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Supporting information:
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Interaction intensity and strength of selection
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In this section we first provide a formal description of the relationship between mean
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pollination intensity, in terms of the proportion of flowers pollinated, and pollinator-mediated
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selection for single-flowered plants and discrete variation in attractiveness. We then continue
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to provide relationships also for plants with more than one flower, for antagonistic
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interactions, and for a continuously varying attractiveness trait. When interactions with
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pollinators are entirely absent (pollination intensity = 0) and when seed predators consume all
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seeds (predation intensity = 1), then there is no selection. However, as soon as there are some
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pollinator visits (pollination intensity > 0) and some seeds escape predation (predation
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intensity < 1), strength of selection can attain high values. Below, we examine the relationship
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between interaction intensities and selection strength for pollinator intensities above zero and
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predation intensities below one.
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Dimorphic attractiveness trait, pollination, and single-flowered plants
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Consider first a scenario with no resource limitation and no seed predation, and a population
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consisting of single-flowered plants, which is dimorphic for attractiveness to pollinators and
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antagonists, and which has a morph ratio of 1:1. Mean intensity of pollination (meanpoll) and
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the pollinator-mediated selection coefficient (pollcoef) are defined as:
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meanpoll = (FruitsA + FruitsUA) / 2
(1)
pollcoef = 1 – FruitsUA / FruitsA
(2)
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where FruitsA and FruitsUA are the mean number of initiated fruits (i.e., pollinated flowers) in
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the attractive and the unattractive morph, respectively. The increase in the proportion of
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pollinated flowers with increasing number of pollinator visits to the population can then be
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described by a saturating function. The increase occurs at a slower rate for the unattractive
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morph (the relative rates depending on the relative attractiveness to pollinators of the
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unattractive morph, pref). The proportions of flowers initiating fruit development, in the
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attractive and unattractive morph, respectively, increase according to:
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FruitsA = 1 – (1 – pA) i
(3)
FruitsUA = 1 – (1 – pA × pref) i
(4)
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where i is the number of pollinator visits to the population, and pA is the probability of
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pollination for the attractive morph per pollinator visit. Substituting these functions into the
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functions for mean intensity of pollination and selection yields:
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meanpoll = (2 – (1 – pA) i – (1 – pA × pref) i ) / 2
(5)
pollcoef = 1 – (1 – (1 – pA × pref) i ) / (1 – (1 – pA) i )
(6)
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Solving these functions numerically for different values of i results in a non-linear
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relationship between pollinator-mediated selection and mean intensity of the interaction, in
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terms of the proportion of flowers pollinated (solid lines in Fig. 1 A-C).
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Many-flowered plants
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Plants with several flowers will have only unpollinated flowers in the absence of pollinators,
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but with increasing number of pollinator visits to the population individuals in each trait
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category will be distributed among a number of classes representing different numbers of
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pollinated and unpollinated flowers. With increasing number of pollinator visits, each class
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will receive plants from the class with more unpollinated flowers and lose plants to the class
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with fewer unpollinated flowers:
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𝑖
PropAk = ( ) × pA k × (1 – pA) i-k
𝑘
(7)
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where PropAk is the proportion of attractive plants in the class with k pollinated and n – k
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unpollinated flowers for i > k (for i ≤ k, PropAk = 0), i is the number of pollinator visits to the
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population, and pA is the probability of pollination for the attractive morph per pollinator
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visit. However, from the class with n = k, i.e., with all flowers pollinated, no plants can be lost
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to a class with fewer unpollinated flowers (the saturation effect) and this class can be
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calculated by subtracting 1 by the sum of PropAk for all k < n. A similar function can be used
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for the unattractive morph (but multiplying pA with pref):
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𝑖
PropUAk = ( ) × (pA × pref) k × (1 – pA × pref) i-k
𝑘
(8)
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For each i it is now possible to calculate the mean number of initiated fruits in each morph,
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and hence a selection coefficient and a mean interaction intensity. This will again result in a
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non-linear relationship (dashed lines Fig. 1).
