How Geitonogamous Selfing Affects Sex Allocation in Hermaphrodite Plants

How geitonogamous sel®ng affects sex allocation in hermaphrodite plants

T. J. DE JONG, P. G. L. KLINKHAMER & M. C. J. RADEMAKER University of Leiden, Institute of Evolutionary and Ecological Sciences, PO Box 9516, 2300RA Leiden, The Netherlands

Keywords: Abstract dioecy; Does the mode of self-pollination affect the evolutionarily stable allocation to geitonogamy; male vs. female function? We distinguish the following scenarios. (1) An pollination; `autogamous' species, in which sel®ng occurs within the ¯ower prior to sel®ng; opening. The pollen used in sel®ng is a constant fraction of all pollen grains sex allocation. produced. (2) A species with `abiotic pollination', in which sel®ng occurs when pollen dispersed in one ¯ower lands on the stigma of a nearby ¯ower on the same plant (geitonogamy). The sel®ng rate increases with male allocation but a higher sel®ng rate does not mean a reduced export of pollen. (3) An `animal-pollinated' species with geitonogamous sel®ng. Here the sel®ng rate also increases with male allocation, but pollen export to other plants in the population is a decelerating function of the number of simultaneously open ¯owers. In all three models sel®ng selects for increased female allocation. For model 3 this contradicts the general opinion that geitonogamous sel®ng does not affect evolutionarily stable allocations. In all models, the parent bene®ts more from a female-biased allocation than any other individual in the population. In addition, in models 2 and 3, greater male allocation results in more local mate competition. In model 3 and in model 2 with low levels of inbreeding depression, hermaphroditism is evolutionarily stable. In model 2 with high inbreeding depression, the population converges to a ®tness minimum for the relative allocation to male function. In this case the ®tness set is bowed inwards, corresponding with accelerating ®tness gain curves. If the sel®ng rate increases with plant size, this is a suf®cient condition for size-dependent sex allocation (more allocation towards seeds in large plants) to evolve. We discuss our results in relation to size-dependent sex allocation in plants and in relation to the evolution of dioecy.

Introduction reduces the evolutionarily stable (ES) allocation to sons below 50%. By analogy, Charlesworth & Charlesworth In his paper on sex ratios of insects, Hamilton (1967) (1978, 1981) and Lloyd (1987a) modelled the effects of showed that, if the breeding structure of a population sel®ng on the allocation to male and female function in forces sons into competition with one another, this cosexual plants. They showed that sel®ng reduces the ES allocation to male function. These conclusions were used to explain the facts that species with high sel®ng rates show low pollen±ovule ratios (Cruden, 1977) and, within Correspondence: Dr T. J. de Jong, University of Leiden, Institute of species, genotypes with high sel®ng allocate less to pollen Evolutionary and Ecological Sciences, PO Box 9516, 2300RA Leiden, The Netherlands. production (Schoen, 1982; Lloyd, 1984; Charnov, 1987; Tel: 31±71±5275118; fax: 31±71±5274900; Parker, 1995). However, different modes of sel®ng exist e-mail: [email protected] (Lloyd, 1979; Lloyd & Schoen, 1992; Schoen & Lloyd,

