EFFECTS OF REPRODUCTIVE COMPENSATION, GAMETE DISCOUNTING AND REPRODUCTIVE ASSURANCE ON MATING-SYSTEM DIVERSITY IN HERMAPHRODITES

Hermaphroditic organisms exhibit diverse mating systems, ranging from complete selfing to exclusive outcrossing (Goodwillie et al. 2005; Jarne and Auld 2006). Two opposing genetic features have long been recognized as key influences on this continuum. Each selfed offspring contributes two copies of maternal alleles to the next generation in contrast to one copy for outcrossed offspring, which strongly favors selfing (Fisher 1941). This genetic-transmission advantage is offset in populations with appreciable genetic load by poor performance of selfed offspring, whose high homozygosity increases the chance of expression of deleterious recessive traits compared to more heterozygous, outcrossed offspring (Charlesworth and Charlesworth 1999). Consideration of these two factors alone leads to the expectation that complete selfing evolves in diploids when selfed offspring are at least half as fit as outcrossed offspring, on average, otherwise selection favors complete outcrossing (Nagylaki 1976; Lloyd 1979; Lande and Schemske 1985). These contrasting outcomes represent the extremes of the mating-system continuum, but do not explain the common occurrence of mixed mating, the production of combinations of selfed and outcrossed offspring (Goodwillie et al. 2005; Jarne and Auld 2006). Consequently, additional processes that modulate the genetic aspects of mating and contribute to mixed mating have received considerable recent attention (reviewed by Goodwillie et al. 2005).

Mixed mating can be adaptive only if it provides a compromise that capitalizes on the contrasting advantages of selfed and outcrossed offspring. Three mating processes that can promote mixed mating illustrate such compromises. First, reproductive assurance, or increased fecundity caused by autonomous selfing when outcrossing opportunities are limited, can be favored even though the resulting offspring may be genetically inferior to outcrossed progeny (Lloyd 1979; Goodwillie et al. 2005; Jarne and Auld 2006). Second, gamete discounting, which occurs when the use of gametes in self-mating that otherwise could have been involved in outcrossing (Nagylaki 1976; Lloyd 1992), erodes the benefits of selfing, selecting for some outcrossing when selfing is genetically advantageous (Holsinger 1991; Johnston 1998; Porcher and Lande 2005a). The third process, reproductive compensation, occurs when individuals produce more eggs (ovules) than the number of offspring that they can provision. These “extra” eggs reduce the fecundity cost of offspring deaths during development (Minchella and Loverde 1981) resulting from predispersal (early-acting) inbreeding depression, and thereby increase the production of offspring of sufficient genetic quality to reach independence. Such compensation for embryo deaths mitigates the disadvantages of self-mating in populations with high genetic load (Porcher and Lande 2005b; Harder and Routley 2006).

These three processes may be the most common mechanisms responsible for mixed mating by hermaphrodites. Hermaphrodites tend to have limited mobility and often fail to realize their reproductive potential (Ashman et al. 2004; Jarne and Auld 2006). Reproductive assurance alleviates this shortcoming (Eckert et al. 2006), if low fecundity results from insufficient, rather than poor-quality, mating. Indeed, reproductive assurance is the most commonly invoked adaptive explanation for mixed mating (Lloyd 1992; Goodwillie et al. 2005; Jarne and Auld 2006). In contrast, male-gamete discounting is recognized as a common nonadaptive explanation for mixed mating (Lloyd 1992; Goodwillie et al. 2005; Jarne and Auld 2006). In particular, for plants, self-pollination between flowers on the same individual involves the same interactions with pollen vectors as cross-pollination, so the resulting pollen discounting has been interpreted as an inevitable consequence of outcrossing (Lloyd 1992). The contribution of reproductive compensation to the incidence of mixed mating is less appreciated, because its influence on mating-system evolution has been recognized only recently (Porcher and Lande 2005; Harder and Routley 2006). Nevertheless, this mechanism may be widespread. For example, a survey of seed production following excess cross-pollination for 65 plant species found that only 60% of ovules become seeds (Harder and Routley 2006), suggesting that plants typically produce considerably more ovules than they can mature into seeds.

