THE CONTEXT-DEPENDENT EFFECT OF MULTIPLE PATERNITY ON EFFECTIVE POPULATION SIZE

Authors


  • This article was published online on March 8, 2011. An error in the author's name was subsequently identified. This notice is included in the online version to indicate that it has been corrected September 16, 2011.

Abstract

Effective population size (Ne) is important because it describes how evolutionary forces will affect a population. The effect of multiple sires per female on Ne has been the subject of some debate, at the crux of which is the effects of monandry and multiple-paternity (MP) on male variance in reproductive success. In both mating systems, females mate with several males over their lifetimes, but sire offspring with one male at a time in the former and have several sires per clutch in the latter. First, I theoretically show that whether the annual male variance in reproductive success in an MP population is greater or less than that of a monandrous population depends on the distributions of within-clutch paternity. Then, I simulated different distributions of within-clutch paternity under a range of parameters that characterize natural populations to show that an MP population can have an Ne smaller or larger than that of a monandrous population with otherwise equal dynamics. The Ne(MP):Ne(Monandry) ratio increased with mating frequency and female variance in reproductive success, was equalized by long generation times, and was affected by the distribution of within-clutch paternities. The results of this model provide a unifying framework for the debate.

The effective size of a population (Ne) is defined as the size of an ideal population, with random union of gametes and no mutation, that would lose genetic variation as a result of random processes at the same rate as the actual population (Wright 1931). This measure is of interest to both population and conservation biologists because it reflects the degree of inbreeding, genetic drift, and potential for adaptation (Crow and Kimura 1970). Also, increased population persistence is associated with higher levels of genetic diversity (Saccheri et al. 1998). Developing an understanding of the factors that influence Ne is of fundamental importance for understanding the evolutionary history of finite populations, as well as for making evolutionary predictions.

The concept of effective population size as originally developed by Wright (1931) considered the size of an ideal population that had the same probability of allele identity as the observed population, later dubbed the inbreeding effective number (Crow and Denniston 1988). Wright's concept was subsequently formalized in various types of monoecious or dioecious populations (Kimura and Crow 1963; Crow and Denniston 1988). In addition, a distinction was made between the inbreeding (size of ideal population that has the same probability of allelic identity as the observed population) and variance (size of an ideal population that has the same amount of gene-frequency drift as the observed population) effective population sizes (Kimura and Crow 1963; Crow and Denniston 1988). More recently, Ne theory has been expanded to take into account population subdivision (Whitlock and Barton 1997), overlapping generations (Hill 1979; Engen et al. 2007), age structure and fluctuating population size (Waples 2006), nonrandom mating (Caballero and Hill 1992), and large variation in fecundity or reproductive success (Nunney 1996; Hedrick 2005). In contrast to the plethora of models that examine how life-history traits influence effective population size, relatively little attention has been paid to the effect of multiple paternity (MP) on Ne.

Mating strategies in nature are extremely diverse, ranging from strict monogamy or strict polygyny to monandry (a female's clutch is sired by one male per season, also referred to as serial monogamy or temporal polyandry) to various types of MP, defined here as the siring of a female's clutch by more than one male within a reproductive season. MP can result from a number of processes, for example extra-pair copulations in socially monogamous or polygynous systems (Matocq 2004; Benedict 2008; Ophir et al. 2008; Sefc et al. 2008; Cohas and Allaine 2009) or promiscuity (Levitan 2008; Borkowska et al. 2009). No matter how it occurs, the effect of MP on genetic diversity and effective population size in natural populations has been debated (Robbins et al. 1987a,b; Waples 1987; Karl 2008; Pearse and Anderson 2009). A parallel and related debate has been ongoing in the sexual-selection literature, on the difference in Ne between monogamous and promiscuous lines (Wigby and Chapman 2004; Rice and Holland 2005; Snook et al. 2009). These debates have arisen because of conflicting views of the effect of multiple mating on male variance in reproductive success and genetic diversity.

On the one hand, MP is predicted to increase male variance in reproductive success, because variance will occur among males in competition for mates in addition to variance in competition for the brood (Karl 2008), but the two types of variance are not additive (Pearse and Anderson 2009). Generally, increasing variance in reproductive success decreases Ne (Wright 1931; Kimura and Crow 1963; Hedrick 2005). Nunney (1993) showed that, when females mated with one to few males (lottery polygyny), Ne decreased with respect to a monogamous mating system because of increased male variance in reproductive success. This type of argument has been used to claim that Ne is decreased in right whales where females mate with multiple males (Frasier et al. 2007).

