logic underlying his prediction.
]), and therefore any reduction in E opt can only be a1
reduction to 0. In a similar vein, we note that, under the
assumptions made by Iwasa (1991) concerning the trade-
off between current and future reproduction, the optimal
reproductive effort is effectively restricted to be 0 or 1 (T.
Day, personal communication). We also note that data
from a field study by Warner (1984a) measuring growth
and survival rates in T. bifasciatum do not appear to sup-
port the assumptions of the early-sex-change models. Fi-
nally, we argue that alternative explanations for early sex
change exist (e.g., Aldenhoven 1986; Rogers 1998; L. Rog-
ers, unpublished manuscript), and thus trade-offs between
life-history components may not be a selective force be-
hind the evolution of early sex change.
Below, we briefly review sex-change models, their ev-
olution, and their explicit and implicit restrictions. Mo-
tivated by the models of Hoffman et al. (1985) and Iwasa
(1991), we then develop a new model applying dynamic-
programming methods (Bellman 1957; Mangel and Clark
1988) that permits E to vary between 0 and 1.
Sex-Change Theory
Sex change is a taxonomically widespread, albeit uncom-
mon, pattern of sexual expression (Policansky 1982; War-
ner 1984b) in which individuals reproduce first as one sex
and later as the other sex. Models of sex change have
historically examined the adaptive significance of two as-
pects of this pattern. First, these models sought to un-
derstand the conditions under which sex change may be
Reproductive Effort under Sex Change 547
favored. Second, given that sex change is indeed favored,
these models explored the factors affecting the timing of
the shift from the initial to the terminal sex.
The principal model of sex change, the size-advantage
hypothesis (Ghiselin 1969), proposes simply that selection
will favor sex change when an organism can reproduce at
a higher rate as one sex when small or young and at a
higher rate as the other sex when large or old, while all
other aspects of the life history (e.g., growth and survival)
are equal between the sexes. Analytical models by Warner
and his colleagues (Warner 1975; Warner et al. 1975; Leigh
et al. 1976; Charnov 1982) confirm the selective advantage
of sex change under these circumstances and show that
this advantage is maximized when sex change occurs at
the point where the curves describing the age- or size-
dependent male and female reproductive rates intersect.
For mathematical tractability, these original formula-
tions of the size-advantage model (Warner 1975; Warner
et al. 1975; Leigh et al. 1976) adopt the convention of
assuming the sex-specific reproductive rates and survival
probabilities to be functions of age (e.g., Pianka and Parker
1975). Thus, as in demographic studies, the reproductive
rate at age x is denoted mx, and the probability of survival
from age 0 to age x is denoted lx. While this assumption
greatly simplifies the models and does not change their
fundamental results (Warner 1975, 1988), this formulation
can implicitly restrict such models in two interrelated ways.
The first restriction results from the fact that, while it is
logical to treat survival (lx) as a function of age, the re-
productive rate (mx) is often more appropriately a function
of size. Although treating mx as a function of age may be
justifiable because size and age are correlated in organisms
with indeterminate growth, this assumption implicitly
forces size and age into synonymity. This becomes a prob-
lem when, for mathematical tractability, lx and mx are as-
sumed to be simple functions of age (for example, l px
and ) because growth rate cannot vary
axe m p bx cx
over x. That is, the result of this fusion of size and age is
that models adopting this formulation are incapable of
addressing trade-offs between growth and other compo-
nents of the life history, such as survival and reproduction.
Related to this is the fact that lx and mx are the realized
outcomes of age- and size-specific potential survival and
reproductive rates and the level of effort directed toward
these potentials. Thus, while models using these terms
allow the conceptual illustration and investigation of ideas
such as the size-advantage model (Warner 1975; Warner
et al. 1975; Leigh et al. 1976; Hoffman et al. 1985; Charnov
1986; Iwasa 1991), because lx and mx are measures of out-
put and not input, they are inadequate when the goal is
to model the pattern of allocation to growth, reproduction,
and survival across an organism?s lifetime.
The first limitation, the implicit fusion of size with age,
was recognized and to a certain extent corrected by Hoff-
man et al. (1985) and Iwasa (1991). Hoffman et al. (1985)
and Iwasa (1991) recast the size-advantage model to per-
mit examination of the effect of trade-offs between life-
history components on sex-change timing. These authors
note that when reproduction imposes a sufficiently high
cost to survival or growth, both of which increase future
reproductive potential, selection may favor a nonrepro-
ductive period. While such a nonreproductive period
could hypothetically occur at any point in an organism?s
lifetime, both Hoffman et al. (1985) and Iwasa (1991)
assume that it intervenes between reproduction in the
initial and terminal sexes. Thus, the phenomenon referred
to as ?early? sex change, in which individuals of many
species of sex-changing fish appear to change sex ?too
early? (i.e., before they are capable of reproducing in the
second sex), may be an adaptive life-history strategy (see
also Thresher 1979; Jones 1981; Moyer and Zaiser 1984;
Aldenhoven 1986).
