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The evolutionary forces shaping within- and across-species variation in the investment in male and female sex function are still incompletely understood. Despite earlier suggestions that in plants the evolution or cosexuality vs. dioecy, as well as sex allocation among cosexuals, is affected by seed and pollen dispersal, no formal model has explicitly used dispersal distances to address this problem. Here, we present a game-theory model as well as a simulation study that fills in this gap. Our model predicts that dioecy should evolve if seeds and pollen disperse widely and that sex allocation among cosexuals should be biased towards whichever sex function produces more widely dispersing units. Dispersal limitations stabilize cosexuality by reinforcing competition between spatially clumped dispersal units from the same source, leading to saturating fitness returns that render sexual specialization unprofitable. However, limited pollen dispersal can also increase the risk of selfing, thus potentially selecting for dioecy as an outbreeding mechanism. Finally, we refute a recent claim that cosexuals should always invest equally in both sex functions.
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- The model
- Supporting Information
Sex allocation theory aims at explaining within- and across-species variation in the investment in male and female function between individuals (Charnov, 1982; Schärer, 2009). Most plant species are cosexual, meaning that each individual fulfils both male and female sexual functions. This contrasts sharply with animals, in which only an estimated 5–6% of species are hermaphroditic (although this figure increases to about 30% if insects, which are never hermaphroditic, are excluded; Jarne & Auld, 2006). Despite this general trend, dioecy (the botanical term for the separation of sex functions in male and female individuals) has evolved many times throughout the plant kingdom (Bawa, 1980), and it occurs in ca. 6% of flowering plants (Renner & Ricklefs, 1995).
Theory suggests that the optimal sex allocation depends on how efficiently fitness can be gained by investing in either sex function (Charnov et al., 1976). Altering mating opportunities thus can select for changed sex allocation, and support for this argument has been found in both animals (Baeza, 2007) and plants (Dorken & Pannell, 2008). The plant case (Dorken & Pannell, 2008, 2009) is of special interest because a follow-up paper documented microevolutionary change and not merely plastic allocation: hermaphrodites of Mercurialis annua mating in the absence of males evolved greater male allocation (Dorken & Pannell, 2009).
In general, optimal male and female allocations depend on the shape of the (sex-specific) mathematical curves mapping fitness gains on investment level. Saturating fitness gain curves make high investment less profitable, which selects against specialization in only one sex function. This makes cosexuality more likely (Charnov et al., 1976). In contrast, when fitness gains are accelerating, high investment levels are required to gain the highest payoffs. This favours sexual specialization, i.e. dioecy. This theoretical framework also predicts that sex allocation among cosexuals should be biased towards whichever sex function has a steeper (less saturated) fitness gain curve (Charnov, 1982).
Although this theory can account for the evolution of both dioecy and cosexuality, the question of why fitness gain curves should be shaped in a certain way in the first place is surprisingly understudied. To explain the prevalence of cosexuality in plants, Maynard Smith (1978) suggested that saturating fitness gain curves may arise from dispersal limitations of both seeds and pollen. According to his argument, additional pollination success will be increasingly difficult to achieve once a focal plant’s neighbours are pollinated already. Similarly, additional seeds produced will face reduced recruitment chances once a maternal parent plant’s vicinity is already well stocked with seeds. These effects will be most notable when dispersal distances are short, leading to strong interference between the spatially clumped dispersal units from the same source.
Previous theoretical work suggests that this link between dispersal and sexual system may be of great general importance. Existing models fall into two categories: one class of models focuses exclusively on reproductive success via the female function (Heilbuth et al., 2001; Wilson & Harder, 2003; Barot & Gignoux, 2004; Vamosi et al., 2007), which provides no adequate framework for explaining the evolution and maintenance of dioecy. Another set of models accounts for both male function and female function, but treats dispersal rather implicitly, by considering, on the one hand, the numbers of seed-producing individuals whose seeds compete locally, and, on the other hand, the numbers of individual seed-producers that each pollen-producer can reach and fertilize (Lloyd & Bawa, 1984; see also Charnov, 1982). Although these models have lent analytical support to Maynard Smith’s (1978) verbal arguments, their simplicity does not wholly capture the role of dispersal for the evolution of sexual systems. First, it is difficult to relate their assumption of fixed competitor numbers to a realistic view of dispersal in terms of dispersal kernels. Second, they do not account for a possible feedback between sexual system and spatial configuration, which arises from the fact that cosexuality implies a greater density of seed and pollen sources for a given population density. Third, none of these models has accounted for the possibility that the risk of selfing, which may be an important cost of cosexuality, may also depend on dispersal patterns.