On the other hand, MP is predicted to increase Ne by two other arguments. The first argument is that females that mate with multiple males will increase the genetic diversity of their broods and can be traced back as far as Murray (1964), who suggested that multiple mating and sperm storage in the land snail Cepaea nemoralis would be beneficial because the population underwent frequent genetic bottlenecks. Murray (1964) did not, however, explicitly test whether multiple or single paternity had different abilities to counteract the reduction in Ne due to a bottleneck. The second argument is that increased mating frequency by females will result in more equal sharing of paternity among males and therefore can potentially lower the male variance in reproductive success. Some empirical evidence supports this prediction, because average paternity share of the most successful male (within a brood) has been shown to be inversely related to the number of sires for that brood in both sea urchins (Levitan 2008) and salmon (Weir et al. 2010). But for this second argument to work, this decreased reproductive variance within a brood must offset the variance among broods in the population. A few models predict that the Ne of an MP population should always be higher than that of a monogamous (Sugg and Chesser 1994; Balloux and Lehmann 2003) or single paternity (Pearse and Anderson 2009) population, but these models assume that the comparison between mating systems is in lifetime reproductive success and ignore the large potential effects of overlapping generations on Ne (Nunney 1993). The claim that MP should increase effective size or maintain genetic diversity has also been made for Atlantic salmon (Martinez et al. 2000) and Pacific rockfish (Hyde et al. 2008).

At least in some circumstances, though, MP can be expected to increase Ne, although under others it may decrease Ne. This conclusion is supported by theory that demonstrates that the effective size of X-linked loci may be increased or decreased relative to Y-linked loci, depending on the differences between the sexes in variance in reproductive success (Charlesworth 2001). At the crux of the current debate is the distinction between multiple sires within the lifetime of the female (but one male per clutch) and multiple males per clutch. Both of these cases represent multiple sires per female, but do the two mating strategies have consistently different effects on Ne? The primary solution to this debate is to understand how the effects of MP (multiple males to a single clutch) on male variance in reproductive success differ from those of monandry (one male per clutch but multiple males in the lifetime of the female). The effect may depend largely on how uncertainty in the environment affects variance in reproductive success of each sex and how this uncertainty interacts with the mating system. In stable environments, most reproductive females will probably have offspring that survive to reproduce; therefore, competition among males can be expected to increase male variance in reproductive success and therefore decrease Ne. In uncertain environments few females in the population will successfully contribute to the next generation, but the benefit to the population Ne of carrying broods sired by multiple males may overwhelm the negative effect of competition among males. These predictions may be complicated by long generation times, which tend to suppress any effect of mating system on Ne (Nunney 1993).

Here I show theoretically that, in stable populations with equal sex ratios, whether an MP population has higher or lower annual variance in reproductive success than a strictly monandrous population depends on the distribution of within-clutch paternities across all females. I used simulations to calculate the male variance in reproductive success of MP populations under different distributions of within-clutch paternities and compared it to that of an otherwise identical monandrous population, under a range of parameters that characterize natural populations. The results from the simulations were used to calculate Ne from a model with separate sexes and overlapping generations (Engen et al. 2007). I have used this framework to show that an MP population can have an Ne higher than, lower than, or equivalent to that of a monandrous population. The results depend on generation time, the mean and variance in offspring production by females, the mean number of mates, and the distribution of paternity within a brood.

Methods

Variance in reproductive success is defined as the square of the deviation of the number of offspring per individual from the expected mean number of offspring for the population. First, I will show that under equal sex ratios, the mean number of offspring is the same for males and females, but the variance in the number of offspring for males will change depending on how paternities are distributed among females. Because the model of effective population size that I am using is based on annual demographic data (Engen et al. 2007), this proof only considers the comparison of male and female variances within a season. Then, I simulated populations to explore how different distributions of paternities may increase or decrease Ne through effects on male variance in reproductive success for a variety of parameters.

For a population of Nm males and Nf females, the total annual reproductive output of a population with Nf females and Nm males is given by inline image where bi is the number of offspring of female i that survive to reproduce and ρij is the proportion of female i’s brood sired by male j in that year. Therefore the ρij coefficients can be thought of as a description of within-clutch paternities. By definition, for each reproductive season and each female i, the sum of the ρij coefficients over all males is 1: inline image.

The mean number of offspring per female is inline image and the variance in the number of offspring per female is given by the sum of squared deviations:

image(1)

The mean number of offspring per male is: inline image. Under equal sex ratios (Nm=Nf), the expected number of offspring per male equals the expected number of offspring per female (μfm), but the variance of reproductive success of males is not equal to that of females and is given by

image(2)

The total number of offspring in the population can be expressed as: inline image. Under equal sex ratios, Nm= Nf,μmf, and using the previous equality it can be seen that female (eq. 1) and male (eq. 2) variances in reproductive success are equal when the sum of squares in the number of offspring are equal in the two sexes and overdispersion

image(3)

This equality is true when for each female i there is only one ρij= 1, which is the case when each male sires offspring with only one female in a season, but a different female in the next season (strict monandry). Whether MP results in a higher or a lower Ne than does monandry population therefore depends on how the distribution of within-clutch paternities (the ρij coefficients) changes the relative distributions of offspring in males when compared to females.

