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Keywords:

  • evolutionarily stable strategy;
  • game theory;
  • life history evolution;
  • models;
  • reproductive strategies;
  • sequential hermaphroditism;
  • sex allocation

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Theory predicts that the ‘size advantage’ (rate of increase in male and female fitness with age or size) determines the direction and the timing of sex change in sequential hermaphrodites. Whereas the size advantage is generated by the mating system and would be expected to vary within and between species, the shape or form of the size advantage has rarely been estimated directly. Here, we ask whether theoretical predictions about the timing of sex change hold under different types of size advantage. We model two biological scenarios representing different processes generating the size advantage and find that different types of size advantage can produce patterns that qualitatively differ from classic predictions. Our results demonstrate that a good understanding of sequentially hermaphroditic mating systems, and specifically, a direct assessment of the processes underlying the size advantage is crucial to reliably predict and explain within-species patterns of the timing of sex change.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Understanding within- and between-species patterns of sex allocation is a major theme in life history theory. Within this theme, the ‘size advantage hypothesis (SAH)’ (Ghiselin, 1969), which was developed to explain when sequential hermaphroditism, or sex change, would be favoured, is usually raised as a primary example of theory successfully predicting empirical patterns. SAH predicts that sex change is advantageous when males and females gain fitness with age or size at different rates (Fig. 1). Whereas further research identified other factors such as differential growth or mortality that also can result in sequential hermaphroditism (Charnov, 1982; Iwasa, 1991), SAH remains as the main body of theory that explains the presence and direction of sex change in many hermaphroditic species (reviewed in Munday et al., 2006).

image

Figure 1.  The classic visual presentation of the Size Advantage Hypothesis (SAH). In each panel, curves labelled as M and F give size-specific fitness for a male and a female, respectively. The size that corresponds to the intersection point of male and female curves denote the optimal size of sex change. (a) Protogynous sex change is favoured. (b) Protandrous sex change is favoured.

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In addition to explaining why sequential hermaphroditism is favoured in some species and not others, SAH has also been used to predict within-species patterns of the timing of sex change. In the classic presentation of SAH, individuals of a species are selected to change sex at the age or size where male and female fitness curves intersect (Warner, 1975a; Fig. 1). Consequently, a change in the relationship between fitness and a male's or a female's size or age, owing to changes in physical characteristics of the mating environment such as the abundance of territories and in life history parameters such as growth and mortality, would cause a shift in the timing of sex change (Warner, 1975a). This basic intuition led to several theoretical predictions about the timing of sex change, which we will refer to as ‘classic predictions’ throughout this paper. First, life history theory predicts that a stronger male size advantage would delay sex change in protogynous (i.e. female-first) species. For example, a decrease in the number of mating territories would increase the competition among males of protogynous species and enhance fitness benefits of attaining a large size and favour individuals that change sex at a greater size (Breitburg, 1987). Similarly, a weaker female size advantage is predicted to result in earlier sex change. Accordingly, individuals in a population of a protogynous species, where females gain fitness with age at a lower rate owing to slower growth, would be expected to change sex earlier (Warner, 1975a,b). Finally, greater mortality is predicted to favour earlier sex change in protogynous species. For example, greater mortality of large males (e.g. due to fishing pressure) would increase mating opportunities for smaller males, weaken the effect of size on male fitness and favour earlier sex change (e.g. Sattar et al., 2008). Also, individuals would change sex earlier when chances to survive to a larger size is low because of higher lifetime mortality (Warner, 1975a; Charnov, 1982).

Although many studies found empirical support for the classic predictions about the timing of sex change, a number of hermaphroditic species were shown to exhibit a relatively fixed timing of sex change despite significant changes in life history parameters such as growth and mortality within and between populations (Munday et al., 2006; Table 1). This disparity between theoretical predictions and empirical patterns has been suggested to result from cues for sex change such as a sex ratio threshold (Shapiro, 1981) that may not have been exceeded, or by characteristics of the fishing regime that may not have created opportunities for a shift in the timing of sex change (Buxton, 1993; Lizaso et al., 2000). Nevertheless, a conclusive explanation does not exist for why the timing of sex change in some species does not respond to changes in life history parameters (Munday et al., 2006).