Despite their significance in the evolution of hermaphroditic mating systems, the relative importance and possible interacting effects of inbreeding depression, reproductive assurance, gamete discounting, and reproductive compensation have not been examined. All mating-system theory considers inbreeding depression explicitly, but the implications of variation in inbreeding depression during the life cycle have received little attention (but see Porcher and Lande 2005 a, b; Harder and Routley 2006). In particular, because recessive lethal traits tend to be expressed earlier than less deleterious traits (Husband and Schemske 1996; Charlesworth and Charlesworth 1999), inbreeding depression may affect selfed offspring more strongly when they depend on parental resources than after they are independent. Mating-system theory also implicitly incorporates female-gamete discounting, or a trade-off between the production of selfed and outcrossed offspring, but this trade-off may have been represented incompletely, because mating-system theory seldom considers consumption of maternal resources by developing offspring (although see Porcher and Lande 2005b; Harder and Routley 2006). Within a framework of nonspecific inbreeding depression and a simple trade-off in the production of selfed versus outcrossed offspring (and perhaps insufficient mating), mating-system theory usually considers the consequences of only one other influence on reproduction (reviewed for plants by Goodwillie et al. 2005). Porcher and Lande (2005 a, b) provided the most complete analyses for plant reproduction; however, they considered pollen discounting and reproductive compensation separately and focused on the consequences of different selfing rates for the evolution of genetic load, so the converse problem focusing on mating-system evolution has not been explored fully.

In this article, we consider the theoretical joint effects of inbreeding depression, reproductive assurance, gamete discounting, and reproductive compensation on the evolution of hermaphroditic mating systems, specifically those of angiosperms. We begin by examining the deterministic influence of reproductive compensation on mating-system evolution when selfing does not alter male outcrossing opportunities, individuals produce many female gametes, and selfed and outcrossed embryos compete equally for maternal resources. We then consider more realistic situations of limited female reproductive potential, stochastic offspring production, and an advantage of outcrossed embryos over selfed embryos during competition for maternal resources. Finally, we return to the deterministic model and allow male-gamete discounting. Although our analysis incorporates features specific to plant mating, the conclusions should apply generally to hermaphroditic organisms.

Mating Models

To explore the consequences of reproductive compensation and male-gamete discounting for mating-system evolution, we model plants that produce P pollen grains and O ovules and consider the entire reproductive cycle from pollination to seedling establishment (Fig. 1; see Table 1 for definitions of additional parameters and variables). We assume that ovule and pollen production per individual remain constant during mating-system evolution, leaving sex-allocation implications for subsequent examination. Mating outcomes within a population depend on the fates of pollen, so we define a plant's phenotype as its contributions of pollen to self-pollination and potential pollen export. Because male-gamete discounting occurs during mating, we focus on selfing that occurs simultaneously with outcrossing. Reproductive assurance can enhance fecundity if insufficient pollen import limits seed production. The plants we model have enough resources to mature only a fraction, m, of their ovules into seeds, so they produce (1 −m)O compensatory ovules that can fail after fertilization for various reasons (e.g., genetic death, predation, maternal choice) without affecting maximal seed production. For simplicity, we assume a fixed genetic load in the population, which is not unreasonable given that reproductive compensation allows deleterious alleles to persist in populations, despite the purging effect of inbreeding depression (Charlesworth 1994; Hastings 2001; Porcher and Lande 2005b). This outcome results because compensation allows heterozygous parents to perpetuate recessive lethal alleles by producing more (viable) heterozygous offspring than they would have if compensation had not absorbed genetic deaths of homozygous embryos. Given the preceding assumptions, we seek the evolutionarily stable mating system for which any possible variant phenotype realizes lower or equal fitness to the resident phenotype.

BASIC MODEL OF DETERMINISTIC MATING WITH MANY OVULES AND EQUAL RESOURCE COMPETITION

Our deterministic model considers plants that produce many pollen grains and many ovules per ovary, so the related results are asymptotically true. Suppose that during pollination proportions p_s and p_x of a plant's pollen grains leave its anthers and have the potential to fertilize the plant's own ovules and those on other plants, respectively. The remaining proportion of pollen p_n= 1 −p_s−p_e stays in the anthers (removal failure) and/or is lost without being carried away by a pollen vector (removal loss) and cannot contribute to the plant's fitness (Fig. 1A; see Harder and Routley 2006). We refer to the common resident phenotype as p= (p_s, p_x) and the phenotype of a solitary invader as p′= (p_s′, p_x′). Various processes either claim pollen as it moves between anthers and stigmas or cause pollen-tube attrition (see Harder and Routley 2006), so only proportions q_s and q_x of the potential self- and cross-pollen contribute to the pollen tubes that enter an ovary and can participate in fertilizing ovules (Fig. 1A). Thus, ovaries of plants that allocate proportion p_s of their pollen to self-pollination receive q_sp_sP self-pollen tubes. Correspondingly, all ovaries receive an average of q_xp_xP cross-pollen tubes, regardless of phenotype, because the invader is so rare that it contributes a trivial fraction of the total cross-pollination in the population. Note that the latter outcome occurs regardless of the dispersion of cross-pollen among recipient plants. Specifically, if each donor contributes q_xp_xP pollen tubes to ovaries of n neighboring plants, each recipient will receive q_xp_xP/n pollen tubes from each of its n neighbors for a total of nq_xp_xP/n=q_xp_xP pollen tubes.