Although the models of Hoffman et al. (1985) and Iwasa
(1991) are improvements over the simplest treatments of
the size-advantage model (e.g., Warner 1975; Warner et
al. 1975; Leigh et al. 1976), there are theoretical and em-
pirical problems with these models of early sex change as
well. First, despite its ability to explain the basic pattern
of reproductive effort (i.e., reproduce as the initial sex,
cease reproduction for some period, and finally begin re-
production again in the terminal sex) across a sex-
changing individual?s lifetime, the model by Hoffman et
al. (1985) artificially dichotomizes reproduction as either
on or off. The functions describing the reproductive effort
at age x in this model are oversimplified because they do
not include a term describing the level of allocation to
reproductive effort at x. Second, the model by Iwasa (1991)
assumes the trade-off between future and current repro-
duction to be linear. If this function is not convex but
rather linear or concave, there can be no intermediate
optimum (Pianka and Parker 1975), and the optimal life
history will be a so-called bang-bang transition from one
reproductive state to another (e.g., fully reproductive to
fully nonreproductive).
Empirically, if the nonreproductive period exists as the
result of a shift in allocation from current to future re-
production, then life-history components correlated to fu-
ture reproduction (e.g., growth and survival rates) are pre-
dicted to increase at this point. In agreement with this
prediction, Warner (1984a) found that growth rates of
terminal-phase male Thalassoma bifasciatum that were
newly sex-changed but not yet territorial (i.e., early sex-
changed) were approximately 50% higher than compa-
rably sized initial-phase females. While this is a substantial
increase, Warner (1984a) also shows that mortality rates
among terminal-phase males are two to three times higher
548 The American Naturalist
than males and females in the initial phase. Thus, it is not
obvious that the early sex-change strategy does in fact
result in an increase in future expected reproduction suf-
ficient to balance the cost of a nonreproductive period.
Reproductive Effort Model
Methods
In order to determine the optimal schedule of effort in-
vested toward current reproduction (E opt) across an or-
ganism?s lifetime, one must be able to determine the effect
of a given level of effort on the expected contributions to
fitness through both current and future reproduction. The
difficulty here is that, while determining the effect of some
level of effort on current reproduction is relatively straight-
forward, establishing its effect on future reproduction can
be very complicated.
One method used to solve such problems is dynamic
programming (Bellman 1957; Mangel and Clark 1988).
Dynamic programming (DP) is an optimization technique
used to find the optimal sequence of decisions (e.g., patch
choice, allocations to a physical structure or to reproduc-
tion, foraging decisions, etc.) along a time axis (see Mangel
and Clark 1988 for a more complete description of these
methods). The dynamic-programming equation applied to
life histories can take the general form
V(x, t, T) p
max m(x, i, t) {p(x, i, t)V[x g(x, i, t), t 1, T]} ,( )
i
(1)
where V(x, t, T ) is the current reproductive value of an
individual of condition x at time t adopting the optimal
decision or behavior i (maxi) given a time horizon of T
(see Sargent 1990; Ydenberg et al. 1995; St. Mary 1997 for
similar treatments of the DP equation). The terms
m(x, i, t), g (x, i, t), and p(x, i, t) are, respectively, the
contribution to fitness through current reproduction, the
increment to condition (e.g., size), and the probability of
survival to the next time ( ). The second term on thet 1
right-hand side (within braces) represents the residual re-
productive value (i.e., the expected contribution to fitness
through future reproduction). It is interesting to note that
by partitioning reproductive value into two elements rep-
resenting current and future expected reproduction, Wil-
liams (1966b) and Bellman (1957) independently derived
solutions that have the same logical form (Schaffer 1974;
Rogers 1998). A DP model is solved by backward iteration
beginning at the last time step to yield the optimal tra-
jectory over time (Bellman 1957; Mangel and Clark 1988).