SIMULATIONS

To determine how the Ne of an MP population differs from that of a monandrous population, I simulated different distributions of ρij coefficients for a variety of life-history and demographic parameters. Previous theoretical models have compared multiple mating to the case in which a female mates with one male in her lifetime (Nunney 1993; Sugg and Chesser 1994; Balloux and Lehmann 2003), so this situation was not reexplored. The distribution of offspring among females was assumed to be independent of mating system. The simulations focused on the overall distribution of sires in a female's brood and not on the frequency of multiple mating. The link between multiple mating and within-clutch MP is not straightforward, because of processes such as cryptic female choice (Thornhill 1983) and sperm displacement (Parker et al. 1990). Through simulations, I explored how Ne(MP):Ne(Monandry) depended on the distribution of offspring among females, the distribution of sires per female, the distribution of offspring per sire, and the generation time.

The MP simulations are depicted in Figure 1. The simulations under monandry were conducted identically, except that every female in the population had only one mate. The simulations assumed a population of constant size, equal sex ratios, and constant probability of survival s independent of age or sex. Generation time T for a population of a constant size is defined as 1/(1 −s) (Engen et al. 2007). Under MP, for each reproductive season the number of sires per female was chosen from a zero-truncated Poisson distribution with a mean m, which ensured that every female had at least one opportunity to mate and simplified comparisons between mating systems. A Poisson-distributed, or nearly Poisson-distributed, number of sires has been observed in nature (Murray 1964; Makinen et al. 2007). Sires were chosen randomly from the population with equal probability. Note that a mean of 1 indicates that, although most females will have one mate, several females in the population will have offspring sired by two or more mates, so this situation is not equivalent to monandry (Fig. 1A).

Figure 1.

An example of the multiple paternity simulations in a population of five males and five females. (A) At time t for each female, her number of mates was chosen from a Poisson distribution with mean m. (B) The number of offspring per female was chosen from a negative binomial distribution with mean b and overdispersion Ω. Offspring were multinomially distributed among males with probabilities given by a geometric distribution with parameter p. (C) Adults were chosen to survive to time t+ 1 on the basis of the probability of survival s. In the picture, each adult that did not survive is marked with a gray “X”; offspring were chosen at random to recruit into the adult population (underlined and marked with arrows in the picture) and became reproductive at age 1. Estimates of male and female reproductive success were based on parentage of offspring that survive to reproduce. The simulations under monandry were conducted identically, except that females mated with only one male in each time step.

In the simulations, the mean offspring production and its variance among females were modeled as a negative binomial distribution. These offspring were then chosen at random to recruit into the adult population. Poisson-distributed reproductive success is probably rare in nature, and some evidence in natural populations indicates that reproductive success is negative-binomially distributed (Clutton-Brock 1988; Araki et al. 2007). Large variation in offspring production among females may also be caused by an increase in fecundity with size (Howard and Kluge 1985) or age (Berkeley et al. 2004). The distribution of offspring among females for each year was randomly drawn from a negative binomial distribution for a mean number of offspring per year (the birth rate, b; see “Calculation of effective size” below). The negative binomial distribution with parameters r and n has mean r(1 −n)/n and variance r(1 −n)/n2. Overdispersion (Ω, the variance divided by the mean) was used to explore how increasing female variance in offspring production affected Ne. Overdispersion was varied from 1 (Poisson) to 30 for a given mean number of offspring. For a given overdispersion and mean, the parameters r and n were calculated. Sex ratios at birth were assumed to be equal.

In the monandrous population simulations, every year each female mated with one male that was chosen randomly without replacement from the population. Because the males were chosen randomly as mates, for the monandrous population each male had a small chance (1/Nm) of being chosen as a mate in more than one mating season. In the MP population simulations, ρij coefficients were defined as the probabilities that male j sired offspring with female i and were distributed as follows: each female i had mi mates chosen from the population as described above. The number of offspring for each mate was multinomially distributed, and the probabilities for each of her mates (k= 1, 2, 3, … ,mi) were given by a geometric distribution with parameter p and scaled to sum to 1: inline image. Mates were randomly chosen and were ordered from the most to the least successful. As p approached 0, all mates approached making equal contributions to the brood. As p approached 1, one mate came to dominate the brood. This distribution has been shown to fit the empirical distribution of paternities in Drosophila when p= 0.5 (Snook et al. 2009). The appropriateness of this distribution for organisms with external fertilization was determined from data on broadcast-spawning organisms (D. R. Levitan, unpublished analysis of data presented in Levitan 2005a). Although mate order matters in Drosophila (Snook et al. 2009), and other factors such as distance and timing between mates are important in broadcast spawning organisms (Levitan 2005b), broadcast spawners also fit a geometric distribution when ordered from the most successful to the least successful sire (Fig. 2). Therefore the use of this distribution has broad applicability. Only the distribution of offspring among mates, not mate order, will affect the calculation of Ne.