166 J. EVOL. BIOL. 12 (1999) 166±176 Ó 1999 BLACKWELL SCIENCE LTD Geitonogamy and sex allocation 167

1992) and Lloyd (1987a) pointed out that the mode of self-pollination in¯uences the ES allocations. We distinguish between three modes of sel®ng. First, in Charlesworth & Charlesworth's (1978, 1981) model it is assumed that sel®ng occurs before the outcross pollen arrives and that the few pollen grains used in sel®ng do not signi®cantly reduce the pollen export of the plant. This model (model 1 in what follows) seems appropriate for an autogamous species, when the anthers and stigma touch brie¯y before the ¯ower opens. Second, geitonogamous (between-¯ower) sel®ng may occur in species with wind or water pollination. The self pollen then competes with outcross pollen. In wind- pollinated plants the architecture of the plant and physical factors such as wind speed or turbulence may determine what fraction of the pollen is dispersed (Freeman et al., 1997). Some pollen grains may land on the stigmas of neighbouring ¯owers but their numbers are probably low compared with the total pollen released. Therefore, the sel®ng rate is expected to increase with the number of open ¯owers, but the fraction of all pollen dispersed is hardly affected by the number of open ¯owers (model 2). Third, in animal-pollinated species, geitonogamy oc- curs when the pollinator moves between ¯owers. Here outcross pollen and the self pollen from neighbouring Fig. 1 Outline of the three models. In model 1 (`autogamy') the ¯owers are applied simultaneously to the stigma. Bumble sel®ng rate (S) does not depend on the number of ¯owers (n) and bees, honey bees and hummingbirds will usually tend to pollen export is a constant fraction of the pollen produced (solid line) visit more ¯owers in succession when more open ¯owers or no pollen is used for sel®ng (broken line). In model 2 (`abiotic are available (Klinkhamer et al., 1989; Snow et al., 1995). pollination') the sel®ng rate increases with the number of ¯owers, Higher sel®ng rates on plants with many ¯owers have but not at the expense of pollen export. In model 3 (`animal pollination') sel®ng also increases with the number of ¯owers. Pollen been shown for a number of animal-pollinated plants export is a saturating function of the number of open ¯owers. As a (DommeÂe, 1981; Crawford, 1984; Dudash, 1991; Schoen result of the pollen discounting an inverse relation exists between & Lloyd, 1992; Harder & Barrett, 1995a; Snow et al., the sel®ng rate and the fraction of the pollen that is dispersed to 1995; Vrieling et al., 1997 and unpublished results). other plants in the population. When a bee moves between successive ¯owers on the same plant, it constantly loses pollen. This loss occurs partly on the stigma, but more important losses are made explain this result. Next we examine how a shift to self- in ¯ight, on the corolla and through grooming (Thom- incompatibility would affect ES allocation to male func- son, 1986; Rademaker et al., 1997). Thus if the plant tion in the three models. Our calculations have assumed presents more simultaneously open ¯owers, this induces that all plants in the population are of equal size. This is pollinators to visit more ¯owers in succession and this clearly unrealistic. Plants differ greatly in size within gives more opportunities for the pollen of the ®rst visited populations, and genotypes may be phenotypically plas- ¯ower to be lost before the pollinator leaves the plant. tic, i.e. adjust sex allocation to their size. This problem Pollen export is then a decelerating function of the has been discussed for outcrossing species (Klinkhamer number of simultaneously open ¯owers (Hessing, 1988 et al., 1997). For sel®ng species, the factors affecting for Geranium caespitosum). Because of the pollen dis- ®tness gain curves have been listed (de Jong & counting (Holsinger, 1991) in model 3, a negative Klinkhamer, 1994), without reaching a de®nite conclu- correlation is expected between the sel®ng rate and the sion. Here we sketch the simplest model without fraction of the pollen exported, as shown by Harder & inbreeding depression and with the sel®ng rate increas- Barrett (1995a). ing with the number of ¯owers. Is it then adaptive to be The different modes of sel®ng and their consequences phenotypically plastic and to adjust gender to size, i.e. for the relation between the number of ¯owers, the allocate more to female function when large and more to sel®ng rate and the pollen export per ¯ower are depicted male function when small? How do wind- and animal- in Fig. 1. In this paper we use the differences between pollinated species compare in this respect? Finally, we wind- and insect-pollinated plants to show how geitono- brie¯y discuss in which model dioecy is most likely to gamy reduces male allocation to levels below 50% and to evolve.

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The general model wm ˆ 1 Sm† T m† ‡ 1 d†Sm T m†

Consider a hermaphrodite plant species in which seed ‡ 1 d†Sm T m† ‡ Em=En† 1±Sn† T n†: production is not pollen limited. Each individual plant 1b† has T units of resource. Plants may allocate this resource freely between seeds and ¯owers, so that there exists a wn can be scaled to 2 and represents the number of copies complete trade off or full `compensation' (Charnov, of the (haploid)genome that an individual passes on to 1982; Lloyd, 1987a; Seger & Eckhart, 1996). For the next generation through seeds. The ®rst term in simplicity, assume that one unit of resource is required eqn 1b denotes the female ®tness contribution through to construct a ¯ower with pollen, but without seed, and outcrossing. The second term is the female ®tness that a single seed also costs one unit of resource. We contribution through sel®ng. The third term is the male assume that plants can vary the ratio of seeds to ¯owers, ®tness contribution through sel®ng. We write the second but that each ¯ower contains the same amount of and third terms separately to be able to discriminate pollen. The alternative is, of course, to assume that between the female and male contribution to ®tness later pollen production per ¯ower varies but this poses further on in this paper. The fourth term denotes outcross siring problems which will require additional calculations in success. The mutant sires its `competitive share' (Lloyd, the future. If plants allocate a fraction r of the T resources 1984) of the available (nonselfed) ovules in the popula- to ¯owers, they produce rT ¯owers and (1 ± r)T seeds. tion according to the ratio of pollen export per plant for There is no limit to the number of seeds per ¯ower, mutant and resident (Em/En). except that construction of a single ¯ower is needed By the usual methods (Parker, 1984; take wm, differ- before any seeds can be formed: with T ˆ 100 resources, entiate with respect to m, denoted by primes, and ®nally a plant may construct 1 ¯ower with 99 seeds. At the set m ˆ n) we ®nd that the ESS-value of n follows from other extreme it may construct 100 `empty' ¯owers with 0 2d 1† Sn ± 1 ‡ T n† 1 2d†S pollen but containing no seeds. The assumption of full n 2† ‡ 1 S † T n†E0 =E ˆ 0: compensation is also used by the Charlesworths for n n n calculating the ES value of r. When we later calculate the The ESS value n*, and consequently r*, was computed