The critical feature of a dynamic program is the time
horizon (T ) that serves as a temporal benchmark for
terminal-fitness determination (e.g., the time of occur-
rence of dominant life-history events such as maturation,
the end of reproductive season, the date on which an
ephemeral pond will dry up, etc.). It is the time horizon,
or rather an organism?s proximity to it, coupled with the
fitness expectation of a given condition level at T (the
terminal-fitness function) that drives the dynamics of DP
models. When the time horizon represents something like
the point in a season at which maturation occurs, the
terminal-fitness function may be set such that expected
fitness at T is an increasing function of condition. Alter-
natively, when the time horizon represents something like
a killing freeze, the terminal-fitness function may be set
to 0, regardless of condition.
In the model developed here, we assume that there is
no important point-in-season effect imposed by the en-
vironment and that individual reproductive, growth, and
mortality rates are not functions of time (or therefore age)
but of size. Thus, while time would exist coincidentally in
such a model, a critical point in time, such as a time
horizon, has no meaning but a critical size does. In this
case, logical replacement of the time horizon with a size
horizon (S) representing the size at sex change permits
solution for the optimal decision trajectory by backward
iteration across size. Because this model has a single axis,
the terminal-fitness function can also be replaced by a
terminal-fitness value; the reproductive value at the size
at sex change, VS .
Dropping the variable time (t) and the time horizon
(T) from the DP equation above yields
V(s) p max m(s, E) {p(s, E)V[s g(s, E), S]} , (2)( )
E
where E (proportion of total effort [e.g., resources or en-
ergy] allocated to current reproduction) replaces i from
equation (1). The terms V(s), m(s, E), g(s, E), and p(s, E)
represent, respectively, the size-specific reproductive value,
reproductive contribution, growth increment, and survival
probability for an individual of size s, allocating a level of
effort E to current reproduction. The reproductive value
in the next size given a size horizon of S is V [ (s, E),s g
S]; the product of this future reproductive value and the
probability of survival to the next size (p(s, E)), is the
residual reproductive value (within braces).
In this model, we assume that reproductive output at
size s is an increasing linear function of both size and
reproductive effort. Conversely, we assume that somatic
output (i.e., growth) is a linearly increasing function of
size and a linearly decreasing function of reproductive ef-
fort. The rationale for these functions is that it seems
Reproductive Effort under Sex Change 549
Figure 3: The schedule of optimal reproductive effort (E) over size for six levels of reproductive value at sex/phase change (VS). If the reproductive
value at the size at sex/phase change (S), approximately 80 mm (in standard length) in Thalassoma bifasciatum, is sufficiently high (e.g., curve c),
the optimal level of reproductive effort will decline as S is approached. If not, Eopt is predicted to increase with size (curve a).
simplest to assume that somatic and reproductive output
will be the product of size-specific physiological potentials
as well as the levels of allocation (E) dedicated toward
each. We assume that the probability of survival (p(s, E))
is a nonlinear, decreasing power function of reproductive
effort ( , where and ). Thisdp(s, E) ? 1
 E d 1 1 0 ! E ! 1
assumption is made to force the model to exhibit an in-
termediate optimum over E (Gadgil and Bossert 1970).
When there is no intermediate optimum, there are two
local optima: (a complete allocation to futureE p 0opt
reproduction) and (a complete allocation to cur-E p 1opt
rent reproduction at a suicidally high level [i.e., semel-
parous reproduction]). As a biological example justifying
this assumption, imagine a fish making a slight overin-
vestment in current reproduction resulting in its being
slightly slower than its schoolmates. Here, a linear increase
in E, resulting in a linear decrease in escape speed, will
result in a nonlinear increase in the probability of mortality
to the slowest member of the school.
Results
The results of this model show that the optimal schedule
of reproductive effort in an iteroparous organism will
initially increase, plateau, and eventually decline over some
range when the size-specific increase in residual repro-
ductive value outpaces the size-specific increase in current
reproductive potential (here, when the reproductive value
at S, VS is sufficiently high [fig. 3, curve c]). Thus, this
model confirms the intuition that inverting Williams?s
(1996a) assumption concerning a size-specific (or age-
specific) decline in RRV results in the inversion of the
predicted schedule of allocation to current reproduction.
In such cases, the effect of very high reproductive value
at the size horizon is that the residual reproductive value
increases dramatically relative to the value of current re-
production (i.e., the fitness contribution through current
reproduction) as S is approached, favoring an increased
proportional investment to RRV. In contrast to this result,
when VS is low, RRV declines with increasing size, and the
schedule of reproductive effort increases monotonically
(fig. 3, curve a). This corresponds to the circumstance
envisioned by Williams (1966a) and is included here for
the sake of comparison. At intermediate values of VS , the
allocation of reproductive effort may remain more or less
constant over size because the contributions to fitness
through current and expected future reproduction remain
in balance (fig. 3, curve b). Significantly, we note that in
none of our simulations were we able to generate a pattern
in which the optimal allocation of effort drops to 0 as
predicted by Hoffman et al. (1985) and Iwasa (1991).
