Introduction

Predators must find and capture prey in order to survive, and have evolved a wide variety of prey-capture strategies. One way to capture prey is to remain stationary and attack passing prey, and individuals using this strategy are known as sit-and-wait predators (Anderson and Karasov 1981; Olive 1982). Clearly, this strategy is only effective when sufficient numbers of prey pass within striking distance of predators, and as a result sit-and-wait predators are under strong selection to hunt in areas of high prey density (Olive 1982; Linton et al. 1991; Shafir and Roughgarden 1998; Higginson 2005; Morse 2007). Flowers make particularly attractive sites for insectivorous sit-and-wait predators since they have evolved to attract pollinators, at least some of which may be potential prey for the predator (Morse 1986, 2007; Dukas and Morse 2005). Hence, it is no surprise that flowers are frequently inhabited by a range of sit-and-wait predators, such as mantids, phymatids and spiders (Schmitz et al. 2000; Halaj and Wise 2001); and that many of these species have evolved striking resemblances to their host plants (or parts thereof) in order to avoid detection by their prey (Chittka 2001; Théry and Casas 2002), and can even make flowers more attractive to prey (Heiling et al. 2003).

A number of studies have attempted to test how the presence of sit-and-wait predators influences plants’ reproductive success (reviewed in Schmitz et al. 2000). These studies have produced mixed results, with some studies demonstrating that sit-and-wait predators impose fitness costs on their host plants (Suttle 2003; Muñoz and Arroyo 2004; Dukas 2005; Knight et al. 2006; Goncalves-Souza et al. 2008), and others demonstrating that they confer benefits (Romero and Vasconcellos-Neto 2004; Whitney 2004). These apparently conflicting results can be explained by considering what the predators eat: predators can reduce the level of pollination by consuming plant pollinators (Morse and Fritz 1982; Morse 1993, 1999; Dukas 2001; Knight et al. 2006), but can increase the number of viable seeds a plant produces by consuming plant antagonists such as nectar-robbers, florivores, and consumers of seeds and fruit (Ruhren and Handel 1999; Snyder and Wise 2001; Romero and Vasconcellos-Neto 2004).

Flower-dwelling predators (and sit-and-wait predators in general) are generalists, and seldom prey solely on either plant mutualists or plant antagonists (Hurd 1999; Matsura and Inoue 1999; Morse 2007). Most predators eat both pollinators and pests whilst plants are in flower (Hurd 1999; Morse 2007), and even after pollination has occurred the former site of the flower may remain attractive to the predator if this site attracts consumers of seeds and fruit that the predator can prey on (e.g. Riechert and Bishop 1990; Carter and Rypstra 1995; Denno et al. 2002; Moran and Scheidler 2002; Suttle 2003). Thus, a single type of flower-dwelling predator can have a complex of both positive and negative effects on its host plant through its interaction with plant mutualists and antagonists. For example, Louda (1982) demonstrated that the presence of the green lynx spider Peucetia viridans reduced the proportion of flowers pollinated on its host plant Haplopappus ventetus by a third. This cost was offset, however, because the release of viable undamaged seeds was higher from inflorescence branches with spiders than those without.

There is currently a high level of interest in trophic cascades mediated by top-down effects (e.g. Schmitz et al. 2000; Halaj and Wise 2001) and many studies suggest that top-down effects of flower-dwelling predators on producers are likely to be profound. Despite this, temporal and spatial variation in the direction of the effect is underappreciated. For example, since the structure of prey communities is likely to change over time, it is important to study the entire reproductive season of plants in order to fully understand the impact of flower-dwelling predators on their hosts. These complex effects are also likely to influence the stability of food webs (Memmott 1999; Memmott et al. 2004), and hence improved understanding of them could lead to improved conservation strategies. Furthermore, if we are able to predict the conditions under which predators will be beneficial to plants we will have greater understanding of the co-evolutionary processes between the three trophic levels. Theoretical and empirical studies have led to significant progress in understanding situations where two or more prey species compete for the consumption of lower trophic levels (reviewed in Chase et al. 2002), but less studied are situations where prey species have differing roles (Melián et al. 2009), such as consumption and mutualism. In order to encourage further integrative studies, here we present a simple but general model of both the costs and benefits to the plant of flower-dwelling predators that prey on both mutualists and consumers. Our model provides a comprehensive set of predictions of the ecological circumstances under which we would expect there to be a net cost or benefit to the host plant.

