O R I G I NA L A RT I C L E
doi:10.1111/j.1558-5646.2009.00916.x
SEX ALLOCATION BASED ON RELATIVE
AND ABSOLUTE CONDITION
Lisa E. Schwanz,1,2,3 Fredric J. Janzen,1,4 and Stephen R. Proulx1,5,6
1
5
Department of Ecology, Evolution and Organismal Biology, Iowa State University, Ames, Iowa 50011
2
E-mail: Lisa.Schwanz@gmail.com
4
E-mail: fjanzen@iastate.edu
Ecology, Evolution & Marine Biology, University of California, Santa Barbara, Santa Barbara, California 93106
6
E-mail: proulx@lifesci.ucsb.edu
Received June 25, 2009
Accepted November 11, 2009
Traditional models predict that organisms should allocate to sex based on their condition relative to the condition of their
competitors, tracking shifts in mean condition in fluctuating environments, and maintaining an equilibrium sex ratio. In contrast,
when individuals are constrained to define their condition absolutely, environmental fluctuations induce fluctuating sex ratios
and the evolutionary loss of condition-dependent sex allocation in short-lived organisms. Here, we present a simulation model of
temperature-dependent sex determination (TSD) in fluctuating environments that specifically examines the importance of relativity
in defining individual condition. When relativity in condition is allowed to evolve, short-lived organisms evolve switchlike TSD
reaction norms and define their condition relative to the annual temperature distribution, thus preventing biased cohort sex ratios
in extreme years. Long-lived organisms also evolve switchlike reaction norms, but define condition less relatively and experience
biased cohort sex ratios. The predictions are supported by data from painted turtles, where TSD reaction norms exhibit pivotal
temperatures of sex determination that partially track mean annual temperature. Examining relativity in amniotic vertebrates
provides a conceptual framework for multifactorial sex determination and suggests new ways of exploring adaptive hypotheses
of sex allocation by focusing on the importance of frequency-dependent selection on sex.
KEY WORDS:
Charnov–Bull, climate change, information theory, mutual information, Trivers–Willard.
Sex allocation theory has provided a remarkably successful framework for predicting investment in offspring sex or in egg versus
sperm for invertebrates and plants (Charnov 1982; Wrensch and
Ebbert 1993; Hardy 2002), yet sex ratio biases in amniote vertebrates seem unsystematic and often are explained a posteriori.
The frequent mismatch between theory and empirical data in amniotes raises doubts as to whether adaptive sex allocation occurs
in these taxa and highlights the need for expanded study focusing
on long-lived organisms.
Many hypotheses of sex allocation for amniote vertebrates
are based on condition dependence of sex allocation (Charnov
3Current
address: School of Marine and Tropical Biology, James
Cook University, Townsville, Queensland 4811 Australia.
C
1331
1982; Cockburn et al. 2002; West 2009). An individual invests
in sons versus daughters or eggs versus sperm depending on individual condition, such as resource availability, body size, or
incubation temperature. That a sex-specific fitness advantage of
body size might select for size-dependent sex allocation was first
proposed for sex-changing invertebrates (Ghiselin 1969). Subsequently, the Trivers–Willard hypothesis (TW; Trivers and Willard
1973) proposed that offspring sex in mammals may be linked
to maternal condition if condition influences the lifetime fitness of one offspring sex more than the other. Charnov and Bull
(1977) (see also Bull 1981) expanded condition-dependent sex
allocation to environmental sex determination, providing an adaptive hypothesis for the occurrence of temperature-dependent sex
determination (TSD) that proposes developmental temperature is
C 2010 The Society for the Study of Evolution.
2010 The Author(s). Journal compilation Evolution 64-5: 1331–1345
L I S A E . S C H WA N Z E T A L .
the fitness-altering condition that determines investment in ovaries
or testes (Gutzke and Crews 1988; Janzen 1995; Shine 1999;
Warner and Shine 2008). Condition dependence in sex allocation
has been expanded to sex biases based on individual size, paternal quality, parental rank, hatching order, and season (Werren and
Charnov 1978; Charnov 1979a; Burley 1981; Silk 1983; Bednarz
and Hayden 1991; Daan et al. 1996).
According to sex allocation theory, sex allocation should be
based on individual condition relative to the condition of competitors, not on absolute condition (Charnov et al. 1978, 1981;
Charnov 1982; West 2009). The TW hypothesis predicts that
mothers of relatively good condition should invest more in sons
than mothers of relatively poor condition (Trivers and Willard
1973). This is because each sex competes for lifetime reproductive success only with members of the same sex, so that the
quality of a son or daughter is defined only with respect to their
sexual competitors (other sons and daughters in the population).
If sex allocation were based on absolute condition, a year with
extreme condition values would vastly overproduce one sex, and
frequency-dependent selection on sex would favor a mutant that
produced the rare sex. The resulting prediction is that the optimal sex allocation versus condition reaction norm should be
relative, shifting in response to changes in the population distribution of condition, and that an equilibrium cohort sex ratio
would be maintained. When individuals are not allowed to define
their condition relatively, substantial fluctuation in condition distributions and, therefore, cohort sex ratio is predicted to lead to
the loss of condition-dependent sex allocation (Bull and Bulmer
1989; van Dooren and Leimar 2003; Leimar et al. 2004; Schwanz
and Proulx 2008).
Charnov et al. (1978) provided the first empirical demonstration of relativity in condition-dependent sex allocation (i.e.,
dependence of sex allocation on relative condition rather than absolute condition) with the timing of sex change in a protandrous
shrimp. Sex change provides a simple framework for theory and
has yielded some of the best empirical examples of relativity in
condition-dependent sex allocation (Warner et al. 1975; Leigh
et al. 1976; Charnov 1979b, 1982; Charnov and Bull 1989a;
Charnov and Skuladottir 2000; Kuwamura et al. 2002; Chen
et al. 2004). The importance of relative condition for sex allocation has also been addressed for egg and sperm production
in simultaneous hermaphrodites (St. Mary 1994; DeWitt 1996;
Angeloni and Bradbury 1999; Ohbayashi-Hodoki et al. 2004), for
sex determination in organisms with environmental sex determination (Blackmore and Charnov 1989; Conover et al. 1992), and
for parasitoid wasps expressing sex allocation in a TW fashion
(Charnov et al. 1981; Ode and Heinz 2002; West 2009). These
examples of relativity in invertebrates and fish reveal the signature of frequency-dependent selection on sex and, thus, support
the adaptive nature of sex allocation in these taxa.