Another pattern that emerges from this model is that,
despite the differences in the schedule of reproductive ef-
fort as VS varies, the effect of VS on the predicted schedule
is generally restricted to the larger sizes and has very little
effect on the pattern at small sizes. Intuitively, although it
is the size (or time) horizon that drives the behavior of a
DP model, the influence of this point-in-size diminishes
with distance. In standard DP models, this commonly ob-
served phenomenon is referred to as ?stationarity? (Mc-
Namara and Houston 1982; Mangel and Clark 1988; Sar-
gent 1990).
550 The American Naturalist
A sensitivity analysis was performed to examine the ef-
fects of three key parameters on the results of the model.
The first of these is the background probability of survival,
which, in conjunction with the schedule of reproductive
effort, determines the schedule of survival and thus an
organism?s eventual probability of reaching the size ho-
rizon. The second parameter is the value of the exponent
describing the decrease in survival as a function of in-
creasing E. Changes in both of these parameters subtly
affect the curves describing the optimal levels of repro-
ductive effort over size. However, the fundamental result
(that the optimal level of reproductive effort is predicted
to decline when VS is sufficiently high) is unaffected by
changes in these parameters. The third parameter exam-
ined is the shape of the function describing the availability
of resources over size. This effectively sets the size-specific
physiological upper limit for growth and reproduction.
Specifically, in addition to the default case in which re-
source income increases as a linear function of size, we
considered the cases where resource income increases ex-
ponentially over size and where it remains constant over
size. We found that as in the two previous cases, the shape
of the schedule of resource income does not qualitatively
affect the optimal schedules of effort (i.e., input) over the
lifetime. Moreover, although the schedules of growth and
reproduction (i.e., output) are strongly affected by this
schedule, their basic shape (e.g., linear, convex, or concave)
is only affected by resource levels when the resource in-
come rate is strongly curvilinear. For example, if the re-
source income increases at an accelerating rate over size,
a decreasing proportional allocation of effort toward re-
production can still result in an increasing level of repro-
ductive output.
Discussion and Predictions
Our model illustrates that the optimal level of reproductive
effort (E opt) is predicted to decline with increasing size (or
age) when the value of future reproduction increases rel-
ative to the value of current reproduction (figs. 2, 3). As
such, this result simply mirrors Williams?s (1966a) con-
jecture of ever-increasing levels of E opt when RRV falls
relative to the value of current reproduction with age or
size (fig. 1). The biological example we point to is that of
protogynous sex-changing fishes in which the residual re-
productive value increases dramatically relative to current
reproductive potential as the critical size at sex (or phase)
change is approached.
This model contrasts with previous models of early sex
change in which reproduction is effectively an on/off de-
cision (Hoffman et al. 1985; Iwasa 1991). In Hoffman et
al. (1985), this restriction results from the explicit consid-
eration of only two reproductive states: active and inactive.
Iwasa (1991), on the other hand, does allow for a full
range of allocation from a total investment to current re-
production at one extreme ( ) to a total investmentE p 1
in future reproduction at the other ( ). However,E p 0
Iwasa?s (1991) assumption of a linear trade-off between
the two components of reproductive value results in the
optimal reproductive effort being functionally restricted
to the active and inactive states as well (see Gadgil and
Bossert 1970; Pianka and Parker 1975).
The model we present permits the proportional allo-
cation to current reproduction to vary semicontinuously
between 0 and 1 over size (i.e., discretely but in greater
detail). It also follows earlier optimal life-history models
in assuming that the trade-off between current and future
reproduction is concave (Williams 1966a, 1966b; Gadgil
and Bossert 1970; Pianka and Parker 1975; Schaffer 1983).
This difference in the structure of the models results in
distinct differences in the predictions. The results of both
Hoffman et al. (1985) and Iwasa (1991) predict that when
the value of future reproduction is sufficiently high, cur-
rent reproductive output will fall to 0 for some period
between bouts of reproduction in the initial and in the
terminal sex. The model we develop here does not predict
a nonreproductive period under any circumstances (fig.
3). It is for this reason that we suggest that the complete
cessation of reproduction leading to a protracted nonre-
productive period exhibited in such species is not due to
trade-offs between current and future reproduction. In
further support of this perspective, we note that at least
in Thalassoma bifasciatum, individuals in the physically
male but nonreproductive interval produce sperm and
thus have some nonzero allocation to both current
and future reproduction (K. Clifton, personal commu-
nication).