The model

We assume that the predator can interact with two prey species: a plant mutualist (the pollinator) and a plant antagonist (the granivore); the density of pollinators (P) and the density of granivores (G) are indicated as a proportion of their maximum possible density. We explore the effect of combinations of these two prey densities, with initial densities of each prey in increments of 0.01 in the range zero to one.

Under some conditions, we assume that the predator tends to reduce the density of one prey type more than the other, but in all cases prey removal increases with prey density. In the absence of bias, we assume that the density of prey removed increases linearly with total prey density δ(P + G), up to some maximum value (D max), for some constant (δ; assumed to take the value 0.5 by default). This maximum might be set by, for example, predator satiation. Greater predation of one prey type could be due to a pre-existing bias (e.g. pollinators are temporally first). Alternatively, predator impact on prey might be prey density dependent, such that predators attack the less abundant or more abundant prey more (e.g. because predators prefer a varied diet or because prey handling time reduces with learning, respectively).

In the absence of the predator, the effective density of pollinators experienced by the plant (P e) is simply the same as P, and likewise the effective density of granivores (G e) is simply G. Given that γ controls the pre-existing bias, and is positive if predators consume more pollinators, and λ controls the frequency-dependent predation, and is positive if predators kill more of the more abundant prey type, then the densities of each prey type are given by:

$$ \begin{gathered} P_{\text{e}} = P - \theta P\delta [\gamma + \lambda (P - G)] \hfill \\ G_{\text{e}} = G - \theta G\delta [2 - \gamma - \lambda (P - G)] \hfill \\ \end{gathered} $$
(1)

where

$$ \theta = {\frac{{D_{\max } }}{\delta P[\gamma + \lambda (P - G)] + \delta G[2 - \gamma - \lambda (P - G)]}}; \, \theta \, \le { 1} $$

so that total prey killed is no greater than D max.

In general the proportion of potential seeds that are pollinated should be expected to increase with increasing effective number of pollinators. However, the shape of the increasing function may be complex and differ between situations. Where there is competition between pollinators such that each individual pollinates more potential seeds when at a lower P, then we might expect a concave (saturating) function. Alternatively where there is social facilitation or other mechanisms acting to increase per capita pollination rates at higher pollinator densities, we might expect a convex (accelerating) function. Where pollination rates are density independent we might expect a simple linear response. We select a functional form that can represent all these different situations. The saturating function is most likely because the number of stigma pollinated is likely to be subject to diminishing returns as the remaining unfertilised stigma become scarcer. Tentative evidence for a saturating function comes from several studies that indirectly indicate diminishing pollination returns from increasing plant density, where the latter also correlates with P, albeit confounded by the greater number of possible pollen donors (Waites and Agren 2004; Feldman 2006; Steffan-Dewenter et al. 2001).

The proportion of seeds pollinated increases with P according to:

$$ F \, = \left( {\pi \, P_{e} } \right) \, ^{\alpha } $$
(2)

where π is the efficiency of pollinators at pollinating and α controls the shape of the function. We use three values for α: 0.5 (concave), 1 (linear), 2 (convex). In all cases the pollination of seeds is limited by the plant’s physiology (F ≤ 1).