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Do amniote vertebrates show similar reliance on relative condition for sex allocation? Confirmation would provide compelling
support for sex ratio patterns as adaptive strategies by demonstrating the central role of frequency-dependent selection on sex. In
predicting the occurrence of relativity in condition-dependent sex
allocation, three assumptions are implicit. First, organisms must
experience temporal or spatial fluctuations in condition distributions. If the environment is invariant, it is unlikely that a relative
strategy would arise, and one condition-dependent strategy based
on absolute condition may be adaptive (Charnov et al. 1981).
Second, a physiological mechanism must be possible for translating information about the environment into sex allocation. Third,
knowledge (even if indirect) must exist of the condition distribution of competitors. In gonochoristic organisms, where generations overlap, the condition distribution of competitors may not
be known if individuals are incapable of sampling competitors
of the same cohort, or if competitors derive from past cohorts of
different condition distributions or from future cohorts of unpredictable distributions. In this case, an optimal “average” strategy
based on the normal range of distributions may arise. For many
amniote vertebrates then, relativity in sex allocation decisions
may be limited by overlapping generations (overlap of different
cohort condition distributions in competitors), physiological limitations of sex ratio manipulation, and lack of information (West
and Sheldon 2002).
Previous theoretical work shows that overlapping generations
per se do not change how offspring fitness is measured to predict
sex ratios (Schwanz et al. 2006). Still, no explicit theoretical
examination of the importance of relative versus absolute condition for sex allocation has been conducted (West 2009). Here,
we model sex allocation when condition-dependent strategies are
allowed to vary from “completely absolute” to “completely relative.” We substantially expand a simulation model (Schwanz and
Proulx 2008) to explore how life span and environmental fluctuation influence the evolution of relativity in reaction norms of
TSD. Finally, we demonstrate how relativity can be examined empirically by utilizing data from a long-lived organism with TSD,
the painted turtle. With this focal species we test the falsifiable
prediction from our model that partial relativity should operate
even in long-lived species.
Model Methods
We examine the evolution of a sex versus temperature reaction
norm in a simulated population (using MATLAB 7.1, MathWorks,
Natick, MA). Each generation of the simulation, an annual mean
temperature for the population, T ann , is chosen from a global tem2
perature distribution with mean T glob and variance σglob
. Offspring
in that generation experience a clutch-specific developmental temperature, t, chosen from the annual distribution with mean T ann
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
2
and variance σann
. The amount of environmental fluctuation is
described by the ratio of the interannual (global) variation to the
2
2
intraannual variation, σglob
/σann
.
Each individual in the population has a sex versus temperature reaction norm given by an equation for a sigmoidal curve
(Girondot 1999; van Dooren and Leimar 2003)
Pr(male) = g1 +
(g2 − g1 )
1+e
−(t−(g3 +g5 (Tann −Tglob )))
[
]
g4
.
(1)
In this equation, g 1 is the probability of being male at extremely low temperatures, g 2 is the probability of being male at
extremely high temperatures, g 3 is the inflection point of the curve
(the pivotal temperature, T piv ), g 4 is the inverse of the “slope” of
the curve (essentially no TSD occurs with g 4 > 10; Schwanz and
Proulx 2008), and g 5 is the component that represents the degree
of relativity and multiplies the annual “shift” in temperature (the
difference between that year’s mean temperature and the global
mean temperature, T ann − T glob ). When g 5 = 1, the pivotal temperature (g 3 ) moves laterally in equal magnitude to the departure
of the annual mean from the global mean, and the strategy defines
developmental temperature completely relatively. When g 5 = 0,
the pivotal temperature is the same every year, regardless of the
annual mean temperature, and the strategy defines developmental
temperature completely absolutely. For 0 < g 5 < 1, the pivotal
temperature partially tracks annual fluctuations in mean temperature. g 5 > 1 may evolve and represents an overresponse to shifting
annual mean temperatures. The evolution of this functional form
of condition-dependent sex allocation requires individual knowledge of the mean annual condition and a mechanism to vary the
pivotal condition (i.e., to have g 5 > 0). For TSD, we propose that
breeding females have knowledge of the mean annual temperature
and can vary nesting or egg traits to alter the pivotal temperature
of their embryos (see Empirical Test).
Schwanz and Proulx (2008) provide detailed description of
the simulation model. Population size remains constant at N =
1000, with density-dependent survival of juveniles. Each clutch
per generation is assigned a developmental temperature given
the chosen annual mean temperature, and individual sex is determined by developmental temperature and individual reaction
norm. Juvenile survival, adult survival, and adult reproductive
success (fecundity or fertility) may be influenced by incubation temperature (t). Suppose that male and female juvenile
survival to adulthood, S m and S f , can be described by sigmoid
curves,
Sm = Sm min +
(Sm max − Sm min )
1 + e(−(t−Tglob )βm )
S f = S f min +
(S f max − S f min )
,
1 + e(−(t−Tglob )β f )
and
because they must be bounded by zero and one. Survival to firstyear adults is determined by sampling individuals from clutches
based on relative juvenile survival. In the simulations presented
here, male and female adult survival, p m and p f , and reproductive
success, F m and F f are maintained at specified values.
Mutation of all reaction norms components (g 1 , g 2 , g 3 , g 4 ,
and g 5 ) occurs prior to sex determination for 2% of offspring. For
these mutants, change in the value of g 1 , g 2 , g 3 , and g 5 is chosen
randomly from a normal distribution with mean 0 and variance
0.01 (g 1 , g 2 , and g 5 ) or 0.02 (g 3 ). Change in the value of g 4
is chosen from a uniform distribution (range: 0–5) and randomly
subtracted or added (Schwanz and Proulx 2008). g 5 is constrained
to be greater than zero, and mutations that lead to values less than
zero are assigned g 5 = 0.
We explored evolution of the TSD reaction norm under
four values of annual adult survival, p (0, 0.5, 0.75, 0.95),
2
2
/σann
= 0.5,
and three environmental fluctuation regimes (σglob
2
2
2, 3; σglob + σann = 1.5). The results presented here are for simulations where we provide selection for TSD by specifying a
sex-differential fitness advantage of incubation temperature in
juvenile survival only—females but not males gain in juvenile
survival as t increases (β f = 0.5, β m = 0). At the start of each
simulation, the population does not demonstrate TSD or relativity
and is homogenous for genotypic components (g 1 = 0.5, g 2 =
0.5, g 3 = 22, g 4 = 1, g 5 = 0). All simulations were run for
10,000 iterations. For some of the p = 0.95 simulations, directional movement in the reaction norm components was observed
near the 10,000th iteration. For this parameter value, all simulations were continued an additional 10,000 iterations. The effect
of the additional time was minimal but discernable, so results for
p = 0.95 after 20,000 iterations are presented here. At the end of
each simulation, the mean values for the genotypic components
in the population were calculated. Additionally, we followed the
mean and variance of incubation temperatures seen in the pool of
2
)
adults following new recruitment each year (T Adults and σann,Ad
for 1000 additional generations following the completion of each
2
2
/σann
= 2. We then calculated the variance in mean
run where σglob
adult temperature among years (σ2glob,Ad ). Ten replicate runs were
completed for each parameter value set, and the means of these
runs are presented.