®tness of alternative phenotypes as a rare mutant, we numerically from eqn 1b after specifying Em, En, Sm and retain this assumption. On this point Charlesworth & Sn. Substituting integer values for n and m in eqns 1a and Charlesworth (1978, 1981) take the opposite view by 1b, n* was considered an ESS if it could not be invaded by assuming that when a pure female is introduced into a mutants with a single ¯ower more or less than n. hermaphrodite population it produces as many seeds as the hermaphrodite and similarly for the pure male. Specifying S and E In what follows, ¯ower production is used synony- n n mously with male allocation, and seed production with In model 1 both pollen export per ¯ower and the sel®ng female allocation. This is reasonable for species that rate are constant. Charlesworth & Charlesworth (1978, produce many `empty' ¯owers, even if pollination is 1981) assumed that a negligible amount of pollen is assured (for instance, Cynoglossum of®cinale, de Jong & necessary for sel®ng. Using a fraction D of the pollen for Klinkhamer, 1989). Alternatively, costs of attraction self-pollination is irrelevant for the model, as long as this (nectar production, corolla) may be kept separate and is a constant fraction of the pollen produced and affect male and female ®tness differentially (for instance, independent of the number of ¯owers. In both cases: Lloyd, 1987a,b). For our present purpose, however, Em/En ˆ m/n. which is to demonstrate that geitonogamy affects alloca- In model 2 pollen export per ¯ower is constant but the tion and study how this occurs, attraction is irrelevant. sel®ng rate increases with n. To facilitate comparison of Assume that all ¯owers are open simultaneously. We the models, we will use the functional relationship de®ne n as the number of open ¯owers, so that a plant between Sn and n that we derive in the following with n ¯owers produces T ± n seeds, and by de®nition paragraphs for animal-pollinated plants. r ˆ n/T. A fraction S of seeds is selfed, and these progeny In model 3 we adopt the so-called `exponential decline' suffer inbreeding depression d. model of pollen dynamics, which is frequently used for

In a large population, the total individual ®tness (wn) animal-pollinated plants. The model has been treated of a resident phenotype (n) reproducing with n ¯owers most extensively in several recent papers (de Jong et al., and T ± n seeds, is (Charlesworth & Charlesworth, 1978, 1992; Harder & Barrett, 1995b; Snow et al., 1995; 1981; Lloyd, 1987a) Rademaker et al., 1997) elaborating ideas set out by Bateman (1947) and Crawford (1984). In its simplest

wn ˆ 2 2Snd† T n†: 1a† form the pollinator looses a fraction k of the pollen it is carrying when visiting a ¯ower, so that 1 ± k is carried And for a rare mutant (m) with m ¯owers and T ± m over to the next ¯ower. Under a number of simplifying seeds its ®tness is assumptions (de Jong et al., 1992), a convenient formula

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can be derived for the sel®ng rate as a function of the (Charnov, 1982) is linear. The pure female lies outside number of ¯owers. The sel®ng rate for a plant with n the line connecting all other phenotypes: when d > 0.5, ¯owers can then be calculated as (de Jong et al., 1992) the female, which cannot self, bene®ts from an `out- n breeding advantage' (Lloyd, 1975). Sn ˆ 1 ‰1 1 k† Š=nk: 3†

Although in previous papers we equated k to the fraction Model 2: pollen export proportional to the number of pollen lost on the stigma of the ¯ower, it is now clear of ¯owers, sel®ng depends on number of ¯owers (n) that other losses of pollen (passive or through grooming of the pollinator) are much more important (Rademaker At a given resource level, the population converges to a et al., 1997) and these losses contribute most to k. single value for n and a corresponding level of sel®ng. To The function for the sel®ng rate is continuous in n and examine how the level of sel®ng affects optimal sex increases from zero, on plants with a single ¯ower, to allocation, it is therefore necessary to choose different close to one on plants with an in®nite number of ¯owers. resource levels and plot the unique, single point for n* For low n and k it can be approximated by a linear and S, corresponding to the different populations, each function, as assumed by Charlesworth & Charlesworth with a certain resource level. We assumed the deceler- (1981). Conveniently, the sel®ng rate does not depend ating relation between Sn and n in eqn 3. When d < 0.5, on the number of open ¯owers on other plants in the selection produces a higher r* value than in model 1 population and can therefore be used to estimate the (compare Fig. 2B with A). For d ˆ 0.5 the curves in sel®ng rate of a mutant with a different number of Fig. 2(A) (d ˆ 0.5) and Fig. 2(B) (d ˆ 0.5) are identical. ¯owers. This result stems from the assumption that The slight difference at low sel®ng rates on the left-hand pollen production per ¯ower and the pollen load on the side of the graphs, corresponding to a low resource level, pollinator are constant, resulting in equal stigmatic loads arises only because in our calculation (Fig. 2B) ¯owers n only come in discrete numbers. These results are, as for for all ¯owers. It can be shown that En ˆ (1 ± (1 ± k) )/k (de Jong et al., 1992), which is a decelerating function of model 1, in the Charlesworth & Charlesworth (1981) the number of ¯owers n. Combining this with eqn 3 paper with minor differences in the assumptions (we dropped the exponents in the gain curves, i.e. m ˆ 1 in gives Sn ˆ 1 ± En/n. Thus, in this model the fraction of the pollen that is dispersed is negatively related to the the terminology of their paper). They noticed that at sel®ng rate with slope ±1. large values of d an ESS often does not exist. We add that, Of course, reality is more complex and some of our with large inbreeding depression (d > 0.5), starting from assumptions may not hold. Pollinators rarely visit all r ˆ 0.5 mutants with rm < 0.5 can invade the popula- ¯owers on the plant and they may revisit ¯owers. tion until it reaches an attractor ra. At the attractor there Nevertheless, the simple model outlined here captures exists, contrary to that seen when d £ 0.5, a ®tness the essence of geitonogamy, which is that pollinators visit minimum: any mutant has a higher ®tness than the more ¯owers in succession on plants with many ¯owers. resident. The ®tness set (Fig. 3B) illustrates this phe- In doing so they lose outcross pollen and accumulate self nomenon. At d < 0.5 the ®tness set is convex, corre- pollen, inducing more sel®ng in the ¯owers that are sponding with decelerating ®tness gain curves and stable subsequently visited and reducing the fraction of pollen hermaphroditism. At d ˆ 0.5, all mutants fall on the exported to other plants. same straight line as in model 1. When d exceeds 0.5 the ®tness set is concave; all mutants have higher ®tness than the resident population and gain curves are accel- Results erating.

