The effective size, Ne, of a population is generally defined as the size of an ideal population, with constant finite size, no mutation and random union of gametes in each generation, in which the population genetic dynamics are comparable with that of the actual population under study (Wright 1931). An important departure from this ideal is caused by ‘clustered reproduction’, i.e. the production of multiple offspring by a female in a single reproductive bout (clutch, brood, litter, etc.). In such cases, it is possible that a single male will sire all the offspring in a brood. This combination of clustered reproduction and single paternity (SP) has the potential to reduce a population's Ne by increasing the variance in male reproductive success, especially in the case of highly polygynous species (e.g. Storz et al. 2001) or when the variance in female fecundity and offspring survival is high. By contrast, if the father of each offspring in the brood is chosen randomly from the population, then clustered reproduction would have no influence on the variance in male reproductive success. Between these two extremes lies multiple paternity (MP), in which the clustered offspring of a polyandrous female are descended from multiple males, each siring a portion of the clutch. Although in most cases there is still a significant departure from the ideal of random mating, it is generally accepted that MP reduces the variance in male reproductive success and hence increases Ne relative to SP in the face of clustered reproduction.
In a recent paper, Karl (2008) reviewed the literature on the relationship between MP and Ne and concluded that, under most natural conditions, MP does not increase Ne relative to SP in an otherwise identical mating system. Karl (2008) admits that this conclusion runs contrary to both theoretical expectations (Sugg & Chesser 1994; Balloux & Lehmann 2003) and the often stated empirical inference (Davis et al. 2001; Shurtliff et al. 2005; Pearse et al. 2006; Moore et al. 2008) but points out that these studies are effectively comparing multiple mating with strict monogamy, which is rare in animals. As multiple mating may also occur in the context of temporal polyandry, in which a female produces several single-sired broods of offspring over its lifetime and may remate between them, Karl (2008) argues that it is more appropriate to compare MP with SP within this context, and that in such a comparison MP will not increase Ne.
We agree with Karl's (2008) excellent point that MP and temporal polyandry can have similar effects on Ne, with temporal polyandry providing an effective equivalent to MP over the lifespan of any female. This similarity has important genetic implications for conservation, yet it is commonly unrecognized in the literature. Notably, studies that detect MP during a single reproductive season, but which do not sample in a manner that allows evaluation of temporal polyandry, do demonstrate an increase in Ne relative to SP within broods (Davis et al. 2001; Shurtliff et al. 2005; Pearse et al. 2006; Moore et al. 2008). However, the importance of this per-season MP should be evaluated in terms of how it affects the variance in male lifetime reproductive success, which will also be influenced by temporal polyandry. Such an assessment is rarely made; so, it does appear that many comparisons between MP and SP implicitly equate SP in a single clutch with lifetime monogamy.
We disagree, however, with Karl's (2008) assertion that MP in the presence of temporal polyandry will not decrease (and may actually increase) the variance in male reproductive success and hence will not increase Ne (and may decrease it). To prevent the spread of this misconception in the literature, we provide a mathematical demonstration that, for a simple class of models, MP will decrease the variance in male reproductive success relative to an otherwise identical mating system with SP, even in the presence of temporal polyandry. This reduced variance in male reproductive success will increase Ne, all else being equal. This conclusion is consistent with the theoretical and simulation studies of Sugg & Chesser (1994) and Balloux & Lehmann (2003), both of which found that multiple mating by females resulting in MP and/or temporal polyandry leads to an increase in Ne. It is likely that Ne is greater under MP than under SP in a much larger class of models, but it is beyond the scope of this brief communication to seek this generality. Rather we aim to show in brief that Karl's (2008) assertions fail to hold for very simple models, and that considerable work is still needed to delineate the situations (if any) under which SP could increase Ne relative to MP, all else being equal.
We consider a class of models defined by the following set of assumptions for a dioecious population with a mating system of temporal polyandry. For each year t, let there be Ft reproductively mature females indexed by f = 1,…,Ft and let denote the number of offspring produced by female f in year t. Let males have a reproductive lifespan of r years and let there be Mt reproductively mature males in each year t, and assume that each male has an equal chance of siring the offspring of any female f, independently of its reproductive status with any other females in the same or different years. Now, we will consider the lifetime reproductive success (i.e. number of offspring produced) of a typical male who becomes reproductively mature at time t = 1 and becomes non-reproductive at time t = r. We first consider the case with no MP, then the case with MP, and finally show that MP always decreases the variance in male reproductive success in this simple model.
Under strict temporal polyandry with no MP, the number of offspring produced by a male is found by simply adding up the clutch sizes of all the females he mated with. Let the variable be 1 if the male mated with female f in year t, and 0 if he did not. The number of offspring produced by this male in its lifetime under SP is thus:
Because all males have an equal chance of mating with any female, and . Thus, conditional on the female clutch sizes, denoted collectively by CSP, the expected value of QSP can be found as:
Because is independent of for f ≠ h, the conditional variance of QSP can also be easily computed:
the variance of a Bernoulli random variable with expected value 1/Mt.
