Behavioral Ecology Advance Access originally published online on September 10, 2007
Behavioral Ecology 2007 18(6):1106-1115; doi:10.1093/beheco/arm083
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
Sex ratio schedules in a dynamic game: the effect of competitive asymmetry by male emergence order
a Department of Systems Sciences, University of Tokyo, Meguro, Tokyo 153-8902, Japan b Faculty of Applied Biological Sciences, Gifu University, Yanagido 1-1, Gifu 501-1193, Japan c Institute of Evolutionary Biology, School of Biological Sciences, University of Edinburgh, West Mains Road, Edinburgh, EH9 3JT, UK d Graduate School of Agriculture, Hokkaido University, Kita 9, Nishi 9, Kita-ku, Sapporo 060-8589, Japan
Address correspondence to Jun Abe. E-mail: jun.abe{at}ed.ac.uk.
Received 26 September 2006; revised 6 May 2007; accepted 10 August 2007.
 |
ABSTRACT |
|---|
 TOP ABSTRACT INTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Studies of sex allocation have provided some of the most successful tests of theory in behavioral and evolutionary ecology. For instance, local mate competition (LMC) theory has explained variation in sex allocation across numerous species. However, some patterns of sex ratio variation remain unexplained by existing theory. Most existing models have ignored variation in male competitive ability and assumed all males have equal opportunities to mate within a patch. However, in some species experiencing LMC, males often fight fiercely for mates, such that male mating success varies with male fighting ability. Here, we examine the effect of competitive ability on optimal sex allocation schedules using a dynamic programming approach. This model assumes an asymmetric competitive ability derived from different mortalities according to the timing of male emergence. If the mortality of newly emerging males is larger than that of already emerged males, our model predicts a more female-biased sex ratio than expected under traditional LMC models. In addition, females are predicted to produce new males constantly at a low rate over the offspring emergence period. We show that our model successfully predicts the sex ratios produced by females of the parasitoid wasp Melittobia, a genus renowned for its vigorously fighting males and lower than expected sex ratios.
Key words: dynamic game model, lethal male combat, local mate competition, Melittobia, parasitoid wasps, sex allocation.
|
INTRODUCTION |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Sex ratio studies have been very successful in behavioral and evolutionary ecology, providing an elegant test of theory (Charnov 1982
; Godfray 1994; Hardy 2002; West et al. 2005). One of the most frequently studied areas is local mate competition (LMC) theory, which predicts a female-biased sex ratio when the offspring of only one or a small number of females mate among themselves within a patch (Hamilton 1967). This is because a female-biased sex ratio reduces the competition among related males and increases the number of potential mates for those males (Taylor 1981). In haplo-diploids, an additional bias is favored because inbreeding makes females relatively more related to their daughters than to their sons (Frank 1985; Herre 1985). Theory predicts that females should adjust their sex ratio depending on the level of LMC, and there is widespread support for this prediction across a wide range of species from a variety of taxa (West et al. 2005). Variation in the sex ratio across populations and species have been explained by factors such as the number of females laying eggs on a patch (Frank 1985; Herre 1985), female clutch size (Stubblefield and Seger 1990; Suzuki and Iwasa 1980; Werren 1980; Yamaguchi 1985), and the proportion of males dispersing from the patch (Fellows et al. 1999; West and Herre 1998). However, along with these successes, there remain questions regarding unresolved variation in sex ratios (Herre et al. 1997; Kinoshita et al. 1998; Moore 2002; Reece et al. 2004; Shuker et al. 2004, 2005, Shuker, Pen and West 2006; Shuker, Sykes, et al. 2006) and even large deviations from the predicted pattern of sex allocation (Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; Cooperband et al. 2003; Innocent et al., forthcoming).
Most currently existing models with regard to LMC have ignored variation in competitive ability among males and assumed that all males within a natal patch have equal opportunities to mate with females (but see Abe, Kamimura, Ito, et al. 2003; Shuker et al. 2005, Shuker, Pen and West 2006). However, male individuals may differ in their competitive ability depending on their state (e.g., size, age, physical condition, and genotype). Moreover, in some species, males exhibit lethal combat under LMC conditions (Hamilton 1979; West et al. 2001; Sabelis et al. 2002; Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005), leading to the possibility of asymmetrical competition among the males. Evolutionary stable strategy (ESS) models have demonstrated that when females lay eggs asynchronously and the mating success of their sons is influenced by their oviposition order, the ovipositing females should adjust their offspring sex ratios according to the order of oviposition (Abe, Kamimura, Ito, et al. 2003; Shuker et al. 2005). Emergence order of male offspring is another possible factor that causes variation in competitive ability (Abe et al. 2005). This might result in asymmetric mate competition even among male offspring produced by the same female. In species where offspring emerge sequentially over a prolonged period, the competitive ability of sons is likely to be influenced by emergence time. Therefore, the optimal offspring sex ratios at each time point may depend on the emergence order, and the resulting overall sex ratio may differ from the predictions of existing LMC models.