The outcome of fertilization depends on the numbers of pollen tubes and ovules in an ovary. If fewer pollen tubes enter the ovary than the number of ovules, all pollen tubes fertilize ovules and fertilization is pollen-limited, so that reproductive assurance could be beneficial. If instead the total number of pollen tubes exceeds the number of ovules, fertilization is ovule-limited, in which case we assume that competing self- and cross-pollen tubes fertilize ovules in proportion to their relative frequencies. We denote the proportions of a plant's ovules fertilized by self- and cross-pollen by  f_s and f_♀, respectively (Fig. 1B; subscript ♀ denotes the female component of outcrossing). Therefore, the proportions of ovules that are self- and cross-fertilized for a plant with phenotype p′ in a resident population of plants with phenotype p are:

((1a))

and

((1b))

respectively. In addition, the expected number of ovules fertilized by the variant on other plants, relative to its own ovule production (i.e., relative siring success) is,

((1c))

Whether fertilization is pollen- or ovule-limited depends on a plant's own self-pollination and the export realized by resident individuals.

We represent seed development as a two-phase process. First, zygotes with lethal traits die shortly after fertilization, before consuming appreciable resources. In particular, selfed and outcrossed zygotes survive genetic death with probabilities g_s and g_x, respectively (Fig. 1B), with g_s < g_x, because the higher homozygosity of selfed zygotes should result in more frequent expression of recessive lethal alleles (Charlesworth and Charlesworth 1999). During the second phase of seed development the surviving selfed and outcrossed embryos draw maternal resources, with proportions k_s and k_x surviving to become seeds, respectively (Fig. 1B). These survival probabilities could be affected by both competition for maternal resources, if the number of surviving embryos exceeds the number that can mature into seeds (mO), and predispersal seed predation; however, we consider only competition explicitly. For convenience in this section we assume that selfed and outcrossed embryos compete equally for maternal resources, so k_s=k_x=k, whereas the finite-ovule model presented in the following section incorporates asymmetric resource competition. Thus, the survival probability for an embryo in an ovary of a plant with phenotype p′ in a resident population of phenotype p is,

(2)

As we model simultaneous self- and cross-pollination, the probabilities of embryo survival do not depend on the relative timing of pollination (for examples of prior and delayed self-pollination see Harder and Routley 2006, M. B. Routley and L. D. Harder, unpubl. ms.).

This characterization identifies three possible limits on seed production. In general, seed production is fertilization-limited if too few embryos survive genetic death to compete for maternal resources, g_s f_s(p′, p) +g_xf_♀(p′, p) < m. However, fertilization limitation arises from either of two causes, so when it occurs we describe seed production as being pollen-limited if, in addition, some ovules remain unfertilized,  f_s(p′, p) +f_♀(p′, p) < 1 (Fig. 2A, light gray area), or ovule-limited if fertilization is complete,  f_s(p′, p) +f_♀(p′, p) = 1 (Fig. 2A, heavy line). Alternatively, seed production is resource-limited when more embryos survive genetic death than can mature given the available maternal resources, g_s f_s(p′, p) +g_xf_♀(p′, p) ≥m, regardless of the limit on fertilization (Fig. 2A, dark gray area). (This condition illustrates that m < 1 is necessary for resource limitation, but it is not a sufficient condition, as implied by Porcher and Lande[2005b: c < 1 in their notation].)

After seed development, proportions g_s f_sk_s and g_xf_♀k_x of the original ovules have become selfed and outcrossed seeds, respectively (i.e., the seed:ovule ratio equals g_s f_sk_s+g_xf_♀k_x). The female outcrossing rate, measured among seed progeny, equals

where G=g_s/g_x and K=k_s/k_x. Because of the generally poorer performance of selfed embryos during seed development, predispersal (early acting) inbreeding depression equals δ_e= 1 −GK. Note that both of these mating parameters may depend on the resident outcrossing strategy, if it causes resource competition among developing seeds.

Proportions d_s and d_x of the selfed and outcrossed seeds, respectively, disperse, establish, and become reproductive adults (Fig. 1B; d_s≤d_x). Consequently, postdispersal (late-acting) inbreeding depression equals δ_d= 1 − (d_s/d_x) = 1 −D, where D=d_s/d_x. Furthermore, total inbreeding depression experienced by self-fertilized embryos is δ= 1 −GKD. Note that although total zygote survival equals the product of pre- and postdispersal survival, total inbreeding depression is not the product of its pre- and postdispersal components (i.e., δ≠δ_eδ_d).