Table 1.   Studies that have investigated within-species changes in the timing of sex change.
StudySpeciesComparisonFactorChange in the timing of sex change
  1. Abbreviations used for the type of population comparisons conducted in a study are BP, between populations; WP, within populations; WP-H, within populations using historical data.

Warner, 1975bSemicossyphus pulcherBPSlower growth rateEarlier sex change
Jones, 1980Pseudolabrus celidotusBPSlower growth and greater mortalityEarlier sex change
Charnov, 1981Pandalus borealisWP-HGreater mortalityEarlier sex change
Breitburg, 1987Coryphopterus nicholsiBPPresence of nest competitorLater sex change
Cowen, 1990Semicossyphus pulcherBPGreater mortalityEarlier sex change
Buxton, 1993Chrysoblephus cristicepsBPGreater mortalityEarlier sex change
Buxton, 1993Chrysoblephus laticepsBPGreater mortalityNo difference
Coleman et al., 1996Epinephelus morioWP-HGreater mortalityNo difference
Coleman et al., 1996Mycoteroperca microlepis, M. phenaxWP-HGreater mortalityEarlier sex change
Bergstrom, 1997Pandalus borealisBPVarious mortality ratesNo difference
Harris & McGovern, 1997Pagrus pagrusWP-HGreater mortalityEarlier sex change
Taylor et al., 2000Centropomus undecimalisBPSlower growth rateEarlier sex change
Branch & Odendaal, 2003Cymbula oculusBPGreater mortalityNo difference
Hawkins & Roberts, 2004Sparisoma viride, S. aurofrenatum, Scarus vetula, S. taeniopterusBPGreater mortalityEarlier sex change
Hawkins & Roberts, 2004Sparisoma rubripinne, Scarus isertiBPGreater mortalityNo difference
Schärer & Vizoso, 2003Thalassoma bifasciatumWPPresence of infectionEarlier sex change
Gust, 2004Chlorurus sordidus, Scarus frenatusBPGreater mortalityEarlier sex change
Hamilton et al., 2007Semicossyphus pulcherBPGreater mortalityEarlier sex change
Götz et al., 2008Chrysoblephus laticepsBPGreater mortalityEarlier sex change

Here, we propose an alternative explanation for the disparity between predicted and observed patterns of the timing of sex change and suggest that it may be caused by an incomplete understanding of the biological processes that generate the size advantage. Models that predict the timing of sex change often assume an increasing male and/or female fitness with age or size. As the mating system plays a critical role in generating the size advantage (Munday et al., 2006), the shape or form of this increase would be expected to vary between as well as within species. The shapes of male and female fitness curves in sex-changing species (e.g. fitness as a function of size), however, have rarely been estimated directly. Consequently, we have little information about the exact shape of the size advantage as well as the factors generating it. While the size advantage determines the pattern and the precise timing of sex change, an explicit knowledge about its nature might also be essential to accurately predict qualitative changes in the timing of sex change. For example, in some protogynous mating systems, where male fitness may increase with size because of a competitive advantage of larger males in obtaining a territory (e.g. Warner & Swearer, 1991), a decrease in territory availability would intensify competition among males, increase the advantage of being a large male and favour delayed sex change (Fig. 2, grey line). In other systems, however, male size advantage is generated by a female preference for better territories (e.g. Breitburg, 1987), for which larger males are superior competitors and have greater fitness. Here, a decrease in territory availability can lead to a change in the timing of sex change that conflicts classic predictions. For example, if the remaining territories are of similar quality, large males are equally preferred by females and have similar fitness as intermediate-sized males with territories. In this case, the superior competitive advantage of largest males would not translate into increased fitness benefits, and earlier sex change would be favoured (Fig. 2, dashed line).