Wakano (2005) has recently considered the evolution of sex allocation schedules under LMC. When the effects of both lethal male–male combat and time-dependent control of sex ratio work together, optimal sex ratios are predicted to be more female biased than the original LMC models (Wakano 2005). However, he assumed that all emerged males suffer equally from the mortality caused by lethal male combat, and he did not examine possible asymmetric effects on mate competitive abilities that arise from emerging at different times (including mortality before full adult maturation). Here, we examine the consequences when males have different competitive abilities depending on the timing of their emergence. This situation has shown to be important in Melittobia wasps, where earlier emerging males have a large advantage over later emerging males (Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; Innocent et al., forthcoming).
Female parasitoid wasps from the genus Melittobia are able to lay more than 500 eggs on a single insect host over several weeks, with their offspring emerging asynchronously depending on the oviposition order (Adams 2002; Abe et al. 2005). The males characteristically exhibit lethal combat (Dahms 1984). Later emerging males are killed by earlier ones in almost all cases (Abe, Kamimura, Kondo, et al. 2003; Innocent et al., forthcoming), and this fighting behavior occurs even among close relatives (Abe et al. 2005). This asymmetry in fighting ability is thought to occur because newly emerging males remain vulnerable for several hours after eclosion. Melittobia species show extremely female-biased sex ratios (less than 5% males), as measured before the lethal male combat had not yet taken place (Dahms 1984; Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; González et al. 2004). However, in contrast to LMC theory that predicts a less female-biased sex ratio as the number of females laying eggs on a patch increases, females do not adjust their offspring sex ratio in response to female number (Werren 1987; Abe, Kamimura, Kondo, et al. 2003; Cooperband et al. 2003; Innocent et al., forthcoming). Lethal male combat has been suggested as a possible explanation for this unusual sex ratio pattern (Abe, Kamimura, Ito, et al. 2003, Abe, Kamimura, Kondo, et al. 2005), with females being selected to avoid producing late-emerging males that would be killed by already emerged males. Although asymmetric competitive ability among sons resulting from the oviposition order of females might not apply for the life history of Melittobia species (Abe et al. 2005; Innocent et al., forthcoming), the asymmetry of male competitive abilities depending on the male emergence order could be a possible explanation for the extremely female-biased sex ratios seen in this group (Abe et al. 2005).
To examine the effect of asymmetric male competitive ability resulting from asynchrony in male emergence on female sex allocation, we here develop a model that optimizes sex ratio across a prolonged period of several oviposition events, as opposed to the usual assumption of a single oviposition event. We construct a dynamic programming model (Mangel and Clark 1988; Clark and Mangel 2000), in which females determine the sex ratio during each time period so as to maximize their own total fitness across the whole emergence period of their offspring. Because the fitness consequences depend critically on the tactics of other females ovipositing on the same host, the model takes the form of a dynamic game (Houston and McNamara 1999). ESSs of sex allocation schedules are calculated by varying the asymmetry of competitive ability among males. Finally, we compare the model's predictions with the observed emergence pattern of Melittobia, and evaluate the effects of lethal male combat on the extremely female-biased sex ratios of the species in this genus.
|
MODEL |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Here, we extend LMC theory to include time-dependent decision making, assuming that offspring emerge asynchronously over a prolonged period. Female offspring disperse soon after mating within a natal patch, whereas male offspring stay on the patch. Males engage in lethal combat when competing for mates and have asymmetrical competitive abilities depending on the order in which they emerged. Although we mainly assume the life history of the parasitoid wasp Melittobia, this model can apply to other species in which male offspring do not disperse and asymmetrically compete for mates (see Discussion).
Assumptions of the model
The model assumes that populations are made up of discrete patches (e.g., parasite hosts) and exactly two females lay eggs on the same host, that is, a two-person game. Although Melittobia females can commence oviposition simultaneously or sequentially depending on the timing of finding a host and the condition of the host, we assume the two females simultaneously commence oviposition. Both females keep laying eggs on the host during an oviposition period of a finite length. For mathematical simplicity, the oviposition period is divided into a series of T discrete time steps (e.g., 1 day). Each female lays N eggs at each time step from the start point (t = 1) to the final time step (t = T), that is, each female produces NT eggs over the whole oviposition period. We ignore variation in developmental time among individual offspring and assume that the timing of emergence of each individual exactly depends on the timing of oviposition (as is approximately the case in Melittobia wasps; Abe et al. 2005). Emerged female offspring mate at random with a male that has already emerged on the same patch and then disperse within the same time step. Males do not disperse, remaining on the natal patch. Males achieve reproductive success by mating with emerging females during the time steps they survive. We calculate female fitness as the number of grandoffspring, which is the sum of the reproductive success from their sons and daughters, weighted by the relatedness and reproductive value (see below).
Here we call the males that have emerged at a previous time step "remaining males" and newly emerging males at the present time step "new males," and define their survival rates as Ls(z) and Ln(z)(0
Ls(z),Ln(z)1), respectively (Figure 1). If new males emerge without being killed by other males, they are the remaining males at the next time step. Combat occurs more frequently when there are a larger number of males on the patch because they encounter each other more often (Reece et al., forthcoming). Therefore, the survival probabilities (Ls(z) and Ln(z)) are decreasing functions of the total number of males, z, on the patch. We adopt a concavely decreasing function for the survival rates, Ls(z) and Ls(z), with increasing z (Figure 2)
 | (1a) |
 | (1b) |
0). The parameter b represents the "rate of change of survival," and the survival rates decrease more steeply as b becomes larger (Figure 2). When z = 1 (i.e., there is only one male on a patch), the male is not attacked by other males and thereby Ls(z) and Ln(z) equal 1 (Figure 2). In Melittobia, newly emerging Ls(z) males are frequently killed by older adult males (Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; Innocent et al., forthcoming), such that Ln(z) is always much smaller than at any z (except at z = 1). Therefore, the asymmetric competitive ability between males is determined by the difference in their survival rates. We ignore the natural death of males because it is likely to be negligibly relative to the death from lethal combat.