Here, we present a game-theory model, backed up with individual-based simulations, that fills in these gaps. We derive specific predictions that we then contrast with known across-species patterns. First, we investigate the effect of seed and pollen dispersal distances on optimal sex allocation among cosexuals. Second, we examine the evolutionary stability of cosexuality vs. dioecy under different dispersal regimes.
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- The model
- Supporting Information
We consider an infinitely large population where each individual produces many dispersal units (seeds and pollen) that become distributed in space centred around its own location. Individuals invest X resources in seed production, and 1 – X resources in pollen production, which defines cosexuality as 0 < X < 1. Reproductive success is gained when opportunities for germination and fertilization arise, which happens at a low rate throughout the habitat. Because fertilization events without subsequent germination do not yield reproductive success, such ‘failed fertilizations’ can be neglected for our purposes. Therefore, we consider that successful fertilization and germination events occur at the same rate.
Because kin competition is known to influence sex allocation (Hamilton, 1967) and dispersal (Hamilton & May, 1977; Gandon, 1999; Ronce et al., 2000), and we are primarily interested in effects of competition at the fertilization and recruitment stage (as opposed to the long-term consequences of growing close to kin), we define that to count as an ‘opportunity’ a location has to be a large enough vacancy that one individual can grow in it without harming kin. This does not harm the generality of the model but strengthens the reasoning that opportunities should occur at a low rate. In the Discussion, we will comment on the likely effects of relaxing our assumption of no post-recruitment competition.
Our assumption of a low rate of opportunity formation means that most dispersal units fail to contribute to reproductive success. When a reproductive opportunity arises in a given area, the probability that a certain individual can reap this opportunity depends on the density of its relevant dispersal units in that area, relative to the total density of such dispersal units in the same local area. This means that an individual’s dispersal units, if poorly spread out, compete mostly with each other, whereas if widely spread they compete mostly with those of other individuals.
We consider a spatial scale such that there is on average one individual per unit of space, implying that there are πr2 individuals per circular area of radius r. We envisage a population where almost all individuals use the same sex allocation strategy, which we refer to as the resident strategy. We assume that dispersal distances r are drawn from probability density function D(r, di) with mean distance di, which takes values ds and dp, for seeds and pollen, respectively. Although we develop our model for a general form of D(r, di), our computational examples are based on either of two possible forms (Fig. 1): a half-normal distribution, , which is appropriate when movement arises from small independent steps in random directions; and a negative exponential distribution, , which is appropriate if each dispersal movement has a constant speed and direction until terminated at a constant rate.
Figure 1. Half-normal (a) and negative exponential (b) shape of dispersal kernels with mean distances: di = 2 (dashed) and di = 4 (solid). Kernels specify the proportion of dispersal units settling at a given distance from the source.
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Letting the functions f(X) and g(1–X) denote the numbers of viable seeds and pollen produced per unit of investment, respectively, we consider two cases: in case 1, we assume that a fixed ‘start-up’ cost C has to be paid for providing and maintaining a ‘machinery’ for producing both seeds and pollen before further investment translates into the production of gametes. Accordingly, we let
so that f and g remain 0 up to a threshold before beginning to increase. This formulation implies that unisexuals can produce gametes more efficiently by paying only one type of start-up cost. In case 1 we assume no selfing.
In case 2, there is no start-up cost; functions f and g increase in the entire range from 0 onwards, so that any small investment results in a small number of gametes being produced. Nonlinearity in f arises because of selfing. Selfing generally leads to reduced seed viability (Lloyd, 1979; Charlesworth & Charlesworth, 1990; Porcher et al., 2009). Because the risk of selfing likely depends on the relative numbers of self- vs. allopollen surrounding each ovule, we let the proportion of selfed seeds be . Here, is the focal individual’s number of nondispersed pollen, and allopollen = g(1 – X)bp is the adjusted density g(1–X) of competitors’ pollen, where the coefficient bp adjusts for spatial clumping. We assume that selfed seeds are unviable with probability k, so that the production of viable seeds is given by f(X) = X(1 – k · selfed).