image

Figure 2.  A hypothetical example where a change in the number of available territories can favour an increase or decrease in the optimal size of sex change. Male and female curves are denoted as F and M. A decrease in the number of territories can further increase fitness benefits of being a large male and select for later sex change (solid grey line and the corresponding intersection point). Alternatively, a decrease in the number of territories could also decrease the variance in territory quality, and select for earlier sex change (dashed grey line and the corresponding intersection point).

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Although the size advantage has usually been implemented as a fitness curve where fitness increases as a simple function of age or size, we do not have an explicit knowledge about the shape or type of the size advantage in nature. Here, we revisit SAH and use a game-theoretical framework to ask whether classic predictions about the timing of sex change hold regardless of the type of size advantage. To this end, we first implement the male size advantage using a power function, where, for example, a male's age or size is a simple scale of his fitness such that larger males are better competitors and have greater chances of obtaining a territory or securing a mate. Alternatively, we use a threshold function to model the male size advantage, which, for example, may be caused by a female preference for males that are greater than a critical size. In each case, we alter life history parameters and investigate whether the predictions of the two versions of our model implementing different scenarios for the male size advantage are qualitatively similar to each other and to classic predictions about the timing of sex change.

The model

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

To calculate the evolutionarily stable age of sex change, we use a classic life history model based on a framework we used previously to investigate the evolutionary stability of sex change vs. dioecy (Kazancıoğlu & Alonzo, 2009). Here, however, we focus on an outcrossing population with overlapping generations that is composed only of sex changers, calculate the stable timing of sex change and ask how it responds to changes in the strength of the male and female size advantage. Variables of the model are summarized in Table 2. We assume that the period prior to maturation affects all individuals similarly and therefore restrict our analyses exclusively to post-maturation (‘reproductive lifetime’). Ages 0 and t denote the beginning and the end of an individual's reproductive lifetime, respectively. In this framework, individuals reproduce as one sex (female for protogyny, male for protandry) until age a and then switch to reproducing as the other sex until the age t. The Evolutionarily Stable Strategy (ESS) value for a represents the evolutionarily stable protogynous or protandrous (i.e. male-first) life history. For simplicity, we link an individual's size to its age and assume deterministic growth such that older individuals are also larger. We denote the age-specific fecundity and fertility by f(x) and m(x), respectively, which we expand below (eqns 7–9). Finally, we assume that the age-specific probability of survival is equal for both sexes, denoted by ezx, where x is age, and z is the instantaneous mortality constant. Assuming that the population is not increasing in size, ezx also describes the age distribution.

Table 2.   Variables used in the model.
VariableDefinition
xAge
mcMale fertility constant
mpShape parameter of the power function for male fertility
TThreshold age of the threshold function for male fertility
hSteepness parameter of the threshold function for male fertility
fcFemale fecundity constant
fpShape parameter for fecundity function
aAge of sex change
tFinal age of reproduction
zMortality constant
EAverage number of eggs produced in the population
SAverage number of sperms produced in the population
FAverage female fitness of a sex-changing individual
MAverage male fitness of a sex-changing individual

With these assumptions, in a population of protogynous hermaphrodites that change sex at age a, the average numbers of eggs, E, and sperms, S, produced by an individual over its entire lifetime are:

  • image(1)
  • image(2)

E and S are individual measures of total egg and sperm production, respectively. Assuming that sperm competition follows a fair raffle (Parker, 1998), the fraction inline image denotes the average number of eggs fertilized per sperm produced. We also use this fraction to introduce a term that denotes the average probability that an egg is fertilized as a function of the relative egg and sperm production in the population. Empirical data (e.g. Warner et al., 1995) suggests that this fertilization probability is expected to approach 0 (i.e. fertilization highly unlikely), if there are many more sperms than eggs, and to asymptotically approach 1 (i.e. fertilization highly likely), as the number of sperms relative to the number of eggs increases. Accordingly, we modelled the fertilization probability using the function inline image that conforms these biologically relevant conditions.