We define an individual's fitness as the expected number of haploid genomes passed on to the next generation per ovule, which must lie between zero and two. Thus, the fitness of a plant with phenotype p′ in a resident population of plants with phenotype p is,

(3)

where the three terms on the right-hand side represent contributions through selfed offspring, a plant's own outcrossed seeds, and outcrossed seeds sired on other plants, respectively.

Phenotype p* is a local evolutionarily stable strategy (ESS) if no alternative phenotype realizes greater fitness when all resident plants exhibit p*, or W(p′, p*) ≤W(p*, p*) for all possible invader phenotypes p′ near p* (Lawlor and Maynard Smith 1976). Phenotype p* is also a continuously stable strategy (CSS) if a population with a resident phenotype similar to p* can be invaded by a phenotype that is even more similar to p* (Eshel 1983). We seek phenotypes that are both ESS and CSS, as they will be favored by natural selection. In addition to the analytic derivations of p* presented in the Appendix, we investigated evolutionary dynamics toward p* by numerically evaluating the gradient of W with respect to p_x and p_s, given the constraint that p_x+p_s≤ 1 −p_n.

We derived all analytic results by considering absolute transition probabilities (e.g., q_s, q_x, g_s, g_x: see Appendix); however, most outcomes depend on the ratios of transition probabilities for selfing and outcrossing (e.g., Q and G), rather than their absolute values. Therefore, we present the results in terms of relative transition probabilities whenever possible. Note that Q=q_x/q_s is an outcrossing:selfing ratio, whereas D, G, and K are selfing:outcrossing ratios; see Table 1.

Results

ESS MATING SYSTEM WITH NO POLLEN DISCOUNTING AND MANY OVULES

The effects of reproductive compensation on mating-system evolution are appreciated most easily by considering situations in which self-pollination does not discount the pollen available for export (i.e., p_x=p_x′= a fixed positive constant) and all fertilized ovules compete equally for maternal resources (i.e., k_s=k_x, so K= 1). The absence of pollen discounting has several implications. First, as all plants are assumed to export an equal proportion of pollen, all plants must receive an equal amount of imported pollen. As a result, any fitness difference between the resident and invading phenotypes depends only on the self- and cross-fertilization of a plant's own seeds. In particular, increased self-pollination alters the mating system only by diluting the proportion of ovules fertilized by imported pollen. Consequently, mating outcomes are not frequency dependent and the ESS can be found simply by maximizing the sum of the selfing and female components of fitness given in equation (3) (see Appendix). Second, in the absence of a trade-off with pollen export, increased self-pollination must reduce pollen that otherwise would not be removed from flowers or would be lost during pollen removal (i.e., p_n= 1 −p_s−p_x). As the incidence of cross-pollination, p_x, is assumed to be a positive constant, exclusive selfing (i.e., p_s= 1) is impossible and instead “complete selfing” is said to occur when p_n= 0 and p_s= 1 −p_x. However, results presented in the Appendix indicate that when “complete selfing” is favored selection also favors a reduction in the fixed value of p_x. Thus for such cases, we assume subsequent evolution of p_x and eventual evolution of exclusive selfing ( f*_s= 1).

As a pollen-limited mating system never maximizes fitness (see Appendix), we focus on situations that result in complete ovule fertilization (i.e.,  f_s+f_♀= 1). With no pollen discounting, the optimal mating system depends on a hierarchy of dichotomous criteria, which progressively incorporate more influences on female fertility (Fig. 3). First, exclusive outcrossing ( f*_s= 0) is favored if postdispersal inbreeding depression is so strong that selfed seeds are less than half as likely to become reproductive adults than outcrossed seeds (D < 0.5: i.e., δ > 1 −G/2 in Fig. 4A). This outcome occurs regardless of the relative success of selfed and outcrossed embryos during seed development. Previous models of mating-system evolution have not identified this condition, because it is subsumed by a more rigorous outcrossing threshold (δ > 0.5) in the absence of reproductive compensation (Porcher and Lande 2005b considered only D= 0.75 in their compensation model).

Unlike the previous case, a diversity of mating systems is possible when selfed seeds have relatively high postdispersal survival (D > 0.5: Fig. 4A for δ < 1 −G/2, and 3, 5). Exclusive self-fertilization (f*_s= 1) is favored if self-fertilized zygotes are at least half as likely to join the next generation as outcrossed zygotes (GD > 0.5: i.e., δ < 0.5). Because selection of exclusive selfing in this situation occurs regardless of the details of ovule production (m), it is also predicted by mating-system models that ignore resource competition (Lloyd 1979; Lande and Schemske 1985; Charlesworth et al. 1990; Jarne and Charlesworth 1993).

