SPERM COMPETITION GAMES: A GENERAL MODEL FOR PRECOPULATORY MALE–MALE COMPETITION

Authors


Abstract

Reproductive males face a trade-off between expenditure on precopulatory male–male competition—increasing the number of females that they secure as mates—and sperm competition—increasing their fertilization success with those females. Previous sperm allocation models have focused on scramble competition in which males compete by searching for mates and the number of matings rises linearly with precopulatory expenditure. However, recent studies have emphasized contest competition involving precopulatory expenditure on armaments, where winning contests may be highly dependent on marginal increases in relative armament level. Here, we develop a general model of sperm allocation that allows us to examine the effect of all forms of precopulatory competition on sperm allocation patterns. The model predicts that sperm allocation decreases if either the “mate-competition loading,”a, or the number of males competing for each mating, M, increases. Other predictions remain unchanged from previous models: (i) expenditure per ejaculate should increase and then decrease, and (ii) total postcopulatory expenditure should increase, as the level of sperm competition increases. A negative correlation between a and M is biologically plausible, and may buffer deviations from the previous models. There is some support for our predictions from comparative analyses across dung beetle species and frog populations.

All else being equal, reproductive males should maximize sperm numbers when facing sperm competition from other males. However, males are likely to experience a trade-off between precopulatory competition for mates and sperm competition: males spending more on sperm may have less to spend on competing for matings. Some evidence exists for this trade-off. For instance, male bluehead wrasse, Thalassoma bifasciatum, with higher mating success release fewer sperm per mating, with consequent reduction in fertilization success; feeding experiments suggested that high mating success males diverted energy from gamete production into mate-competition activities such as mate guarding (Warner et al. 1995).

Sperm competition games (Parker 1998; Parker and Pizzari 2010) assume a trade-off between ejaculate expenditure (i.e., on postcopulatory competition such as testes size, sperm number, sperm size) and expenditure on acquiring matings (i.e., on precopulatory competition such as mate searching). Under sperm competition, increased sperm expenditure increases a male's fertilization success per mating, v, but decreases his expected number of matings, n. Analyses use evolutionarily stable strategy (ESS, Maynard Smith 1982) approaches because the fitness, w, of a given sperm allocation strategy depends on the strategies played by other males in the population. ESSs are found by competitive maximization of the product, w=nv.

Previous sperm competition game models extensively investigated how predictions vary when the expected value of a mating depends on the information available to males. For example, sperm allocation changes if males can detect prior mating by a female, or how much sperm her sperm stores contain (Simmons 2001; Wedell et al. 2002; Parker and Pizzari 2010). But how do predictions change under varying assumptions about male–male competition? Though not usually stated explicitly, a male's number of matings is assumed to increase linearly with expenditure on precopulatory competition, as in processes such as scramble competition through mate searching, where the expected number of females found and mated increases linearly with male search rate or time. Though useful for generating simple, tractable models, the question remains whether the conclusions are robust under other forms of precopulatory competition. Darwin (1871) stressed that competition often involves intense male–male combat. Small unilateral increases in armament above the population mean may here yield high increases in winning probability, and hence large increases in a male's number of matings. Does contest competition alter predictions about sperm allocation?

Here we develop a general model of sperm allocation incorporating the trade-off between precopulatory and sperm competition, encompassing the range of male–male competition from contest to scramble. We compare effects of male–male (contest) competition with those of mate searching (scramble competition) on evolutionarily stable (ES) sperm allocation. Previous models (Parker and Pizzari 2010) that assume males acquire matings by scrambles are shown to be a special case of our model. We review data supporting our assumption of a trade-off between expenditures on sperm production and on armaments or ornaments used during precopulatory contest competition, and provide preliminary evidence for our predictions in species characterized by contest competition between males for mates.

The Model

We first propose a more general form for models of sperm allocation under sperm competition (for a review, see Parker and Pizzari 2010). As earlier (e.g., Parker and Ball 2005), we assume that (i) females determine how often they mate, and hence the sperm competition level faced by males (male control would yield ESS sperm allocations different from those under female control, Fromhage et al. 2008; see Discussion), and (ii) males face a trade-off between precopulatory competition (male–male mate competition) and postcopulatory competition (sperm competition). This trade-off occurs because, in the time it takes each female to complete one reproductive cycle, each male has a fixed total energy budget for reproduction, R, which he divides between his allocation, T, to precopulatory competition (searching, armament, etc.), and his allocation, U, to postcopulatory competition (ejaculate, testes, etc.), so that:

image(1)

where a hat (e.g., inline image) indicates the population level of a variable, and an unmodified symbol (e.g., T) indicates the value for a mutant playing a different strategy from the rest of the population.

A male's success in male–male competition, and hence his number of matings gained, n, is determined by both his own precopulatory expenditure and that of the rest of the population. A male playing the population-level precopulatory allocation inline image, gains inline image matings, that is, inline image is the population average, whereas a mutant playing T in a population playing inline image gains inline image matings. Note that the function inline image encompasses all forms of precopulatory competition from scramble (e.g., random search) to contest (e.g., male–male combat).