The size-horizon model makes very specific predictions
about the pattern of reproductive effort over size in a
hypothetical iteroparous organism that experiences inde-
terminate growth and a step function increase in repro-
ductive potential with increasing size. For all levels of re-
productive value at the size horizon (VS) , the model
predicts an initial period of immaturity. That is, even
though reproduction at very small sizes is permitted in
the model, the optimal sequence of reproductive effort
includes a juvenile period. Likewise, at all levels of VS ,
small reproductive individuals are predicted to have a low
investment to current reproduction, while intermediate-
sized individuals over a broad range of sizes are predicted
to have higher, and relatively constant, levels of repro-
ductive effort. This is in contrast to the predicted pattern
of E opt at very large sizes, which is strongly dependent on
the reproductive value at sex/phase change (i.e., at the size
horizon). If VS is sufficiently high, the value of this future
reproduction may push the current level of reproductive
Reproductive Effort under Sex Change 551
effort down as the size at sex/phase change is approached,
which results in either accelerated growth, increased sur-
vival, or both. In this case, the predicted relationship be-
tween reproductive rate and size should be curvilinear and
convex and bend down at large sizes (i.e., the slope of the
first derivative is negative). Correspondingly, the predicted
relationship between growth rate and size should also be
curvilinear but concave and bend up at large sizes.
Conclusions
In 1966, Williams showed that reproductive value is max-
imized at the level of reproductive effort that balances the
marginal gain in fitness through current reproduction
against the marginal losses in fitness through expected
reproduction in the future. Based on this conclusion, Wil-
liams (1966a) speculated that because the expected con-
tribution to total lifetime fitness through future repro-
duction generally declines with increasing age, the optimal
proportion of effort allocated to current reproduction
(E opt) should always increase. Fagen (1972), Charlesworth
and Leo?n (1976), and Roff (1992) offer three counter-
examples in which E opt decreases with age, but these are
biologically unrealistic (see Roff 1992). In this article, we
develop a life-history model based on the natural history
of the sex/phase-changing bluehead wrasse, Thalassoma
bifasciatum, that illustrates that Williams?s (1966a) con-
jecture can be reversed to predict that when the potential
contribution to fitness through future reproduction in-
creases relative to the potential through current repro-
duction, the optimal level of E should decrease.
In many ways, the model we develop resembles previous
ones addressing the phenomenon of early sex change
(Hoffman et al. 1985; Iwasa 1991). These models show
that a nonreproductive period can be favored when the
fitness benefits in the future outweigh fitness costs in the
present. However, we propose that because both of the
models of early sex change effectively treat reproduction
as an on/off decision, the predicted nonreproductive pe-
riod results from implicit restrictions in these models. That
is, any reduction in E opt with age or size can only be a
reduction to 0 in these models. Our model permits E to
vary semicontinuously and therefore contrasts with the
predictions of Hoffman et al. (1985) and Iwasa (1991).
We interpret this inconsistency to suggest that the observed
nonreproductive period is not a result of trade-offs be-
tween life-history components. In support of this per-
spective, we note that despite the fact that growth rates
increase in T. bifasciatum at sex/phase change, there is a
concomitant decrease in survival rates at this transition
(data in Warner 1984a). Finally, we note that an alternative
explanation for early sex change is that environmental
uncertainty in the timing of successful acquisition of a
mating territory in the terminal phase drives early sex
change (Moyer and Zaiser 1984; Hoffman et al. 1985; Al-
denhoven 1986; Rogers 1998; L. Rogers, in review).
Acknowledgments
We would like to thank our colleagues at the University
of Kentucky for many interesting discussions about the
ideas in this manuscript, especially G. Adkison, P. Crowley,
L. Hartt, K. Jackson, D. Miller, V. Rush, A. Sih, T. Sparkes,
D. Westneat, B. Wisenden, and J. Wolf. We thank P.
Abrams, S. Austad, K. Clifton, T. Day, M. Holyoak, D.
Houle, M. Mangel, H. Rodd, L. Rowe, C. St. Mary, J.
Travis, M. Tseng, R. Warner, and several anonymous re-
viewers for additional discussions and comments on earlier
drafts. This research was supported in part by a Smith-
sonian Tropical Research Institute short-term fellowship
to L.R. and by a National Science Foundation grant to
R.C.S.
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Associate Editor: Steven N. Austad