In this population, the average fitness through female function of a protogynous mutant changing sex at age a′ is given by:

  • image(3)

Similarly, the average fitness through male function of a protogynous mutant changing sex at age a′ is:

  • image(4)

The term inline image in eqn (4) links male and female fitness and ensures a ‘self-consistent’ modelling framework (Houston & McNamara, 2002 ), where the total fitness in a population through male function is equal to the total fitness through female function due to the biological fact that every offspring has one mother and one father (Düsing, 1884; Fisher, 1930). A self-consistent framework is crucial to build a biologically sound model and to make reliable predictions about the timing of sex change. In a graphical presentation of SAH for protandrous species that is not self-consistent, for example, a shallower female fitness curve intersects the male fitness curve at a later age and predicts delayed sex change (Fig. 3a), which contradicts the classic prediction of earlier sex change (Warner, 1975a). The classic prediction is obtained by a self-consistent presentation (Fig. 3b), where a change in female fitness curve results in a corresponding change in the male fitness curve. Similarly, the male fitness curve in a protogynous population, where large males have decreased fitness due to greater mortality, intersects the female fitness curve at a later age, predicting delayed sex change (Fig. 3c) and contradicting the classic prediction of earlier sex change (Warner, 1975a). Again, a self-consistent framework, where fitness lost by large males will be gained by small males (Fig. 3d), is needed to correctly predict earlier sex change.

image

Figure 3.  A self-consistent treatment of size advantage hypothesis (SAH) is necessary to accurately predict changes in the timing of sex change. (a) A shallower female fitness curve (F1 to F2) predicts delayed sex change and contradicts classic predictions, if self-consistency is not considered. (b) In a self-consistent scenario, male fitness curve changes with the female fitness curve (M1 to M2), leading to the classic prediction of earlier sex change. (c) Similarly, decreased fitness for old males predicts for delayed sex change, contrary to classic predictions, when self-consistency is not considered. (d) With self-consistency, fitness lost by old males is gained by young males, resulting in earlier sex change, as predicted by SAH.

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In this framework, we find the evolutionarily stable age of sex change using the ESS criterion (Maynard Smith, 1982), which states that a strategy is evolutionarily stable in a population, if no other strategy has higher fitness when rare. Accordingly, a* is the evolutionarily stable age of sex change, if:

  • image(5)

where W(a′,a*) is the fitness of a mutant changing sex at age a′ in a population of individuals that change sex at age a*, and:

  • image(6)

Female size advantage

We assume that older (and larger) females have greater resources to produce eggs, and consequently, a greater fecundity and a fitness advantage over younger (and smaller) females. To implement this ‘female size advantage’, we assume that fecundity is an increasing function of age. We give the age-specific fecundity as:

  • image(7)

where fc is the fecundity constant, fp is the shape parameter, and x is age. We incorporate this fecundity function into the model by replacing f(x) in eqns (1) and (3) with eqn (7). We explore various scenarios for the fecundity increase with age by employing a range of values for fp. For example, a physiological constraint could limit the maximum fecundity a female can achieve and result in a decelerating fecundity increase with age (0 < fp < 1). Alternatively, older or larger females may be better at accessing food resources, which would accelerate the rate of increase in fecundity with age (fp > 1). In all these cases, greater values of fc and fp correspond to a steeper increase in fecundity with age, and consequently, a stronger female size advantage.

Male size advantage

We also implement a ‘male size advantage’ such that older (and larger) males have a competitive advantage over younger (and smaller) males, for example, to obtain mating territories or securing mates, and consequently, have greater fitness. Here, older (or larger) males fertilize a greater proportion of eggs relative to their younger (or smaller) competitors and therefore have a fitness advantage. We consider two possibilities for how male fitness could increase with age.