Model 1: pollen export is proportional to the number of ¯owers, S is ®xed Model 3: both pollen export and S depend on n

If there is a ®xed sel®ng rate and no pollen discounting, If d ˆ 0 we ®nd by substitution of Sn ˆ 1 ± En/n in the population converges to an ES value of r, r*, that eqn 2 that declines with sel®ng (Fig. 2A). This ES value for the rà ˆ nÃ=T ˆ 1 S†=2: 5† relative allocation to male function was ®rst derived by Charlesworth & Charlesworth (1981; see also Charnov, Even for d ˆ 0, plants should have female-biased sex 1982, eqn 14.8 with n ˆ 1, or Lloyd, 1987a). It follows ratios. Without inbreeding depression the predicted rela- tion between S and r* is the same for models 3 and 1 directly from eqn 2 with Sn¢ ˆ 0 and En ˆ (1 ± D)n, so (eqn 4). With inbreeding depression the predictions that En¢ ˆ 1 ± D, and is equal to differ markedly (Fig. 2C). For all levels of inbreeding à à r ˆ n =T ˆ 1 S†= 2 ± 2Sd†: 4† depression, male allocation decreases with S. With Figure 3 shows the ®tness values of different mutants increased inbreeding depression, selection against sel®ng when the population is at the ES value of r, assuming full becomes stronger, resulting in a more female-biased compensation. Note that for model 1 the ®tness set allocation pattern (Fig. 2C). In models 2 and 3, high

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Fig. 2 Evolutionarily stable allocation to male function vs. the sel®ng rate. d denotes inbreeding depression: squares d ˆ 0, crosses d ˆ 0.5, triangles d ˆ 0.9. (A) Model 1: sel®ng rate is ®xed, sel®ng is costless. (B) Model 2: sel®ng rate increases with male allocation (decelerating relation, see text), sel®ng is costless. Note that at d ˆ 0.9 in this model no ESS exists, and instead the ®tness minimum to which the population converges is shown. (C) Model 3: sel®ng rate increases with male allocation, pollen export per ¯ower decelerates with male allocation. (D) Effects of self-incompatibility in models 1±3. The seven symbols in (B), (C) and (D) correspond with, going from left to right, T ˆ 10, 20, 50, 100, 200, 500 and 1000. k ˆ 0.1. inbreeding depression lowers r*, while in model 1 it number of ¯owers n, as in models 2 and 3, results do increases r*, resulting in a more equal allocation between change. Phenotypic plasticity (a conditional strategy the sexes. adjusting r to the size of the individual) can then invade a ®xed strategy (the same r on plants of different sizes). For simplicity we will assume that the population consists Effects of self-incompatibility only of small (V resources) and large (W resources) plants, with frequencies p and 1 ± p. The strategy of the Self-incompatibility eliminates sel®ng and in models 1 mutant is to form m ¯owers when the genotype and 2 this results in r* ˆ 0.5. In model 3, pollen export is s encounters poor conditions and the phenotype is small, a decelerating function of the number of ¯owers: the and to form m ¯owers when conditions are good and the male gain curve levels off and r* is smaller than 0.5. l ¯owering plant is large. The resident genotype reproduc-

es with ns and nl ¯owers when the phenotype is small Size-dependent sex allocation and large, respectively. The ®tness criterion in such a population is analogous to eqn 1: So far we have assumed that all plants in the population have the same amount of resources T. How is r* affected wn ˆ p 2 2Sn;sd† V ns† ‡ 1 p† 2 2Sn;1d† W n1†; if T differs for different plants in the same population? In 6a† model 1, r* does not depend on T, so the results remain unchanged. If the sel®ng rate S increases with the and

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Fig. 3 Fitness sets for the different modes of self-pollination. (A) d ˆ 0, T ˆ 100, k ˆ 0.1. Model 1: the ®tness set is discontinuous; the ®tness of a pure female in populations with different rates of sel®ng are indicated by the dots on the y-axis. Model 2: rà ˆ 0.28, corresponding to 66% sel®ng. Model 3: rà ˆ 0.21, corresponding to 57% sel®ng, k ˆ 0.1. (B) d ˆ 0.9, T ˆ 100, k ˆ 0.1. Model 1: the ®tness set is discontinuous; the ®tness of a pure female in populations with different rates of sel®ng are indicated by the dots on the y-axis. Model 2: r ˆ 0.23, corresponding to 60.5% sel®ng; an ESS does not exist. Instead the population converges to the shown value of r, which is an unstable ®tness minimum. Model 3: r* ˆ 0.12, corresponding to 40.7% sel®ng. Note that for d ˆ 0.5 the ®tness set for model 2 is linear and coincides with the line drawn for model 1.