By similar arguments, the ability of individual granivores to predate seeds might be positively dependent on effective G, negatively dependent or independent, hence we require similar flexibility in the form of the function describing the effect of effective G on seed survival. The proportion of seeds eaten increases with G according to the same function and values of two parameters: β, and the effectiveness of granivores ϕ.

$$ C \, = (\phi G_{\text{e}} )^{\beta } $$
(3)

where ϕ is determined by the extent to which granivores consume more seeds when more are available according to:

$$ \phi \, = { 1 } + \, \varepsilon \left( {F - 0. 5} \right). $$
(4)

Thus, when ε = 0, G does not respond to seed density (F) and when ε is positive granivores consume more seeds when there is higher F. Therefore β controls the effect of G on per capita seed consumption, while ε controls the appetite or foraging intensity of granivores.

All parameters and their default values are shown in Table 1. We use as our fitness measure the fraction of potential seeds (S) that are created by successful pollination and survive granivory, which is given by:

$$ S \, = \, F \times \left( { 1- C} \right). $$
(5)

We calculate this in the absence of the predator (S NP) and in the presence of the predator (S p) for all combinations of prey densities. Thus, for each combination of starting densities of the two prey types, we find the change in seed success if a predator is added. We assume that the predator has a positive effect on the plant if the number of seeds surviving is greater than 2% more in the presence of the predator than in the absence of the predator (S P > S NP + 0.02). We assume that the predator has a negative effect on the plant if the number of seeds is smaller by more than 2% in the presence of the predator (S P < S NP − 0.02). Otherwise we consider the effect of the predator to be approximately neutral.

Table 1 Parameters in the model and their values
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Results

Magnitude of effect of predator

In general, P controls the magnitude of the effect of the predator, whilst G controls the direction of the effect (Fig. 1). When G is low, the magnitude of the negative effect of the predator increases with P, and when G is high, the magnitude of the positive effect increases with P. When predation is positively density dependent, at high P fewer granivores are killed as P increases, and so the optimal value of P for the plant is at an intermediate P. This optimal P increases as G increases because more pollinators are needed to cancel out the increasing consumption by the granivores.

Fig. 1
figure 1

Net effect of predators on seed set for the range of pollinator density (P) explored for various densities of granivores (G) (lines). Parameter values were set to the default values shown in Table 1, with the exception of λ = 0.25, so that predation is slightly positively density dependent (when λ is zero, the lines are vertically symmetrical about S P − S NP = 0, where S p is presence of the predator and S NP absence of the predator). Increasing P increases the magnitude of the effect of the predator, whilst increasing G makes the effect more positive

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Density dependence in seed success

First we explored the impact of varying α and β, controlling the shape of the functions relating pollinator and granivore densities to seed numbers (Fig. 2). At the default values (Table 1), when the pollination function is concave (α = 0.5) and the granivore function is linear (β = 1), the predator has a positive effect on plant fitness when G is high, whereas P has little influence on the direction of the net effect (Fig. 2d). When G is high, a positive effect occurs even when P is low because individual pollinators are very effective and few pollinators are removed by predators, because prey types are removed in proportion to their relative density. A positive effect occurs when P is high because removing equal numbers of both prey types has less impact on pollination because of the concave pollination function. When G is low, there is always a negative effect of predator presence on plant fitness because when P is high pollinators are removed by the predator more than granivores, and when P is low each pollinator has more impact because of the concave pollination function.

Fig. 2a–i
figure 2

Overall positive (+) or negative (−) effect of predators on seed set in response to variation in P and G. Light grey areas represent less than 2% seed success both with and without a predator, dark grey areas represent less than 2% difference between predator and no-predator situation. Parameter values for α and β are shown alongside figure parts. All other parameter values are shown in Table 1. The default values of α and β are used in d

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Increasing α, and thereby changing the pollinator function through linear to convex, causes the area of a positive effect to reduce upwards and to the right. That is, it is now limited to when both pollinators and granivores are super-abundant, because a high density of pollinators is required to pollinate all available seed and losing pollinators is more costly at higher densities if the pollinator function is convex than when the function is concave or linear. The area of no effect on the left gets larger, because low densities of pollinators are insufficient at pollinating so there are few surviving seeds.