|
|
The females are confronted with the decision of what sex ratio to produce to maximize fitness, given this pattern of male survival. If they produce females, they constantly gain reproductive success as a linear function of the number of females. We assume that one male can potentially mate with all females on the patch and that mating interactions are not temporally constrained by male–male interactions. However, the reproductive success of sons is largely influenced by the number of other males and females on the patch because their survival rate is determined by total male number at each time step and they compete with other males for females (see Equation 2). The strategy at a time step is a vector
= (p(0), p(1), ..., p(n), ..., p(N)), where p(n) is the probability that a female produces n males (0 n N) at the time step. Then, the sex ratio is represented by n/N and the average sex ratio of a population is
 |
(t, s) is a function of time step, t, and the expected number of remaining male offspring of the focal female, s. In this model, each female has no information about the strategy (the sex ratio at each time step) of her opponent.
Calculating the ESS
We determine the ESS sex ratio using a dynamic game model based on the damping procedure introduced by Houston and McNamara (1999). In essence, the damping procedure is a process that updates a resident population strategy with the best mutant one responding to it until the ESS is reached. Here we describe that the process of a strategy k(t, s) reaches to the ESS, where k is the number of times the resident population is updated.
- (i) Arbitrarily chosen initial values are assigned to all strategies 0(t, s) at any t and s. In this study, we adopted an initial population strategy that the females lay n males (0 n N) with an equal probability at each time step: p(0) = p(1) =
p(n) = = p(N) = 1/(N+1), but the initial values do not affect the final results.
- (ii) The expected frequencies F(t, s) for which females have s remaining males at time step t are calculated by assuming that all females follow the current population strategy
k(t, s). Given that the two females do not have remaining males at the initial time step, all possible frequencies at later time steps are identified by forward iteration from the first time step (t = 1) to the terminal time step (t = T). If a female produces males and the sons emerge without being killed (with the probability Ln(z)), the number of remaining males will increase (Figure 1). In contrast, lethal male combat decreases the remaining male number with the probability 1 – Ls(z) (Figure 1). Detailed methods are given in Appendix A.
- (iii) The best response strategies Bk(t, s) to the current population strategy
k(t, s) are calculated by backward iteration based on the expected frequency F(t, s) from step (ii). The best response is also a vector B = (p'(0), p'(1), ..., p'(n), ..., p'(N)) and is determined so as to maximize the total fitness by considering the expected fitness gained in future time steps as well as the present time step. We define W(t, s, n) as the expected fitness of a female that gained between time step t and terminal period T, when the female having s remaining males lays n males at the time step t:
- (ii) The expected frequencies F(t, s) for which females have s remaining males at time step t are calculated by assuming that all females follow the current population strategy
 | (2) |
), the hat means the sons of the opponent female, and Wstart(t+1) is the fitness gained after the start of the next time step:
 | (3) |
) and strategy (
) of the opponent female. The summed term over and in Equation 2 is the reproductive success through sons and the expected fitness gained during the subsequent time steps. If the sons survive, they also gain reproductive success mating with females that emerge subsequently. Under a best response strategy, an optimal strategy that gives the highest fitness (highest W(n)(0 n N)) is chosen with the highest probability (p'(n)). In order to converge the sequence of the strategies to a global optimum, we incorporated errors into the choice of action (McNamara et al. 1997). Iterating Equations 2 and 3, the best response strategies are identified at all time steps from the terminal time step (t = T) to the beginning time step (t = 1). The detailed procedure relating to the backward iteration with error is described in Appendix B.- (iv) The new strategies for the next population k + 1 are calculated from the current population strategy k and the current best response Bk. In order to avoid an oscillatory solution, a weighted average between them is given as
 | (4) |
(0< 1) determines the degree of damping and is set at = 0.1 in our model.- (v) The ESS can be obtained by repeating step (ii)–(iv). If the best response to a strategy is the strategy itself, this is the ESS. Therefore,
 | (5) |
n N) in relation to all time steps and females' states were less than 0.0001, and the resulting strategies were identified as the ESS.