wm ˆ ‰p 1 Sm;s† V ms† ‡ 1 p† 1 Sm;1† W m1†Š conditions will be more female biased than in a similar, ‡ 2 2d†‰pS V m † ‡ 1 p†S W m †Š genetically separate, population adapted to poor growing m;s s m;1 1 conditions (V ˆ 10 and W ˆ 50). We can compare a pE ‡ 1 p†E † ‡ m;s m;1 plant of similar size (50 units of resources) in these two pEn;s ‡ 1 p†En;1† populations, each with its own genotype adapted to the  ‰p 1 Sn;s† V ns† ‡ 1 p† 1 Sn;1† W n1†Š: local growing conditions. Then under the conditions of 6b† model 3, r* is predicted to differ for the two genotypes (Fig. 4). In model 2 the ESS of the plant with 50 units of In the last term we sum, on the left-hand side, the pollen resources is independent of the population in which it produced by a genotype when it grows to a small and to a grows (Fig. 4). large plant, to calculate the `competitive share' (Lloyd, 1984) of the mutant. This ratio then determines the Discussion fraction of seeds that m sires from the total number of seeds produced by the resident (sum of small and large What's going on? plants). Figure 4 was constructed for large populations consisting of 50% small and 50% large plants (p ˆ 0.5). We have shown that the mode of sel®ng matters for It is optimal for a genotype to emphasize seed production allocation. Thus it is problematic to estimate the rate of when large, and ¯ower production when small. This sel®ng for a given species and then use, regardless of the predicted gender change with size is smaller for model 2 mode of sel®ng, the Charlesworth equation (eqn 4) for than for model 3. Figure 4 shows that in the latter case, predicting r. the ESS can have a male-biased sex allocation when To see how different mechanisms work and why plants small (V ˆ 10) and a female-biased sex-allocation when should bias allocation towards female function, it is large (W ˆ 50). In a population with good growing instructive to consider ®rst a case where sel®ng does not conditions (V ˆ 50 and W ˆ 100) plants have, on affect r*. Imagine a small population of 10 wind- average, more resources and it is more dif®cult to avoid pollinated plants. The plants are fully self-compatible geitonogamous sel®ng. Therefore, the average allocation and there is no inbreeding depression. We do not specify at the ESS in a population adapted to good growing Sn beforehand. Instead, the pollen is dispersed by wind

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female-biased mutant m with sel®ng rate S, by de®nition, a fraction S of its seeds is sired by the parent individual. A fraction 1 ± S of the seeds of this mutant plant m is sired by resident individuals. Under homogeneous pollen distribution, however, a negligible fraction of this pollen comes from a given n-type individual, so that the parent individual bene®ts more from the bias than any n-type individual. In this situation, female bias will be selected for. The greater bene®t to the parent from female-biased allocation is the only mechanism working in model 1, but is also present in models 2 and 3. In model 1 the ®tness set was linear (Fig. 3): when the population is at the ESS there exists for the mutants a linear relation between r and ®tness gained as a male. Harvey (1984) de®ned local mate competition (LMC) loosely as diminishing returns from investing more in male function. Under this de®nition LMC does not operate in our model 1 and the biased sex allocation is solely due to the greater bene®t to the parent from female-biased allocation. Note that our model 1 is identical to the case of optimization of the sex ratio in dioecious organisms when a ®xed fraction of the females Fig. 4 Evolutionarily stable, relative allocation to male function are mated by their brothers, as discussed by Maynard à à (r ˆ n /T) for plants growing in isolated populations consisting of Smith (1978, pp.160±161). 50% large and 50% small plants, which differ in resource availability In model 2, the ®tness returns from pollen dispersed to T. In one population under unfavourable conditions half the plants other plants still depend in a linear fashion on r. has V ˆ 10 and the other half has W ˆ 50 (circles), in another isolated population under more favourable conditions, half the Diminishing returns exist on the ®tness contribution individuals has V ˆ 50 and the other half has W ˆ 100 (triangles). from sel®ng (Fig. 3) so that LMC operates. On the one Equations 3 and 6 were used for calculating ®tness; k ˆ 0.1, d ˆ 0. hand, a mutant with a higher r value than r* has a higher Model 2: broken line; model 3: solid line. sel®ng rate. On the other hand, the trade-off between male and female investment implies that, when r increases, less seeds are produced, so that the ®tness and homogeneously distributed over the whole popula- gain from selfed seeds becomes zero for r ˆ 1. tion, so that 10% of the pollen is returned to the parent r* values are somewhat higher in model 2 than in plant where it is deposited on the stigmas. In this case model 1, resulting in more sel®ng. Therefore (with low there is assumed to be 10% sel®ng and applying eqn 4 inbreeding depression) the individuals bene®t somewhat would predict r* ˆ 0.45. It can, however, easily be from the Fisherian or automatic advantage of sel®ng shown that equal allocation to male and female function (Fisher, 1941), as can be seen from eqn 1, for the case is evolutionarily stable. For instance, an individual has 20 without extra costs for sel®ng (Em ˆ En). A completely units of resource available for ¯owers and seeds that each sel®ng genotype (Sm ˆ 1) then has a 50% higher ®tness cost a single unit of resource. Suppose one mutant plant when rare in an outcrossing population (Sn ˆ 0) if lowers its ¯ower number from 10 to 9 and produces an d ˆ 0. It is well known that this advantage depends on d, additional seed. The probability of sel®ng this extra seed, the frequency of m, and on pollen discounting (Holsing- or any other seed on the plant, now becomes 9/99 for the er, 1991; Schoen et al., 1996). Nevertheless in model 2 a mutant m. The probability that the extra seed is sired by genotype may bene®t to some extent from the automatic any of the nine other plants is 90/99. Therefore, for an advantage if it produces more ¯owers, so that it enjoys a individual resident plant the probability of siring the seed higher sel®ng rate. on the mutant is (1/9) ´ (90/99) ˆ 10/99. Per individ- In model 3 the ®tness gain through sel®ng also suffers ual, the resident bene®ts more from the female bias in from diminishing returns on r, as in model 2. In addition, the mutant than does the mutant itself. As a result there the number of pollen grains dispersed to other plants exists, in this speci®c example, a disadvantage of alloca- saturates with r, which further compromises male tion away from 50%, and r* ˆ 0.5. reproductive success. In essence, the way we modelled This example illustrates that it is not the sel®ng or sib pollinator behaviour and the resulting sel®ng violates the competition per se that selects for female biased sex- assumption of homogeneous pollen distribution. A mu- allocation. If we increase population size from 10 to an tant then bene®ts more from its female bias than another in®nite number of plants, the probability of sel®ng by individual in the same population. In retrospect, it is thus homogenously dispersed pollen drops to zero. On a obvious that geitonogamy affects sex allocation, contrary