Similar modifications to f(X) could also be caused by mechanisms other than selfing. For example, if seed-dispersing animals are disproportionally attracted to high-yield plants (Vamosi et al., 2007), seeds on low-yield plants may perish because of dispersal failure. As perished seeds do not yield any fitness to their mother, this effectively reduces the value of f(X) at the low end of X.
We let the pollen available for export follow a linear function, g(1 − X) = 1 − X, implying that selfing uses up a negligible amount of pollen (no pollen discounting).
It is important to note that the linearity or nonlinearity in f refers to the stage of seed production and is thus independent of our central question which is to analyse the pathway to nonlinearity envisaged by Maynard Smith (1978): whatever the scaling rules that determine the numbers of seeds and pollen, they subsequently will have to compete locally over germination or fertilization opportunities, and this can yield saturating fitness gains depending on competitor numbers.
Sex allocation in a purely cosexual population
Figure 2. Outline of the spatial situation envisioned in our model. Dark blotches represent areas of high seed density (seed shadows) around individual plants. If dispersal distances from the focal individual (marked x) are short (a), seeds compete mostly with sibling seeds, making high investment in seed production unprofitable. If dispersal distances are long (b, c), any competition faced is likely from nonsiblings, making seed success chances nearly independent of their mother’s total investment. An analogous argument applies to pollen.
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If we consider a narrow ring-shaped area (such that r1 ≈ r2), all mutant seeds within this area have approximately the same chances of success (i.e. we can neglect the effect that mutant seeds face somewhat stronger competition from each other on the inner part of the ring). The relative success of mutant seeds in this area is then
In the limit, where r2 → r1, we obtain the distance-dependent seed success rate
Taking into account that reproductive opportunities available at distance r increase in proportion to the circumference 2πr of a correspondingly sized circle, the mutant’s fitness gain from seed production is
The success of outbreeding pollen is developed exactly analogously. The mutant’s distance-dependent pollen success rate is thus
To ensure internal consistency of our model, we must now implement two logical constraints. First, in a population at density dependent equilibrium, average fitness is independent of the strategies being adopted. Second, in a sexually reproducing population, the population-wide total fitness gained through the male function must equal the population-wide total fitness gained through the female function (the so-called Fisher condition, Houston & McNamara, 2005). These constraints are met when
In other words, any cosexual strategy, when competing against itself, attains unity fitness from seed production and unity fitness from pollen production. Because the Fi functions are monotonously decreasing in bi and are independent of X given that , there is a unique pair of values satisfying eqn (5) for given parameter values. These ‘clumping coefficients’bi lie in the range 0 < bi < 1, are easily found numerically, and take lower values as the degree of spatial clumping increases. This means that for a focal resident, competing against a background of clumped dispersal units is equivalent in effect to competing against a smaller number of homogeneously distributed units, as adjusted by bi. Based on the assumption that a focal mutant is similarly affected by its surrounding spatial configuration, the coefficients bi can be used to approximately account for the effect of spatial clumping. Essentially, this approach amounts to comparing the performance of mutants and residents against the same simplified background.
A mutant’s total fitness gain from both seed and pollen production is given by:
We can find a stationary point in this function by setting
and solving for X. This is a fitness maximum and hence a candidate evolutionary stable strategy (ESS) (Maynard Smith, 1982) to which we refer as X*, if
Now a population may contain three types of individuals: a proportion p of cosexuals using sex allocation X, a proportion (1 – R) (1 – p) of females, and a proportion R (1 – p) of males. Here, R is the sex ratio among unisexuals, i.e. the proportion of dioecious individuals that are males. To find the best cosexual strategy X* in this population, we generalize eqn (1) to
which has in its denominator the contribution of the (1 – R) (1 – p) females to seed production. Note that females, producing no pollen, do not incur a cost of selfing. Similarly, eqn (3) generalizes to
taking into account the contribution of the (1 – p) R males to pollen production. To ensure that population-wide per capita fitness gain from each the female and the male function is unity, we choose coefficients satisfying
Using these values, eqns (9) and (10) allow us to derive a fitness function analogous to eqn (6). For a given sex ratio R among dioecious individuals and for a given strategy X of resident cosexuals, the best cosexual mutant strategy achieves fitness
which we can approximate numerically by computing solutions at . Note that mutant fitness becomes independent of X when cosexuals are very rare. Assuming that unisexuals occur at their optimal sex ratio R = R*, their fitness is given by
Note that this optimum R* will depend on the sex allocation X of cosexuals. We can numerically approximate R*, and hence Wdioecy, by computing solutions at .
