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We examine models for evolution of sperm size (i.e. mass m) and number (s) under three mechanisms of sperm competition at low 'risk' levels: (i) raffle with no constraint on space available for competing sperm, (ii) direct displacement mainly by seminal fluid, and (iii) direct displacement mainly by sperm mass. Increasing sperm mass increases a sperm's 'competitive weight' against rival sperm through a diminishing returns function, r(m). ESS total ejaculate expenditure (the product m(*)s(*)) increases in all three models with sperm competition risk, q. If r(m), or ratio r'(m)/r(m), is independent of ESS sperm numbers, ESS sperm mass remains constant, and the sperm mass/number ratio (m(*)/s(*)) therefore decreases with risk. Dependency of sperm mass on risk can arise if r(m) depends on competing sperm density (sperm number / space available for sperm competition). Such dependencies generate complex relationships between sperm mass and number with risk, depending both on the mechanism and how sperm density affects r(m). While numbers always increase with risk, mass can either increase or decrease, but m(*)/s(*) typically decreases with risk unless sperm density strongly influences r(m). Where there is no extrinsic loading due to mating order, ESS paternity of the second (i.e. last) male to mate (P(2)) under displacement always exceeds 0.5, and increases with risk (in the raffle P(2)=0.5). Caution is needed when seeking evidence for a sperm size-number trade off. Although size and number trade-off independently against effort spent on acquiring matings, their product, m(*)s(*), is invariant or fixed at a given risk level, effectively generating a size-number trade off. However, unless controlled for the effects of risk, the relation between m(*) and s(*) can be either positive or negative (a positive relation is usually taken as evidence against a size-number trade off)., (Copyright (c) 2010 Elsevier Ltd. All rights reserved.)

We examine the risk model in sperm competition games for cases where female fertility increases significantly with sperm numbers (sperm limitation). Without sperm competition, sperm allocation increases with sperm limitation. We define 'average risk' as the probability q that females in the population mate twice, and 'perceived risk' as the information males gain about the sperm competition probability with individual females. If males obtain no information from individual females, sperm numbers increase with q unless sperm limitation is high and one of the two competing ejaculates is strongly disfavoured. If males can distinguish between virgin and mated females, greater sperm allocation to virgins is favoured by high sperm limitation, high q, and by the second male's ejaculate being disfavoured. With high sperm limitation, sperm allocation to virgins increases and to mated females decreases with q at high q levels. With perfect information about female mating pattern, sperm allocation (i) to virgins that will mate again exceeds that to mated females and to virgins that will mate only once, (ii) to virgins that mate only once exceeds that for mated females if q is high and there is high second male disadvantage and (iii) to each type of female can decrease with q if sperm limitation is high, although the average allocation increases at least across low q levels. In general, higher sperm allocation to virgins is favoured by: strong disadvantage to the second ejaculate, high sperm limitation, high average risk and increased information (perceived risk). These conditions may apply in a few species, especially spiders.

There are various ways to estimate ejaculate expenditure. Ejaculate size or sperm number (s) is an absolute number of units of ejaculate. Relative ejaculate expenditure (E) is the expenditure on the ejaculate as the proportion of the total expenditure on all aspects of the mating, including finding and acquiring a female, and so on. Relative testis size or gonadosomatic index (sigma) is testes mass divided by body mass; it is assumed to reflect the product of mating rate (M) and ejaculate mass (s). In a new model, where mating rate, sperm competition and sperm allocation interact, and where the female's inter-clutch interval is assumed to be independent of s or M, we show that sigma is directly proportional to the mean E for a species; across species sigma and E increase monotonically with sperm competition. However, the relation between s and sperm competition across species depends on the range of sperm competition (low risk or high intensity): s increases with sperm competition at low risk levels, but decreases with sperm competition at high intensity levels. This situation arises because s alpha E/M; both E and M increase with sperm competition, but E increases differently with sperm competition in its two ranges.