The power function

First, we consider a scenario where a male's age or size is a simple scale of his mating success, such that, for example, older or larger males are also better competitors for available mating territories. In this case, we give male fertility as a power function of age:

  • image(8)

where mc is the fertility constant, mp is the shape parameter, and x is age. We incorporate this function into the model using eqns (2) and (4). Similar to the female size advantage, different values for mp correspond to different scenarios for the increase in male fitness with age. For example, the increase in male fitness could accelerate (mp > 1), owing to effects of experience or female choice in addition to an intrinsic competitive advantage of being older or larger, or decelerate (0 < mp < 1) owing to some physiological constraints. A smaller value of mp represents a shallower increase in fitness with age, and a weaker male size advantage.

The threshold function

Alternatively, we consider the situation where a threshold age exists for increased competitive ability. Here, males past this age have greater mating success, for example, because of a female preference for older males. We implement this biological scenario using a sigmoidal function:

  • image(9)

where mc is the fertility constant, T is the critical age, h is the steepness parameter, and x is age. We incorporate this function into the model in eqns (2) and (4). Here, smaller values for h corresponds to a shallower increase in male fitness around the critical age, T, and consequently, to a weaker male size advantage. Similarly, a smaller critical age, T, decreases the relative fitness benefit of being an older (and larger) male, and corresponds to a weaker male size advantage.

Using this modelling framework, we manipulate the strength of the female and male size advantage by changing values of shape parameters fp, fc, mp, mc, h, or T as well as mortality z in eqns (7–9) and ask how the evolutionarily stable timing of sex change is affected. Furthermore, we compare the results of two versions of the model that use a ‘power function’ and ‘threshold function’ to implement male size advantage, respectively, and ask whether different types of male size advantage lead to similar predictions about the timing of sex change.

Analysis of the model

We implemented this game-theoretical model using Mathematica 6 (Wolfram Research Inc., 2007). The Mathematica code is available on request. As equations could not be solved analytically because of mathematical complexity, we analysed the model numerically using a wide range of parameter values. First, we explored the effect of the strength of the female size advantage by varying the shape parameter of the fecundity function, fp, between 0 (i.e. static fecundity, no size advantage) and 2 (i.e. accelerating increase in fecundity with age, stronger size advantage). Similarly, we manipulated the strength of the male size advantage by changing the shape parameters mp, for the power function, and h and T, for the sigmoidal function. We explored mp from very small (i.e. 0.1; shallow, decelerating increase; weaker size advantage) to very large (i.e. 15; steep, accelerating increase; stronger size advantage) values, with increments of 0.1. We also varied the sigmoidal steepness parameter, h, from very small (i.e. h = 0.001; shallow increase; weaker size advantage) to large values (i.e. h =0.2; steep increase; stronger size advantage). Expanding the range for h did not significantly affect conclusions. In all analyses, we used large values for the fecundity constant (fc = 500), fertility constant (mc = 106), and the age of last reproduction (t = 1000) to implement biologically relevant scenarios with a large number of opportunities to gain fitness. Different values for these parameters yielded qualitatively similar results. Finally, we investigated a biologically realistic range of values for the instantaneous mortality constant, z, where only a small fraction of the population (here, < 1%) can reach the end of the reproductive lifetime, t (see Fig. S1). For example, for t = 1000, we used mortality values that range from 5×10−3 (‘low mortality’) to 10−2 (‘high mortality’). Similarly, based on this range of mortality values, we investigated values for the threshold age (T) where at least some proportion of individuals (here, more than 1%) is alive (see Fig. S1). Accordingly, we used values for T that range from T = 100 to T = 450. In all analyses, adopting different biologically realistic sets of parameter values yielded qualitatively similar results.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