To describe the degree of interdependence of sex and temperature in a given year, we used an index of the information shared
between sex and the annual temperature distribution, known as
annual mutual information (AMI). This measure provides a single
index of TSD that incorporates the entire shape of the reaction
norm (g 1 , g 2 , g 3 , g 4 , and g 5 ) as well as the temperature distribution
(Schwanz and Proulx 2008). Due to its reliance on natural temperature distributions, AMI is ecologically relevant. In addition, AMI
evolves as the reaction norm evolves, describing in a single measure the relationship between sex and temperature. AMI is highest
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L I S A E . S C H WA N Z E T A L .
when the reaction norm is switchlike and the pivotal temperature is
near T ann . To compute AMI, we modified calculations for mutual
information from Schwanz and Proulx (2008). Whereas our previous calculation of mutual information was based on the global
2
2
+ σann
)
temperature distribution (mean T glob and variance σglob
and is a universal measure across environments, AMI is based
on an annual temperature distribution (mean T ann and variance
2
σann
) and provides an environment-specific measure of mutual
information (Appendix A). This distinction allows us to explore
how relativity influences the relationship between temperature
and sex in different environments. The influence of g 4 (slope of
2
(annual variance in develreaction norm), g 5 (relativity), σann
opmental temperatures), and T ann (mean annual temperature) on
AMI is illustrated in Figure S1.
Model Results
RELATIVITY
For all parameter space examined, individuals showed some degree of relativity (mean g 5 > 0; Fig. 1). In semelparous populations (p = 0), individuals had a completely relative strategy, almost
perfectly tracking the mean annual temperature (T ann ) with their
T piv (g 3 : Fig. 2). The degree to which individuals defined their
incubation temperature relative to the annual mean diminished
as average life span increased (Fig. 1), although even long-lived
organisms (p = 0.95) demonstrated a small degree of relativity
(Fig. 2). The degree of environmental fluctuation had little effect
on the degree of relativity (Fig. 1).
In our simulations, the mean incubation temperature seen
in the pool of adults (T Adults ) fluctuated annually, and the degree
of fluctuation among years depended on life span (Fig. 2). In
semelparous organisms, fluctuations in T Adults matched fluctua2
= 1, σ2glob,Ad = 1.08), whereas the degree of
tions in T ann (σglob
fluctuation in T Adults in populations with p = 0.5, 0.75, and 0.95
was a fraction of fluctuations in T ann (σ2glob,Ad = 0.36, 0.15, and
0.03, respectively), and less than the evolved degree of relativity.
Because individuals compete for lifetime fitness with the pool of
adults they encounter over their life span rather than with just their
cohort, the degree of relativity that evolves is likely responding
to fluctuations in T Adults rather than T ann . Analytical explorations
confirm that the optimal degree of relativity should decrease as
life span increases (Appendix B). Individuals face selection to
produce the rare sex and selection to maximize survival given the
temperature effects on female survival. Maximizing the geometric
mean of the product of male and female offspring production is
equivalent to finding a local evolutionarily stable strategy (ESS)
(Charnov 1986, 1988; Tuljapurkar 1990). Without relativity, this
leads to a trade-off between achieving “optimal” allocation at
different mean annual temperatures and a shallow reaction norm
for TSD. With relativity, this leads to a sharp reaction norm for
TSD that tracks mean annual temperature. In contrast, very longlived organisms spread their reproduction over a large sample
of years and compete in mating pools over a large sample of
years. These factors combine to cause the operational sex ratio
(OSR) to be stable and to cause the offspring sex ratios of individual parents to be averaged over the exact distribution of yearly
temperatures. This means that individual parents benefit by increasing the product of male and female offspring production
over their life span, which can be accomplished by using a fixed
reaction norm for TSD. Thus, for species with effectively infinite
life span, no relativity is the unconstrained optimal solution for
TSD.
Based on invasion analyses (Appendix S1), the degree of relativity that evolved in our simulated populations of semelparous
( p = 0) and short-lived ( p = 0.5) individuals had a strong selective advantage over an absolute strategy, whereas the relative
strategies that evolved in populations of moderate- and long-lived
individuals (p = 0.75, 0.95) appeared to be selectively neutral.
This was reflected by the tendency over time of mean g 5 to increase and then be maintained at constant levels in short-lived organisms but to vary over time for long-lived organisms (Fig. S4).
However, relativity in long-lived organisms did not appear to be
solely an artifact of the high mutation rate specified in the simulation. Across adults, g 5 largely displayed a normal distribution,
even for long-lived organisms (Fig. S5), suggesting that stabilizing selection may be operating. In addition, when mutation was
removed from the simulation and evolution continued, g 5 > 0 was
most often maintained in all replicates (Fig. S6).
SHAPE OF THE REACTION NORM
Evolved relativity (g 5 ) as a function of annual adult
survival and environmental fluctuation.
Figure 1.
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AMI when T ann = T glob provides a measure of how switchlike
the reaction norm is; that is, it describes the relationship between sex and temperature in an average year (Fig. 3A,B). When
the reaction norm could not be defined by relative temperature
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
Figure 2.
The mean evolved temperature-dependent reaction norms (columns one and two) and annual temperature fluctuations from
representative simulations (columns three and four) for different adult annual survivals: p = 0 (first row), p = 0.5 (second row), p = 0.75
(third row), and p = 0.95 (fourth row). The first two columns show the probability of developing into a male as a function of temperature
when the evolution of relativity is constrained (g 5 = 0; first column) or unconstrained (second column), for three different years (fifth
row; T ann = T glob [22; solid line], T glob + 1 [dashed line], and T glob + 2 [dotted line]). Panels in the first column show the single reaction
norm (solid line) that is expressed in every year. Panels in the second column show three different reaction norms expressed depending
on T ann (T glob = solid line; T glob + 1 = dashed line; T glob + 2 = dotted line). Columns one and two show mean genotypes from 10 replicate
runs. The third column shows deviation from T glob in mean T Adults (black lines) and mean T ann (gray lines) for 1000 generations following
2
the end of four representative runs. The mean σann,Ad
of 10 replicated runs is shown. From the same 1000-generation simulations, the
change between two consecutive generations in T Adults is plotted against the change in T ann , and the mean slope of the relationship
2
2
/σann
= 2.
from ten replicate runs is shown in the fourth column. For all, σglob
(g 5 constrained at 0), AMI at T ann = T glob was high for longerlived organisms but was low in short-lived organisms (Fig. 3A),
indicating that TSD was lost in semelparous organisms and in
short-lived organisms in highly fluctuating environments. When
the reaction norm was allowed to be defined by relative temperature (g 5 unconstrained), a high and fairly uniform AMI in the
average environment (more switchlike reaction norm) evolved
across parameter space (Fig. 3B), indicating the maintenance of
similar a shape of the TSD reaction norm regardless of life span
or environmental fluctuation.
