While males are expected to be promiscuous, the adaptive significance of females copulating with multiple males is less clear. This is because male reproductive success typically relates directly to the number of females inseminated, whereas for females reproduction is usually unaffected by the number of ejaculates received beyond one (Bateman, 1948; Parker, 1992b; Thornhill and Alcock, 1983). Female multiple mating (defined here as females mating with multiple males) may be male driven, but females often directly solicit copulations from a number of males, and it is becoming increasingly clear that many (or most) females in a wide range of taxa are genetically polyandrous (Gowaty, 1994).

The benefits to females of such behavior may be direct, such as nutrients or fertility insurance (e.g., Birkhead and Fletcher, 1995; Sheldon, 1994; Simmons, 1992) or, alternatively, benefits may be genetic. Genetic benefit arguments explaining female multiple mating typically relate to male quality and may require females to be able to choose males with the desired characteristic(s). Hypotheses include compensation for low-quality partners (Kempenaers et al., 1992), avoidance of genetic incompatibility (Zeh and Zeh, 1996, 1997), promotion of sperm competition (e.g., the intrinsic male quality hypothesis, Birkhead et al., 1993; the sexually selected sperm hypothesis, Keller and Reeve, 1995), the increased heterozygosity hypothesis (Brown, 1997; see also Müller and Ward, 1995), the genetic diversity-benefit hypothesis (Tooby, 1982), and genetic bet-hedging (Watson, 1991).

Another possible benefit of multiple mating by females relates to avoiding costs of inbreeding (Stockley et al., 1993). Although Williams (1975) argued multiple mating produces little more offspring diversity than copulating with a single male, this need not be true (Yasui, 1998). For example, models of hymenopteran offspring relatedness and number of mating partners suggest offspring heterogeneity increases steeply when the number of partners increases from one to five and sperm use is random (Page and Metcalf, 1982; see also Yasui, 1998). Moreover, empirical data indicate that multiple mating can potentially increase fitness in at least some instances (e.g., Liersch and Schmid-Hempel, 1998; and see Page and Metcalf, 1982). For example, in bumble bee colonies increasing within-colony genetic heterogeneity led to decreased parasite prevalence, diversity and intensity (Liersch and Schmid-Hempel, 1998). The authors proposed that genetic polyandry (or polygyny) could therefore potentially increase fitness under parasitism (Liersch and Schmid-Hempel, 1998), and this has recently been confirmed (Baer and Schmid-Hempel, 1999). It has also been proposed that female multiple mating reduces the fitness cost(s) of inbreeding in some instances (Stockley et al., 1993), a hypothesis that has been widely discussed (e.g. Watson, 1997; Zeh and Zeh, 1996). The original argument was that if females cannot recognize or cannot avoid copulating with close relatives, especially when dispersal is low, then female multiple mating could increase female fitness by increasing the probability of producing some outbred young (Stockley et al., 1993), and it is the inbreeding avoidance hypothesis that we consider here.

It is clear that outbreeding can be beneficial in fitness terms (e.g., Bateson, 1983; Munson et al., 1996), and Liersch and Schmid-Hempel's (1998) bumble bee study clearly indicated that, in terms of parasitism, females benefited from producing genetically diverse offspring. However, in most species female bumble bees typically copulate only once (and polygyny is uncommon) and the question posed was why are females not copulating more frequently, as there are clear fitness benefits (Liersch and Schmid-Hempel, 1998). Obviously, there can be costs involved in mating multiply (e.g., Thornhill and Alcock, 1983) and, equally obviously, the relative magnitude of the costs and benefits will influence selection for or against female multiple mating and increased offspring heterogeneity. However, what is not intuitively clear is that potential fitness benefits obtained by females from multiple mating is crucially dependent on the shape of the relationship between relatedness of copulatory partners and fitness. Explicitly, we argue here that the overall relationship between partner relatedness or number and fitness can be positive, but increased variance (i.e., more mates) may nonetheless be selected against.

Functions relating fitness to other variables are typically thought to adopt one of three shapes: linear, increasing with diminishing returns, or sigmoid (e.g., Ricklefs, 1979). However, functions relating a character to fitness are unlikely to be linear (e.g., Page, 1980; Ricklefs, 1979; Schmid-Hempel, 1994). Because the relationship between female fitness and number of mates is typically unknown (because of the inherent problems of measuring costs and benefits in fitness terms) but unlikely to be linear (see above), we consider here the effect of multiple mating on fitness when this relationship is either increasing and convex (i.e., f″ > 0) or increasing with diminishing returns (increasing and concave; i.e., f″ < 0).

Consider the relationship between fitness and relatedness of copulatory partners, and note that inbreeding (high relatedness) is typically associated with low fitness (e.g., Wildt et al., 1987). A relationship where fitness benefits asymptotically decrease (i.e., to the right of the point of inflection on our sigmoid curve, Figure 1) would be expected when there is a trade-off between costs (e.g., mate searching) and benefits (see, e.g., Parker, 1992a). Under these conditions random female mating with multiple males will not increase fitness relative to singly copulating females, in spite of the overall positive relationship between fitness and relatedness: a female with less variation in the relatedness of her mates will have a higher average fitness than a female with greater variance in mate relatedness [from Figure 1; female A (with little variation in mate relatedness) = solid lines, mean fitness = y, compared with female B (large variation in mate relatedness) = dashed lines, mean fitness = yd]. Note that mean partner relatedness is the same for both females, as expected if females cannot distinguish between kin and nonkin (i.e., there is a equal chance of the next partner being more or less related).