We analyse a co-evolutionary sexual conflict game, in which males compete for fertilizations (sperm competition) and females operate sperm selection against unfavourable ejaculates (cryptic female choice). For simplicity, each female mates with two males per reproductive event, and the competing ejaculates are of two types, favourable (having high viability or success) or unfavourable (where progeny are less successful). Over evolutionary time, females can increase their level of sperm selection (measured as the proportion of unfavourable sperm eliminated) by paying a fecundity cost. Males can regulate sperm allocations depending on whether they will be favoured or disfavoured, but increasing sperm allocation reduces their mating rate. The resolution of this game depends on whether males are equal, or unequal. Males could be equal: each is favoured with probability, p, reflecting the proportion of females in the population that favour his ejaculate (the 'random-roles' model); different males are favoured by different sets of females. Alternatively, males could be unequal: given males are perceived consistently by all females as two distinct types, favoured and disfavoured, where p is now the frequency of the favoured male type in the population (the 'constant-types' model). In both cases, the evolutionarily stable strategy (ESS) is for females initially to increase sperm selection from zero as the viability of offspring from unfavourable ejaculates falls below that of favourable ejaculates. But in the random-roles model, sperm selection decreases again towards zero as the unfavourable ejaculates become disastrous (i.e. as their progeny viability decreases towards zero). This occurs because males avoid expenditure in unfavourable matings, to conserve sperm for matings in the favoured role where their offspring have high viability, thus allowing females to relax sperm selection. If sperm selection is costly to females, ESS sperm selection is high across a region of intermediate viabilities. If it is uncostly, there is no ESS in this region unless sperm limitation (i.e. some eggs fail to be fertilized because sperm numbers are too low) is included into the model. In the constant-types model, no relaxation of sperm selection occurs at very low viabilities of disfavoured male progeny. If sperm selection is sufficiently costly, ESS sperm selection increases as progeny viability decreases down towards zero; but if it is uncostly, there is no ESS at the lowest viabilities, and unlike the random-roles model, this cannot be stabilized by including sperm limitation. Sperm allocations in the ESS regions differ between the two models. With random roles, males always allocate more sperm in the favoured role. With constant types, the male type that is favoured allocates less sperm than the disfavoured type. These results suggests that empiricists studying cryptic female choice and sperm allocation patterns need to determine whether sperm selection is applied differently, or consistently, on given males by different females in the same population.

Sperm precedence patterns are typically highly variable within (and between) species. Intraspecific variation in sperm precedence (measured as P2, the proportion of progeny fathered by the last male to mate' is frequently seen as a candidate for adaptive interpretation through either male effects (e.g. body size), female effects (e.g. cryptic female choice) or an interaction between the two. Here we show, using computer simulation, that if ejaculates divide into a number of 'packets' and packets from two males mix randomly, then a variety of patterns of sperm precedence may result. We term this process 'sloppy' mixing. If ejaculates break into a small number of packets, bimodal P2 distributions are predicted. As the number of packets is increased, then a complex series of changes through multimodal and flat to unimodal distributions results. Sloppy mixing can thus result in many of the observed P2 distributions. Sloppy mixing is unlikely to change the predictions of adaptive models of sperm competition.

Our main aim is to compare the additive model, due to Mesterton-Gibbons, and the multiplicative model, due to Parker, of sperm allocation under sperm competition, when other influences are treated in the same way. We first review these (and other) models and their foundations, leading to a generalization of the multiplicative model. Sperm is assumed to cost energy, and this constraint is incorporated differently in the two models. These give the same results in the random-roles situation when the males occupy roles (of first and second to mate) randomly: the number of sperm ejaculated in the favoured role is greater than that in the disfavoured role by an amount that depends on the effect of sperm limitation (i.e. the probability that there is insufficient sperm to ensure full fertility). If the latter is negligible, or the fertilization raffle fair, this difference is zero, as Parker found originally. In the constant roles situation (where males of a particular type always occupy the same role) the predictions differ: the additive model has the same predictions as in the random roles case, but the multiplicative model predicts that males of the type occupying the favoured role ejaculate less than males of the type occupying the disfavoured role, in accord with Parker's original conclusion. The fitnesses of the two types of male can be calculated in the multiplicative model: the fitness of the favoured male is usually higher, even if he has to expend more energy in "finding" a female, e.g. through fighting, etc. These conclusions relate to inter-male behaviour (i.e. of different male types), as distinct from intra-male behaviour (i.e. of a given male when in different roles). We analyse situations in which one male type has some probability of acting in its less usual role: calculations with varying amounts of sperm limitation are presented. It is found that the presence of a male of a different type has an effect on intra-male ejaculate behaviour, which also depends critically on the role usually occupied. We conclude that the multiplicative model is the more accurate model and provides more information. Some experimental data on sperm numbers are used to find the effects of sperm limitation. For species which conform to the loaded raffle model, sperm limitation typically has small or negligible effects: in this case, we argue that empiricists should look for equal ejaculates in the two roles when studying random role situations; when roles are occupied non-randomly average sperm expenditure should be greater by male types typically occupying the disfavoured role, but within a male type, expenditure should be greater in the role it typically occupies., (Copyright 2000 Academic Press.)