AMI ACROSS ENVIRONMENTS
Examining how AMI changes as T ann moves away from T glob
provides a measure of relativity. If reaction norms are perfectly
relative, then AMI does not decline across different years. When
reaction norms were constrained to be based only on absolute
condition (g 5 constrained at 0), AMI was higher for longerlived than shorter-lived organisms across different years (i.e.,
T ann = T glob , T glob + 1 or T glob + 2), except in a +2 year,
where AMI was universally low (Figs. 3C,E,G). AMI was low
for shorter-lived organisms across years because the evolved
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L I S A E . S C H WA N Z E T A L .
Annual mutual information (AMI) for TSD reaction norms constrained and unconstrained in the evolution of relativity. (A,
B) AMI across adult survival and environmental fluctuations for constrained (g 5 constrained at 0; A) and unconstrained (B) simulations;
T ann = T glob . (C–H) AMI across years (T ann = T glob , T glob + 1, T glob + 2) for different adult survival values, shown for constrained (C, E, G)
Figure 3.
2
2
2
and unconstrained (D, F, H) simulations and for σann
= 1 (C, D), σann
= 0.5 (E, F), and σann
= 0.375 (G, H). For C–H, line dashing corresponds
to annual adult survival as shown in legend.
reaction norm was shallow (Fig. 2), whereas it was low for longerlived organisms in a +2 year despite switchlike reaction norms
because T piv was far from T ann , producing a biased sex ratio
(Fig. 2).
In contrast, when condition could be defined relative to T ann ,
AMI in extreme environments was higher for shorter-lived organisms than for longer-lived organisms (Fig. 3D, F, H). Shorter-lived
organisms maintained high AMI across years because they had
switchlike reaction norms and tracked T ann strongly (high g 5 ;
Fig. 2). Longer-lived organisms had reduced AMI in extreme
years despite having switchlike reaction norms because they had
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EVOLUTION MAY 2010
only a weakly relative strategy (low g 5 ; Fig. 2), thus T piv was far
from T ann and extreme sex ratios would result.
Empirical Test
Our model suggests that organisms should allocate to sex based
on their condition relative to the annual mean condition rather
than on their absolute condition. Empirical examination of sex allocation according to relative condition has rarely been presented
for amniote vertebrates. Here, we examine the role of relative condition for a moderately long-lived turtle with TSD. This serves to
illustrate the application of a “relativity” approach to empirical
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
data and to test the model prediction that long-lived organisms
should exhibit partial relativity.
If individuals in populations allocate to sex based on relative
condition rather than on absolute condition, then sex versus condition curves for the population should shift laterally among years
dependent on the annual condition distribution, and the intercept
or inflection point of such curves should be related positively to
mean condition, as seen in our simulated populations (Fig. S7).
Under complete relativity, T piv and the annual mean condition correspond 1:1, and the offspring cohort sex ratio does not change
across years. When individuals track the environment incompletely, as predicted for long-lived organisms, the inflection point
and annual mean condition exhibit a shallower correspondence.
Detecting relative strategies in populations with partial relativity
may be more difficult because of this shallower relationship.
In painted turtles (Chrysemys picta), embryos develop as
males under cold incubation temperatures and as females under
warm temperatures (Schwarzkopf and Brooks 1985; Weisrock and
Janzen 1999; L. E. Schwanz, R.-J. Spencer, R. M. Bowden, and F.
J. Janzen, unpubl. ms). The data analyzed here were collected from
a population that nests on an island in the Mississippi River near
Thomson, IL. This population has been studied for 20 years, and
details on its nesting ecology and the field research methods can
be found elsewhere (Janzen 1994a,b; Morjan and Janzen 2003;
Schwanz et al. 2009). Annual adult survival is estimated as 0.83
(R. J. Spencer and F. J. Janzen, unpubl. data). For most nests laid
each year, data are available on nest vegetation cover and nest sex
ratio (for surviving nests). For a subset of nests since 1995, temperature profiles of nests are known. Most embryos experience their
temperature-sensitive phase of sex determination in July, and July
nest temperatures are a good predictor of nest sex ratio (Weisrock
and Janzen 1999; L. E. Schwanz, R.-J. Spencer, R. M. Bowden,
and F. J. Janzen, unpubl. ms). Nest temperatures in July depend
on July air temperature and vegetation cover (Morjan and Janzen
2003). Therefore, annual fluctuation in climate alters the annual
mean of nest temperatures in the population and variation in vegetation cover provides much of the intraannual variance in mean
2
2
/σann
= 1.94). Cohort sex ratios fluctuate
nest temperatures (σglob
among years (0%–100% male hatchlings) and correlate strongly
with mean July air temperature (Janzen 1994a; L. E. Schwanz,
R.-J. Spencer, R. M. Bowden, and F. J. Janzen, unpubl. ms).
We tested whether partial relativity in sex allocation exists
in this population, as predicted by our model. For each year between 1995 and 2006, we plotted nest sex ratio versus mean nest
temperature in July and established sigmoidal curves to estimate
inflection points for each year (Annual T piv ). Annual T piv is the
population-based pivotal temperature that represents the mean
of individual T piv . Sigmoidal curves could not be established in
2004–2006 because of low nest survival in 1 year and mostly male
hatchlings being produced in the other years.
Figure 4. Annual T piv is positively correlated with the mean of
the July nest temperatures in a year, indicating that developmental
temperature is defined at least partially relative to the population
mean.
Two measures of “annual mean temperature” were used as
predictors of Annual T piv . The first was mean July air temperature. The second was the mean of all nest temperatures laid at
the nesting beach. To determine the annual distribution of nest
temperatures, we used vegetation cover and July climate to estimate mean July temperature for each nest that was not directly
recorded (Morjan and Janzen 2003).
Annual T piv was correlated positively with the mean of the
nest temperatures in July (Fig. 4, r2 = 0.62, P = 0.011, slope =
0.47 ± 0.14; estimate ± SE). Because the mean of nest temperatures is correlated with mean July air temperature (r2 = 0.82,
P = 0.0008), Annual T piv was also weakly correlated with annual
climate (r2 = 0.337, P = 0.1012, slope = 0.37). Thus, a nest with
an intermediate temperature (say, 26◦ C) produces mostly females
in a cold year when it is a relatively warm nest (e.g., annual mean
nest temperature of 24◦ C and Annual T piv of 25.5◦ C). In contrast,
a nest of the same temperature produces mostly males in a warm
year when it is a relatively cold nest (e.g., annual mean nest temperature of 27◦ C and Annual T piv of 27◦ C). Intriguingly, Annual
T piv matches T ann at high T ann , but lies above T ann at low T ann .