Increasing β reduces the area of a positive effect to only where G is high, because at intermediate granivore densities a reduction in G by a small amount has a smaller effect on granivory when the function is convex than when it is linear. However increasing β also increases the magnitude of the effects of the predator, mostly because the positive effect is very large when the function is convex and G is large, because granivores have a higher per capita rate of consumption. Decreasing β both widens the range of granivore densities and reduces the range of pollinator densities where we predict a positive effect. The latter occurs because the reduction in G reduces granivory by a large amount as the concave function is steep at this value of G. The former occurs because at low P a positive effect only results if a large number of granivores are killed, and this proportion must be all the greater if granivores have a higher per capita seed consumption rate at low granivore densities (which increases as β decreases). Decreasing β also decreases the magnitude of the positive effects, because when G is high removing granivores has little effect. In general, the sign of the effect depends on the magnitude of the change in prey density (i.e. P − P e and G − G e), and the shape of the functions (Eqs. 2 and 3) below the predator-absent densities (P and G).

Effectiveness of the predator

We varied the maximum effectiveness of the predator (D max), and the effect of prey density on predator effectiveness (δ) (Fig. 3). Generally, as we reduce the effectiveness of the predator (lower D max and/or lower δ), we increase the area of positive effect slightly, and as δ is reduced much of the area of negative effect changes to no effect. This change occurs mostly where P is less than 0.5, because in this space the pollination success changes greatly as predator effectiveness decreases, due to the steepness of the concave function at this point. Hence, reducing the effect of predators on pollinators removes the negative impact. Increasing δ increases the area of negative effect, by increasing the tendency for the predator to remove densities of pollinators that are highly effective, i.e. those in the first third of the density range. Increasing D max has little effect alone, but interacts with increases in δ. The interaction occurs because only when δ is greater than 0.5D max is the mortality of a given prey type influenced by the abundance of the other prey type. If δ is not greater than 0.5D max, the number of prey eaten is only affected by δ, and not D max. If δ is greater than 0.5D max then the density of pollinators impacts on the predator’s effect at intermediate antagonist densities (approximately 0.4 < G < 0.7; Fig. 3g). As P increases from zero the same number of granivores are killed while more and more pollinators are killed, so the net effect of the predator on the plant is negative. However, when P is high (P > 0.5), the impact of the predator is limited by D max, and so a greater proportion of both prey types survive as P increases. However, this effect is greater for pollinators when they are at a higher density, and so the net effect of the predator switches to positive.

Fig. 3a–i
figure 3

Overall effect of predators on host plant for the maximum number of prey that a predator consumes (D max) and the effect of number of prey on the number of prey consumed (δ) (a larger δ results in a greater proportion of the available prey being consumed). All other parameter values are shown in Table 1. Model evaluation and figure creation as in Fig. 2. The default values of D max and δ are used in e

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Unsurprisingly, the magnitude of the predator effects, both positive and negative, increase with both D max and δ because the predator kills more prey. These increases are similar for both positive and negative effects unless both D max and δ are high, whereupon negative effects are much larger because most of the pollinators are being killed.

Predator bias

Figure 4 shows how our predictions change if predators are biased in their effect on prey. Note that we set δ = 0.8, so that at high prey densities predators become satiated. Changing γ, and so making predators have a pre-existing bias towards one prey type, has minor and unsurprising effects on the predictions. That is, the boundary between negative and positive simply moves vertically so that predators have positive effects over a greater range of G when predators prefer granivores (Fig. 4b), and have negative effects over a greater range of G when predators prefer pollinators (Fig. 4h). As γ increases and so predators prefer pollinators more, negative effects become larger and positive effects become smaller. Decreasing λ, and so making predators preferentially attack the rarer prey, rotates the boundary so that the predator has a beneficial effect if P is large. This occurs because in the area at high P and low G most granivores are killed so making this area positive, whilst in the area at low P and high G most pollinators are killed so making this area negative. Increasing λ, and so making predators preferentially attack the more abundant prey, has the effect of rotating the boundary in the opposite direction, so that a positive effect of the predator is predicted mostly when G is greater than P. However, the boundary is not straight because when the predator becomes satiated at high values of P, pollinators benefit from this satiation more than granivores. The magnitude of all effects is much increased as λ increases, but because a positive λ means most pollinators are eaten when G is small the negative effects increase in magnitude more than the positive effects.