In agreement with classic predictions, the version of our model where male fitness increases as power of age (‘the power function’) predicts an increasing age of sex change with the strength of the male size advantage. This result holds across all parameter values for the rate of increase in female fecundity (fp and fc) and mortality (z; two representative sets of results are shown in Fig. 4) investigated. Similarly, when the male size advantage is implemented using a ‘threshold function’, an increase in the strength of the male size advantage through an increase in the critical age (T) leads to delayed sex change. How strongly the critical age affects the timing of sex change, however, depends on the steepness of the increase in male fitness around the critical age (h) as well as the strength of the female size advantage and mortality. For example, a greater critical age does not cause a significant delay in sex change, if the female size advantage is strong (e.g. fp is large), the increase in male fitness around the critical age is steep (e.g. h is large) and mortality is low (i.e. z is small; Fig. 5a, solid lines at large values for h).

image

Figure 4.  The age of sex change increases with the strength of the male size advantage, when it is implemented through a power function. Similarly, a decrease in mortality (dashed vs. solid lines) as well as an increase in female size advantage (grey vs. black lines) is predicted to increase the age of sex change. Grey and black lines represent the scenario where female fitness does not increase (fp = 0) and strongly increases with age (fp = 1.5), respectively. Solid and dashed lines represent low (z = 10−4) and high mortality (z = 10−3) conditions, respectively.

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image

Figure 5.  Changes in the timing of sex change predicted by size advantage hypothesis can be significantly weakened or even reversed, when the male size advantage arises through a threshold function. For example, even though the male size advantage is stronger at point b relative to point a, our model predicts no difference in the timing of sex change and contrasts classic predictions (e.g. see Fig. 4). The threshold age denotes the male age that is critical for fitness increase, while the steepness represents how strongly male fitness increases around the threshold age. Female size advantage denotes the rate of increase in female fitness with age.

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On the other hand, we find that a stronger male size advantage as a result of a steeper increase in male fitness around the critical age (i.e. greater h) can have a qualitatively different effect on the timing of sex change than classic predictions, depending on the critical age (T) and the strength of the female size advantage (fp). For example, the age of sex change increases with h, consistent with classic predictions, when the critical age is large and the female size advantage is weak (Fig. 5, black dashed line). In contrast, increasing h leads to earlier sex change, if the critical age is small and/or the female size advantage is strong (e.g. Fig. 5, light grey, dashed line).

Both versions of the model that implement a male size advantage using a ‘power function’ and a ‘threshold function’, respectively, agree with classic predictions and predict earlier sex change, when mortality is higher (Figs 4 and 5). In contrast to classic predictions, however, we find that female size advantage has no effect on the timing of sex change, if its strength is altered through the fecundity constant (fc; results not shown). On the other hand, in agreement with classic predictions, we find that stronger female size advantage owing to a greater value for the shape parameter (fp) results in earlier sex change (Figs 4 and 5). In our analyses using a ‘threshold function’, we find that a change in mortality or the strength of female size advantage has a small effect on the timing of sex change, when the male size advantage is strong. Accordingly, mortality has a small effect on the timing of sex change, when male fitness increases steeply around a large critical age (Fig. 5, the two solid or dashed black lines compared at high values for h).

In sum, we find that the type of size advantage significantly affects predictions about the timing of sex change. Accordingly, our analyses agree with classic predictions, when male size advantage is implemented by a power function, using a threshold function leads to patterns that strongly contrast classic predictions.

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Size advantage hypothesis predicts that the timing of sex change in sequential hermaphrodites will respond to changes in life history parameters such as growth and mortality, and to changes in the strength of the relationship between male and female age or size and fitness (Warner, 1975a; Charnov, 1982). Our analyses indicate, however, that these predictions do not hold under all types of size advantage. When the size advantage is implemented using a ‘power function’, for example, where the mating success of a male is a simple scale of his age or size, predictions of our model about the changes in the timing of sex change agree with previous studies. In contrast, we find that changes in life history parameters can have a weak or no effect on the timing of sex change, if male size advantage is modelled through a ‘threshold function’ such that females prefer to mate with males greater than a critical age or size. Mortality, for example, which is classically predicted to significantly affect the timing of sex change, becomes largely irrelevant when there is a very strong female preference for old (and large) males (Fig. 5).