Figure 2.

The empirical proportions of eggs sired in broadcast-spawning organisms follow a geometric distribution with p= 0.55, when ordered from the most to the least successful male (error bars represent standard errors). Unpublished analysis of data was presented by Levitan (2005a). The same distribution fits the proportion of offspring sired in Drosophila when p= 0.5 (Snook et al. 2009).

Individuals in the population were chosen to survive on the basis of the probability of survival s, which was constant and independent of age or sex. Offspring were chosen at random to replace dead adults in the population in the next time step (Fig. 1C). Recruits became reproductive at age one.

CALCULATION OF EFFECTIVE POPULATION SIZE FROM SIMULATIONS

The Ne for each mating system was calculated according to the method of Engen et al. (2007) for a fluctuating population with two sexes and overlapping generations. This method calculates the demographic variance of a rare allele in the population per generation, which also predicts the magnitude of genetic drift for genes of intermediate frequencies (Engen et al. 2005) and is therefore a direct measure of variance Ne. The Ne of females is defined as Nef=Nf/(Tfσ2df) where Tf is generation time and σ2df is the demographic variance of females carrying a rare neutral allele (Engen et al. 2007). In a population of constant size, 1:1 sex ratios, and equal survivorship, the demographic variance of the rare allele is a function of the annual mean (bf) and variance (σ2f) of the number of offspring per female, probability of survival (sf), and the covariance between an individual's number of offspring and the indicator variable (0 or 1) for its survival (cf) (Engen et al. 2007): σ2df=bf/4 +σ2f/4 +sf(1 −sf) +cf. The effective size of males is defined similarly. The total effective population size is Ne=N/(Tσ2dg) where inline image, and T and b are the averages of the male and female generation time and birth rates, respectively (Engen et al. 2007).

For a given generation time (Tf=Tm), survival (sf= sm) and birthrate (bf= bm) were calculated for a population of a constant size according to Engen et al. (2007). The parameters σ2m, σ2f, cm, and cf used in the estimate of Ne were calculated from simulations (shown below). Although the parameter b was given as input into the simulations, its estimate from the simulations varied around the theoretical expectation and was therefore also used in the calculation of Ne. The distribution of reproductive success used in the measurements of Ne was based on the number of offspring that survived to reproduce (not initial offspring production). If an adult individual i produces rij recruits in year j, the variances σ2 for each sex were estimated from simulations by the sum of squares:

image(4)

where j= 1, 2, … .t, i= 1, 2, …N, and Rj is the mean number of offspring registered in year j (Engen et al. 2007). Covariance between survival and reproduction was estimated from simulations by the sum of cross-products (Engen et al. 2007):

image(5)

where lij is the indicator of survival of individual i in year j, and Lj is the mean survival indicator across all individuals in year j.

One hundred replicate populations of 1000 individuals (500 males and 500 females) were simulated for 10 generations, under both MP and monandry. In this way each population was paired, such that it underwent exactly the same population dynamics except for mating system. This allowed the calculation of Ne(MP):Ne(Monandry) for each population, which was averaged and is presented in the results with 95% confidence intervals. Combinations of the following parameter sets were explored over values of overdispersion in the number of offspring per female from 1 (Poisson) to 30: generation time T= 2, 5, 10; mean number of mates m= 1, 2, 5, 10; geometric parameter of mate distribution p= 0.001, 0.5, 0.9; and mean offspring b= 2, 10, 100. Results were found to be independent of population size (Appendix 1). Because the focus of this article is on populations with overlapping generations, the results for simulations of discrete generations are presented in the Supporting information (Appendix 2). All simulations were performed with R (version 2.10.1, R Core Development Team 2009).

Results

The annual distribution of offspring among females for a given mean number of offspring (b) and overdispersion (Ω) is shown in Figure 3. When female reproductive success is Poisson distributed (Ω= 1), approximately 13.5%, 0.004%, and 0% of females fail to reproduce for b= 2, 10, and 100, respectively (Fig. 3). When Ω= 30, approximately 79.1%, 30.9%, and 0.0008% of females fail to reproduce for b= 2, 10, and 100, respectively (Fig. 3).

Figure 3.