In contrast, when selfed seeds have relatively high postdispersal survival (D > 0.5), but self-fertilized zygotes have relatively poor prospects of success (GD < 0.5), the best mating system depends on the extent to which “excess” ovules compensate for genetic deaths of selfed and outcrossed embryos (3-5). Exclusive outcrossing evolves if plants do not produce enough ovules to compensate for all predispersal deaths among outcrossed zygotes (m/g_x=M_x≥ 1: 3-5). In this situation maternal resources exceed the demand from the embryos that survive genetic death, so developing seeds do not compete for resources (K= 1). If instead ovule production compensates for all predispersal losses of outcrossed embryos, but only some deaths of selfed embryos (g_x > m > g_s), a mixture of self- and cross-fertilization is favored (3-5). Recall that we are considering cases in which selfed zygotes perform relatively poorly (GD < 0.5), but selfed seeds perform well (D > 0.5), so if selfed seeds are among the lucky few that survive genetic death they are very valuable to the maternal parent because they possess only maternal alleles. When mixed mating is optimal, the combination of self- and cross-fertilization that maximizes fitness lies at the transition between resource and ovule limitation,

(4)

(Fig. 2, open square) and developing embryos do not compete for maternal resources (k_s=k_x= 1). For this situation the optimal proportion of outcrossed seeds, a common measure of the mating system, is

(5)

(Fig. 6), where g_x/m is the relative increase in the proportion of outcrossed embryos during seed development. Equation (5) indicates that the optimal maternal outcrossing rate (t*) increases as the maximum proportion of ovules that can mature into seeds (m) approaches the proportion of outcrossed ovules that survive genetic death, g_x (Fig. 6). Finally, when plants produce enough ovules to compensate for all predispersal deaths (g_x > g_s≥m) exclusive selfing is favored (i.e., f*_s= 1: 3-5). Full compensation allows the maternal plant to benefit from contributing two haploid genomes through selfed seeds whose future prospects are at least half as good as those of outcrossed seeds (recall that D > 0.5), even though many selfed zygotes succumb to predispersal inbreeding depression.

Examination of fitness isoclines close to the maximum reveals that the three limits on seed production differ in their fitness consequences. For example, consider the situation that favors mixed mating, for which the optimal mating system lies at the intersection between pollen, ovule, and resource limitation (Fig. 2B). The dashed line in Figure 2B closest to the optimal mating system illustrates combinations of self- and cross-fertilization for which fitness is 90% of the maximum possible. This line is closest to the optimum in the area of the figure that represents pollen limitation, next closest in the direction of ovule limitation and farthest in the direction of resource limitation. Consequently, pollen limitation results in the steepest fitness gradient, whereas resource limitation results in a shallow gradient, so selection should act more strongly to alleviate pollen limitation and least strongly to counteract resource limitation, with intermediate selection against ovule limitation. These results expose two conclusions. First, the intense selection under pollen limitation places a premium on reproductive assurance; however, increased selfing to provide this assurance rarely produces the ESS mating system. Second, in the absence of pollen limitation (i.e.,  f_s+f_♀= 1) deviations from the optimal mating system toward more cross-fertilization, which aggravates resource limitation, should be more common than elevated self-fertilization, which intensifies ovule limitation.

ESS MATING SYSTEM WITH COMPLETE POLLEN DISCOUNTING AND MANY OVULES

We now consider the consequences of complete pollen discounting, for which each pollen grain involved in self-pollination reduces the pollen available for export. This model assumes that a fixed minimum proportion of pollen is not removed from anthers or lost during removal, p_nmin, so p_x= 1 −p_s−p_nmin when pollen vectors remove all the pollen available for export. Because plants of different phenotypes now contribute unequal numbers of cross-pollen tubes to ovaries, a plant's paternal success now depends on the pollination strategies of other plants in the population. The resulting frequency dependence complicates identification of the ESS (see Appendix).

Pollen discounting allows a broader range of ESSs and mating-system evolution need not follow a direct trajectory to the ESS. The vectors in Figure 7A–C depict the direction and strength of selection (log transformed) for different pollen allocations (p_s, p_x) given the three combinations of reproductive compensation and inbreeding depression indicated in Figure 7D. The long vectors for conditions that cause pollen limitation (white areas) illustrate that in such situations selection strongly favors some self-pollination for reproductive assurance. Note in particular that the selection vectors associated with reproductive assurance rarely point toward the ESS, so that alleviating pollen limitation of fertilization takes (short-term) precedence over offspring quality. In contrast, when seed production is not pollen-limited the direction of selection on pollen allocation depends on whether the current allocation and the ESS invoke the same constraint on seed production. If so, selection acts first to commit all possible pollen to pollination (e.g., Fig. 7C, dark gray area) and then to “tune” the mating system along the p_x+p_s≤ 1 −p_nmin constraint. If instead the ESS involves a different limitation than currently prevails, selection usually first shifts pollen allocation to the transition between resource and ovule limitation (e.g., Fig. 7C, light gray area), then increases the pollen involved in pollination along this transition, and finally moves allocation along the 1 −p_nmin constraint. Thus, the trajectory followed during mating-system evolution is guided by the various constraints on pollination, fertilization, and seed production. We now focus on the ESSs under different conditions (see Appendix), which always involve complete fertilization (i.e.,  f_s+f_♀= 1; Fig. 7A–C).