This pattern is consistent with that predicted in our analytical
explorations when organisms are allowed to vary their degree of
relativity (g 5 ) across T ann rather than having a constant g 5 as in
our simulations.
In accordance with our prediction, this population of painted
turtles appears to exhibit relativity in its condition-dependent sex
allocation, with Annual T piv shifting in response to changes in
climate and the distribution of nest temperatures in the population.
Because the slope between Annual T piv and the annual mean nest
temperature is substantially less than one, this finding represents
only partial relativity. The overall effect for the cohort sex ratio
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L I S A E . S C H WA N Z E T A L .
is to reduce, but not eliminate, sex ratio biases elicited by climate
fluctuations.
MECHANISTIC HYPOTHESES
What could be the mechanisms for relativity? Two mechanistic
components may be involved—the physiological means of altering the “pivotal condition” and the translation of environmental cues to physiology. We suspect mechanisms will be highly
species-specific, so here we discuss potential mechanisms for the
apparent relativity observed in painted turtles.
First, climate could influence sex hormones in egg yolks,
which appear to affect sex determination in painted turtles, as
they are correlated with T piv (Bowden et al. 2000). A decrease in
yolk estradiol:testosterone (E:T) in warm years could shift T piv to
warmer temperatures (i.e., a greater likelihood of developing as
male at a given temperature). Across reptiles with TSD, sex determination is strongly influenced by exogenously applied steroid
hormones, whereas the influence of natural maternal yolk steroids
remains debated (Elf 2003; Radder 2007). This physiological
mechanism in painted turtles could be linked to the environment
via correlations between winter and summer climate. Females deposit yolk in eggs between September and April (Congdon and
Tinkle 1982), and warmer winters (September–April) are positively associated with warmer Julys near the field site (Schwanz
and Janzen 2008). It is plausible that warmer winters are associated with differences in food resources, social interactions, or
physiologies that lead to altered circulating hormones in females,
thus altering hormonal deposition in yolk (Bowden et al. 2000;
Janzen et al. 2002; Bowden et al. 2004). Relationships between
annual climate and yolk hormones have not been investigated.
Alternatively, sex determination may depend on aspects of
temperature that are more integrative (e.g., constant temperature
equivalents; Georges et al. 2004) rather than on mean temperature. For example, if July climate influences the amplitude of the
nest temperature profile, then an average nest in a cold year (e.g.,
24.5◦ C) and an average nest in a warm year (e.g., 26.5◦ C) may
have the same constant temperature equivalent (temperature above
which half of all development occurs), and the same sex probability. Indeed, the underlying thermal dependence of developmental
rates may provide a “built-in” mechanism of relativity that simultaneously contains information about the annual mean (if nest
fluctuations are correlated with annual climate) and the physiology to connect information with phenotype. Testing this hypothesis and its implication for optimal cue use requires analyses
beyond the scope of this article but is a fruitful avenue to pursue.
Discussion
Sex allocation models often describe simple life histories where
individual condition is defined relative to the population distri-
1338
EVOLUTION MAY 2010
bution. The importance of assessing relative condition during sex
allocation has been demonstrated empirically for many invertebrates and fish where condition is compared to a short-term
distribution (e.g., Charnov et al. 1978, 1981; West 2009). Recent
sex allocation models have incorporated the complexities of overlapping generations but have constrained individuals to define
condition according to a global mean condition distribution (Bull
and Bulmer 1989; van Dooren and Leimar 2003; Leimar et al.
2004; Schwanz and Proulx 2008).
In this article, we examined the importance of defining condition based on absolute versus relative terms for the sex allocation
of organisms across different life spans and in environments of
different degrees of annual fluctuation. We found that when relativity is allowed to evolve freely, the degree to which condition is
defined relative to the annual mean condition depends strongly on
the life span of the organism. Semelparous organisms show complete relativity, perfectly tracking changes in mean annual condition. The degree of relativity decreases as life span increases until
organisms with high annual adult survival exhibit only a modest
degree of relativity that appears effectively selectively neutral.
RELATIVITY, EVOLUTION OF TSD, AND
ENVIRONMENTAL CHANGE
The increased importance of relativity in condition for shorterlived organisms is evidenced by the evolution of temperaturedependent reaction norms of sex determination. When organisms
are not allowed to define their condition relative to the annual
mean condition, fluctuating environments lead to extreme biases
in cohort sex ratios (when switchlike TSD reaction norms show
low AMI in extreme environments). The cost of biased cohort
sex ratios is high in short-lived organisms because few cohorts
overlap to reproduce, leading to strong selection for production
of the rare sex each year. This process leads to the evolutionary
loss of TSD (Bull and Bulmer 1989; van Dooren and Leimer
2003; Schwanz and Proulx 2008). For these organisms, the ability
to define condition relative to the annual mean provides a large
fitness advantage in the face of frequency-dependent selection
on sex under condition-dependent sex allocation, and leads to
the persistence of condition-dependent sex allocation (switchlike
TSD reaction norms).
For longer-lived organisms, the cost of fluctuating cohort sex
ratios in fluctuating environments appears to be low due to the
overlap of generations. Thus, when condition must be defined absolutely, switchlike reaction norms are maintained despite biased
cohort sex ratios (low AMI) in extreme years. In addition, there is
a comparatively smaller advantage to defining fitness relative to
the annual mean condition and relativity evolves only to a minor
degree.
Reduced relativity in long-lived organisms is likely due
to two components. First, there is reduced selection to track
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
cohort fluctuations. Individuals experience reduced frequencydependent selection on sex in a given year because overlapping
generations reduce biases in adult sex ratios. Longer-lived organisms compete for lifetime fitness across more cohorts, hence
more condition distributions, and adult condition distributions do
not fluctuate as widely as cohort distributions. That is, longerlived organisms experience a broader competitor distribution that
incorporates the multiple condition distributions seen within the
lifetime of an individual, which effectively lowers the perception of environmental fluctuation. Second, temperature effects
on fitness (here, female juvenile survival) are based on absolute
temperature and lead to selection to optimize female survival. In
longer-lived organisms, the reduced need to produce the rare sex
allows organisms to determine sex in a manner that more closely
maximizes survival gains. That the evolved degree of relativity
in our simulation exceeds the degree of fluctuation in adult distributions may be due to stronger selection for relativity near the
global mean temperature caused by the sigmoidal shape of the
female fitness curve.