Histograms of the number of offspring per female under equal means but different variances in female offspring production. The mean number of offspring (b) equals (A) 2, (B) 10, and (C) 100. Female offspring production is shown as either Poisson (overdispersion = 1, gray bars) or negative binomial (overdispersion = 30, black bars).

The relationship between Ω in female offspring production and annual female variance in reproductive success (measured as which offspring survive to reproduce) is presented in Figure 4. When the number of offspring per female is Poisson distributed (Ω= 1), the effect of the mean is small at all generation times, but at larger values Ω (>3) and when T= 2 years, decreasing b increases female variance in reproductive success because more females fail to reproduce (Fig. 4A). As T gets longer (10 years), both Ω and b have small effects on annual female variance in reproductive success (Fig. 4C).

Figure 4.

Annual female variance in reproductive success (VRS, measured by which offspring survive to reproduce) as a function of mean and overdispersion in female offspring production, estimated from simulations. Generation time (T) equals (A) 2 years, (B) 5 years, (C) 10 years.

Covariance estimates depended solely on generation time and were not affected by mating system, mean, or overdisperson in the number of offspring per female. The sexes did not differ in covariance estimates. Covariance estimates were 0.215, 0.055, and 0.016 for T= 2, 5, and 10, respectively.

Under the Engen et al. (2007) model for the situations considered here, as generation time (T) increases, the Ne of an otherwise ideal population (Poisson-distributed reproductive success and no covariance) will approach 1/2 N (Nunney 1993; Fig. 5). When Ω= 1, b has small effects on Ne (Fig. 5). Also, when b= 100, Ω has little to no negative effect on Ne(Monandry) for a given generation time (Fig. 5), but there is an interaction between T and b at larger values of Ω. When Ω= 30 and b= 100, Ne(Monandry) decreases with T (Ne(Monandry)= 534, 490, and 489 for T= 2, 5, and 10, respectively). When Ω= 30 and b= 2, Ne(Monandry) increases with generation time (Ne(Monandry)= 149, 226, and 299 for T= 2, 5, and 10, respectively). This interaction occurs because longer generation time tends to decrease Ne when female reproductive success is Poisson distributed (Nunney 1993; Engen et al. 2007), but when many females fail to reproduce annually (such as when Ω= 30 and b= 2) longer generation time can “rescue” Ne through increased numbers of seasons during which females can reproduce.

Figure 5.

The effective size of a monandrous population as a function of mean and overdispersion in female offspring production, estimated from simulations. Generation time (T) equals (A) 2 years, (B) 5 years, (C) 10 years.

The Ne of an MP mating system compared to a monandrous mating system for a variety of parameters is presented as the ratio Ne(MP):Ne(Monandry) with 95% confidence intervals in Figures 6 and 7. In Figure 6, the effect of changing b and T on Ne(MP):Ne(Monandry) is shown for the parameter p= 0.5 on the geometric distribution of paternity. Under all scenarios presented, increasing the mean number of mates (m) increases Ne(MP):Ne(Monandry), because it always decreases male variance in reproductive success (Fig. 8), but the magnitude of this increase depends on T, b, and Ω. The positive effect on Ne(MP):Ne(Monandry) of increasing the number of mates is largest when b= 2 and T= 2 (Fig. 6 upper left) and smallest when b= 100 and T= 100 (Fig. 6 bottom right). The reasons for this are discussed below.

Figure 6.

The ratio of effective population size under multiple paternity to that under monandry (Ne(MP):Ne(Monandry)) as a function of overdispersion in offspring production and mean number of mates. Each panel represents a different combination of the mean number of offspring (b= 2, 10, 100 in columns) and generation time (T= 2, 5, 10 in rows), with the probability of paternity geometric distributed with p= 0.5.

Figure 7.

Ne(MP):Ne(Monandry) as a function of overdispersion in offspring production and mean number of mates. Each panel represents a different combination of the parameter on the geometric distribution of paternity. When p= 0.001 (left column) approximately all males have an equal probability of paternity; when p= 0.9 (right column) one male will dominate the brood. Rows show the effect of increasing generation time (T= 2, 5, 10 years), and the mean number of offspring is fixed at b= 10.

Figure 8.

Annual male variance in reproductive success under monandry and multiple paternity, as a function of the mean number of mates (m) in an MP population (1, 2, 5, and 10) and overdispersion in female offspring production. Parameters are p= 0.001, generation time (T) = 2, mean number of offspring (b) = 10 (compare to Ne(MP):Ne(Monandry) plotted in Figure 7A).