Similarly, a male's success in postcopulatory competition, v, depends on his own postcopulatory expenditure and that of the rest of the population, and also on the number of matings per female (also inline image, since we assume a sex ratio of unity), which determines the level of sperm competition. Thus, a mutant playing U in a population playing inline image gains an expected value per mating of inline image. As in earlier models, a male's fitness, w, is the product of his number of matings, and the value per mating:

image(2)

THE TRADE-OFF BETWEEN PRE- AND POSTCOPULATORY COMPETITION

We seek ES allocations to precopulatory competition, T*, and postcopulatory competition, U*, taking into account the trade-off between these allocations. We assume that males produce ejaculates each containing s units of sperm, and that each unit of sperm costs D energy units, so (1) becomes

image(3a)

that is,

image(3b)

and mutant fitness is

image(4)

The ESS value for s* (and hence T* and U*) is given by inline image, evaluated at inline image, subject to inline image. Thus, from (4)

image(5)

(i.e., Parker 1990a, eq. 5). Depending on function inline image, (5) can sometimes be solved for s* analytically, and otherwise by numerical iteration (see below).

PRECOPULATORY COMPETITION

Earlier models used a specific form for male precopulatory competition, inline image, in which a male's number of matings rises linearly with his precopulatory expenditure, as may occur in competitive mate searching. We next introduce a general form which allows investigation of a wide range of categories from contest to scramble, and competitive mate searching becomes a special case.

To find s*, we need a form for inline image that specifies how a mutant male's number of matings (n) varies with his precopulatory expenditure (T) and the total number of males (M, including himself) that compete for each mating opportunity. We assume that:

image(6)

where a (the “mate-competition loading”) is a parameter determining the competitive weights of males in relation to their expenditure on precopulatory competition. This function has the following desirable properties: (a) the number of matings a mutant male can win, n, is bounded. When the average number of matings per male is inline image and M males compete for each mating, n must lie between 0 (never winning) and inline image (winning all mating opportunities for which he competes: if population males mate inline image times on average, and compete in groups of M males, they must each compete for an average of inline image mating opportunities). This also applies for mutant males, because we assume that precopulatory investment affects success in competition, but not the number of bouts of competition entered. (b) When two males compete for a female and have different values of T (call these T1 and T2), reversing the labels, T1 and T2, simply reverses the number of matings that each male achieves. (c) The sum of the probabilities that each of the M males secures the mating is one.

Note that (6) is the familiar result that an individual's gain matches his “competitive weight” (Ta, his precopulatory expenditure raised to the power a) divided by inline image, the summed competitive weights of all contestants. Competitive weights have been used in animal distribution studies (e.g., Parker and Knowlton 1980; Parker and Sutherland 1986).

Note also that our formulation can be interpreted as a developmental trade-off where any developmental armament investment T causes the resulting phenotype's sperm production to be reduced according to U=RT (e.g., because of small testes, or because of having to carry heavy armament).

Equation (6) can represent both scramble and contest competition. For contests between two males, the mutant's expected number of matings will be

image(7)

For scrambles, every male in an infinite population effectively competes for every copulation, that is, M→∞, so:

image(8)

with mutant search rate proportional to Ta.

The meaning of (6–8) is explained by Figure 1, showing how a mutant male's relative number of matings (inline image) varies with his relative precopulatory expenditure (inline image, plotted on a logarithmic scale). Figure 1A shows the effect of varying the mate-competition loading parameter a, using a contest between two males (M= 2) as an example. As a increases, the mutant's relative gain through marginally increasing T above inline image increases, that is, the steepness of the slope of inline image around inline image increases. This slope determines the ESS because it indicates the potential fitness gains to a mutant with a slightly different value of T from inline image.

Figure 1.

The effect of varying (A) the mate-competition loading, a, (with M= 2), and (B) the number of males in precopulatory competition, M, (with a= 1), on the relative number of matings gained by a mutant (inline image= mutant's number of matings/number of matings of a wild-type male) as a function of his relative (on a logarithmic scale) precopulatory expenditure (inline image= mutant's expenditure/wild-type expenditure). In (A) continuous curve is a= 1, broken curve a= 0.2, and dotted curve a= 5. In (B) continuous curve is M= infinity, broken curve M= 5, and dotted curve M= 2.

Figure 1B shows the effect of varying the number of males competing for each available mating, M, on the mutant's relative number of matings. Here, the slope of the curve for T values around inline image increases as M increases. This arises because (when a= 1, as shown in Figure 1B) gains are proportional to self's T divided by the total T for all competitors: with just two males, increases in self's T inflate the total T (the denominator of the right-hand side of (7)), reducing benefits; with vast numbers of males, self's T does not affect this denominator (see (8)).

In dyadic contests, the opponent with greater body size or armament commonly wins; high a therefore probably correlates with low M (contest competition typically has M= 2). In contrast, although scramble competition may also involve low M, it more typically involves many males (high M) and will often be characterized by much lower a (around or even below 1). For mate searching, having a= 1 means that the number of mates encountered is directly proportional to T, as applies when each search time unit has a fixed energetic cost. Alternatively, males might search harder by using more energy. This would be expected to yield diminishing returns (i.e., a < 1), because if it gave accelerating returns (i.e., a > 1), males would always use the most expensive search method. Because the slopes in Figure 1 for T values around inline image increase with both a and M, which are probably negatively correlated (M highest for extreme scrambles, and a highest for dyadic contests), there is no straightforward prediction as to whether this slope should generally be higher for scramble or contest competition.