Furthermore, modelling the male size advantage through a threshold function results in patterns that qualitatively differ from classic predictions. For example, we find that increasing the strength of the male size advantage through the steepness of the male fitness around the critical age can cause earlier as well as later sex change (Fig. 5). As the steepness of the threshold function increases, and, for example, female preference for older males strengthens, males younger than the critical age lose fitness, whereas those older than the critical age gain. A delay in sex change is intuitively predicted when, for example, large males gain greater reproductive advantage relative to smaller males. The increase in the age of sex change observed at smaller values of the steepness parameter (Fig. 5) is consistent with this intuition.

The threshold function we used to model male size advantage, however, also puts an upper limit on male mating success. Consequently, as the increase in male reproductive success around the critical age becomes steeper, the mating success of oldest males hits this upper limit and ceases to increase. Because of the self-consistent framework, this results in a fitness increase for males of intermediate age relative to oldest males and males that are younger than the critical age and favours earlier sex change. While this pattern is partly an outcome of the functional form we used, it clearly demonstrates that changing the way size advantage is modelled can reverse classic predictions about the timing of sex change. In this case, for example, while an increase in the critical age for male mating success favours delayed sex change and agrees with classic predictions, we find that an accompanying increase in the steepness of the preference around the critical age can lead to the lack of the classically predicted change in the timing of sex change (Fig. 5, going from point a to point b). This theoretical example is analogous to the empirical scenario in the sex-changing goby species Coryphopterus nicholsi, where a male size advantage is generated by a female preference for better territories (e.g. Breitburg, 1987). Here, a decrease in the number of available territories would increase fitness benefits of being a large male (‘increasing the critical age’), but could also decrease the variation in territory quality, and consequently, in fitness of large males (‘increasing the steepness’). Field experiments in this species that manipulate both territory abundance and quality would enable a direct empirical investigation of the effect of changes in the size advantage on the timing of sex change.

Size advantage hypothesis predicts that changes in the female fitness curve would also affect the timing of sex change (e.g. Warner, 1975a). In contrary to this classic prediction, however, we find that a change in the strength of the female size advantage through the fecundity constant (e.g. the slope in a linearly increasing female fitness) does not affect the age of sex change. This pattern contradicts classic predictions and demonstrates that self-consistency ensures that our model not only complies with biological facts, but also is essential to be able to make reliable predictions about the timing of sex change. Here, self-consistency links male and female fitness curves such that any change in the female fitness curve would result in a corresponding change in the male fitness curve. If the female size advantage is altered through the fecundity constant, the fitness of each female and, consequently, male age class is changed by the same proportion. Consequently, the age where male and female fitness curves intersect, that is, the timing of sex change, would be unaffected.

Classic predictions about the timing of sex change have been empirically tested in various protogynous and protandrous species. In contrast to many studies that found support for theoretical predictions, the timing of sex change does not seem to respond to significant changes in growth or mortality in other species (Table 1). Although several explanations have been proposed for these observed deviations from theoretical predictions (Shapiro, 1981; Buxton, 1993; Lizaso et al., 2000), a conclusive explanation does not exist for why some species have a relatively fixed timing of sex change (Munday et al., 2006).