In an ideal female population (Poisson offspring production, Ω= 1), Ne(MP):Ne(Monandry)≤ 1 for all T, m, and b investigated when p= 0.5 (Fig. 6). As Ω is increased, Ne(MP):Ne(Monandry) is also increased, but it may or may not exceed 1 depending on the parameters. Increasing b both decreases Ne(MP):Ne(Monandry) at values of Ω > 1 and decreases the positive effect of increasing m on the Ne(MP) populations. When b= 100, Ne(MP):Ne(Monandry) < 1 for all values of Ω simulated (Fig. 6 right column), but much larger values of Ω will eventually cause Ne(MP):Ne(Monandry) to exceed 1 (results not shown). This occurs because Ne(MP):Ne(Monandry) is driven largely by the variance in reproductive success among females. As the mean number of offspring increases, female variance in reproductive success decreases for a given value of Ω (Figs. 3 and 4), and a monandrous population will have a higher Ne than an MP population.

The mean number of offspring (b) and generation time (T) interact in their effects on Ne(MP):Ne(Monandry) (Fig. 6). When b= 2, increasing T will decrease Ne(MP):Ne(Monandry) for a given m. For example, when b= 2, m= 2, and Ω= 5, Ne(MP):Ne(Monandry) will decrease (Ne(MP):Ne(Monandry)= 1.018, 1.010, and 1.006 for T= 2, 5, and 10, respectively; Fig. 6 left column), but when b= 100, increasing T will increase Ne(MP):Ne(Monandry) for a given m (when m= 2 and Ω= 5, Ne(MP):Ne(Monandry)= 0.945, 0.974, and 0.986 for T= 2, 5, and 10, respectively; Fig. 6 right column). This interaction occurs because the effect of mating system becomes compressed at long generation times (Nunney 1993) and tends to equalize Ne(MP):Ne(Monandry).

The effect of changing the parameter p is shown in Figure 7 for generation times T= 2, 5, and 10, while b is held constant at 10. When p= 0.001, all males will have approximately equal probabilities of contributing to the female's brood. In this case, Ω and m are important in determining Ne(MP):Ne(Monandry), although their importance decreases at longer T (Fig. 7, left column). When p= 0.9, one male will dominate the brood and other mates have a small probability of siring offspring. This situation results in Ne(MP):Ne(Monandry) < 1 for the values of T and m simulated (Fig. 7, right column). The parameters p, m, and T interact in their effects on Ne(MP):Ne(Monandry). When p= 0.001 (all males equal), Ω= 1, and m= 2, increasing T will increase Ne(MP):Ne(Monandry) (0.952, 0.978, 0.988 for T= 2, 5, and 10, respectively), but at Ω= 15 and m= 5, increasing T will decrease Ne(MP):Ne(Monandry) (1.068, 1.033, 1.018 for T= 2, 5, and 10, respectively). However, when p= 0.9 (one male dominates), increasing T will increase Ne(MP):Ne(Monandry) across all values of Ω and m, although the ratio is always predicted to be less than 1. These interactions again occur because the effect of mating system becomes compressed at long T (Nunney 1993) and drives Ne(MP):Ne(Monandry) toward one.

Whether Ne(MP):Ne(Monandry) is larger or smaller than 1 depends on the relative effects of MP and monandry on male variance in reproductive success, because female variance in reproductive success is simulated to be independent of mating system. This situation is shown in Figure 8 for p= 0.001, T= 2, and b= 10 (compare to the ratio plotted in Fig. 7A). Increasing m will always decrease male variance in reproductive success for a given set of parameters, because males have more opportunities to produce offspring. Under monandry, increasing Ω has a direct relationship to male variance in reproductive success. Under MP, however, the intercept of this relationship is increased, but the slope is decreased (Fig. 8). The increase in intercept depends on m represents how multiple mating increases competition among males, and the decrease in slope represents how the variance in reproductive success among males in an MP population increases at a lower rate with female variance in reproductive success than it does in an monandrous population. When Ω= 1 and m= 1, such that most females have one mate but a few have multiple mates, the male variance in reproductive success is higher under MP than under monandry, and Ne(MP):Ne(Monandry) < 1 (Fig. 8; compare to the ratio plotted in Fig. 7A). When m= 10 and offspring production among females is somewhat overdispersed (Ω > 2), the variance in reproductive success among males in an MP population is lower than that in a monandrous population, causing Ne(MP):Ne(Monandry) > 1. Finally, as variance in reproductive success among females increases (regardless of the frequency of multiple mating), the variance in reproductive success among males in an MP population rises more slowly than it does in a monandrous population, causing Ne(MP):Ne(Monandry) to increase (Fig. 8).

Although not the focus of this article, simulations were also performed for populations with discrete generations, and similar results were found (Appendix 2).