Complete pollen discounting greatly expands the range of conditions that favor some outcrossing (compare Fig. 4A, B). Indeed, exclusive selfing is never an ESS mating system with complete pollen discounting (Fig. 4B) if fertilization is not pollen-limited (also see Holsinger 1991). This outcome arises because once stigmas receive enough pollen to fertilize all ovules, increased allocation of pollen to self-pollination both intensifies competition between a plant's own pollen grains for access to its own ovules and reduces potential siring opportunities on other plants. The synergistic impacts of local mate competition and pollen discounting on siring success are sufficient to counteract the twofold genetic benefit of self-fertilization, even in the absence of inbreeding depression.

Compared to the nondiscounting case, the inbreeding-depression threshold that favors exclusive outcrossing now depends on the relative success of self- and cross-pollen, decreasing by q_x/2q_s (=Q/2: compare Fig. 4A, B). This change is probably relatively small for self-compatible species, as a pollen grain with export potential has a much smaller chance of entering an ovary than a potential self-pollen grain (i.e., q_x≪q_s). Thus, although the details of pollination and pollen-tube growth influence the evolution of exclusive outcrossing, their effects are limited compared to those caused by postfertilization differences between selfing and outcrossing. Note that even when conditions favor exclusive outcrossing, some self-pollination is advantageous if it alleviates pollen limitation (Fig. 7A).

Pollen discounting greatly expands both the conditions that promote mixed mating and the diversity of ESS mixed-mating patterns. As with the nondiscounting case, the transition between ovule- and resource limitation can be an ESS mixed-mating system, which occurs if

((6a))

or

((6b))

(see Appendix). If q_x is very small, as is likely for most species with granular pollen, then Q≅ 0 and equation (6b) describes essentially the same conditions for mixed mating as observed in the nondiscounting case (compare Fig. 4A, B). Thus, the effect of reproductive compensation on the evolution of an ESS at the transition between resource- and ovule limitation is largely unaffected by pollen discounting. In contrast, pollen discounting creates conditions that additionally allow mixed-mating ESSs with ovule or resource limitation.

An ESS mixed-mating system that involves chronic ovule limitation of seed production evolves when

(7)

These conditions include two situations. First, if ovule production does not compensate for all genetic deaths of outcrossed zygotes (i.e., m≥g_x) resource limitation is impossible and a mixed-mating ESS must involve ovule limitation (see Fig. 2A). Indeed, mixed mating with ovule limitation can be an ESS even when ovule production does not compensate for any embryo deaths (i.e., m= 1: along the top of Fig. 4B: also see Holsinger 1991). Second, pollen discounting promotes ovule-limited mixed mating for most values of g_s < m < g_x if inbreeding depression is less than (1 −Q)/2 (Fig. 4B). In particular, if q_x is very small, so Q≅ 0, the right-hand side of the first condition of equation (7) exceeds g_s only slightly. With ovule-limited mixed mating, the ESS depends on all aspects of reproduction, except the proportion of ovules that can mature into seeds (m: Table 2). Over most of the range of inbreeding depression that allows ovule-limited mixed mating the ESS involves predominant selfing (see dotted and dashed lines in Fig. 4B), with a minimum proportion of outcrossed seeds of t=Q/(G+Q[1 −G]) when δ= 0.

Table 2. Characteristics of the mixed ESS mating systems when self-mating discounts opportunities for outcross siring and the ESS involves limitation of seed production by resources or ovule production, or the transition between these constraints. Equations 6 and 7 detail the conditions that favor these different outcomes (also see 4). See Table 1 and 1 for parameter definitions.

Limitation on seed production under ESS	Proportion of pollen allocated to self−pollination (p*s)	Proportion of ovules that are self-fertilized ( f*s)	Proportion of outcrossed seeds (t*)
Resources
Transition
Ovules

A resource-limited mixed-mating system evolves when

(8)

(Fig. 4B). As with equation (7), the right-hand side of the first condition of equation (8) approximates g_s if Q is very small. When m < g_s complete ovule fertilization always leads to resource limitation (see Fig. 2A), so any mating system must be resource-limited, whereas very restrictive conditions allow a resource-limited ESS when m > g_s (Fig. 4B). Again, the ESS depends on all aspects of reproduction, except the proportion of ovules that can mature into seeds (Table 2). Resource-limited mixed-mating ESSs typically involve predominant selfing (see dotted and dashed lines in Fig. 4B), with a minimum t=Q/(2 −G) when δ= 0.