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to the view held by Lloyd (1987a) and Brunet (1992). pollen carry-over may differ when a pollinator visits a Lloyd (1987a) noted that in model 1 optimal allocations male or a female ¯ower. If bees groom more after visiting to the sexual functions match the relative sizes of their the pollen-containing male ¯owers (compare Thomson, ®tness rewards and extrapolated this result to ge- 1986), the pollen loss is greater and carry-over is smaller itonogamous pollination. This shortcut fails in this more for male ¯owers. In such cases pollen export is a complex case, where the sel®ng rate is not ®xed but decelerating function of r and the situation is exactly as depends on male allocation. outlined for model 3. Geitonogamy is not well studied in In model 3, when pollen discounting is complete, it monoecious species. Therefore, we tailored our model to costs as much pollen to sire a seed on another plant, as to a hermaphrodite plant that produces excess ¯owers. self: thus the Fisherian advantage of sel®ng does not exist (see also Schoen et al., 1996). With inbreeding depres- Empirical data on sex allocation sion, it is then always better to outcross than to self a seed (Lloyd, 1987a). Geitonogamous sel®ng requires the same We argued that geitonogamous sel®ng in animal-polli- structures and costs as outcrossing, yet lacks the bene®ts nated plants results in female-biased sex allocation. As (no costs) of autogamy and is not adaptive (Lloyd, 1992). many sel®ng species have a mixture of autogamy and Yet the advantage to the parent of a female-biased sex geitonogamy, this is dif®cult to test. In C. of®cinale, sel®ng allocation remains in model 3, and this will drive occurs mainly through geitonogamy (Vrieling et al., selection for r* values below 0.5. The automatic advan- 1997; Vrieling unpublished result). By preventing polli- tage of sel®ng (absent under model 3) and the greater nation, we found for C. of®cinale that per seed that the bene®ts to the parent of a female-biased sex allocation plant did not produce, it compensated by making, on (present in model 3) are clearly different. average, 3.6 extra ¯owers (Klinkhamer & de Jong, 1987). Under natural conditions C. of®cinale plants abort many seeds and produce approximately one seed per ¯ower. Means of modifying gender This would correspond to a relative allocation to ¯owers In our models, the plant can vary the number of of 21.8%. The actual relative allocation r to male maturing seeds per ¯ower. This occurs in the Bora- function must be less, as this calculation attributes costs ginaceous plant Cynoglossum of®cinale; many ovules are for both sexual functions (nectar, corolla) to the male fertilized and later aborted (De Jong & Klinkhamer, function only. Dried seeds of C. of®cinale weigh about 1989), so that `empty' ¯owers (with no seeds) mainly 14.7 mg and the corolla, anthers included, weighs serve a male function. This species may vary r by making 2.8 mg. This yields an estimate of r ˆ 0.16, which is in fewer, or more, seeds per ¯ower with a maximum of four the same range as the above estimate. The female-biased seeds per ¯ower. sex allocation in this species is consistent with the model In model 1 the sel®ng rate is ®xed. When plants vary r prediction that geitonogamy affects allocation. Lloyd by making more seeds per ¯ower, keeping pollen (1984) reviewed data to suggest that in fully outcrossing production in the ¯ower constant, this model may apply. species r is close to 0.5 in wind-pollinated species and Alternatively, pollen production per ¯ower varies. It then lower than 0.5 in insect-pollinated species. In addition, seems unlikely that phenotypes with different pollen Poot (1996) carefully collected stamens and seeds from production per ¯ower will shed exactly the same number the self-incompatible, wind-pollinated ribwort plantain of pollen grains onto the stigma and have the same (Plantago lanceolata), which he weighed and analysed for sel®ng rate. It is more likely that producing more pollen nitrogen. Measured in both currencies, allocation to per ¯ower results in more autogamous sel®ng, analogous ¯owers was roughly equal to allocation to seeds. These to model 2. ®ndings are in accordance with Fig. 2(D), which predicts In monoecious species, male and female ¯owers r* ˆ 0.5 for a wind-pollinated plant without sel®ng and correspond to male and female allocations. Plants with a smaller value for self-incompatible, insect-pollinated many male ¯owers are likely to have higher sel®ng rates plants with geitonogamous pollination. Note, however, than plants with few male ¯owers. Whether pollen that we ascribe the difference between fully outcrossing, export per ¯ower is also reduced for plants with many wind- and insect-pollinated species to the fact that pollen male ¯owers is then less clear. In the simple model we export is a decelerating function of the number of proposed for hermaphrodites, a ¯ower and a seed are ¯owers. In the literature (e.g. Charnov, 1982) the equally costly. If, in monoecious species, a male ¯ower emphasis was rather on pollen distribution (assumed to costs as much as a female ¯ower (costs of seeds included), be more homogeneous in wind-pollination) and on then each plant will produce the same number of ¯owers pollinator saturation. (male plus female) regardless of r. If pollen export per ¯ower depends on the total number of ¯owers, it is then Size-dependent sex allocation independent of r, and model 2 applies. However, male ¯owers may be cheaper to produce, so that they come in For monocarpic, insect-pollinated plants a decrease of the greater numbers for a given investment of resources. Also relative male allocation with the size of the plant