Cosexuality is evolutionarily stable if
Dioecy is evolutionarily stable if
Although our game-theory model is flexible in certain respects (e.g. it can be derived for any shape of a dispersal kernel), it does not account for some other aspects, such as stochasticity, and kin competition after the stage of germination. To account for such processes, we have also performed individual-based simulations to examine the robustness of our results. The details of these simulations are given in the Electronic Appendix S1 – Figs S1–S4.
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Confirming and expanding on previous arguments (Maynard Smith, 1978; Lloyd & Bawa, 1984), our model shows that dioecy evolves more easily in plant species with widely dispersing seeds and pollen. Conversely, dispersal limitations tend to stabilize cosexuality. This is because limited dispersal makes ‘sibling’ dispersal units (seeds or pollen) compete for the same germination and fertilization opportunities, such that increasing seed or pollen production gives diminishing returns (Fig. 3) and a better strategy is to invest some effort in producing both.
Another prediction from our model is that cosexuals should bias their investment towards the sex function associated with wider dispersal (cf. Lloyd & Bawa, 1984), which is typically the male function (Levin & Kerster, 1974; Hu & Ennos, 1997; Latta et al., 1998). As with the evolution of dioecy, such an allocation bias can be interpreted as a strategy to reduce self-competition, i.e., competition among dispersal units from the same parent. This finding contradicts Iyer & Roughgarden’s (2008) recent claim that cosexual individuals should always invest equally in both sex functions. As detailed in Appendix A, Iyer and Roughgarden reached this (incorrect) conclusion by re-evaluating a model by Charnov (1982), but in doing so failed to recognize how competition via male and female function occurs at different scales.
In addition to spatial clumping, the net benefits of dioecy vs. cosexuality are also influenced by any ‘start-up’ costs of possessing the machinery for producing both seeds and pollen and by the cost of selfing. High start-up costs and selfing costs both select for dioecy (Figs 5 and 6). Cosexuals respond to selfing costs by investing less in pollen (Fig. 4), which reduces the extent of selfing. Moreover, because selfing interferes especially with cosexuals’ female function, it facilitates the invasion of females (Fig. 6) who can avoid these costs. The effect of pollen dispersal in our model depends on the type of cost assumed. Low pollen dispersal increases self-competition of the male function, thus potentially stabilizing cosexuality (Fig. 5). On the other hand, low pollen dispersal may increase the risk of selfing, thus selecting for dioecy as an outbreeding mechanism (Fig. 6).
According to our model, high population density has an effect equivalent to wide dispersal in facilitating the evolution of dioecy. This is because our results are governed by the relative strengths of competition at two levels: first, between the same plant’s dispersal units; second, between dispersal units from different plants. Between-plant competition becomes relatively more important when more competitors are ‘within reach’, which may be the case either because they are nearby, or because their seeds and pollen disperse widely. This argument can also be used to extend the results to dispersal kernels that we did not explicitly consider. Fat-tailed kernels, for example, are likely to have a similar effect as wide dispersal in general, because they similarly increase the number of competitors ‘within reach’.
An interesting feedback between sexual system and spatial configuration arises from the fact that, for a given population density, cosexuality implies a greater density of seed and pollen sources. If (say) pollen is distributed by twice as many individuals under cosexuality than under dioecy, this causes a more even pollen distribution across the habitat. Reduced spatial clumping relaxes self-competition, leading to less saturating fitness gains from each sex function. This association between cosexuality and less saturating gains, combined with the result that saturating gains stabilize cosexuality, can lead to negative frequency-dependent selection. For example, if cosexual mutants invade a dioecious population, this may change the sex-specific fitness gain functions such that it eliminates their initial advantage. This negative frequency-dependence is reflected by narrow areas of parameter space where cosexuality and dioecy can mutually invade each other (Fig. 6). The possibility of such mutual invasibility contradicts earlier findings of a nonspatial model (Wolf & Takebayashi, 2004) and suggests a novel explanation for the rare maintenance of all three sex morphs (trioecy) in some species (Geber et al., 1999). Consistent with Charlesworth & Charlesworth (1978), selfing costs in our model facilitate the invasion of female (rather than male) mutants (Fig. 6), which may explain the more frequent occurrence of gynodioecy as compared to androdioecy in nature.