Fig. 4a–i
figure 4

Overall effect of predators on host plant for prey preference (γ) and density-dependent hunting (λ). When γ < 1 predators prefer granivores and when γ > 1 predators prefer pollinators. When λ < 0 predators prefer the less abundant prey type and when λ > 0 predators prefer the more abundant prey type. All other parameter values are shown in Table 1 except δ = 0.8 (so that predators were satiated at high prey densities). Model evaluation and figure creation as in Fig. 2. The default values of λ and γ are used in e

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Efficiency of prey

Increasing pollination efficiency π (Fig. 5) has minor effects on the areas of positive and negative effect at low pollinator densities, but increases the slope of the boundary at high pollinator densities, so that a positive effect occurs at lower granivore densities. This results from the fact that pollinators are now less important, since a small number is sufficient to provide high plant fitness. Decreasing π has remarkably minor effects, merely increasing the area of no effect, because P does not strongly affect plant fitness anyway. As granivore consumption ε increases, granivory becomes correlated with F resulting in the boundary “rotating” clockwise (cf. Fig. 3b, h). That is, at low P and intermediate G the effect of the predator changes from no effect to negative because granivores eat less when there are fewer seeds, so there is no benefit from removing granivores that would otherwise compensate for the loss of pollinators. At high P and low G the effect changes from negative to positive because granivores eat more at high F, so removing them has a greater benefit. Also, π and ε interact additively, so that when both are large the negative effect is restricted to an area where total prey density is not large. Increasing π tends to increase the magnitude of the positive effects and decrease the magnitude of the negative effects of the predator, because effective pollination compensates for pollinator loss. Increasing ε has very little impact on effect sizes, because overall consumption of seeds is largely unaffected.

Fig. 5a–i
figure 5

Overall effect of predators on host plant for pollinator efficiency (π) and granivore distribution (ε). As π increases the seed success per pollinator increases, and as ε increases granivores consume more seed where there is more pollination. All other parameter values are shown in Table 1. Model evaluation and figure creation as in Fig. 2. The default values of π and ε are used in b

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Discussion

In summary, the net effect of the flower-dwelling predator on the host plant can be positive, negative or neutral, and the direction of the effect is dependent on the relative densities of mutualists and antagonists. The flower-dwelling predator has a strong positive effect on the plant when both pollinators and granivores are at high density. When granivores are at high density, the pressure they exert on plant fitness is high and the presence of the predator has a substantial positive benefit on the fitness of the plant. However, this benefit comes at a cost of reduced pollinator availability. This cost is lowest for the plant when pollinator numbers are high and so the loss of some pollinators still allows high levels of successful pollination. Our work provides both a novel approach for studying top-down effects that are conditional on community structure, and a clear example of how the top-down effects of predators are dependent on all available prey.

Predictions of the model

In the absence of clear data in the literature we have assumed simple functional forms and plausible parameter values. This means that our results should be interpreted with caution, but also that our study should prompt further progress through empirical testing of our assumptions and predictions. Under our default parameter settings (Fig. 2d), we predict that the direction of the impact of the predator on plant fitness will only be influenced by the density of antagonists. This is firstly because the mortality of mutualists is independent of that of antagonists. As a result, increasing pollinator abundance does not affect the proportion of pollinated seeds that are eaten. Secondly, increasing P increases the number of pollinators that are killed, which means that any increase in the number of pollinators is not sufficient to significantly alter the reproductive success of the plant. The proportion of pollinated seeds that survive obviously decreases with increasing numbers of antagonists, and in the absence of a predator will lead eventually to zero seed success. When a predator is present the maximum proportion of pollinated seeds that will survive is 0.5 (driven by the value given to the prey capture parameter, δ), resulting in the predator having a negative impact on the plant. However the minimum proportion of seeds that will survive is one half (or δ) of those pollinated, since half (or δ) of the antagonists are killed. Hence, the decrease in seed success with increasing antagonists is one quarter of that in the absence of a predator, and at an antagonist density determined by the density dependence of prey effectiveness the predator becomes beneficial to the plant. This central result is amenable to empirical test by a three-way full factorial experiment with two densities of pollinators (e.g. using exclusion by netting) and two of granivores (e.g. by removal and addition to specific flowers), and the presence or absence of a predator such as a crab spider.