Our analyses suggest that this disparity between classic predictions and observed patterns could also be caused by a general lack of explicit knowledge about the nature or form of size advantage. Many empirical tests of classic predictions, for example, focus on the effect of mortality on the timing of sex change (e.g. Buxton, 1993; Hawkins & Roberts, 2004). Classic predictions, however, implicitly assume that, when older (or larger) males become rare in a population (e.g. owing to size-selective mortality), earlier sex change would be ‘adaptive’, where younger (or smaller) individuals would gain fitness by changing sex. We suggest that this may not always be the case, and implementations of SAH that do not correctly capture processes that shape male and female size advantage can produce misleading predictions about the timing of sex change. For example, a female preference for males above a certain age or size favours sex change by establishing a size advantage for older or larger males, but also restricts changes in the timing of sex change (Fig. 5). If the male size advantage is strong enough, the decrease in the chances of surviving past the critical age due to greater mortality would be offset by the increased fitness benefits through female preference. In this case, in contrast to the classic prediction that greater mortality favours earlier sex change, the timing of sex change would be relatively unaffected by mortality.

Consequently, our analyses demonstrate that an empirical and direct assessment of male and female size advantage curves will be essential to reliably predict within-species patterns of the timing of sex change. Direct estimation of the size advantage is a challenging task that requires assessing the fitness of individuals that may simply be missing in a population, such as young males or old females in purely protogynous species. With the help of an increasing understanding of proximate mechanisms that control the timing of sex change (e.g. Godwin et al., 2003; Perry & Grober, 2003; Munday et al., 2006), it can be possible to establish a population with a range of male and female age or size classes and calculate their relative fitness (Warner, 1988). This approach would not only elucidate processes that generate the size advantage in different systems, but also enable one to test whether a sex change schedule observed in a population is adaptive and provides the greatest fitness benefits to the individuals that adopt it.

It is important to note that most empirical tests of predictions about the timing of sex change investigated the effect of size-dependent mortality caused by, for example, size-selective harvesting of larger males (Table 1). While our model implements instantaneous mortality that remains constant throughout the reproductive lifetime, the intuition we gain from our analyses also apply to a scenario where mortality increases with age or size. Here, mortality benefits of being a younger or smaller male could be offset by a substantial decrease in reproductive success due to female preference. Consequently, strong female preference could prevent a decrease in the age or size of sex change despite an increase in the age- or size-dependent mortality. This prediction, however, assumes a static female preference for males older or larger than a specific age or size. Alternatively, females might choose mates according to relative male age or size (Bateson & Healy, 2005). In this case, as largest males become rare in the population as a result of increased size-specific harvesting, female preference would shift towards smaller males. As a result, smaller individuals would gain fitness from changing sex, and earlier sex change would be predicted. While theoretical patterns arising from this scenario would agree with classic predictions, this example further demonstrates the main message of our study that it is essential to have an explicit understanding of the processes underlying the size advantage to be able to make accurate predictions about the timing of sex change.

Life history theory predicts that the timing of sex change in sequential hermaphrodites will respond to changes in life history parameters. Here, we confirm these classic predictions, but also find that the predicted change in the timing of sex change strongly depends on the type of size advantage. Finally, we suggest that a good understanding of sequentially hermaphroditic mating systems and specifically a direct assessment of the size advantage and processes that generate it are crucial to be able to reliably predict and explain within-species patterns of the timing of sex change.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

We would like to thank members of Alonzo Lab and two anonymous reviewers for their helpful suggestions. The material in this paper is based upon work supported by the National Science Foundation under grant numbers EF-0827504 and IOB-0450807.

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  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Supporting Information

Figure S1 A graph used to determine biologically-realistic values for the length of the reproductive lifetime, t, and the threshold age, T for mortality values that range from 10−3 to 10−2. Each line gives the age at which the proportion P of the population is alive. Given a mortality value, a value for t is biologically realistic if only a small proportion of individuals is alive by that age. Similarly, a value for T is biologically realistic, if at least some proportion of individuals is alive by that age. Assuming that P represents the proportion of individuals that is critical to determine realistic parameter values, for a given P and mortality, values of t that lie above the line and values for T that lie below the line are biologically realistic.

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