Another possible feedback, not accounted for here, may exist between sexual system and inbreeding depression (e.g. Lande et al., 1994). Frequent selfing in cosexual populations may facilitate purging of deleterious recessive alleles, thus limiting the strength of inbreeding depression. On the other hand, less effective purging may lead to higher degrees of inbreeding depression under dioecy. In the light of the present model, this effect would stabilize dioecy once evolved.
As we derive very similar results from a game-theory approach (that necessitates some approximations) as well as from an individual-based simulation (electronic Appendix A), the predictions appear robust with respect to the modelling approach used. We will therefore now turn to examining how well these predictions hold in nature. Dispersal and dioecy have been linked by various lines of empirical evidence. Dioecious plants often have fleshy fruits dispersed by birds (Bawa, 1980; Vamosi et al., 2003). This is thought to be a particularly effective dispersal mode (Hubbell, 1979), which may also explain why dioecious species are overrepresented in island floras (Bawa, 1982; Lloyd, 1982). In terms of our model, this pattern could be interpreted such that bird dispersal facilitated the evolution of dioecy by relaxing competition between maternal siblings. An alternative (but not mutually exclusive) explanation for the same pattern has been proposed by Bawa (1980) and Givnish (1980): if birds are disproportionally attracted to high-yield plants, this will generate an incentive for plants to maximize their fruit yield, thus favouring the evolution of specialized females. In other words, there will be an accelerating fitness gain curve for the female function as a direct result of bird behaviour. However, such disproportional bird attraction lacks empirical support (Howe, 1980, 1981; Russo, 2003).
Correlations between dioecy and certain pollination modes have also been reported. Dioecious plants often have small, inconspicuous, greenish flowers pollinated by small generalist bees and flies (Thomson & Brunet, 1990). Although such insects have been suggested to be highly responsive to flower number, potentially leading to accelerating returns from male function (Bawa, 1980), there appears to be no evidence to support this view. Neither do these small insects typically cover distances large enough to greatly reduce competition between pollen from the same plant (Bawa, 1980). However, as already speculated by Lloyd (1982), the habit of small generalist insects to visit the same plant repeatedly may create a high risk of selfing, potentially selecting for dioecy as an outbreeding mechanism.
Dioecy is also associated with wind pollination and with a woody growth form (Vamosi et al., 2003). These features may favour the evolution of dioecy as follows. Our model predicts that conditions where selfing costs are likely to arise will favour dioecy as such conditions select against the production of both seed and pollen in the same area. The undirected movement of pollen by wind, and the spatial concentration of flowers on a (typically large) woody plant, may both elevate the risk of selfing, which then favours the evolution of dioecy as an outbreeding mechanism. On the other hand, both wind dispersal and large size of adult plants may facilitate long-distance pollen movement (Klinkhamer et al., 1997; De Jong & Klinkhamer, 2005), and, as shown in our model, this makes saturating gain functions less likely. Indeed, there is accumulating evidence for wind pollination yielding less saturating gain curves than biotic pollination (De Jong & Klinkhamer, 2005). It could also be argued that woody, large plants, with their typically long generation times, are under increased pressure to produce varied, outbred offspring that present new challenges to their faster-evolving parasites. This would effectively increase the viability cost k of selfing for woody plants. Although this last hypothesis is rather speculative, the net effect of woody growth form on fitness gain functions clearly deserves much more attention.
One limitation of our game-theory model is that it accounts for population kin structure only up to the point of germination. It is therefore reassuring to note that the results of our individual-based simulations (electronic Appendix A) are in good qualitative agreement with the game-theory model. In individual-based simulations, kin structure and kin competition can arise automatically, without any need to formulate its action explicitly.
To conclude, and contrasting with some other recent work (Iyer & Roughgarden, 2008), we have shown that sex allocation can respond to unequal dispersal distances of seeds and pollen. There is some support for the predictions, e.g. that sex allocation is biased towards the longer dispersing sex. However, the precise shape of the gain curves has rarely been measured in nature. It is clearly a future challenge to do so.