Discussion

The models described above provide a general framework for investigating the evolution of hermaphroditic mating systems and synthesize the results of many previous analyses. Our models provide additional support for claims since Darwin (1876) that selfing often persists in populations when it provides reproductive assurance (reviewed by Eckert et al. 2006), as the strongest selection identified by our models occurred when too few ovules (eggs) were fertilized to maximize offspring production (2, 7). We also found the well-known dichotomy between selfing and outcrossing (e.g., Lloyd 1979; Lande and Schemske 1985) when selfing does not affect male outcrossing and ovule (egg) production does not exceed an individual's resource capacity to produce offspring (along top axis of Fig. 4A). Male-gamete discounting and production of compensatory ovules (eggs) modify this outcome differently and largely independently. On one hand, male-gamete discounting promotes some outcrossing to convert local mate competition for self-fertilization into outcrossing opportunities when weak inbreeding depression otherwise favors selfing (Fig. 4B: Holsinger 1991; Johnston 1998; Porcher and Lande 2005a). On the other hand, production of “extra” eggs (m < 1) can allow individuals to benefit from the genetic-transmission advantage of selfing when strong predispersal inbreeding depression would otherwise favor outcrossing (Fig. 4: Porcher and Lande 2005b).

Our models also expose new insights on mating-system evolution and identify specific conditions favoring particular ESS outcomes. Significantly, these models illustrate that the prevailing constraint on offspring production governs the course of mating-system evolution (Fig. 7) and the ultimate ESS (Table 2, Fig. 2B). The overall ESS mating system never involves pollen limitation (2, 7), so mixed mating associated with reproductive assurance always represents a suboptimal solution to insufficient outcrossing. Furthermore, the widespread observations of pollen limitation among angiosperms (reviewed by Ashman et al. 2004) seem enigmatic from an evolutionary perspective, unless they represent common misinterpretation of the effects of poor-quality pollen, rather than insufficient pollen (see Harder and Routley 2006; Aizen and Harder 2007).

Given that selection strongly promotes sufficient mating to fertilize all female gametes, or at least more than can mature into independent offspring given available maternal resources (2, 6), mating-system evolution should typically occur in a context of ovule- and/or resource limitation. Most previous mating-system models assumed complete fertilization and ignored resource limitation, so they implicitly involved ovule limitation. As we have defined it, ovule limitation of seed production occurs when all ovules are fertilized, but too few embryos survive genetic death to compete for maternal resources. If most embryos survive, then mating-system evolution can do little to improve seed production, although it may increase the proportion of selfed offspring that contribute two gene copies to the next generation. In this case, ovule limitation should also favor reallocation of resources from potential seed production to increased ovule production. In contrast, poor embryo survival implies extensive fertilization by poor-quality pollen, which should select for floral mechanisms that promote high-quality pollination. In most mating-system models, including those presented here (e.g., 2, 7), this response involves selection for increased outcrossing. In contrast, resource limitation occurs only when a parent produces more female gametes than can mature into offspring (i.e., m < 1: Fig. 2A), which intrinsically allows compensation for deaths caused by predispersal inbreeding depression. Consequently, resource limitation favors selfing when postdispersal inbreeding depression is weak (Fig. 3), because competition among developing embryos can exclude valuable selfed embryos that contain only maternal genes and have survived predispersal inbreeding depression. These contrasting effects of ovule and resource limitation create opportunities for mixed mating to be an ESS, if the occurrence of one limitation or the other depends on the proportions of outcrossed and selfed embryos.

Our models indicate universal and distinct roles for the relative, intrinsic performance of selfed and outcrossed embryos (G) and seeds (D) in mating-system evolution (Table 2, 3, 4). The genetic consequences of self-fertilization for reduced offspring performance have been recognized as significant influences on mating-system evolution since Darwin's assertion that “nature abhors perpetual self-fertilisation” (1876, p. 8); however, the contrasting natures of predispersal inbreeding depression (1 −G) and postdispersal inbreeding depression (1 −D) have been appreciated only recently (Husband and Schemske 1996; Charlesworth and Charlesworth 1999) and the implications of this contrast remain to be explored fully. Our analyses reveal that intense postdispersal inbreeding depression (D < 0.5) by itself is sufficient to select for exclusive outcrossing (Fig. 3: Fig. 4A, δ > 1 −G/2: Fig. 4B, δ > 1 −[Q+G]/2). Consequently, self-incompatibility should be especially prevalent in species subject to strong postdispersal inbreeding depression. More diverse mating systems are possible with weak postdispersal inbreeding depression (D > 0.5). Thus, mating-system evolution depends fundamentally on the dynamics of inbreeding during the life cycle.