Having specified how a male's number of matings varies with his precopulatory expenditure (6), we can combine it with the energetic constraint (3a) to give an equation for inline image

image(9)

We cannot derive a general expression for inline image from (9) to enable ESSs to be derived analytically via equation (5), though this can be done for certain specific cases. One is where a= 1 and M→∞, representing the simplest assumptions for scramble competition: that all males in an infinite population compete for all matings, and the number of females a male encounters (and mates, since there is no other form of precopulatory competition) is directly proportional to his relative precopulatory expenditure, T. Setting a= 1 and M=∞ in equation (9), the denominator then approximates to inline image, and we obtain

image(10)

This case is, in fact, the classic model (reviewed in Parker and Pizzari 2010), which was framed in terms of C, the cost of obtaining a mating, so that for a wild-type male inline image. By substituting inline image into equation (10), we recover the classic model:

image

(e.g., Parker et al. 1996, eq. 4). Similarly, substituting T*=n*C* into equation (10) yields:

image(11)

(e.g., Parker et al. 1997, eq. 6). We also note that substituting T=nC and inline image into equation (8) shows that, for scrambles in an infinite population with a= 1, inline image; that is, the costs of obtaining a mating are equal for mutant and wild-type males. This was an assumption of the classic model: that a mutant deviating in sperm allocation experiences no change in the costs of finding a receptive female.

POSTCOPULATORY COMPETITION

To find s*, we also need to specify the value of a mating, inline image, to a mutant male in relation to his own ejaculate size and that of any wild-type males with whom his sperm compete. For mathematical tractability, we assume inline image has the same form as in many previous treatments (Parker and Pizzari 2010); that is, that (i) females are not sperm limited, so full fertility is achieved with an arbitrarily tiny amount of sperm, , where ∂→ 0, (ii) females control the number of matings they make (i.e., males investing more in precopulatory competition win more matings, but do not influence the number of times a female mates, and hence the number of mating opportunities), and (iii) we use two models that differ in how matings are distributed across females. In the “risk” model, for fertilization of a given clutch, females mate either once with probability (1 −q), or twice with probability q. This is used for low levels of sperm competition (q≤ 1; i.e., inline image≤ 2). For higher levels of sperm competition, we use the “intensity” model, which assumes that before fertilization, all females mate with N males (N=inline image≥ 2). N represents the mean number of ejaculates that females in a population receive. Though this must be an integer for individual females, using N yields ESS values approximating to the mean ESSs for the population.

We used the risk and intensity models to generate a single continuous range, and to allow comparison with previous results (see Parker and Pizzari 2010). We could have used just the intensity model throughout, with the range from N= 1 to 2 replacing the risk range (q= 0 to 1). Results in this zone differ little between risk and intensity assumptions (e.g., see Parker and Ball 2005), and are identical at the extremes of the risk range (N= 1, q= 0, and N= 2, q= 1). The risk model is, however, more realistic for low probabilities of sperm competition, as occurs in many species, and readily enables effects of sperm precedence and varying information (e.g., virgin female, once-mated female) to be included. It offers a simple, analytically tractable way of incorporating variation between females in the number of matings. Some analyses assume that females receive a probability distribution of matings, the population sperm competition level increasing with the mean of this distribution (see Parker and Pizzari 2010); we do not consider such models here. In Figures 2–4, risk and intensity ranges are separated by a thick broken vertical line.

Figure 2.

ESS (A) expenditure per ejaculate, Ds*, and (B) postcopulatory expenditure (hence relative testes size), U*/R (=E*) (left axis), or precopulatory expenditure, T*/R (right axis) in relation to sperm competition level, inline image, when male competition, M, occurs independently of the incidence of twice mating by females. The sperm competition level (inline image) is shown on a log scale so that details of the risk range (to the left of the vertical broken line) are clearer. Continuous curves show results for a= 1, broken curves a= 0.2, and dotted curves a= 5. The upper curve of each pair in (A) and (B) is for dyadic contests (M= 2), and the lower curve for a scramble with infinite numbers of males (M=); thus, the heavy continuous curve gives the ESSs for the classical model. Results are for a fair raffle (r= 1), with D= 1 and R= 10.

We make the further simplification that the sex ratio is unity. Under risk, the average number of matings per female cycle is inline image, hence with constraint (3a)

image(12a)

cf. Parker and Ball (2005), Parker et al. (2010), who use inline image. Under the intensity model, the average matings per female cycle is inline image, hence from (3a)

image(12b)

Risk

This model allows “loaded raffles” to be modeled (see Parker and Pizzari 2010). If a female mates with two different males, the probability of each egg being fertilized by male 1 in the favored role (e.g., being the second male to mate) is s1/(s1+rs2), and by male 2 in the disfavored role is rs2/(s1+rs2), where r (0 < r < 1) is the loading factor against the disfavored male during fertilization. For a mutant male playing inline image:

image

arising as follows: proportion (1 −q) of females mate only once. Since there is no sperm competition, the mutant male fertilizes all their eggs. Proportion q of females mate twice, and here he has an equal probability of being in the favored or disfavored role. Summing these three mating conditions for the male gives (1 +q), hence division by (1 +q) gives his expected fertilization probability per mating. ESS equation (5) therefore contains

image(13a)

In the models presented below, we assume that r= 1 (a fair raffle), so that at the ESS:

image

and hence

image(13b)

Intensity

Here, the raffle loading does not affect the average value of a mating:

image

and ESS equation (5) contains

image(14a)

that is,

image(14b)

cf. (13b). Thus, although the value of a mating to a population male is equivalent (=inline image) under risk and intensity assumptions (it must, because it is controlled by females), the value of [dv(s, s*)/ds]/v(s*, s*), and hence ESS values, s*, T*, and U*, usually differ between risk and intensity models (cf. eqs. 13b and 14b). ESSs do not differ when q= 1, N= 2 (i.e., inline image= 2).

Previous models (Parker and Pizzari 2010) analyzed cases where males have information about sperm competition, which allow conditional ESSs that change the average value of a mating, v. Our interest here is to investigate how varying forms of precopulatory competition affect sperm allocation, so we examine the simplest case, where males have no information. In sperm competition games where males allocate sperm conditionally in response to information, the average ESS allocation sometimes equals that for no information (Parker et al. 1997).