Since the direction of the net effect of predators on plant fitness only depends on antagonist density (at least in the absence of predator satiation or bias) we might expect that if plants can selectively attract predators, they should do so only when G is high. Furthermore, P should not influence plants’ attempts to attract predators. Whether plants are capable of responding to granivory is under explored, but they are known to respond to antagonists such as leaf herbivores that influence plant fitness earlier in the reproduction process (Marquis 1984; Mothershead and Marquis 2000). Often this response includes attracting enemies of their antagonists (Vet and Dicke 1992; Unsicker et al. 2009). The magnitude of the influence of the predator on plant fitness increases with P (Fig. 2), and it is only the boundary of the region of no net effect that is insensitive to P. This raises the question: will plants evolve responses only to negative effects, or to increase the positive effect of the predator? Since the magnitude of both positive and negative effects increases with the effectiveness of the predator, then the accuracy of the plant at identifying the times predator attraction will be beneficial becomes more critical. Hence, we might expect greater adaptations in plants that harbour more voracious predators, which might be tested by comparative analysis of a plant taxon that commonly hosts predators.

It is easy to understand why the range of density combinations that yield a positive net outcome for the plant increases when the performance of pollinators is negatively density dependent (low α), such that the loss of a fraction of pollinators has a less than proportionate cost to the plant. It seems that this should only be true at high pollinator densities: at intermediate pollinator abundance there is no difference between different α values in the incremental importance of pollinators (the slope of P on F is always 0.5 when P = 0.5). However, seed production is higher when pollination is negatively density dependent than when density independent, and so the decrease in seed survival decreases with antagonist abundance at a higher rate (since there is more to eat). However, the predator reduces seed production by a proportionally smaller amount when there are fewer pollinators, while still reducing G by the same proportion. Hence, the range of G for which a predator is beneficial increases as α decreases.

The impact on plant fitness of the density dependence of the effect of antagonists is less dramatic. The range of density combinations that yields a positive outcome for the plant is increased when the per capita effectiveness of granivores is negatively density dependent (low β). The point at which the predator is beneficial is at lower G because the impact of the granivores is proportionally greater, so fewer are required to make removal by the predator beneficial. This reduction in the range of G where the change to the predator being beneficial occurs is smaller when we make granivory negatively density dependent than when we make pollination negatively density dependent. This is because the proportionally greater impact of granivores at low abundance is partly off-set by the fact that negative density dependence makes granivores in the altered area less effective. Thus, predation on granivores has a diminished effect on net fitness. It is precisely because pollinator abundance does not affect the direction of the net effect that the density dependence of pollinators has more effect on the results. Our assumption is that pollination is negatively density dependent but that antagonism is density independent, and clearly these assumptions are important to test empirically. The former assumption could be tested by excluding pollinators by netting flowers for various proportions of the day and measuring the rate of pollination.

Effects of predator behaviour

Increasing D max and δ can both be interpreted as increasing the effectiveness of the predator. When both of these values are low, then the predator has a low impact on both prey types and so it is no surprise that for a wide range of prey density combinations, a neutral effect of the predator is predicted. As both parameter values increase, and the impact of the predator on prey density increases, so the range of prey density combinations that yield a positive benefit for the plant declines. This is because we assume that pollination is negatively density dependent, and so increasing the impact of predators means that predators kill pollinators that have a high per capita effect on seed set. When δ is high and D max is not high the abundance of one prey type affects the survival of the other, because predators kill more of the abundant prey. Thus, only when the predator is able to kill most of the prey that it encounters, but is limited as to how many prey it attacks, do we predict that P will partially determine the net effect. When predators have a low success rate or very high satiation threshold, P will not determine the net effect.