At step (ii), the frequencies F(t, s) were calculated under the assumption that all females in the population adopted the same strategy k (t, s). This assumption is not actually possible in a two-person game, although it can be approximated in a game that has a large number of players. If the strategy of a mutant is largely different from that of the resident, a discrepancy arises between the real and the calculated frequencies on a patch in which a mutant and a resident lay eggs. However, if the strategies approach a convergence point by the damping procedure, the difference between the real and expected frequencies gradually decreases. When the damping completely converges in Equation 5, all players take the same strategy and the frequency differences disappear. Therefore, if the strategies converge to a point, we can regard the final converged strategy as the ESS. In all cases we examined here, the sex ratio strategies converged to such a point. We confirmed that the average ESS sex ratio converged at 0.25 and 0.214 in diploid and haplo-diploid genetic systems, respectively (the ESS sex ratio with two females in Hamilton 1967, 1979; Taylor and Bulmer 1980) when our model did not have discrete time steps (i.e., T = 1) and did not assume lethal male combat.
|
RESULTS |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Although the ESS strategies
(t, s) and the expected frequencies F(t, s) were calculated for all possible female states (i.e., the number of remaining male offspring, s), the results below are represented by the averages of sex ratios (the proportion of males) and the expected number of remaining male offspring for each time step (see Equations A6 and B6 in appendix, respectively). All our results assume haplo-diploid organisms—the same qualitative pattern is predicted for diploid species, but with slightly less female-biased sex ratios. Sex ratios declined over the oviposition period, regardless of the specific parameters used (Figure 3). The highest sex ratios are predicted to occur at the beginning of oviposition (Figure 3A), when new males are able to survive with a high probability owing to reduced male combat. In addition, females favor early production of sons rather than daughters because sons stay in the patch and acquire reproductive success during later time steps, whereas daughters' reproductive success is independent of time. However, sex ratios are still female biased at the beginning (Figure 3A).
|
As the remaining males are accumulated over the time steps (Figure 3B), females produce more female-biased sex ratios. Newly produced males have little chance of survival to the subsequent time step. Females, however, do not stop male production and continue to produce males at a low rate over time. The accumulation of remaining males stops at a marginal value where the advantage of having more remaining males with a high death rate is balanced by the advantage of having females that constantly gain reproductive success (Figure 3B). After this time, females continue producing just enough new males to maintain this balance. As the emergence period draws to the last time step, the advantage of new male production decreases because remaining males do not acquire reproductive success after the period, and so the females preferentially produce daughters rather than sons. In all cases, the model predicts that sex ratios should approach 0 at the final time step (Figure 3A).
The survival rate of new males has a large influence on the predicted sex ratio (Figure 3). When the survival rate of new males steeply decreases with increasing male number (large bn; Figure 2), females are predicted to produce more female-biased sex ratios over the emergence period up until the last few steps (Figure 3A). As a result, the marginal number of remaining males decreases (Figure 3B).
The survival rate of remaining males also influences the predicted sex ratios (Figure 4). In the early part of the emergence period, females produce almost the same sex ratio independently of the survival rate (Figure 4A). As the time step progresses, a larger number of remaining males accumulates with higher survival rate (Figure 4B), such that new males suffer increased mortality, favoring the production of daughters (Figure 4A). As the survival rate of remaining males decreases, females produce higher sex ratios and continue producing new males, even near the last time step (Figure 4A). When the survival rate of remaining and new males is very similar (i.e., there is no asymmetry; bs = bn = 0.25), the optimal sex ratios of each time step are predicted to be close to 21.4% (Figure 4; Hamilton 1979).
|
When females are able to produce a large number of offspring at each time step (higher N), the optimal sex ratio is reduced (Figure 5). Because the survival rates of both types of males are assumed to be influenced by the total male number on the patch (Figure 2), optimal numbers of new males are mainly determined by male number (not proportion). Females should produce almost the same number of new males regardless of their offspring number (Figure 5B). If females can lay a large number of offspring in a given time step, they invest in females (Figure 5A,B).
|
Finally, the effect of the total number of time steps, T, had little effect on the total sex ratios (Figure 6). Optimal sex ratios are similar in the early part of the emergence period regardless of the length of the oviposition period with females favoring female production at the end of the oviposition period (Figure 6A,B). As a result, more female-biased total sex ratios are predicted for shorter time steps, but the effect is small, except over very small numbers of time steps.
|
|
DISCUSSION |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
General predictions from the model
In our model, the difference in survival between newly emerged males and their already emerged counterparts drives the production of extremely female-biased sex ratios. Females favor a low rate of male production because this large asymmetry in survival reduces the value of producing sons and hence increases the relative advantage of daughters. This effect does not vary to any great extent with the number of time steps that make up the oviposition period or clutch size. However, females do not completely stop producing sons, even when new male offspring are highly likely to be killed. This is because, if a male survives, they can gain an extremely large reproductive success due to the highly female-biased sex ratio. Therefore, females maintain a low level of male production, until close to the end of the oviposition period, when male production tails off. As the asymmetry of competitive ability decreases, females invest in more males and the sex ratios at each time step approach 21.4% males (Figure 4) as predicted by the original LMC model that assumes symmetric mate competition without temporal schedules in haplo-diploid species (Hamilton 1979
).
Several other models have examined different types of competitive asymmetries among adult males on a local patch under conditions of LMC. Godfray (1986) examined the influence of asymmetric competition between siblings of the two sexes. When the sexes have different effects on individual fitness through resource competition, the sex ratio is predicted to be biased toward the sex that reduces the negative effect of the competition (Godfray 1986; Sykes et al., forthcoming). Asynchronous emergence on a patch also influences the extent of LMC. When females produce their broods asynchronously and emerging males remain on the natal patch, the extent of LMC increases and the optimal sex ratio is more female biased than predicted by the original LMC model (Nunney and Luck 1988). Although this model assumes that all individual females behave the same, other models allow individuals to produce different sex ratios (Abe, Kamimura, Ito, et al. 2003; Shuker et al. 2005). Because males emerging from later broods more often compete for mates with unrelated males (relatively weaker LMC), optimal sex ratios are predicted to be less female biased in later broods (Shuker et al. 2005). Shuker et al. (2005) and Shuker, Pen, et al. (2006) termed this situation asymmetrical LMC and showed that the parasitoid wasp Nasonia vitripennis produces different offspring sex ratios in the directions predicted by their theoretical model.