FINDING THE ESS

To examine how different forms of precopulatory competition affect sperm allocation patterns, we have predicted ESS levels of three variables (see Parker and Ball 2005): (i) “expenditure per ejaculate,”Ds*, (ii) “postcopulatory expenditure,”E*= U*/R, the ESS proportion of reproductive expenditure spent on the ejaculate (across species, E* has been shown to be proportional to relative testes size; Parker and Ball 2005), and (iii) “precopulatory expenditure,”T*/R, that is, the additive inverse of postcopulatory expenditure (T*/R= 1 −E*). We found ESS values either analytically using equation (5) or by numerical iteration.

Analytical solutions: The basic scramble model (a= 1, M→∞)

Where the mate-competition loading, a, equals 1, and the number of males competing for each mating, M, is vast, ESS postcopulatory expenditure can be solved analytically:

image

For the risk model, calling α≡ 2r/(1 +r)2 and setting (13a) = (11),

image(15a)

and for a fair raffle where r= 1, α= 0.5, and so inline image.

Similarly for the intensity model, setting (14a) = (11) gives

image(15b)

For the risk model, ESS expenditure per ejaculate (derived using (3a) and (13a)), is

image(16a)

and for the intensity model (derived using (3a) and (14a)),

image(16b)

All these solutions have been obtained for the classic model (see Parker et al. 1996, 1997; Parker and Ball 2005). Under the classic model (where a effectively equals 1), E* rises linearly with risk (inline image 15a), and monotonically with decreasing gradient with intensity (inline image, 15b). Ds* (inline image, 16a) rises across the risk range (0 < q <1), but falls (inline image, 16b) across intensities of N > 2. [Correction added after online publication August 17, 2012: In the previous sentence, E* was added before the word “rises”.]

We conclude that the classic model, on which most previous analyses were based, assumes a form of competition resembling scramble by mate searching in a large population, and where a male's number of matings rises linearly with his search effort relative to the mean search effort for the male population.

Iteration of numerical solutions

Because no analytical solution for inline image could be obtained from (9) when a≠ 1 and competition for local matings involves a small number M of male competitors, we found ESSs by iteration. Taking an initial value for the population value inline image, we used software package pro Fit to derive Twmax, that is, the mutant T value that yields the maximum value, wmax, of fitness inline image in the range between 0 and R. Depending on the values of q or N, and inline image, inline image has either a single peak or increases up to the maximum possible value of T (i.e., R). If Twmax>inline image, we repeated the calculations with gradually increasing values of inline image until Twmax converged on inline image. If Twmax<inline image, we decreased inline image until Twmax converged on inline image. In either case, at convergence wmax=w*= 1, inline image equals the ESS value T*, and s* and U* can then be found from equations (3a) and (1), respectively.

Results

We summarize results for our three ESS measures: expenditure per ejaculate, Ds*, postcopulatory expenditure, E*=U*/R (and hence its equivalents, relative ejaculate expenditure, relative testes size), and precopulatory expenditure, T*/R= (1 −U*/R).

ALL PARAMETERS INDEPENDENT

We first assess the effect of each of parameters a, inline image, and M acting independently. Figure 2 shows how each ESS measure varies with the level of sperm competition. Each pair of curves shows the limits for competing males (M= 2, M→∞), and the three pairs of curves each represent different values for the mate-competition loading (a= 1, 0.2, 5), which increases the advantage of a marginal increase in a mutant male's precopulatory expenditure above the population level (Fig. 1A).

Absolute expenditure per ejaculate, Ds*, increases across the lowest risk levels, and decreases across high intensity levels (Fig. 2A). The classic model (i.e., M→∞, a= 1) has its peak Ds* at q= 1, N= 2 (thick continuous curve). However, if a is low, Ds* peaks at lower values of q or N (especially if M is also low), and if a is high, Ds* peaks at higher values of q or N. Thus viewed across the entire sperm competition range, Ds* responds in a qualitatively similar way to the classic model, but different forms of precopulatory competition generate small deviations in the peak Ds*. At any given sperm competition level, Ds* increases with unilateral “decreases” in either a or M.

Postcopulatory expenditure (E*=U*/R, and hence its analogue, relative testes size) increases across both risk and intensity ranges (Fig. 2B), following classic results (Parker and Ball 2005; Parker and Pizzari 2010). As with Ds*, at any given sperm competition level, E* is increased by unilateral “decreases” in either a or M. Precopulatory expenditure, that is, T*/R*= 1 −U*/R, therefore declines across both risk and intensity ranges (Fig. 2B), and increases with unilateral “increases” in a or M.

However, we urge caution when predicting how ESS expenditure changes in the range between dyadic contest competition (M= 2) and extreme scramble competition (large M). We expect a negative correlation between a and M (see “Precopulatory competition”); the relation between these two parameters therefore requires empirical establishment before their effect on sperm expenditure can be determined.

COVARIANCE BETWEEN PARAMETERS

Our model assumes that each time a female mates, M males drawn at random from the population compete with each other, the winner mates with the female, and then they disperse. (Under extreme scramble competition, M is effectively infinity.) This means that it is possible for the levels of pre- and postcopulatory competition (respectively, M and inline image) to be independent; for example, females might mate only once with one male from a competing group of 10 (M= 10, inline image= 1), or, conversely, they might mate 10 times, each time with a male randomly chosen from the population, who has not had to compete with other males (M= 1, inline image= 10). However, although M and inline image need not covary, they might do so—for example, females may be more likely to mate multiply if more males engage in precopulatory competition, creating a positive correlation between M and inline image. Related complications are that the mate-competition loading (a) may co-vary with the sperm competition level (inline image) or the number of males (M) competing for a female.