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occurred in 37 of the 44 species listed in Klinkhamer et al. species is wind-pollinated include the US ¯ora (Freeman (1997), consistent with the model prediction (Fig. 4). et al., 1980; Fox, 1985; Muenchow, 1987), the ¯ora of Most wind-pollinated plants for which data are available South Africa (Steiner, 1988), woody species in the are monoecious. In only six out of 14 species was the Hawaiian ¯ora (Sakai et al., 1995), several tropical ¯oras ¯oral sex ratio shifted towards fewer male ¯owers in (Renner & Feil, 1993) and the global ¯ora analysed at large plants (Bickel & Freeman, 1993). The remaining both the family and the genus level (Renner & Ricklefs, eight species allocated more to male function when large 1996). Recently, Bullock (1994) pointed out that for and do not ®t the model predictions. Figure 4 also many tropical trees small, ¯ower-visiting bees are not indicates the relations expected when comparing two pollinators and that, apparently, wind pollination is quite isolated populations in which the plants differ in size. For common among dioecious tropical trees. the animal-pollinated Cynoglossum of®cinale we previously Bawa (1980) noted that ¯owers of the few animal- found, for a plant of a given size, greater allocation to pollinated dioecious species are typically `relatively small, seeds in populations on fertile than on infertile soil unspecialized and of white, pale yellow, or pale green' (Klinkhamer & de Jong, 1987). and that pollinators are small in size. Bawa (1994) further emphasized that dioecy is extremely rare among taxa pollinated by large pollinators such as bats, birds or The evolution of dioecy medium- to large-sized bees. All these pollinators are With full compensation, a male in model 1 has the same ef®cient foragers and are likely to respond to ¯ower ®tness as a hermaphrodite when the population is at an number as assumed in model 3. Small pollinators or ESS (Fig. 3A). When d > 0.5, pure females can establish accidental visitors to ¯owers are, in our opinion, less through an outbreeding advantage (Lloyd, 1975). This likely to respond to ¯ower number because they ®nd criterion is, not surprisingly, less stringent than the enough food in a single or in a few ¯owers, con®ning criterion Sd > 0.5 derived by Charlesworth & Charles- mostly to model 1. Although in general these data worth (1978), who assumed no compensation if pure support the predicted association between dioecy and males and females are introduced in a hermaphrodite abiotic pollination, exceptions do exist. For instance, in population. Values for cumulative inbreeding depression the Dutch ¯ora wild asparagus (Asparagus of®cinalis), red that are greater than 0.5 have been reported for some campion (Silene dioica) and creeping thistle (Cirsium partially sel®ng plant species (Husband & Schemske, arvense) are all dioecious and their ¯owers are frequently 1996). In model 3, pure males and females are always at visited by bumble bees. Clearly, these species do not ®t a disadvantage compared with hermaphrodites and our model and case studies may help determine how well unlikely to become established (Fig. 3A), even with full they ®t model 3. The alternative explanation for dioecy is compensation. Males and females can establish in model that in these species specializing in one sexual function 2 (Fig. 3B) if d > 0.5. Figure 3(B) shows that for high results in greater ef®ciency (Willson, 1979; Bawa, 1980). levels of inbreeding depression the ®tness of pure males The mechanism could be that a greater fraction of the and females is quite high, so that they can establish with fruits (or pollen) on individuals that emphasize one partial compensation or even without compensation. sexual function is dispersed. However, the empirical In the Charlesworth & Charlesworth (1978) model of support for this mechanism is, despite considerable effort the evolution of dioecy from hermaphroditism, gyno- (see Charlesworth & Morgan, 1991 or de Jong & dioecy is an intermediate stage. In model 2 with d > 0.5 Klinkhamer, 1994 for a review), still lacking. the population moves towards an attractor which is a For wind-pollinated species, Freeman et al. (1997) ®tness minimum (Abrams et al., 1993; Geritz et al., emphasized the different requirements for pollen capture 1998). At this point it is possible that dioecy evolves on the stigma and pollen dispersal from the anthers, with small steps and not through gynodioecy as an promoting sexual specialization. They further argued that intermediate stage. We will detail this process in a future dioecy has frequently evolved in families that already paper, using the methods of Geritz et al. (1998). have an antisel®ng device. As an alternative, especially From the comparison of the three models, the evolu- for wind-pollinated plants, our model underscores the tion of dioecy seems most likely in species with abiotic view from the 1950s and 1960s (summarized in Thomson pollination (model 2), provided that inbreeding depres- & Barrett, 1981; Freeman et al., 1997) that dioecy can sion exceeds 50%. The models then predict an associa- evolve primarily as an outbreeding mechanism. We have tion between abiotic pollination and dioecy. Dioecy is shown that the conditions for this route are less stringent unlikely to evolve with complete pollen discounting and than generally believed (Charlesworth & Charlesworth, animal pollination, especially with species such as bum- 1978), chie¯y because of the accelerating ®tness gain ble bees, honey bees or hummingbirds, that adjust their curves in model 2 with d > 0.5. If effective antisel®ng foraging bouts to ¯ower number. devices exist in the population as Freeman et al. (1997) Several authors have emphasized the consistent asso- have claimed, this stops evolution towards dioecy and ciation between abiotic pollination and dioecy. The data reverts the situation to that shown in Fig. 2(D). The sets in which a disproportionate number of dioecious effectiveness of these devices would need to be estab-