The female reproductive tract can be particularly aggressive towards ejaculates, often leading to the death of large numbers of sperm. It has been suggested that males can respond to these actions by investing more in sperm and donating larger ejaculates. Such counteractions may lead to arms races, which can have significant implications for the mating system. In a series of simple models we first show that arms races are not necessarily supported: in fact, sperm killing may even favour no change or reductions in sperm allocation. Second, we identify a simple mechanistic rule for sperm killing that determines whether an arms race or sperm reduction will be favoured. Which of these responses is favoured by selection depends on whether a certain number, or proportion, of sperm are killed. When a specific number is killed, larger investment in sperm is favoured and when a specific proportion is killed, no change or lower investment in sperm is favoured. Both of these mechanisms are biologically plausible.

We aim to interpret sperm displacement in relation to male size in the yellow dung fly, Scatophaga stercoraria, and to compare the general properties of indirect and direct size-dependent sperm displacement in insects. We examine the hypothesis that male size-dependent sperm displacement in dung flies can be explained by size-dependent increases in the ejaculatory apparatus, allowing greater sperm flow rates in larger males. We expect sperm flow rates to be proportional to the diameter of the aedeagus duct to the power x, where x lies between 2 and 3. We test this hypothesis using a simulation model of indirect sperm displacement that has been developed to accommodate recent observations on sperm transfer, in which sperm flow from the male into the female bursa and are then transferred to the spermathecae by movements of the female tract. The indirect model approximates to the pattern of size-related sperm displacement, with scaling power 3 giving a better fit than power 2. Copula duration shows a male size-dependent decrease in this species. We apply the indirect model of sperm displacement, in conjunction with parameters obtained from field and laboratory data, to predict size-dependent changes in optimal copula duration from the male perspective. This model concurs with the observations by predicting a size-dependent decline in optimal copula duration, as did an earlier model in which displacement was direct (new sperm displace previously stored sperm directly from the sperm stores). Our new approach gives a better fit than the earlier direct model. Thus, both results (displacement rates and copula duration) can be explained by size-dependent changes in the ejaculatory apparatus of the male with the female's exchange rate of sperm (from bursa to spermathecae) remaining constant with respect to male size, although we discuss the possibility that this female process may accelerate with increased male size. In general, where the sperm input rate is around the same magnitude as the exchange rate, indirect displacement will be dependent on the size of the male, as in dung flies, but this dependency is lost if the input rate is very high relative to the exchange rate across the entire range of male size. Size-dependent displacement should always apply for males with direct displacement.

Three sperm competition games against relatives are examined. In the first, a male has no information at the time of mating as to whether or not his ejaculate will face sperm competition from a related or unrelated male. Sperm expenditure increases with overall sperm competition risk q and declines with the probability rho that the competitor shares the same allele for sperm expenditure. In the second game, males have almost perfect information: they 'know' whether there will be sperm competition and, if so, whether this involves a related or unrelated male. Sperm expenditure is reduced by a factor rho when competing with a relative. In the third game, males 'know' when they compete with relatives, but have no information for other matings whether they will face sperm competition from unrelated males. A male without information expends less on his ejaculate than a male competing with a close relative if the overall risk of sperm competition is low, but more if the overall risk is high. The average relative ejaculate expenditure is the same in all three games so that, if this determines testis size, data is required only on the overall sperm competition risk, the probability of competing with a relative and the average rho in order to perform comparative analyses.

We develop the logic of assessment of sperm competition risk by individual males where the mechanism of sperm competition follows a 'loaded raffle' (first and second inseminates of a female have unequal prospects). Male roles (first or second to mate) are determined randomly. In model 1, males have no information about the risk associated with individual females and ejaculation strategy depends only on the probability, q, that females mate twice. Evolutionarily stable strategy (ESS) ejaculate expenditure increases linearly from zero with q, and reduces with increasing inequality between ejaculates, though the direction of the loading (which role is favoured) is unimportant. In model 2, males have perfect information and can identify each of three risk states: females that will (1) mate just once ('no risk'), (2) mate twice but have not yet mated ('future risk'), and (3) mate twice and have already mated ('past risk'). The ESS is to ejaculate minimally with 'no risk' females, and to expand equally with 'past' and 'future' risk females; the direction of the competitive loading is again unimportant. Expenditure again increases with risk, but is now non-zero at extremely low risk. Model 3 examines three cases of partial information where males can identify only one of the three risk states and cannot distinguish between the other two: they therefore have just two information sets or 'contexts'. Expenditure in both contexts typically rises non-linearly from zero with q, but (whatever the loading direction) expenditure is higher in the context with higher risk (e.g. if contexts are 'mated' and 'virgin', males spend more with mated females). However, in highly loaded raffles, sperm expenditure can decrease over part of the range of risk. Also, the direction of the loading now affects expenditure. Biological evidence for the predictions of the models is summarized and discussed.