Comparison with empirical data in Melittobia wasps
Melittobia species exhibit extremely female-biased sex ratios that cannot be explained by LMC or other preexisting models (Werren 1987; Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; Cooperband et al. 2003; Innocent et al., forthcoming). Lethal male–male combat means that later emerging males are killed (Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005), yielding a large asymmetry in competitive abilities depending on emergence time. Our model predicts that the sex ratio decreases from 21.4% toward extreme female bias as the asymmetric competitive abilities between remaining and new males increases. The observed emergence patterns in Melittobia, with a small number of males emerging across the emergence period (Abe et al. 2005) and a slightly less female-biased sex ratio at the beginning of the emergence period (Innocent et al., forthcoming), are also consistent with our model. Moreover, males that win the fights are also larger (Hartley and Matthews 2003; Innocent et al., forthcoming; Reece et al., forthcoming). Because individual offspring tend to be smaller over the course of their emergence period, the asymmetric competitive ability between early and late males is expected to be even greater. A crucial next step will be to estimate the parameters in our model and especially the survival rate functions when there are multiple males in a patch.
Wakano (2005) similarly analyzed time-dependent sex allocation and predicted a low rate of male production over the emergence period. He assumed that all emerged males suffered an equal risk of death from lethal male combats. To generate biased sex ratios, males must have exclusive territories and superfluous males need to be strictly killed. However, such behavior has not been recorded from any Melittobia species (Dahms 1984; González et al. 2004; González and Matthews 2005). Our models, which rely on asymmetrical competitive ability among emerging males (Abe, Kamimura, Kondo, et al. 2003; Abe et al. 2005; Innocent et al., forthcoming), therefore have more plausible underlying biological assumptions to explain the highly female-biased sex ratios observed in species of wasps such as Melittobia.
Applications of the model
Trivers and Willard (1973) suggested that females should adjust their offspring sex ratios in response to different environmental conditions. If oviposition order differently influences the reproductive successes of sons and daughters, different sex ratios can be expected, depending on the order in which eggs hatch. Several empirical reports show that female birds actually adjust the sex of their eggs according to the order within broods (Cockburn 2002). In these species, females can maximize fitness by avoiding oviposition of the less suitable sex depending on subsequent brood interactions between or within sexes (Bortolotti 1986; Nager et al. 1999; Legge et al. 2001; Badyaev et al. 2002; Janssen et al. 2006). For example, in a carpenter bee species where female offspring guard the nest after their emergence, females are preferentially laid earlier than males (Stark 1992).
Our model would apply equally well to other species whenever asymmetrical mate competition occurring among males and females can estimate the competitive ability of their sons before oviposition. Nonpollinating fig wasps from several different families show severe lethal combat when males compete locally for mates (Hamilton 1979; West et al. 2001). Whereas pollinating fig wasps oviposit synchronously, nonpollinating fig wasps lay eggs from the outside of the fruit over a longer period. Asymmetries might arise from the differential development and emergence of males depending on their oviposition time. In fact, there are some nonpollinating fig wasp species that show more female-biased sex ratios than expected under LMC predictions (Herre et al. 1997; West and Herre 1998). In addition, Cardiocondyla ants have so-called ergatoid males that do not disperse from their maternal nest and instead fight aggressively for access to females. In several species, adult ergatoid males kill all other newly emerging ergatoid males, and only one or a small number of ergatoid males survive and monopolize mates in the nest (Heinze et al. 1998; Cremer and Heinze 2002; Schrempf et al. 2005). In this situation, sex allocation is likely to have evolved in response to similar asymmetries. Indeed, different production schedules of ergatoid males do occur in the predicted direction, with ergatoid males produced earlier in multiqueen colonies than in single-queen colonies (Yamauchi et al. 2006). Finally, our model could apply to a broad range of species, such as Ozopemon bark beetles or various mite species in which offspring sex ratios are extremely female biased and males fight fiercely under LMC conditions (Jordal et al. 2002; Sabelis et al. 2002).
|
FUNDING |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Research Fellowship of the Japan Society for the Promotion of Science for Young Scientists (181102 to J.A.).
|
APPENDIX A |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Calculation of expected frequencies at all possible states
Expected frequencies of females having s remaining males, F(t, s), can be calculated using forward iteration from t = 1 to t = T, assuming all females in the population adopt the same strategy
k (t, s) (although this approximation is not exactly achieved in a two-person game; see the main text for a more detailed explanation). Because the frequency F(t, s) is the probability distribution of the female's state (i.e., the number of her own remaining males, s),
 | (A1) |
At the beginning of time step (t = 1) in which the two females start laying offspring, they do not have any remaining males:
 | (A2) |
First, let Pr(z, s, n, x) denote the probability that a total of x males survive at a time step in which a female has s remaining males and lays n new males in a patch containing z males. If we suppose v males survive out of the s remaining males, x – v males need to survive out of n new males (x – v = n). Then, the probability is given by
 | (A3) |
+ n + ), where a hat designates the sons of the opponent female.