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lished. This would be possible in dioecious species in Crawford, T.J. 1984. What is a population? In: Evolutionary which occasional cosexual genotypes are found (Kay & Ecology (B. Shorrocks, ed.), pp. 135±173. Blackwell, Oxford. Stevens, 1986). If these cosexual genotypes are self- Cruden, R.W. 1977. Pollen±ovule ratios: a conservative indicator compatible, experience high levels of geitonogamous of breeding systems in ¯owering plants. Evolution, 31: 32±46. sel®ng and produce selfed seeds of inferior quality, this DommeÂe, B. 1981. Roà les du milieu et du geÂnotype dans le reÂgime de la reproduction de Thymus vulgaris L. Acta Oecol. supports our hypothesis that dioecy evolves as an Oecol. Plant. 2: 137±147. outbreeding mechanism. If, on the other hand, the Dudash, M.R. 1991. Plant size effects on female and male cosexual genotypes are self-incompatible, have low function in hermaphrodite Sabatia angularis (Gentianaceae). sel®ng rates and produce high-quality seeds, this goes Ecology, 72: 1004±1012. against our hypothesis. Fisher, R.A. 1941. Average excess and average effect of a gene Our models con®rm Lloyd's (1987a) idea that the mode substitution. Ann. Eugen. 11: 53±63. of sel®ng is important for optimal sex allocation. They give Fox, J.F. 1985. Incidence of dioecy in relation to growth form, new insight in how accelerating and decelerating gain pollination and dispersal. Oecologia, 67: 244±249. curves arise (compare Thomson & Brunet, 1990) and Freeman, D.C., Harper, K.T. and Ostler, W.K. 1980. Ecology of could serve as a basis to which other factors (for instance, plant dioecy in the intermountain region of western North America and California. Oecologia, 44: 410±417. local resource competition between seedlings around the Freeman, D.C., Lovett Doust, J., El-Keblawy, A., Miglia, K.J. and parent plant, de Jong & Klinkhamer, 1994) affecting the McArthur, E.D. 1997. Sexual specialization and inbreeding shape of the gain curves, can be added. avoidance in the evolution of dioecy. Bot. Rev. 63: 65±92. Geritz, S.A.H., Kisdi, E ., MeszeÂna, G. and Metz, J.A.J. 1998. Acknowledgments Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12: 35±57. We appreciated comments from Carl Freeman, Yoh Hamilton, W.D. 1967. Extraordinary sex ratios. Science, 156: Iwasa, Hans Metz, John Pannell and the referees on 477±488. previous drafts. M.C.J.R. was supported by the Life Harder, L.D. and Barrett, S.C.H. 1995a. Mating costs of large Sciences Foundation (S.L.W.), which is subsidized by the ¯oral displays in hermaphrodite plants. Nature, 373: 512± 515. 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