Temporal variation in the severity of inbreeding depression is an essential feature of the adaptive role of reproductive compensation, which influences mating-system evolution primarily when selfed zygotes survive poorly compared to outcrossed zygotes (GD < 0.5), but selfed seeds realize high relative survival (D > 0.5: 3, 4). In this situation, production of extra female gametes allows a heterozygous maternal hermaphrodite to produce families comprised of both outcrossed offspring with relatively good predispersal survival and the small proportion of selfed offspring that are heterozygous for early acting lethal alleles and also have relatively good postdispersal survival. As angiosperms commonly produce more ovules than they can mature (Harder and Routley 2006) and predispersal inbreeding depression is often much stronger than postdispersal inbreeding depression (Husband and Schemske 1996), reproductive compensation may frequently contribute to adaptive mixed mating.

When self-mating discounts outcrossing opportunities, the relative success of male gametes involved in self- versus cross-mating (Q=q_x/q_s) also influences mating-system evolution (Table 2, Fig. 4). This parameter incorporates both the dispersal of male gametes and their survival in female reproductive tracts. It influences mating-system evolution by determining the relative impacts on paternal success of local mate competition for self-fertilization and pollen discounting. In the absence of male-gamete discounting, increased self-mating augments an individual's genetic contributions to its own offspring without affecting its contributions to offspring produced by other individuals, so the resulting aggravation of local mate competition bears no siring costs. In contrast, when self-mating reduces outcross siring opportunities, a specific male gamete may have a higher probability of fertilizing an egg on another individual than of succeeding in local mate competition to fertilize a sibling egg. We argued above that Q is probably quite small for angiosperms with granular pollen, because of the vagaries of pollen dispersal; however, this term may be larger for plants with more efficient pollen transfer (e.g., orchids: Harder and Johnson in press) and hermaphroditic animals with internal fertilization. In general, large Q elevates the ESS proportion of outcrossed offspring (Fig. 4B), because of the greater probability that a male gamete diverted from self-mating actually contributes to cross-fertilization.

In contrast to male-gamete performance and the intrinsic performance of embryos and seeds, the relative competitive ability of selfed embryos during competition for maternal resources (K) had little impact on the ESS mating system in our stochastic model (Fig. 5) and is probably a general outcome. This insensitivity results for several reasons. When exclusive selfing or outcrossing is favored, the presence of only one class of embryos precludes competition among classes. When the ESS mixed-mating system involves either ovule limitation, or the transition between ovule and resource limitation, the optimal number of embryos surviving genetic deaths is less than or equal to the maximum number that can be developed, given the available resources. Consequently, all surviving ovules should develop, regardless of their relative competitive ability. The possible exception to a limited impact of relative competitive ability could occur when the ESS mating system involves resource limitation. However, we expect this situation to occur rarely and to be transient for two reasons. First, resource-limited mixed mating is favored primarily when individuals produce more female gametes than are needed to compensate for all genetic deaths, even with exclusive selfing (i.e., m < g_s). Such profligate production of female gametes should be countered by selection to redistribute maternal resources to offspring development. Second, mixed-mating ESSs that involve resource limitation also involve predominant self-fertilization and Porcher and Lande (2005b) concluded that “high selfing rates purge most embryonic lethals …, regardless of the opportunity for compensation” (p. 680). Thus, regardless of the ESS mating system, the relative competitive abilities of selfed and outcrossed embryos seem unlikely to play a major role in the evolution of hermaphrodite mating systems, as long as self- and cross-fertilization occur simultaneously.

Our models demonstrate that the complete spectrum of hermaphrodite mating systems could arise as evolutionary stable strategies in contrasting mating environments. Clearly, many circumstances could cause deviations from the ESS mating system, especially insufficient pollination, which favors reproductive assurance (Fig. 7). Nevertheless, all of the requirements for adaptive mixed mating incorporated in our models are probably common features of reproduction by hermaphrodites, including: male-gamete discounting (e.g., Harder and Barrett 1995); production of compensatory female gametes (Harder and Routley 2006); and heterogeneity in the nature of inbreeding depression during the life cycle, with lethal traits expressed early when homozygous (Husband and Schemske 1996; Charlesworth and Charlesworth 1999). Thus, the diversity of hermaphrodite mating systems (Goodwillie et al. 2005; Jarne and Auld 2006) may often bear the imprint of selection that maximizes genetic contributions to the next generation, rather than simply the sloppiness of mating, or convenient remedies for insufficient fertilization.

Associate Editor: T. Meagher