The expected frequency F(t, s) is derived from the frequency F(t – 1, s) and the strategy (t – 1, s) at the last time step. Summing all possible situations with regard to expected own state s, opponent state , own new male number n, and opponent new male number at the last time step t – 1, the total frequency is calculated by
 | (A4) |
To facilitate alleviation of arithmetic processing, we used smaller values for M (e.g., M = 20) and employed the "chop" function (Mangel and Clark 1988):
 | (A5) |
 | (A4') |
From the initial distribution of Equation A2, the frequencies at all subsequent time steps in all possible states are calculated in a stepwise process using Equation A4'. Although we calculated all possible F(t, s)(1 t T, 0 s M), only the averages of remaining male numbers for each time step were represented in the results:
 | (A6) |
|
Appendix B |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Calculation of the best response strategy
Best response strategy B(t, s) can be calculated with backward iteration from t = T to t = 1, given the current frequencies F(t, s) and the current population strategy
k(t, s). First, we define W(t, s,n) as the expected fitness of a female that gained between time step t and terminal period T, when the female having s remaining males lays n new males at the time step t. Then, W(T, s, n) is the fitness gained only at the final time step, so that
 | (B1) |
). When s + n > 0, the expected fitness is obtained from both the sons and daughters. If the focal female does not have any remaining males and does not produce new males (s + n = 0), she acquires fitness only from her daughters under the condition that the opponent female has at least one remaining male or produces one new male. Because we assume species are haplo-diploid here, vf = 2/3 and vm = 1/3 (Taylor 1996), Rf = (1+3f)/(2+2f), where f is the inbreeding coefficient, and Rm = 1 (Taylor, 1993). The inbreeding coefficient, f, depends on the proportion of sibmating. When two females lay eggs on a patch and there is no asymmetry between them, f = (1/5) (Taylor 1993).
When t T – 1, W(t, s, n) can be divided into the fitness element gained at time step t and after the subsequent time steps:
 | (B2) |
 | (B3) |
).
To facilitate convergence, we introduced errors in decision making (McNamara et al. 1997). If there are no errors, the best response strategy always chooses an action that maximizes fitness (maximal W(n) (0 n N)). In contrast, the error-prone best response B = (p' (0), p '(1),..., p'(n),..., p'(N)) is calculated as follows
 | (B4) |
represent the degree of error and C(i) is a measure of the loss in fitness as a result of choosing action i:
 | (B5) |
make convergence easier, whereas smaller values yield a more accurate response strategy (there is no error when = 0). We adopted = 0.1 throughout this study.
The chop function was also used for the calculation of best response (see Equation A5 in Appendix A). Then, Equations B1 and B2 are substituted with
 | (B1') |
 | (B2') |
By repeating the process with Equations B2', B3, and B4 from the terminal time step (Equation B1'), the best response at each time step is identified. The strategies for each time step were represented as the average sex ratio in the results:
 | (B6) |
|
ACKNOWLEDGEMENTS |
|---|
We are very grateful to H. Ito and M. Setoyama for their helpful programming suggestions and to S. A. West, D. M. Shuker, A. Gardner, S. A. Reece, J. Y. Wakano, N. E. Pierce, and two anonymous referees for their valuable comments on previous versions of the manuscript.
|
REFERENCES |
|---|
|
TOPABSTRACTINTRODUCTIONMODELRESULTSDISCUSSIONFUNDINGAPPENDIX AAppendix BREFERENCES |
|---|
Abe J, Kamimura Y, Ito H, Matsuda H, Shimada M. Local mate competition with lethal male combat: effects of competitive asymmetry and information availability on a sex ratio game. J Evol Biol (2003) 16:607–613.[CrossRef][Web of Science][Medline]
Abe J, Kamimura Y, Kondo N, Shimada M. Extremely female-biased sex ratio and lethal male-male combat in a parasitoid wasp, Melittobia australica (Eulophidae). Behav Ecol (2003) 14:34–39.
Abe J, Kamimura Y, Shimada M. Individual sex ratios and offspring emergence patterns in a parasitoid wasp, Melittobia australica (Eulophidae), in relation to superparasitism and lethal combat between sons. Behav Ecol Soc (2005) 57:366–373.[CrossRef][Web of Science]
Adams RF. Dynamics of daily progeny production in Melittobia digitata Dahms: baseline studies. Undergrad Sci Bull (2002) 3:17–21.
Badyaev AV, Hill GE, Beck ML, Dervan AA, Duckworth RA, McGraw KJ, Nolan PM, Whittingham LA. Sex-biased hatching order and adaptive population divergence in a passerine bird. Science (2002) 295:316–318.
Bortolotti GR. Influence of sibling competition on nestling sex ratios of sexually dimorphic birds. Am Nat (1986) 127:495–507.[CrossRef][Web of Science]
Charnov EL. The theory of sex allocation (1982) Princeton (NJ): Princeton University Press.
Clark CW, Mangel M. Dynamic state variable models in ecology: methods and applications (2000) New York: Oxford University Press.
Cockburn A, Legge S, Double MC. Sex ratios in birds and mammals: can the hypotheses be disentangled? In: Sex ratios: concepts and research methods—Hardy ICW, ed. (2002) Cambridge (MA): Cambridge University Press. 266–286.