We develop a model of a continuous fertilization process in which eggs and sperm are shed simultaneously, and in which the eggs are fertilized at a rate proportional to sperm density surrounding the egg mass. The model derives the ESS size and number of sperm in an ejaculate of an externally-fertilizing animal such as a fish species, in which the probability or intensity of sperm competition varies. It also predicts the ESS level of infertility (eggs remaining unfertilized after all sperm have died). Sperm size is assumed to increase sperm competitive ability (e.g. by increasing speed) and is also assumed to affect sperm longevity (either positively or negatively). Ejaculate expenditure is traded off against expenditure on obtaining further spawnings, and size and number of sperm can vary independently. The model predicts that the ESS ejaculate expenditure (product of sperm size and number) should increase, and that the ESS infertility should decrease with sperm competition intensity measured across species. Other results depend on the way that sperm size affects longevity. The available biological evidence suggests that longevity decreases with sperm size, probably because the main increase is in tail length which increases sperm energy expenditure. In this case, sperm size should increase with sperm competition intensity from an optimum at zero competition which maximizes the total distance travelled by the entire ejaculate in its lifetime, to an optimum for maximum sperm competition which maximizes the product of speed and sperm number. However, if longevity increases with sperm size, then the non-competitive optimal sperm size is greater than that for maximum competition, so that sperm size decreases with sperm competition intensity. Sperm numbers typically increase with sperm competition intensity, and always so if sperm competition is high enough, though decreases are possible over a range of low sperm competition intensity if (i) sperm longevity decreases with sperm size, and (ii) infertility is high enough.

Evolutionary games of sperm competition in which two males mate with the same female have previously considered sperm size to be fixed at some (small) constant level. Although male gametes in multicellular organisms are typically small compared with ova, they vary greatly both between and within groups, and sperm size sometimes correlates with the probability of sperm competition. This paper examines 'raffle principle' sperm competition games in which both size and number of gametes can be varied strategically under control of the diploid parent. If ejaculate investment trades off against the number of matings that a male can achieve, the evolutionary stable strategy (ESS) ejaculate expenditure (as a fraction of reproductive effort per mating) approximates to p/4 when the probability, p, of sperm competition is low. Sperm size may either: (i) increase a sperm's competitive weight (a measure of its advantage in the fertilization raffle), or (ii) influence its mortality rate in the female tract. On the simplest model, size is optimized after the marginal value theorem, and may be large or small depending on how size influences competitive weight or survivorship. Further, sperm size is independent of the risk of sperm competition, and only sperm numbers increase with this risk. However, some recent studies show sperm size to increase with sperm competition. The present analysis offers the following possibilities: (i) there are unidentified constraints on sperm number, so that ejaculate mass can increase only by increase in sperm size; (ii) competitive benefits of size become more important as sperm numbers increase; (iii) size mainly increases survivorship, and sperm competition risk increases with the mean duration between mating and fertilization; and (iv) size increases competitive ability at the expense of survivorship, and sperm competition risk decreases with time between mating and fertilization. These conclusions relate to advantages conferred by size on sperm before fertilization; they do not affect the prediction of previous models that no component of sperm size should evolve for provisioning the zygote.

This paper examines sperm competition games in which both size and number of gametes in an ejaculate can be varied strategically, but under the control of the haploid sperm. It is a companion to the previous paper (Parker (Proc. R. Soc. Lond. B 253, 245 (1993))) for the case where the strategic control is by the diploid parent. Under haploid control, the optimal balance between size and number of sperm in an ejaculate becomes quite different from that predicted for diploid control; there is a conflict between parental and gametic interests over sperm size and number. This occurs because both intra-ejaculate and inter-ejaculate sperm competition are present with haploid control, whereas intra-ejaculate competition is absent under diploid control. The magnitude of this conflict is affected by the risk of inter-ejaculate sperm competition, and by the fact that gametic interests depend on the nature of available mutations. If ejaculate expenditure trades off against numbers of matings achieved, with haploid control there is escalation towards maximal expenditure on the ejaculate. If the ejaculate expenditure is fixed by the diploid parent, but there is a size-number trade-off under haploid control, solutions depend on the mode of action of the mutation affecting sperm size or number. If mutant-bearing sperm deviate entirely at the expense of non-mutant sperm, any increase in size or number will spread. If mutant sperm deviate entirely at their own expense, the evolutionarily stable strategy (ESS) is the same as for diploid control; there is no conflict.(ABSTRACT TRUNCATED AT 250 WORDS)