In many systems the occurrence of the pollinator and the granivore may be temporally separated, since pollinators will stop visiting flowers when all seeds have been set. Hence, if predators become satiated they may consume proportionally more pollinators. Conversely, if predators grow in size they could develop larger appetites, and so consume proportionally more granivores: thus predator growth rate relative to the length of time plants are in flower could affect the impact of predators on plant reproductive success. Furthermore, if predators have to learn to handle certain prey, we would expect predation on these prey types to be positively density dependent. Under these conditions, the presence of the predator would dampen the benefit of high pollinator visitation rates, and decrease the magnitude of the positive effects. At very high P, further pollinators might only reduce the predation of plant antagonists. This leads to the prediction that if plants are very likely to harbour predators that show positive density-dependent prey choice, plant fitness will be maximised at an intermediate P. This could be studied by excluding pollinators from flowers for a range of periods during the day and measuring seed success after granivory. Also, the assumption that predators reduce visitation to a greater extent for the more abundant prey type is amenable to test by adding and removing predators from flowers with differing visitor communities.

Effects of pollinator behaviour

Increasing the effectiveness of the prey has some interesting effects on how the presence of the predator influences plant reproductive success. Increasing π increases the effectiveness of pollinators and so allows both a net benefit to the plant under a wider range of circumstances and greater magnitude of the positive effects when they occur. Increasing ε means that seeds experience some protection from granivores when seeds are relatively uncommon (when pollination has not been successful), and this protection again allows a small increase in the range of circumstances over which a plant benefits from the presence of the predator. We might expect that ε would differ between mobile granivores that should be ideally distributed and larval granivores that are placed as eggs by a parent in highly favourable locations, and so the predator effects on seed success should differ between granivore life stages.

Since the effect of the predator will vary with the effectiveness of pollinators we might expect to find different effects when pollinators are specialists compared to when they are generalists, and/or when a plant and its pollinator have a close evolutionary relationship. This is because specialised pollinators are likely to be more effective pollinators. Many pollinators, especially social bees, show specificity to a particular plant species or even flower patch, a phenomenon termed ‘flower constancy’ (Darwin 1876; Waser 1986; Chittka et al. 1999; Raine and Chittka 2005). Such pollinators will only carry compatible pollen, and so are also likely to be effective pollinators (Chittka et al. 1999; Goulson 2003). When pollinators are highly effective, we would predict that predators will only have negative effects on plant reproductive success when the number of granivores is low and the number of pollinators is not high. We would also expect the positive effects of predators to be larger than the negative effects. As a result, flower-constant pollinators will be associated with large beneficial effects of predators on host plants under most conditions. Furthermore, when plants are at high abundance pollinators will be more likely to carry the pollen of the plant’s conspecifics, and predators will be more likely to have beneficial effects on plants.