Figure 1

A sigmoid curve describing the relationship between relatedness of mating partners (x-axis) and fitness (y-axis). On the concave portion of the curve (f″ < 0, i.e., to the right of the point of inflection), a relationship expected when there is a trade-off between benefits and costs, a female (B) with a greater number of mating partners (as shown by the width of B compared with A) will have reduced fitness (yd) compared to a female with fewer (female A; fitness = y), all else being equal (see text). On the increasing convex portion of the curve (f″ > 0, i.e., to the left of the point of inflection), a relationship that may be found when populations expand into new habitats, a female (D) with a greater number of mating partners (as shown by the width of D compared with C) will have increased fitness (y + d) compared to a female with fewer (C; fitness = y), all else being equal (see text).

Conversely, if the fitness function is exponentially increasing (i.e., to the left of the point of inflection on our sigmoid curve, Figure 1), as may occur when a population is expanding into new habitat, females inseminated by few males have a mean fitness of y (Figure 1; female C = solid lines), while females mating with many males has a mean fitness of y + d (Figure 1; female D = dashed lines). (Note: if the relationship between fitness and relatedness is linear, then multiple mating will not alter fitness relative to that of more monogamous females, but see above.) The underlying assumption is that variation on the x-axis (relatedness at insemination) is translated into variation in fitness via genetic variation in the offspring and that females mate nonselectively. When this occurs it is clear that the fitness outcome of multiple mating will be influenced by the shape of the fitness curve (convex or concave), or by a female's location on the x-axis relative to the point of inflection when the curve is sigmoid (Figure 1). For example, if a female is closely related to her previous mating partner compared to the general population (i.e., to the left of the point of inflection), she would probably benefit from more random mating, and vice versa. The essence of this argument has been applied in several other biological contexts (e.g., Parker, 1984; Sherman et al., 1988), including risk-sensitive foraging (e.g., Caraco et al., 1980), insect growth under variable temperature (e.g., Blanckenhorn, 1997), and Gillespie's (1977) classic variance reduction principle.

Although the exact relationship between fitness and relatedness of mating partners is often unknown, the curve(s) described above is a versatile estimator covering a range of possibilities (Page, 1980), and changing the slope while retaining the shape of the curve(s) only alters the magnitude of the results. We have shown that for multiple copulations to increase female fitness, without pre- or postcopulatory selection and when mating is more or less random (i.e., females cannot detect/choose high-quality males which by definition increase female fitness; e.g., Petrie, 1994; Welch et al., 1998; see also Bateson, 1983), the relationship between fitness and multiple mating must be convex and increasing. However, note that because fitness obviously has an asymptote, the convex relationship would typically be found to the left of the inflection point of a logistic curve. Of course, if some pre- or postcopulatory selective mechanism is operating, then the potential for fitness benefits increases. This is true even when fitness is increasing with diminishing returns relative to relatedness, as long as mean relatedness is shifted far enough to the right to compensate for the decreased mean fitness generated by the increased variation. This could be achieved by mate choice (e.g., Petrie, 1994), selecting against self/like-self sperm, a process described in ascidians, lizards, and many plants (e.g., Bishop, 1996; Olsson et al., 1996; Willson, 1990), or if the competitive ability of sperm is greater in outbred males (e.g., Wildt et al., 1987). In addition, it has been argued that increased offspring diversity may be favored even when it decreases mean offspring fitness (i.e., genetic bet hedging; Phillipi and Seger, 1989). However, it appears the evolution of female multiple mating via the genetic bet hedging mechanism alone is highly unlikely (Yasui, 1998; see also Birkhead and Møller, 1992).

Returning to the shrew study above and the relationships between fitness and genetic relatedness considered here, for Stockley et al.'s (1993) hypothesis regarding shrews to be accurate, the relationship of fitness to relatedness must be increasing and convex, or there must be some postcopulatory influence(s) on paternity. Note however in the latter case, there is no overt indication of sperm selection by female shrews (Stockley, 1997).

In conclusion, although it is intuitively clear that the magnitude of the costs and benefits of multimale copulations will influence selection for or against female multiple mating, the function relating female multiple mating to fitness will also have an effect. In spite of a positive relationship between fitness and female multiple mating, increased mate/offspring diversity can still be detrimental and therefore selected against under conditions that may be quite common in nature. Moreover, with all else being equal, we would expect female promiscuity to be more prevalent in species likely to face an increasing and convex fitness curve, such as those with high baseline costs of reproduction but negligible cost changes thereafter and/or those exhibiting semelparity. Although mate choice can increase fitness regardless of the fitness function, random multiple mating may also have fitness advantages under some conditions.

We thank Max Reuter, François Balloux, Paul Ward, Bart Kempenaers, Leigh Simmons, Paul Schmid-Hempel, Georgina Bernasconi, Paula Stockley, Fredrik Widemo, and the anonymous referees for comments and discussion. This work was supported by grants from the Swiss National Science Foundation.

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