Biological reasons for covariance between M and inline image are easy to imagine. For example, females might be denied the opportunity to mate multiply when only one male is present, generating a positive relation between M and inline image. The intensity of both precopulatory and postcopulatory competition may covary because of density dependence (Emlen and Oring 1977; Kokko and Rankin 2006). As population density increases, the encounter rate between individuals should increase, increasing male–male contests and mating opportunities for both males and females. Alternatively, but less plausibly, if females that mate multiply act as “hotspots,” attracting most males in the locality, a negative relation between M and inline image could be generated.

Covariance between a and inline image or M is also plausible. As population density increases, the efficiency of male–male contests in securing females may decrease and the cumulative costs increase (Knell 2009), selecting instead for scramble competition. We expect a to be highest in dyadic contests (M= 2), reducing to 1.0 or less as extreme scramble competition (high M) is approached. Density dependence may thus influence the direction of selection acting on male armaments and sperm production. In dung beetle communities, for example, the probability that weapons evolve depends strongly on population density, with horns more likely to evolve in species occurring at low densities (Pomfret and Knell 2008). Moreover, in the dung beetle Onthophagus taurus, increased population densities in novel environments have been linked to evolutionary reductions in horn expression (Moczek 2003). In Soay sheep, male expenditure on contest competition (horn size, body size and condition) and sperm competition (testes size) both contribute to siring success, but as the number of estrus females increases, the competitive advantage obtained via contest expenditure declines, and siring success obtained via sperm competition increases: when estrus females are rare, males obtain greater fitness through contest competition, but when they are common, males obtain greater fitness through sperm competition (Preston et al. 2003). Increased density may also provide females with greater opportunities for multiple mating (Kokko and Rankin 2006), with sperm competition (inline image) increasing with population density, as in laboratory studies of invertebrates (Gage 1995; Martin and Hosken 2003) where males may respond to increased population density by increased postcopulatory expenditure (He and Tsubaki 1992; Gage 1995; He and Miyata 1997; Tan et al. 2004). Comparative studies both among (Hosken 1997) and within species (Brown and Brown 2003; Dziminski et al. 2010) have found that population density correlates positively with increased male postcopulatory expenditure.

Simultaneous decreases in a and increases in inline image and/or M in response to increasing population density would generate negative covariance between a and inline image and/or M. An alternative possibility is that increased encounter rates may increase the relative benefits of armament by increasing the probability that females are guarded by the best-armed set of males (“perfect resource structuring”Parker 1983), leading to positive covariance between a and inline image.

We consider the effects of covariance between parameters by using contour plots in which ESS expenditures (Ds* or U*/R) are given as the contour height within the x, y coordinate space (Figs. 3–5). Figure 3A shows the effects on Ds* of covariance between the mate-competition loading, a, and sperm competition level, inline image, for dyadic contests (M= 2). If a is independent of inline image (e.g., line 1), expenditure per ejaculate (Ds*) rises in the low-risk range to reach a peak value (white curve) and then decreases (see Figure 2A). Because the peak value is relatively invariant with inline image, this qualitative prediction does not change if a either increases (e.g., line 2) or decreases (e.g., line 3) with inline image; a peak at intermediate sperm competition occurs provided that the relationship between a and inline image always cuts across the contour lines in the manner shown in lines 1, 2, 3. This prediction is only lost for extreme relationships between a and inline image, for example, if a accelerates more steeply with inline image than the contour lines.

Figure 3.

The effect of covariation between the mate-competition loading, a, and sperm competition level, inline image, on (A) the ESS expenditure per ejaculate,Ds*, and (B) ESS postcopulatory expenditure (hence relative testes size), U*/R (=E*), when M= 2. Contours show the values of Ds* or U*/R (color code shown on bar at right). The ridge of peak Ds* values is shown by the white curve. Line 1 =a independent of inline image; line 2 =a increasing with inline image; line 3 =a decreasing with inline image; line 4 =a rapidly accelerating with inline image. The sperm competition level (inline image) is shown on a log scale so that details of the risk range (to the left of the vertical broken line) are clearer. Results are for a fair raffle (r= 1), with D= 1 and R= 10.

Figure 4.

The effect of covariation between the number of males in precopulatory competition, M, and sperm competition level, inline image, on (A) the ESS expenditure per ejaculate (Ds*), and (B) ESS postcopulatory expenditure (hence relative testes size), U*/R (=E*), for a= 1. Contours show the value of Ds* or U*/R (color code shown on bar at right). The ridge of peak Ds* values in (A) approximately follows the vertical broken line separating the risk and intensity sperm competition ranges. Line 1 =a independent of inline image; line 2 =a increasing with inline image; line 3 =a decreasing with inline image. The sperm competition level (inline image) is shown on a log scale so that details of the risk range (to the left of the vertical broken line) are clearer. Results are for a fair raffle (r= 1), with D= 1 and R= 10.

Figure 5.

The effect of covariation between the mate-competition loading, a, and the number of males in precopulatory competition, M, on ESS expenditure per ejaculate,Ds*, for the case q= 0.5 (i.e., inline image= 1.5). Contours show the value of Ds* (color code shown on bar at right). The Ds* contours are also weakly negative for other values of inline image (not shown here). Line 1 =a independent of M; line 2 =a increasing with M; line 3 =a decreasing with M. Postcopulatory expenditure (hence relative testes size), U*/R (=E*), exhibits the same pattern as shown here for Ds*, but with the contour heights reduced to 15% of the values shown (see text). Results are for a fair raffle (r= 1), with D= 1 and R= 10.