This paper extends the sperm displacement model of Parker et al. (Behav. Ecol. Sociobiol. 27, 55 (1990)), in which sperm displacement is viewed as a process in which one unit of sperm introduced displaces one unit of sperm from the female's sperm stores. Here this process is envisaged in terms of the change in density of sperm in the sperm stores. In matings with virgin females, only sperm store fluid is displaced at the start of sperm transfer, but if there is swift random mixing of seminal and sperm store fluid, the fluid displaced will contain sperm at the same average density as that in the sperm stores (random displacement). In mating of the same female by two or more males, the sperm density of the last male to mate is assumed to be independent of the presence of previous sperm; P2 (the proportion of eggs fertilized by the last male) thus equals the density of the last male's sperm divided by the current total density of sperm in the sperm stores. Once the sperm stores have reached the asymptotic density (equivalent to the input density, i.e. the density of sperm in the seminal fluid), the present model becomes equivalent to that of Parker et al. (1990). Predictions for this model are tested using all available data from the dung fly, Scatophaga stercoraria. They are based on the assumption that sperm are transferred at a constant rate with copulation time. The data concur with this model, and we conclude that it is better than various other simple alternatives for explaining P2 in Scatophaga.(ABSTRACT TRUNCATED AT 250 WORDS)

Evolutionary theory for expenditure on gonads attracted little attention until studies in the past 3-4 decades of allocation to male and female function in hermaphrodites, and of relative testes size (RTS) in animals with separate sexes. RTS appears to have varied enormously over evolutionary time, from extremely high (over 40%) in some broadcast spawners to less than 1% in some taxa with copulation and internal fertilization. Reasons for this wide evolutionary diversity in testes expenditure deserve attention. Sperm demand (the product of expenditure per ejaculate and mating rate) increases expenditure on testes, and is moulded by a number of selective forces: sperm competition, mating rate and sperm limitation are identified as the three typically most powerful. Trade-offs between somatic and testes expenditures can be complex, particularly those between pre- and post-copulatory expenditure. Theoretical models predict that RTS should correlate positively with sperm demand, and there is much evidence from many taxa that RTS increases with sperm competition level. The relation between RTS and expenditure on each ejaculate is predicted to be positive over low sperm competition (risk) levels, and negative over high (intensity) sperm competition levels. Sexual cascade logic suggests that gonad expenditure should be high and equal for the two sexes soon after the early evolution of anisogamy, but should diverge widely through evolutionary time, with RTS reducing notably in those taxa that have evolved advanced mobility and internal fertilization. [ABSTRACT FROM AUTHOR]

Trivers proposed that, if parental care by both sexes is advantageous, males should practice a "mixed" strategy of seeking extrapair copulations, while restricting their parental investment to offspring of social mates. We explore circumstances under which males should limit their parental care in the predicted manner. We find that Trivers's "mixed" strategy will generally be evolutionarily stable so long as either socially monogamous or polygynous males usually sire more offspring per brood from a social mate than they typically sire in broods of extrapair mates. Polygynous males should spread investment across their home nests unless the expected number of chicks sired in them differs widely. Whether polygynous males should restrict paternal care to social mates' offspring hinges additionally on resident male investment in broods containing extrapair young: if resident males contribute minimally, some investment by a polygynous extrapair male becomes more advantageous. Recently reviewed data on extrapair fertilization distributions within monogamous and polygynous passerines suggest that extrapair offspring often predominate numerically within their broods, consistent with sperm expenditure theory. Nevertheless, most species conform to the model's criterion regarding relative parentage levels in broods of social versus extrapair mates. Patterns of extrapair parentage thus appear sufficient to stabilize biparental care systems. [ABSTRACT FROM AUTHOR]

Presents a study which examined the effect of male size variation on indirect sperm displacement in dung flies, Scatophaga stercoraria. Methodology; Results; Discussion.