Cooperband MF, Matthews RW, Vinson SB. Factors affecting the reproductive biology of Melittobia digitata and failure to meet the sex ratio predictions of Hamilton's local mate competition theory. Entoml Exp Appl (2003) 109:1–12.[CrossRef]
Cremer S, Heinze J. Adaptive production of fighter males: queens of the ant Cardiocondyla adjust the sex ratio under local mate competition. Proc R Soc Lond B (2002) 269:417–422.[Medline]
Dahms EC. A review of the biology of species in the genus Melittobia (Hymenoptera: Eulophidae) with interpretations and additions using observations on Melittobia australica. Mem Old Mus (1984) 21:337–360.
Fellowes MDE, Compton SG, Cook JM. Sex allocation and local mate competition in Old World non-pollinating fig wasps. Behav Ecol Soc (1999) 46:95–102.[CrossRef][Web of Science]
Frank SA. Hierarchical selection theory and sex ratios. II. On applying the theory, and a test with fig wasps. Evolution (1985) 39:949–964.[CrossRef][Web of Science]
Godfray HCJ. Models for clutch size and sex ratio with sibling interaction. Theor Popul Biol (1986) 30:215–231.[CrossRef][Web of Science]
Godfray HCJ. Parasitoids: behavioral and evolutionary ecology (1994) Princeton (NJ): Princeton University Press.
González JM, Matthews RW. An annotated bibliography of Melittobia (Hymenoptera: Eulophidae). Caribb J Sci (2005) 8:1–41.
González JM, Terán JB, Matthews RW. Review of the biology of Melittobia acasta (Walker) (Hymenoptera: Eulophidae) and additions on development and sex ratio of the species. Caribb J Sci (2004) 40:52–61.
Hamilton WD. Extraordinary sex ratios. Science (1967) 156:477–488.
Hamilton WD. Wingless and fighting males in fig wasps and other insects. In: Sexual selection and reproductive competition in insects—Blum MS, Blum NA, eds. (1979) New York: Academic Press. 167–220.
Hardy ICW. Sex ratios: concepts and research methods (2002) Cambridge: Cambridge University Press.
Hartley CS, Matthews RW. The effect body size on male-male combat in the parasitoid wasp Melittobia digitata Dahms (Hymenoptera: Eulophidae). J Hymeno Res (2003) 12:272–277.
Heinze J, Hölldobler B, Yamauchi K. Male competition in Cardiocondyla ants. Behav Ecol Soc (1998) 42:239–246.[CrossRef][Web of Science]
Herre EA. Sex ratio adjustment in fig wasps. Science (1985) 228:896–898.
Herre EA, West SA, Cook JM, Compton SG, Kjellberg F. Fig-associated wasps: pollinators and parasites, sex-ratio adjustment and male polymorphism, population structure and its consequences. In: The evolution of mating systems in insects and arachnids—Choe JC, Crespi BJ, eds. (1997) Cambridge: Cambridge University Press. 226–239.
Houston AI, McNamara JM. Models of adaptive behaviour: an approach based on state (1999) Cambridge (MA): Cambridge University Press.
Innocent TM, Savage J, West SA, Reece SE. Lethal combat and sex ratio evolution in a parasitoid wasp. Behav Ecol (2007) 18:709–715.
Janssen K, Erikstand KE, Bensch S. Offspring sex ratio allocation in the parasitic jaeger: selection for pale females and melanic males? Behav Ecol (2006) 17:236–245.
Jordal BH, Beaver RA, Normark BB, Farrell BD. Extraordinary sex ratios and the evolution of male neoteny in sib-mating Ozopemon beetles. Biol J Linn Soc (2002) 75:353–360.[CrossRef][Web of Science]
Kinoshita M, Kasuya E, Yahara T. More highly female-biased sex ratio in the fig wasp, Blastophaga nipponica Grandi (Agaonidae). Res Popul Ecol (1998) 40:239–242.
Legge S, Heinsohn R, Double MC, Griffiths R, Cockburn A. Complex sex allocation in the laughing kookaburra. Behav Ecol (2001) 12:524–533.
Mangel M, Clark CW. Dynamic modeling in behavioral ecology (1988) Princeton (NJ): Princeton University Press.
McNamara JM, Webb JN, Collins EJ, Székely T, Houston AI. A general technique for computing evolutionarily stable strategies based on errors in decision-making. J Theor Biol (1997) 189:211–225.[CrossRef][Web of Science][Medline]
Moore JC, Compton SG, Hatcher MJ, Dunn AM. Quantitative tests of sex ratio models in a pollinating fig wasp. Anim Behav (2002) 64:23–32.[CrossRef][Web of Science]
Nager RG, Monaghan P, Griffiths R, Houston DC, Dawson R. Experimental demonstration that offspring sex ratio varies with maternal condition. Proc Natl Acad Sci USA (1999) 96:570–573.
Nunney L, Luck RF. Factors influencing the optimum sex ratio in structured populations. Theol Popul Biol (1988) 33:1–30.[CrossRef]
Reece SE, Shuker DM, Pen I, Duncan AB, Choudhary A, Batchelor CM, West SA. Kin discrimination and sex ratios in a parasitoid wasp. J Evol Biol (2004) 17:208–216.[CrossRef][Web of Science][Medline]
Reece SE, Innocent TM, West SA. Forthcoming. Lethal male-male combat in the parasitoid Melittobia acasta: are size and competitive environment important? Anim Behav.