Figure 3B shows that the prediction that relative postcopulatory expenditure, U*/R, (hence relative testes size) increases with inline image applies for most relationships between a and inline image (e.g., lines 2 and 3 in Fig. 3A). However, extreme forms of the relationship can again change this behavior; for example, curve 4 in Figure 3B shows a relationship that would generate a peak E* value at an intermediate inline image level.

Figure 4 shows the effects of covariance between the number of males in the group competing for the mating, M, and sperm competition level, inline image, for the case where the mate-competition loading, a= 1. Figure 4A shows contours for Ds*: here the ridge of peak Ds* values is close to the vertical broken line separating the risk and intensity ranges (i.e., at q= 1, N= 2). The general prediction of a peak Ds*value at an intermediate sperm competition level for most monotonic relationships between M and inline image (e.g., lines 1, 2, and 3) is therefore at least as robust as in Figure 3A. Figure 4B shows contours for E*, which are nearly vertical across the entire sperm competition range. Thus most monotonic relationships between M and inline image (e.g., lines 1, 2, and 3) will “climb uphill” as inline image increases, that is, postcopulatory expenditure, E*, (hence relative testes size) will generally increase with sperm competition level, as in the classic model.

It seems likely that the mate-competition loading, a, will vary with the number of males involved in each precopulatory competition, M. The relationship between a and M is probably negative (see “Precopulatory competition”), although we know of no empirical data. Figure 5 shows the expenditure per ejaculate (Ds*) contours for a in relation to M at a constant risk level of q= 0.5 (i.e., inline image= 1.5). Since q and n are constants, E* is directly proportional to Ds* (E*=inline image; for the case in Fig. 5, E*= 0.15Ds*), the contours shown also relate to E* but with the scale values on the right reduced to 15% of their values for Ds*. At all inline image levels, Ds* (and hence E*) declines as both a and M increase, generating negative contours: line 1 shows a transect across the landscape representing constant a(M), and demonstrates a weak negative relation between Ds* or E* and M; a vertical transect would show a strong negative relationship between Ds* or E* and a. Positive relationships between a and M (e.g., line 2) thus increase the slope of the negative relation between Ds* or E* and M. However, if, as predicted, the a(M) relationship is more than weakly negative (e.g., line 3), this generates a positive relationship between Ds* or E* and M, reversing the conclusion (see previous section) that sperm allocations Ds* and E* increase as M decreases.

Discussion

THE MODEL AND ITS PREDICTIONS

Here, we present a general form for the male precopulatory competition component of sperm competition game models, which retains the trade-off between expenditure on pre- and postcopulatory competition between males. This new form incorporates parameters for the mate-competition loading (a) and the number of males competing for each mating (M), allowing the continuum from dyadic contest to scramble to be investigated. This enables us to extend the conclusions of earlier models, which implicitly assume that M is very large and a= 1 and thus represent scramble competition (mate searching). It allows us to consider the effects of these parameters independently, but also with covariance between them. Unless covariance takes on rather extreme forms, we predict that across species or populations (i) postcopulatory expenditure, E*=U*/R, increases (hence precopulatory expenditure (T*/R) decreases) with sperm competition level, and (ii) expenditure per ejaculate (Ds*) shows a peak at intermediate sperm competition. Because of the assumed trade-off between pre- and postcopulatory expenditures, as sperm competition increases across populations, relatively more should be allocated to testes and less to contest armaments or mate searching.

Fromhage et al. (2008) showed that predictions about sperm allocation depend on assumptions as to whether males or females control mating frequency. Classic models (see Parker and Pizzari 2010) and those presented here assume that females determine the number of matings they receive per reproductive cycle. In contrast, some recent models (e.g., Williams et al. 2005; Fromhage et al. 2008; Tazzyman et al. 2009) allow the female mating rate to emerge as a consequence of male investment in precopulatory competition: sperm competition levels are under male control. Both approaches seek the ESS s* and assume R to be constant. However, classic approaches have n fixed by females, which then causes the cost C of gaining a mating to vary, while male control models take C as constant, which then causes n to vary. Fromhage et al. (2008) modeled a continuum between male and female control by varying both female unreceptivity after an initial mating, and male success against female resistance. Despite these conceptual differences, the predicted positive relation between E* and n, and negative relation between s* and n in the intensity range, remain generally similar (e.g., Fromhage et al. 2008, Fig. 3). The main difference occurs in the risk range, where male control can generate independence or (typically) a negative relationship between s* and n, with a positive relationship only occurring when the fertilization raffle is highly loaded (e.g., Fromhage et al. 2008, Fig. 2; Williams et al. 2005). As yet, we do not have our current model's predictions for male control, but the fact that it behaves similarly to previous models under female control suggests that this may again apply.

Our results therefore indicate that the predictions of earlier female control models concerning how expenditure per ejaculate and total postcopulatory expenditure vary with sperm competition are rather robust across a wide range of forms of male precopulatory competition. However, it is impossible to predict how male precopulatory competition affects these expenditures at given sperm competition levels without knowing the covariance between precopulatory competition (M) and the mate-competition loading parameter (a). If a and M are negatively correlated (See “Covariance between parameters”), the predictions of the classic model with a= 1 and large M are more likely to hold for other forms of competition; empirical investigation is much needed.

While previous models have assumed that the dominant component of male precopulatory competition is mate searching, our general form allows us to address cases where contest competition is preeminent. Because previous work has considered evidence for the model assumptions and predictions when males compete for mates mainly by searching, we restrict discussion below to contest competition, for which previous models were inadequate.