Discussion

Previous models of Ne have compared the effect of multiple sires in the lifetime of the female with strict monogamy (Nunney 1993; Sugg and Chesser 1994; Balloux and Lehmann 2003). The study reported here is the second to compare explicitly the effect of multiple sires over the reproductive life of the female with that of multiple sires per clutch on variance Ne. The first was by Pearse and Anderson (2009), whose model predicted that an MP population will always have a lower variance in male reproductive success than a single paternity population (in which a male's mating success is indpendent of his previous mating success within and between seasons). The simulations presented here illustrate that MP can produce a variance effective size that is higher than, equal to, or lower than that of a strictly monandrous population, in which a male mates with one female in one season and a different female in the next. The direction and magnitude of this effect depend on how generation time, the frequency of multiple mating, the mean and variance in offspring production, and the distribution of paternities within the brood affect male variance in reproductive success under MP and monandry.

Generally, when annual female variance in reproductive success is Poisson-distributed, competition between males dominates, and variance in male reproductive success is higher under MP than it is under monandry (Ne(MP):Ne(Monandry) < 1). These results are consistent with those of Nunney (1993) for a lottery polygyny. As the variance in reproductive success among females is increased, the variance in reproductive success among males under MP increases at a lower rate than it does under monandry, producing Ne(MP):Ne(Monandry)≥ 1 for some parameter combinations. Variance in reproductive success among females is inversely related to the mean number of offspring and generation time and directly related to overdispersion in offspring production. Parameters that increase opportunities for males to sire offspring, such as higher mating frequency or more equivalent brood-share among males, decrease the variance in reproductive success among males and therefore increase the Ne(MP):Ne(Monandry) ratio. At longer generation times, the effect of the mating system becomes more compressed, in the sense that the range Ne(MP):Ne(Monandry) is smaller for a given set of parameters, and Ne(MP):Ne(Monandry) will approach 1. This result is consistent with theory that predicts that the effect of the mating system will be small under longer generation times (Nunney 1993).

In natural systems that have low variance in reproductive success among females and generation times of less than two years, MP will decrease Ne because of the increased variance in male reproductive success. This scenario is characteristic of systems like that of Drosophila, in which male variance in reproductive success is generally higher than that of females (Bateman 1948) and offspring survival is random (Crow and Morton 1955). Male variance in reproductive success is commonly shown to exceed that of females in several taxa, an effect attributed to competition among males (Shuster and Wade 2003; insects, Finke 1988; McVey 1988; Rodríguez-Muñoz et al. 2010; amphibians, Howard 1988; birds, Bryant 1988; Smith 1988; fish, Tatarenkov et al. 2008), but most estimates of variances in reproductive success come from experimental manipulations or small populations where individuals can be tracked. Also, because of uncertainty about the size of the potential gene pool, standardized variances are often calculated for only the proportion of the population that actually reproduces. This practice will cause published estimates to be lower than actual values.

In general, only when a small percentage of females in the population have offspring that successfully survive to reproduce will MP increase Ne. This result is consistent with the notion that MP can be a variance-discounting strategy in uncertain environments (Yasui 1998; Jennions and Petrie 2000). In this case, effective population size will already be small because few parents will contribute to the next generation. Direct measurements of large variances in reproductive success have been observed or estimated in insects (Cushman et al. 1994), birds (Fitzpatrick and Woolfenden 1988; McCleery and Perrins 1988; Scott 1988), and fish (Waples 2002; Araki et al. 2007). In Florida scrub jays, Aphelocoma coerulescens, about 20% of the breeders in the population were estimated to produce about 65% of replacement breeders, and standardized variance in reproductive success (measured as fledglings that survived to breed) to range from 0.81 to 32.11 in any given year (Fitzpatrick and Woolfenden 1988). In marine organisms, standardized female variance in reproductive success could be much higher, but the difficulty of tracking larvae and juveniles through pelagic development precludes direct measurements. For high-dispersal organisms, “sweepstakes” reproduction has been inferred from patterns expected in cases with few effective breeders, including lower genetic diversity in offspring than in adults (Hedgecock 1994), heterozygote excess (Hedgecock et al. 2007), a large number of siblings in a recruit cohort (Selkoe et al. 2006), and isolation by time among recruit cohorts (Maes et al. 2006). In salmon, family-correlated survival during the marine phase decreased the effective number of breeders by up to 60% in some cases (Waples 2002), but evidence for this sweepstakes phenomenon has not been found in all marine organisms with life histories that one might expect to generate sweepstakes (Flowers et al. 2002). Because many of these marine organisms are also long-lived, the simulations presented here suggest that the mating system will play a small role under generation times longer than 10 years.