We found support for one prediction of our model, namely
that long-lived organisms would show partial relativity, by examining patterns of TSD in a population of painted turtles across
years over which mean temperatures fluctuated. Despite a low
likelihood that developing embryos have direct knowledge of the
annual mean temperature distribution, we found that the pivotal
temperature of the sex versus temperature reaction norm shifted
among years. In comparatively warm years, a warmer July nest
temperature is required to produce a 50:50 sex ratio compared
to cool years. The mechanism underpinning this annual shift is
unknown, but may include responses of maternal physiology or
behavior to climate or a more complex nature of sex determination
that relies on integrating incubation temperature rather than simple averaging. Interestingly, the empirical pattern also suggests
that painted turtles may vary their degree of relativity depending on the mean annual temperature, a possibility that was not
explored with our simulations.
A strong test of the theory would be a comparison of relativity
across species with different life spans. The data are not available
currently to address this question in organisms exhibiting TSD,
but will hopefully become available in the future. For example,
a suitable intraspecific comparison to the Illinois painted turtles
could be provided by a population of painted turtles in Nebraska
that has greater annual adult survivorship than the Illinois population (Iverson and Smith 1993). Interspecific comparisons may
be provided by Blanding’s turtles (Emydoidea blandingii, Emydidae; Congdon et al. 1993) or the more distantly related pig-nosed
turtle (Carettochelys insculpta, Carettochelidae; Georges 1992).
The evolution of relativity in condition-dependent sex allocation suggests that populations should be impervious to directional
environmental change because extreme sex ratios are prevented. If
sex allocation strategies are not based on relative condition, directional change in environment or condition induced by unnatural
situations (e.g., climatic warming, habitat degradation, supplemental feeding) may yield consistently biased sex ratios and decreased population viability. We show that long-lived organisms
may be more vulnerable to directional environmental change. For
these organisms, relativity is predicted to be only partial, to be
relatively neutral, and to fail to prevent extreme cohort sex ratios
in environmentally extreme years. These predictions match our
findings for our painted turtle population (Janzen 1994a; L. E.
Schwanz, R.-J. Spencer, R. M. Bowden, and F. J. Janzen, unpubl.
ms). Thus, the consequences for demography and implications
for conservation differ substantially between relative and absolute reaction norms.
RELATIVITY AND ADAPTIVE SEX ALLOCATION
IN AMNIOTES
The prediction that all organisms, even long-lived ones, should
depend on relative condition at least partially during sex allocation
suggests new approaches to examine sex allocation in the confusing sex ratios of amniote vertebrates. For the study of TSD, this
approach provides a conceptual framework for the emerging data
supporting a multifactorial nature of sex determination (Elf 2003;
Radder et al. 2009). In many species with TSD or temperaturesensitive sex reversal, factors other than temperature, such as
steroid hormones, egg size, and maternal diet, are increasingly
shown to contribute to sexual differentiation (Bowden et al. 2000;
Warner et al. 2007, 2009; Radder et al. 2009). Multiple mechanistic underpinnings may be explained at an ultimate level by
the fitness advantages of defining condition relatively. That is,
reproductive females and developing embryos are predicted to incorporate cues other than absolute incubation temperature when
sex is being determined, so multifactorial sex determination may
be expected.
To examine the importance of relativity in empirical data
from any system of condition-dependent sex allocation (e.g., TW
Hypothesis), sex allocation relative to local-condition distributions may be examined across populations that differ in condition
distributions (all else being equal) or within a population where
shifts in condition distribution occur over time. Where sex allocation decisions occur repeatedly within the lifetime of a single individual, shifts over time in the relative condition of an
individual should lead to sex allocation shifts, even if absolute
condition remains the same (i.e., plasticity in sex allocation). Individual plasticity has provided some of the strongest evidence
for adaptive sex allocation in vertebrates (e.g., Komdeur et al.
1997). In particular, a single physiological condition could lead
to different sex ratios across years if the distribution of physiologies shifts and changes relative condition. This finding refutes the
main nonadaptive hypothesis for sex ratio biases that contends
EVOLUTION MAY 2010
1339
L I S A E . S C H WA N Z E T A L .
female-biased sex ratios produced by mothers of poor condition
may be a physiological side-effect of the vulnerability of sons
to poor developmental conditions (i.e., one sex requires more energy during development; Clutton-Brock et al. 1985). Because our
model assumes that “condition” is randomly distributed (similar
to Charnov 1982; Pen and Weissing 2002; Schwanz et al. 2006;
Wild and West 2007), some caution should be exercised when
applying the predictions to those birds and mammals in which
condition is inherited or in which relative condition is maintained
among breeding individuals across years (Leimar 1996; Wild and
West 2007).
The study of relative condition has rarely been approached directly in birds and mammals (e.g., Blanchard et al. 2005), yet relativity has implications for the structure of empirical data. Analyses
that clump data across diverse years could obscure annual patterns
of sex allocation. In addition, the degree of relativity expressed by
individuals will determine whether the primary sex ratio for the
population reflects a stable evolutionary equilibrium (Frank and
Swingland 1988; Charnov and Bull 1989a) or fluctuates around
a putative equilibrium. Researchers have previously examined
relative condition indirectly by using maternal rank as the condition variable, which is by definition relative in the population.
Maternal behavioral dominance has been linked to sex biases in
many mammals (Clutton-Brock et al. 1986; Hiraiwa-Hasegawa
1993; Hewison and Gaillard 1999) and is a better predictor of
offspring sex than maternal morphological condition across ungulates (Sheldon and West 2004). Changes in maternal glucose
and body condition are better predictors than absolute values in
several mammals (Fisher 1999; Cameron and Linklater 2007;
Cameron et al. 2008), suggesting that offspring sex is linked to
relative changes in condition rather than absolute condition. Thus,
previous work in birds and mammals and our present work in reptiles suggest that organisms may partially define their condition
relatively when making sex allocation decisions. Although the
best test for adaptive hypotheses of sex allocation examines the
underlying fitness structure, demonstrating relativity would provide strong support for adaptive hypotheses of sex ratio biases in
two ways. First, it provides a signature for frequency-dependent
selection on sex, which is the evolutionary mechanism of adaptive
sex allocation theory. Second, it refutes the main nonadaptive hypothesis for sex ratio biases. Thus, directly examining relativity
may be a fruitful avenue of research for exploring the adaptive
value of offspring sex ratios.
ACKNOWLEDGMENTS
We thank O. Bochmann, J. Bragg, H. Hua, and K. Roh for discussion on
the simulation model. We thank the many members of the Janzen Lab
who have contributed to collection of the field data. E. Charnov provided
valuable discussion during development of the ideas in this article. Earlier
versions of the manuscript were improved by comments from J. Brown, E.
Charnov, A. Kodric-Brown, J. Millar, R. Moses, I. Pen, and several anony-
1340
EVOLUTION MAY 2010
mous reviewers. Access to Thomson Causeway was provided by the U.S.