Our model has been couched in terms of a direct effect of predators on their prey. However, predators can have profound influences on the behaviour of their prey in addition to the effect of consumption. Such modifications of prey behaviour have been demonstrated for flower-dwelling predators (Dukas 2001; Reader et al. 2006; Ings and Chittka 2008, 2009) and these behavioural changes have recently been demonstrated to feed through to fitness benefits for the plant (Goncalves-Souza et al. 2008). Such effects may be at the patch level and so affect whole plants or groups of plants (Dukas and Morse 2003; Dukas 2005; Dukas et al. 2005), but it is also possible that pollinators avoid specific flowers (Dukas 2001; Dukas and Morse 2003), especially if they can detect and avoid predators (Dukas 2001; Robertson and Maguire 2005; Reader et al. 2006; Ings and Chittka 2008, 2009). Furthermore, honeybees communicate the presence of predators to conspecifics (Abbott and Dukas 2009), so might be even more deterred by predators. Although couched in terms of direct effects for presentational simplicity, our general model structure (and thus our predictions) would be equally applicable to situations where the effect of the predator on the prey was caused by direct consumption, indirect behaviour modification, or a combination of the two. The most likely influence of enhanced behavioural modification of the prey might be felt in the parameter reflecting the maximum number of prey adversely affected by the predator (D max), this constraint is likely to be of less importance (that is D max would increase) for behavioural modification rather than direct consumption. Thus as a general prediction, we would expect that direct consumption might be expected to more readily provide a net positive outcome for the plant than behavioural modification. A large number of potential pollinators will be able to overcome the direct cost of the predator by simple dilution of risk, but the costs to the plant associated with behavioural modification may be much more substantial if the risk of predation drives a very large number of potential pollinators to switch to alternative plant species that are perceived to present lower risk. Our model could therefore be tested by manipulating pollination services (e.g. by placing honeybee colonies near flowers) and assessing the fitness of flowers (with and without spiders) that are visited by different pollinator communities.

Patch specificity is likely to have further effects when pollinators avoid patches containing high densities of predators. Here, the presence of a predator might have negative effects on the reproductive success of whole plants or even groups of plants. Interestingly, the tendency for predators to deter more prey when prey availability is higher (high δ), has a greater impact on plant fitness than increasing D max. Thus, predators should have negative impacts on plant fitness especially when they are able to move to areas of higher prey density, or if predation is more successful at high density prey. For example, such predators may serve to reduce the benefits of large flower size. These effects may be somewhat reduced by the specialisation of pollinators, but it is clear that this interaction is likely to be complex and worthy of further investigation.

Implications for plant fitness

Studies of the interactions between plants, flower-dwelling predators and their prey have tended to neglect the impacts on plants. However, several studies have demonstrated fitness consequences for the plant of the presence of the predator (Suttle 2003; Muñoz and Arroyo 2004; Romero and Vasconcellos-Neto 2004; Whitney 2004; Dukas 2005; Knight et al. 2006; Goncalves-Souza et al. 2008). Our model predicts that such benefits should occur in some delineated circumstances whereas net costs are predicted to occur in others. These predictions await empirical investigation. If plants of given species find themselves consistently in conditions where they might benefit from the presence of a predator then we might expect the plants to develop traits that increase the likelihood of predator recruitment. Whilst the recruitment of the predators and parasitoids of herbivores by plant-released volatiles have been extensively studied (Vet and Dicke 1992; Unsicker et al. 2009), the potential for luring flower-dwelling predators seems to have been ignored. Whilst flowers may be under stronger selection to attract pollinators than their potential predators, this hypothesis has not been tested empirically. Such a test might involve comparisons of male and female flowers because male flowers do not set seed and so will not be impacted upon by the presence of granivores, so only the rate of pollinator visitation impacts on fitness. Thus, if plants can influence predator behaviour and when most antagonists are granivores there could be differences between male and female flowers in the tendency to harbour predators.

In order to ensure the general applicability of our model, we have assumed that antagonists reduce the seed success proportionally; we assumed a simple multiplicative function. Granivory could be thought of as a special case of antagonism, because pollination must occur before granivores can have an impact. Most plant antagonists, such as herbivores, nectar robbers and florivores, can inflict a fitness cost of plants irrespective of whether pollination occurs. Thus there is an intrinsic link between granivores and pollinators that is weaker for other types of antagonists, and it would be interesting to test whether the effects on plant fitness vary among types of antagonist. Furthermore, there may be non-predator-related interactions between mutualists and some antagonists, since nectar robbers and pollinators often compete for the same resource, and florivores similarly decrease the attractiveness of flowers to pollinators. It is clear that in such complex situations the qualitative and quantitative effects of the predator on the plant could be influenced by the timing and abundance of all the species in the community. The effect of flower-dwelling predators on these interactions and the net outcome for plant fitness are very much worthy of further theoretical and empirical study.