The model presented here assumes that a female's reproductive success is independent of the number of males she mates with and is determined instead by the environment. In Bateman's (1948) classic example, male reproductive success was correlated with the number of mates, but female reproductive success was not. In birds, multiple copulations are not correlated with clutch size, although MP of clutches was not explicitly examined (Birkhead et al. 1987). Independence of female reproductive output from mating frequency has been observed in amphibians (Jones et al. 2004; Croshaw 2010) and fish (Tatarenkov et al. 2008), but recent studies challenge Bateman's paradigm (Snyder and Gowaty 2007; Rodríguez-Muñoz et al. 2010) and suggest that reproductive success of females is increased by multiple mating. This result is expected in marine broadcast spawners such as sea urchins, which release a large number of gametes into the water column to mix and in which sperm can be limiting (Levitan 2008). A review on insects (Arnqvist and Nilsson 2000) reported that MP significantly increased offspring production, although it decreased the longevity of females without nuptial feeding. Higher numbers of offspring from multiply sired broods has been observed in mosquitofish (Robbins et al. 1987a), and positive correlations between the degree of MP and clutch size have also been observed in reptiles and turtles (Pearse et al. 2002; Uller and Olsson 2008). Many of these studies, however, measure reproductive success as survival of recently fertilized gametes or hatchlings, which are then subject to often high mortality from ecological and environmental factors. If the number of offspring that survive to reproduce does indeed increase with the number of sires of a brood, then MP populations may be able to attain larger effective sizes than my simulations suggest.

I examined the effect of multiple mating on Ne as a function of overall fitness and ignored the processes of sexual selection and fecundity selection, which may determine fitness (Arnold 1994). The costs and benefits of multiple mating have been well documented and are likely to be species-specific (Arnqvist and Rowe 2005). In natural populations, the frequency of MP will depend on the balance between these costs and benefits and mate encounter rates (which may vary with density or population size, Jensen et al. 2006).

The results presented here clarify that the variance in male reproductive success due to multiple sires per clutch cannot be assumed to be added on to the variance due to having multiple mates (resulting in a decrease of Ne, Karl 2008). Ultimately however the effect of multiple paternity on Ne will depend on the comparison of interest, whether it be to a monogamous (Nunney 1993; Sugg and Chesser 1994; Balloux and Lehmann 2003), single paternity (Pearse and Anderson 2009), or strictly monandrous (this study) population. Although no model is likely to capture the full complexity of the real world, my results provide a context within which MP can be expected to affect Ne.


Associate Editor: R. Buerger

ACKNOWLEDGMENTS

Many thanks to D. R. Levitan, B. D. Inouye, A. A. Winn, C. P ter Horst, N. P. Fogarty, M. Lowenberg, M. Tomaiuolo, J. Fierst, B. Hollis, A. Thistle, and two anonymous reviewers for comments on earlier versions of this manuscript. Thanks are also due to D. Pearse for helpful discussion on the topic. This work was supported in part by a Graduate Research Fellowship from the National Science Foundation.

Appendices

Appendix 1

The applicability of these simulations is limited if the results change with population size. Although the value of Ne changes with population size, the results for Ne(MP):Ne(Monandry) were independent of population size. This independence was verified by plotting of estimates of Ne(MP):Ne(Monandry) in populations of N= 5000 against those for N= 100 for every parameter combination. The results revealed a 1:1 relationship (Fig. S1). More stochasticity was observed in very small populations (N < 100). A 1:1 relationship was also found when the results from these populations were plotted against the results presented in the main section for N= 1000 (results not shown).

Appendix 2

Because by definition populations with discrete generations cannot be monandrous, these populations were simulated under multiple paternity and monogamy in the same manner as described in the Methods. Because the model of Ne for overlapping generations used in the simulations cannot be extended to populations with discrete generations for all the parameter combinations, Kimura and Crow's (1963) model of variance effective size was used to calculate the Ne of each sex i:

image

where Ni is the number of individuals, μi is the mean number of offspring, and σ2i is the variance in the number of offspring of sex i that survive to reproduce. Total effective size was calculated with Wright's (1931) formula:

image

Ne(MP):Ne(Monogamy) was calculated for each population, which was averaged and presented with 95% confidence intervals. Combinations of the following parameter sets were explored over values of overdispersion in female offspring production (1 ≤Ω≤ 30): mean number of mates = 1, 2, 5, 10; geometric parameter of mate distribution = 0.001, 0.5, 0.9; and mean offspring number = 2, 10, 100. The distributions of offspring in the generations are identical to those presented in Figure 3.

Female variance in reproductive success (based on which of their offspring survive to reproduce) decreases as the mean number of offspring (b) is increased (Fig. S2A) and is estimated to be much larger than under overlapping generations when b= 2 (compare to Fig. 4). Female variance in reproductive success has a direct influence on Ne under monogamy (Fig. S2B). Ne(MP):Ne(Monogamy) increased as the mean number of mates increased, decreased as the parameter on the geometric distribution (p) increased (as p→ 1, one male will approach domination of the brood), and decreased as b increased (Fig. S3).

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