BIOLOGICAL EVIDENCE FOR THE ASSUMPTIONS

There is considerable evidence for our assumptions about precopulatory contest competition:

  • (i) Increased expenditure on armament size or strength increases the probability of winning contests, and contests among males occur frequently over access to females

Since Darwin (1871), male–male combat is recognized as a notable feature of precopulatory sexual selection, and increased armament is known to increase the probability of winning (Andersson 1994; Hardy and Briffa in press). Male traits, such as body size and/or the size of secondary sexual traits that determine the outcome of interactions among males, often correlate with mating success (Preston et al. 2003; Hunt et al. 2009).

  • (ii) Males trade expenditure on winning contests for expenditure on ejaculates

Several studies report short-term changes in sperm quality or quantity associated with social dominance. In domestic fowl, Gallus gallus domesticus, sperm quality declines in males that dominate competitive interactions during a social challenge, but remain unchanged in subordinate males (Pizzari et al. 2007). In Arctic charr, Salvelinus alpinus, males becoming dominant after a social challenge have reduced ejaculate sperm counts and decreased sperm swimming velocities (Rudolfsen et al. 2006). Such patterns could reflect costs of male contest expenditure. In Arctic charr, dominance also correlates with changes in plasma sex steroids (Rudolfsen et al. 2006), which could provide the mechanistic basis for these short-term trade-offs. In humans, physical training has negative consequences on semen quality, with extreme expenditures on physical strength and endurance impacting the hypothalamus-pituitary-testis axis (Arce and De Souza 1993; De Souza et al. 1994; Safarinejad et al. 2009). Ejaculate costs associated with social interactions are not limited to vertebrates. In cockroaches, Nauphoeta cinerea, stress associated with social interactions and the establishment of dominance hierarchies reduces spermatophore size and sperm numbers in spermatophores for both dominant and subordinate males (Montrose et al. 2008).

There is also growing evidence for more long-term trade-offs between contest competition and ejaculate expenditure. In humans, a recent study found that men with lower pitched voices, which predict both attractiveness and social dominance, have lower sperm counts (Simmons et al. 2011). In houbara bustards, Chlamydotis undulata, males engaging in extravagant displays to attract females experience a more rapid deterioration in spermatogenic function than males investing less in sexual displays (Preston et al. 2011). In salmon, Oncorhynchus kisutch, large hooknose males spawn in dominance-based hierarchies where social status depends on development of secondary sexual traits such as body size and intensity of red breeding coloration (Flemming and Gross 1994). Males that invest more in secondary sexual coloration have been found to have lower sperm swimming velocities than males that invest less (Pitcher et al. 2009), and this relationship has been found to have a genetic basis in guppies (Evans 2010). In Onthophagus dung beetles, males are subject to a resource allocation trade-off between weapons and testes (Simmons and Emlen 2006). Following assumption (i), males develop enlarged horns on the head and/or thorax for direct combat with rivals over access to the tunnels within which females breed, and those investing more in horns are competitively superior (Emlen 1997; Moczek and Emlen 2000), and have greater reproductive success (Hunt and Simmons 2001). However, horn expenditure comes at a cost to sperm production: prepupal stage experimental ablation of the imaginal disk that normally produces the thoracic horn of Onthophagus sagittarius results in hornless males with larger testes (Simmons and Emlen 2006). Within species, hornless males (sneaks) have larger testes than horned males (guarders), consistent with the assumption of a trade-off between armament and ejaculate expenditures (Simmons et al. 2007). Nutrient-allocation trade-offs are widespread in this beetle genus, where adult structures compete during pupal development for resources sequestered during larval feeding (Emlen 2001; Moczek and Nijhout 2004). Perhaps unsurprisingly, these trade-offs can have a genetic basis. Yamane et al. (2010) documented a correlated response in testes size to selection on mandible size in flour beetles, Gnatocerus cornutus, where mandibles serve as weapons in male–male contests. Lines selected for enlarged mandibles had reduced testis size and sperm numbers ejaculated, while lines selected for reduced mandible size showed reverse responses. The mechanistic basis for such trade-offs may again relate to hormonal control of trait expression. Thus, in stalk-eyed flies, a male's eye-span predicts his success in male contest competition (Panhuis and Wilkinson 1999); application of juvenile hormone increases male allocation to eye-span width at the cost of reduced allocation to testes (Fry 2006). Collectively, these studies provide good evidence that males face a trade-off between weaponry and ejaculate production.

However, across species, the assumption that increased armament (measured as horn length) will be negatively associated with relative testes size is not supported in onthophagine beetles (Simmons and Emlen 2006). Use of across-species comparisons as evidence for trade-offs can be notoriously problematic (Lessells 1991); for instance, in the present case even though pre- and postcopulatory expenditure (T and U) may trade-off within the fixed energy budget R in a given species, both may increase with R across species. Thus, when variance in acquisition of resources exceeds variance in allocation of those resources to life-history traits, those species with greater ability to acquire resources will be able to allocate more resources to both armaments and ejaculates (Reznick 1985; van Noordwijk and de Jong 1986). The same applies to within-species correlational studies. Indeed, positive associations between pre- and postcopulatory expenditure have been reported (Malo et al. 2005; Locatello et al. 2006; Rogers et al. 2008), making experimental manipulations of T and U critical for revealing trade-offs. A further explanation may be that instead of a fixed reproductive energy budget R, there is a fixed somatic energy budget Q, from which both reproduction and also somatic morphology and maintenance (cost S) are allocated, that is, Q=R+S. This would allow R to increase with sperm competition, and both testes and armament to increase at the expense of survival, or whatever else S determines. Data for onthophagines suggest that species with the greatest selection for increased weaponry have also evolved the most protected or canalized testes development patterns, so that selection may have overcome the constraints imposed by a direct trade-off between armament and testes (Simmons and Emlen 2006). Alternatively, quantifying male weaponry expenditure from crude measures of horn length, thereby treating various horn types (head vs. thoracic) as equivalent, may obfuscate reality, in which case the qualitative evidence might be afforded greater weight.