Now, we consider the case of MP. We let the total clutch sizes of each female in each year, , remain the same but imagine that each clutch is broken into Lf > 1 subgroups of size , and assume that each of these subgroups of offspring will be sired, independently, by a single male, and every male has an equal chance of siring each subgroup. As the total number of individuals in a clutch remains the same, we have
In addition, we let if the male sired the lth subgroup of offspring from female f in year t and 0 otherwise; hence, and . The expected number of offspring for a typical male, conditional on all the s, collectively referred to as CMP, remains unchanged from that of the SP case:
Likewise, the conditional variance of QMP can be computed as:
From eqns 1 and 2 it is now clear that var(QMP|CMP) ≤ var(QSP|CSP) for any value of CMP and the corresponding values of CSP because of the simple fact that
The inequality in eqn 3 follows from the fact that a2 + bc ≤ (a + b)2 for all non-negative integers a and b. var(QMP|CMP) and var(QSP|CSP) are equal only in the degenerate case in which for every f and t, there is a single equal to , which is, by definition, the case of SP. Furthermore, the inequality holds for the marginal variances: var(QMP) ≤ var(QSP). This follows from the fact that var(X) = E[var(X|Y)] + var[E(X|Y)] and the values of the conditional variances and expectations computed above; however, the details are omitted.
The above inequality is strict so long as at least one female in at least 1 year has a clutch with multiply sired members. It holds strictly for any male regardless of its probability of siring a clutch or a subgroup, so long as that probability is neither 1 nor 0 (therefore, the restriction that every male has an equal probability of siring a clutch or subgroup in a year could be relaxed). It is thus clear that, all other things being equal, the occurrence of MP in a range of simple models will, on average, always decrease the variance in male reproductive success, regardless of the variance in female fecundity or the distribution of subgroup sizes, and hence will increase the effective size of a population, even when compared with the case of temporal polyandry.
It is worth noting that from a conservation standpoint, the importance of the effects of MP on Ne is greatest when population size is small. At one extreme is the example of a single female land snail surviving a bottleneck (Murray 1964, as cited by Karl 2008). If that female carries stored sperm from more than one male, Ne will clearly be significantly higher than if she mated with a single male. On the other hand, if thousands of males and females are in a mating system with temporal polyandry, the increased Ne gained by the addition of simultaneous polyandry (MP) may have a less meaningful influence on the maintenance of genetic diversity in the population. Nonetheless, multiple mating by either sex will typically increase Ne and decrease the variance in Ne per generation, significantly improving the maintenance of genetic diversity in a conservation context (Fiumera et al. 2004).
The discrepancy between our conclusions and those of Karl (2008) appears to stem primarily from three differences in the interpretation of the effect of MP on the variance in male reproductive success. First, MP within a single reproductive season occurs when individual females mate with more than one male (or, in some species, may also result from storage of sperm across reproductive seasons even if a female mates only once per season; Pearse & Avise 2001; Uller & Olsson 2008). Assuming that the mates are chosen at random, the increase in mating opportunities introduced by MP will both reduce the number of males that achieve zero reproduction and allow males to acquire fractional paternity of a brood. We have shown that both of these factors reduce the variance in male reproductive success relative to a system in which each female mates only once, while Karl (2008) assumes that the addition of MP will increase the variance in male reproductive success.
One assumption of our model is that a male's chance of mating with a given female is independent of its previous mating success. This assumption was made mostly for the mathematical convenience it confers; however, we do note that it is possible that a strong dependence of mating success between females could lead to scenarios in which Ne under SP is greater than under MP. For example, a mating system model could be devised with an operational sex ratio of exactly 1:1, zero variance in female reproductive success (i.e. equal fecundity and offspring survival) and perfect monogamy (i.e. all males mate with exactly one female and vice versa), such that all males achieve exactly one mating. In this setting, there will be zero variance in male reproductive success and the addition of partial MP would necessarily increase the variance in male reproductive success because some males would now sire only a fraction of a clutch, while others will sire offspring equivalent to more than one full clutch. Thus, it appears that rare combinations of atypically low variance in female reproductive success, equal or close-to-equal sex ratios, and perfect or near-perfect monogamy could lead to higher Ne under SP than under MP; however, we suspect that such conditions will be the norm in few species.
The second difference between our model and that of Karl (2008) lies in how the variance in total male reproductive success is computed. In considering animal mating systems with both temporal polyandry and MP, Karl (p. 3975) states, ‘…there is variance in the per-brood reproductive success of a male in addition to the existing population variance in male mating success. The total variance in male reproductive success…is therefore necessarily larger with MP and would lower Ne relative to what would be seen only with temporal polyandry’. In other words, the variance in total male reproductive success [var(T)] is equated to the sum of the variance in the number of matings obtained by a male [var(P)] and the variance in the number of offspring per mating [var(C)]. As the variance of a sum is equal to the sum of the variances for uncorrelated random variables, Karl's assumption that var(T) can be written as var(P) + var(C) seems to imply incorrectly that the underlying model of total reproductive success is T = P + C. However, as we have shown, the total reproductive success is not the sum of P and C, but rather the sum of the number of offspring sired with each female a male mates with each year, summed over the male's reproductive lifetime (e.g. imagine that all females produce 10 offspring and a given male mates with five females (P) and sires four of each female's offspring (C); then its total reproductive success is not 5 + 4 = 9, but 4 + 4 + 4 + 4 + 4 = 20). Thus, the variance in total male reproductive success under MP is not var(C) + var(P).
Finally, Karl (2008) implicitly assumes that MP adds variance in per-brood reproductive success where none previously existed. In fact, there is a non-zero variance in male reproductive success under any mating system unless male copulation success, female fecundity and offspring survival are all completely equalized. In species with clustered reproduction, MP will tend to reduce this variance relative to SP, all else being equal. Thus, the conclusion that ‘temporal polyandry and MP together are likely to result in a larger variance in male reproductive success [and] the additional variance due to MP is expected to reduce Ne’ (Karl 2008, p. 3976) is incorrect. On the contrary, MP will probably reduce the variance in male reproductive success, and hence increase Ne, relative to SP, both with and without temporal polyandry.