Army Corps and Engineers and collecting permits were obtained from
the U.S. Fish and Wildlife Service and the Illinois Department of Natural
Resources. Support for the field work was provided by National Science
Foundation grants (DDIG BSR-8914686, DEB-9629529, UMEB IBN0080194, LTREB DEB-0089680, IBN-0212935), as well as the ASIH
Gaige Fund, Sigma Xi, and the Department of Zoology and Genetics
at Iowa State University. While conducting the research and writing the
article, LES was supported by a NSF Graduate Research Fellowship and
a NSF Postdoctoral Fellowship in Biological Informatics.
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Associate Editor: A. Badyaev
Appendix A
CALCULATION OF ANNUAL MUTUAL INFORMATION
Annual mutual information, AMI(x), is a measure of the mutual
information between sex and temperature present in a single year.
Thus, AMI(x) is a function of the average nest temperature in a
specific year (x = T ann ) and depends on the level of environmental
2
variance present in the simulations (σann
). This measure is both
an indicator of how effective the reaction norm is in transferring
information from temperature to sex, and how well the relative
nature of the strategy copes with between year variability. AMI(x)
is the difference of the joint entropy between annual temperature
and sex (HTS) and the sum of the entropy associated with sex
(HS) and the entropy associated with annual temperature (HT).
Note that both the distribution of temperatures and the realized
sex of offspring implicitly depend on the mean nest temperature
within the year (x). Specifically,
AMI(x) = HTS − (HT + HS).
The entropies are given by
HT = Pr(t) ln(Pr(t)) dt,
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EVOLUTION MAY 2010
(A1)
(A2)
Pr(male | t) Pr(t) dt × ln
Pr(male | t) Pr(t) dt
+ Pr(female | t) Pr(t) dt × ln
Pr(female | t) Pr(t) dt
HS =
(A3)
and
HTS =
+
Pr(male | t) Pr(t) ln(Pr(male | t) Pr(t)) dt
Pr(female | t) Pr(t) ln(Pr(female | t) Pr(t)) dt,
(A4)
where
Pr(t) = 2
1
( −(t−x)
)
2(σ2 )
e ann ,
2
2π σann
Pr(male) = g1 +
(A5)
(g2 − g1 )
1+e
[
−(t−(g3 +g5 (x−Tglob )))
]
g4
[as in Eq. 1], and
Pr(female | t) = 1 − Pr(male | t).
(A6)
Values presented in the article are normalized to range between 0 and 1 by dividing all values by ln(2). Note that when
strategies are perfectly relative, AMI(x) is actually independent of
x. AMI(T glob ) is a measure of how effective the reaction norm is
at producing females at high temperatures and males at low temperatures. The extent to which AMI(x) decreases as x moves away
from T glob is a measure of the populations’ ability to maintain sex
ratios as the within-year mean nest temperature varies. Comparison of reaction norms across different environmental fluctuations
2
2
2
(σglob
/σann
) is slightly complicated by the role of σann
in calculating AMI (see Fig. S1).
Appendix B
LIMITING CASES FOR THE INVASION DYNAMICS
OF TSD MUTANTS
Here we develop analytical results for limiting cases of population structure in the absence of drift by assuming population sizes
are effectively infinite. This allows us to take expectations over
all possible nest temperatures within years based on the assumed
Gaussian distribution of temperatures. We consider two limiting
cases, semelparity and effectively infinite life span. Under semelparity, the geometric mean of the mutant R 0 is an appropriate
fitness measure (Seger and Brockman 1987).
Evolution of TSD under semelparity, no relativity
Under semelparity the OSR fluctuates every generation because
the mean temperature in the environment is random. To approximate the phenotypic evolutionary dynamics we assume a simple
model where haploid zygotes determine their adult sex and therefore only need to track the number of undifferentiated zygotes
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
produced in the next generation per undifferentiated zygotes in
the current generation. A rare sex ratio mutant will have a stochastic growth rate that is approximately determined by the geometric
mean of this multiplier. The invader spreads into the resident population on average if this exponent is greater than 1. This can be
expressed as
⎛ ⎛
Pr(t | Tann )M I (t) dt
⎜
⎜
⎜ )log
λ = exp ⎜
Pr(T
ann
⎝
⎝
2 Pr(t | Tann )M R (t) dt
⎞⎞
Pr(t | Tann )FI (t) dt ⎟⎟
⎟⎟ dTann ,
+ ⎠⎠
2 Pr(t | Tann )FR (t) dt
(B1)
Evolution of TSD under semelparity, with relativity
Our simulation model assumes that reaction norms follow a specific, albeit flexible, functional form and that the TSD strategy
can be shifted relative to the mean annual temperature whenever
g 5 does not equal 0. These assumptions place some biologically
realistic constraints on the strategic TSD responses that can be
achieved. In this section we examine the features of the unconstrained optimal strategy.
Recall equation B1 which defines the stochastic invasion
exponent for an invader strategy. We can simplify this by defining
the mean production of male and female offspring for specific
annual temperatures as
2 (t) dt,
M̄ I (Tann ) = r I (Tann , t)Sm (t)N Tann ,σann
where Pr(Tann ) represents the probability of a particular annual
temperature, Pr(t | Tann ) is the probability an individual egg experiences temperature t given the annual temperature is T ann , M I (t)
and M R (t) represent the number of surviving male offspring per
invader and resident, and FI (t) and FR (t) are the number of surviving female offspring per invader and resident (e.g., Charnov
1986).
This expression can, in principle, be used to calculate the
invasion exponent for an arbitrary temperature distribution and
offspring production functions and is analogous to equation 9
in Van Dooren and Leimar (2003). It also follows that a small
change in the allocation strategy that increases the geometric mean
of Pr(t | Tann )M R (t) dt · Pr(t | Tann )FR (t) dt will invade (see
Tuljapurkar 1990). Although we cannot in general solve for the set
of parameters g 1 , g 2 , g 3 , g 4 that maximize this expression, we can
use numerical methods. We used a simulated annealing algorithm
implemented in Mathematica and found that TSD reaction norms
similar to those observed in the simulations.
Given this formulation we can consider the effect of a sharp
2
is small. In this case, a population with
reaction norm when σann
a sharp reaction norm would produce male-biased clutches (low
T ann ) or female-biased clutches (high T ann ). Only when T ann is
very close to the pivotal temperature will both sexes be produced.
Thus, the product of male and female function will be very low in
years with extreme temperatures, biasing the geometric mean of
this product downwards. A strategy with a shallow reaction norm
for TSD, in contrast, would produce a similar product of male and
female function regardless of the year and can therefore invade.