Sabelis NW, Nagelkerke CJ, Breeuwer JAJ. Sex ratio control in arrhenotokous and pseudo-arrhenotokous mites. In: Sex ratios: concepts and research methods—Hardy ICW, ed. (2002) Cambridge (MA): Cambridge University Press. 266–286.
Schrempf A, Reber C, Tinaut A, Heinze J. Inbreeding and local mate competition in the ant Cardiocondyla batesii. Behav Ecol Soc (2005) 57:502–510.[CrossRef][Web of Science]
Shuker DM, Reece SE, Taylor JAL, West SA. Wasp sex ratios when females on a patch are related. Anim Behav (2004) 68:331–336.[CrossRef][Web of Science]
Shuker DM, Pen I, Duncan AB, Reece SE, West SA. Sex ratios under asymmetrical local mate competition: theory and a test with parasitoid wasps. Am Nat (2005) 166:301–316.[CrossRef][Web of Science][Medline]
Shuker DM, Pen I, West SA. Sex ratio under asymmetrical local mate competition in the parasitoid wasp Nasonia vitripennis. Behav Ecol (2006) 17:345–352.
Shuker DM, Sykes EM, Browning LE, Beukeboom LW, West SA. Male influence on sex allocation in the parasitoid wasp Nasonia vitripennis. Behav Ecol Soc (2006) 59:829–835.[CrossRef][Web of Science]
Stark RE. Sex ratio and maternal investment in the multivoltine large carpenter bee Xylocopa sulcatipes (Apoidea: Anthophoridae). Ecol Entomol (1992) 17:160–166.
Stubblefield JW, Seger J. Local mate competition with variable fecundity: dependence of offspring sex ratios on information utilization and mode of male production. Behav Ecol (1990) 1:65–80.[Medline]
Suzuki Y, Iwasa Y. A sex ratio theory of gregarious parasitoids. Res Popul Ecol (1980) 22:366–382.[CrossRef]
Sykes EM, Innocent TM, Pen I, Shuker DM, West SA. Asymmetric larval competition in the parasitoid wasp Nasonia vitripennis; a role in sex allocation? Behav Ecol Soc (2007) 61:1751–1758.[CrossRef][Web of Science]
Taylor PD. Intra-sex and inter-sex sibling interactions as sex determinants. Nature (1981) 291:64–66.[CrossRef]
Taylor PD. Female-biased sex ratios under local mate competition: an experimental confirmation. Evol Ecol (1993) 7:306–308.[CrossRef]
Taylor PD. Inclusive fitness arguments in genetic models of behaviour. J Math Biol (1996) 34:654–674.[Web of Science][Medline]
Taylor PD, Bulmer MG. Local mate competition and the sex ratio. J Theor Biol (1980) 86:409–419.[CrossRef][Web of Science][Medline]
Trivers RL, Willard DE. Natural-selection of parental ability to vary sex-ratio of offspring. Science (1973) 179:90–92.
Wakano JY. Evolution of extraordinary female biased sex ratios: the optimal schedule of sex ratio in local mate competition. J Theor Biol (2005) 237:193–202.[CrossRef][Web of Science][Medline]
Werren JH. Sex ratio adaptations to local mate competition in a parasitic wasp. Science (1980) 208:1157–1159.
Werren JH. Labile sex ratios in wasps and bees: life history influences the ratio of male and female offspring. Bioscience (1987) 37:498–506.[CrossRef][Web of Science]
West SA, Herre EA. Partial local mate competition and the sex ratio: a study on non-pollinating fig wasps. J Evol Biol (1998) 11:531–548.[CrossRef][Web of Science]
West SA, Murray MG, Machado CA, Griffin AS, Herre EA. Testing Hamilton's rule with competition between relatives. Nature (2001) 409:510–513.[CrossRef][Medline]
West SA, Shuker DM, Sheldon BC. Sex-ratio adjustment when relatives interact: a test of constraints on adaptation. Evolution (2005) 59:1211–1228.[CrossRef][Web of Science][Medline]
Yamaguchi Y. Sex ratios of an aphid subject of local mate competition with variable maternal fecundity. Nature (1985) 318:460–462.[CrossRef]
Yamauchi K, Ishida Y, Hashim R, Heinze J. Queen-queen competition by precocious male production in multiqueen ant colonies. Curr Biol (2006) 16:2424–2427.[CrossRef][Web of Science][Medline]
CiteULike 
Connotea 
Del.icio.us What's this?
This article has been cited by other articles:
|
|
 |
|
  S. Lebreton, C. Chevrier, and E. Darrouzet Sex allocation strategies in response to conspecifics' offspring sex ratio in solitary parasitoids Behav. Ecol., January 1, 2010; 21(1): 107 - 112. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 M. Suefuji, S. Cremer, J. Oettler, and J. Heinze Queen number influences the timing of the sexual production in colonies of Cardiocondyla ants Biol Lett, December 23, 2008; 4(6): 670 - 673. [Abstract] [Full Text] [PDF]
|
|
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||