BIOLOGICAL EVIDENCE FOR THE PREDICTIONS

Considerably fewer data exist to test our predictions about precopulatory and/or postcopulatory contest expenditure covariation with sperm competition level when there is precopulatory contest competition. Data from comparative studies of Onthophagine dung beetle species offer some support. They are frequently characterized by male dimorphisms, where a subset of the male population does not develop horns or fight for females, but rather adopts alternative tactics for securing matings. These hornless sneaks enter tunnels guarded by horned males and copulate with the females within (Emlen 1997; Moczek and Emlen 2000). Among species of Onthophagus, there is a positive association between sperm competition risk (estimated as the proportion of sneaks) and relative testes size (Simmons et al. 2007), as predicted both by earlier and present models. Evolutionary gains of novel horns on the thorax (in closer proximity to the developing testes and thus more likely to compete for available resource), are found less frequently on branches of the phylogenetic tree where sneaks are present (thus sperm competition greater) (Simmons and Emlen 2006). In many fish species, alternative tactics are associated with differences in ejaculate expenditure, such that males investing in guarding females expend less on ejaculates than males obtaining matings by sneaking (Gage et al. 1995; Leach and Montgomerie 2000; Vladic and Järvi 2001; Neff et al. 2003). An alternative (or additional) explanation here is that greater ejaculate or testes expenditure by sneaks reflects the fact that they always face sperm competition, whereas guarders do not (Parker 1990b). This early model also assumes a trade-off between mate-acquisition and ejaculate expenditures, but relies on different levels of sperm competition faced by sneaks and guarders, rather than their different expenditures on armament. In species with alternative mating tactics, both effects may often plausibly contribute to differences in relative testes size.

Support for our predictions comes from recent studies of the myobatrachid frog Crinia georgiana, which occurs across a broad geographic range throughout southwest Western Australia. Sexual dimorphism in forelimb morphology is common in frogs, where males have longer and/or enlarged forelimbs, reflecting greater development of the flexor carpi radialis muscle (Wells 2007). Males use their forearms in wrestling matches over territory ownership (Howard 1978) and/or to gain possession of females prior to or during spawning (Davies and Halliday 1979; Wells 2007), and larger forearms confer greater mating success (Lee 2001; Wells 2007). Male density at choruses has a major effect on male–male encounter rates and contests; the frequency of wrestling matches increases with male density at choruses, and larger males invariably win wrestling bouts (Byrne and Roberts 2004). Male density also predicts the frequency of group spawning, where two or more males amplex a single spawning female so that sperm competition results in mixed paternity within clutches (Roberts et al. 1999; Byrne and Roberts 2004). This mating system may well conform to the case where sperm competition risk q is positively associated with M through male density. Whatever this relation, we should expect testes size to increase and armament to decrease with sperm competition level (Fig. 2B). Dziminski et al. (2010) sampled males from 10 C. georgiana populations varying in male density, and thus risk of sperm competition, from across their geographic range. Consistent with our models, male expenditure on sperm numbers (Ds*) and testes size (U*/R=E*) both increased with increasing male density (Fig. 6). We also measured male expenditure on forearm development within and among these populations (electronic Supporting Information). As predicted by our models, after controlling for body size, relative forearm enlargement was negatively associated with male density (Fig. 6). Across populations, there was a negative correlation between relative testes size and relative forearm enlargement (Pearson's r=−0.646, N= 10, P= 0.044). These data conform well to the predictions of our present contest-based sperm competition models.

Figure 6.

Patterns of male investment in testes and weaponry across populations of the myobatrachid frog Crinia georgiana. Superimposed on the significant positive association between male density and least square mean (LSM) (controlling for body mass) testes size (±SE) from Dziminski et al. (open circles; 2010) is the negative relation between male density and the major axis of variation in forearm size (filled circles; see electronic Supporting Information). The partial effect of male density on forearm size was significant (F1, 7= 7.85, P= 0.027) after controlling for the effect of body size measured as snout-vent length (F1, 7= 348.8, P < 0.0001) (the data plotted are the LSM (±SE) forearm size, controlling for snout-vent length).

CONCLUDING COMMENTS

More complex models might divide components of male mating success into mate searching, contests, mortality costs, etc. Our analysis is intentionally simplistic, focusing on the contest-scramble range to examine how generally the classic model predictions (based solely on competitive mate-searching scrambles) apply. We show that when extended to include contest competition, qualitatively similar predictions are usually obtained. Relative testes size has been shown to increase ubiquitously with sperm competition across species in many different animal groups (see, e.g., Birkhead and Møller 1998), suggesting that the rather extreme covariances between parameters required to generate declining testes size with increased sperm competition (e.g., curve 4, Fig. 3B) are rarely if ever met in nature. Nevertheless, we suggest that further theoretical investigations of how a male's number of matings varies through trade-offs between expenditures on testes/ejaculate and the different forms of male–male competition for matings would be justified, as would further comparative studies that examine the nature of male contest costs and expenditure on armaments and testes/ejaculates, and empirical studies that estimate how the mate-competition loading (a) varies with number of males competing for matings (M).

Associate Editor: E. Morrow

ACKNOWLEDGMENTS

We are most grateful to two reviewers for helpful suggestions. LWS was supported by Australian Research Council, and GAP by Leverhulme Trust.

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