Figure S2 shows this effect for one set of parameter values. This
can be thought of as a trade-off between matching the production
of each sex to the conditions favorable to it and achieving some
reproduction in years with extreme annual temperatures. Thus,
when between-year variance is relatively high, shallow reaction
norms are favored.
F̄ I (Tann ) =
2 (t) dt.
(1 − r I (Tann , t))S f (t)N Tann ,σann
These equations express the notion that the TSD strategy codes for
a temperature- and relative temperature-specific probability as developing into a male or female combined with a temperature- and
sex-specific probability of surviving. Equation B1 now becomes
F̄ I (Tann )
M̄ I (Tann )
2 (Tann )log
+
dTann .
λ = exp
N Tglob ,σglob
2 M̄ R (Tann ) 2 F̄ R (Tann )
(B2)
Recall that the { M̄ I (Tann ), F̄ I (Tann )} terms depend in turn upon the
r I (Tann , t) function. We would like to find the strategy r I (Tann , t)
that is uninvadable. If we find a strategy whereby the invader
achieves a lower multiplier in every possible annual temperature
then we are guaranteed that λ < 1. The term inside the logarithm
will be less than one whenever
M̄ I (Tann ) · F̄ R (Tann ) + M̄ R (Tann ) · F̄ I (Tann ) < 2 M̄ R (Tann ) · F̄ R (Tann ).
Inequality B3
Under the relativity scenario, the developing offspring have
information about both T ann and their own developmental temperature. Thus, a strategy that makes inequality B3 true for all other
strategies for that specific T ann will be uninvadable. As has been
previously shown, the globally uninvadable strategy in a constant
environment is to choose a critical value tcrit (Tann ) below which
only males are produced and above which only females are produced that maximizes the product of male and female function
(Charnov 1986; Van Dooren and Leimar 2003).
The unconstrained solution in a fluctuating environment simply maximizes the product of male and female function by shifting
the pivotal temperature. Any relativistic strategy that has a pivotal temperature too far from the mean annual temperature will
make very few of one sex and therefore achieve a small product.
The optimal pivotal temperature does not track mean annual temperature exactly, however, because the degree of nonlinearity in
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L I S A E . S C H WA N Z E T A L .
the sex-specific mortality schedule depends on the mean annual
temperature. For the parameters used in this study, the optimal pivotal temperature approaches the mean annual temperature when
mean annual temperature is high (because female survival is relatively constant at higher temperatures) and is shifted above the
mean annual temperature for lower mean annual temperatures
(See Fig. S3).
Because our simulations constrain the pivotal temperature to
linear shifts, the observed TSD reaction norms represent a compromise weighted by the contribution of each annual temperature
to the geometric mean of the mutant growth factor.
The limit of infinite life span
At the opposite end of the spectrum from the limit of semelparity
lies the case where life span is effectively infinite. In this limit,
each individual reproduces over a characteristic sample of annual
temperatures. The appropriate fitness measure is the arithmetic
mean of the number of zygotes produced per mutant zygote.
Because individuals live for many breeding seasons, there are no
temporal fluctuations in the OSR, and there is no selection on
reducing the variance in sex ratios produced. This also means that
there is no selection for relativistic strategies because the mating
competition faced by a developing offspring will not be limited to
individuals who developed in the same annual condition. On the
other hand, there is direct selection against a relativistic strategy
because a greater number of offspring can be produced if females
are produced at temperatures conducive to their survival, even
if this means waiting for the right annual conditions to produce
them.
In this case, the ESS sex allocation strategy is one that
maximizes
H = log[ M̄ · F̄],
where M̄ and F̄ are the mean relative number of male and female
offspring averaged over all clutch temperatures, irrespective of
the mean temperature in each given year. Thus,
2 (t) dt
2 (Tann ) dTann
N Tglob ,σglob
M̄ =
Pr(t, Tann )Sm (t)N Tann ,σann
(B4)
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F̄ =
2 (t) dt
(1 − Pr(t, Tann ))S f (t)N Tann ,σann
2 (Tann ) dTann .
×N Tglob ,σglob
(B5)
For nonrelativistic strategies the probability of developing as
a male does not depend on the annual mean temperature and these
expressions become
2
M̄ = Pr(t)Sm (t)N Tglob ,σglob
2 (t) dt
+σann
F̄ =
2
Pr(t)S f (t)N Tglob ,σglob
2 (t) dt.
+σann
In this formalism, a sex allocation strategy defines the function Pr(t), which is realized as a pair ( M̄, F̄). There is a single
function mapping clutch temperature to sex ratio that maximizes
H. This allocation strategy can be defined as a mapping between
clutch temperature and clutch sex ratio, (t). For a fixed seasonal
distribution of t, the optimal strategy is to produce all one sex below a threshold temperature and the other sex above the threshold,
such that (t) = 0 for t < tcrit and (t) = 1 for t > tcrit (Charnov
and Bull 1989b; Van Dooren and Leimar 2003).
Any relative strategy that depends on Tann in a nontrivial
way will have Pr(t, Tann ) = (t) for some values of (t, Tann ). By
rewriting equations B4 and B5 we have
M̄ = Pr(t)Sm (t)
Pr(t, Tann ) Pr(Tann | t) dTann dt,
F̄ =
Pr(t)S f (t)
(1 − Pr(t, Tann )) Pr(Tann | t) dTann
dt.
If the inner integrals equal (t) and (1 − (t)), respectively, then
we achieve the values of M̄ and F̄ that maximize H. Thus, any
strategy that on average maps (t, Tann ) to (t) will be equivalent
to the optimal strategy. However, the optimal function (t) is
0 or 1 almost everywhere, and therefore any relativistic strategy
will cause a deviation from (t) and will therefore be selected
against.
S E X A L L O C AT I O N BA S E D O N R E L AT I V E A N D A B S O L U T E C O N D I T I O N
Supporting Information
The following supporting information is available for this article:
Appendix S1. Invasion Analysis.
Figure S1. Annual mutual information.
Figure S2. Fitness under semelparity without relativity.
Figure S3. Optimal pivotal temperature.
Figure S4. Mean g 5 over 50,000 generations for individual runs: p = 0 (dashed gray line), p = 0.5 (solid gray line), p = 0.75
(dashed black line), two replicate runs with p = 0.95 (solid black line).
Figure S5. Frequency distributions of g 1 , g 2 , g 3 , g 4 , and g 5 among adults at the end of representative runs (Generation = 10,000
for p = 0, 0.5 and 0.75; Generation = 20,000 for p = 0.95).
Figure S6. The influence of mutation on mean relativity (g 5 ).
Figure S7. Annual T piv as a function of T ann .
Supporting Information may be found in the online version of this article.
Please note: Wiley-Blackwell is not responsible for the content or functionality of any supporting information supplied by the
authors. Any queries (other than missing material) should be directed to the corresponding author for the article.
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