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Seed predation
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In a scenario with full fruit initiation, a dimorphic attractiveness trait, plants with a single
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flower, and an initial morph ratio of 1:1, mean intensity of seed predation (meanpred) and the
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seed predator mediated selection coefficient (predcoef) are defined as:
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meanpred = 1 – (FruitsA + FruitsUA) / 2
(9)
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predcoef = 1 – FruitsA / FruitsUA
(10)
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The proportion of unpreyed plants is reduced by a negative exponential function of the
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number of seed predator visits in the population, but at a slower rate for the less attractive
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morph than for the more attractive morph (the difference in rates depending on the relative
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attractiveness to seed predators, pref):
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FruitsA = (1 – pA) i
(11)
FruitsUA = (1 – pA × pref) i
(12)
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where pA is the probability of predation for the attractive morph per seed predator visit, and i
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is the number of predator visits to the population.
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Substituting the function of fruits depending on the number of predator visits into the
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functions for mean intensity and selection yields:
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meanpred = 1 – ((1 – pA) i + (1 – pA × pref) i ) / 2
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(13)
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predcoef = 1 – (1 – pA) i / (1 – pA × pref) i
(14)
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Solving these functions numerically for different values of i results in a non-linear
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relationship between seed predator-mediated selection and mean intensity of the interaction,
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in terms of the proportion of flowers preyed (solid lines in Fig. 1). We use – predcoef to
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reflect that seed predation select for unattractiveness rather than attractiveness. Note that for
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meanpred = 1 no plants have any fitness, and selection is absent.
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Given that flowers and fruits are attacked one by one, and that attractiveness is not affected by
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previous attacks, depletion occurs faster for few-fruited than for many-fruited plants. With
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increasing number of seed predator visits, each class will receive plants from the class with
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more intact fruits and lose plants to the class with fewer intact fruits.
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𝑖
PropAk = ( ) × pA k × (1 – pA) i-k
𝑘
(15)
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where PropAk is the proportion of attractive plants in the class with k preyed and n – k intact
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fruits for i > k (for i ≤ k, PropAk = 0) and pA is the probability of predation for the attractive
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morph per seed predator visit. However, from the class where n = k, i.e., with all fruits preyed,
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no plants can be lost to a class with fewer intact fruits (the depletion effect) and this class can
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be calculated by subtracting 1 by the sum of PropAk for all k < n. A similar function can be
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used for the unattractive morph (but multiplying pA with pref):
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𝑖
PropUAk = ( ) × (pA × pref) k × (1 – pA × pref) i-k
𝑘
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(16)
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For each i it is now possible to calculate the mean number of intact fruits in each morph, and
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hence a selection coefficient and a mean interaction intensity. This will also result in a non-
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linear relationship (dashed lines Fig. 1).
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Continuous attractiveness trait
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For a continuous normally distributed trait, the predictions are more complicated. Mean
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intensity of pollination (meanpoll) and the pollinator-mediated selection gradient (pollgrad)
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are defined as:
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meanpoll = sum of initiated fruits / sum of flowers in a population
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pollgrad = the regression coefficient for relative fitness regressed on standardized trait values
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Mean intensity of seed predation (meanpred) and the seed-predator-mediated selection
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gradient (predgrad) are defined as:
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meanpred = 1 – sum of intact fruits / sum of fruits in a population
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predgrad = the regression coefficient for relative fitness regressed on standardized trait values
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For single-flowered plants, a negative exponential function describes how both the
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distribution (d) of unpollinated flowers in the pollinator scenario and the distribution (d) of
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intact, unpreyed fruits in the seed predator scenario decrease with the number of pollinator or
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seed predator visits (i), but with a slower speed for the less attractive plants in the distribution:
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d = d0 × (1 – p) i
(17)
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where d0 is the initial (normal) trait distribution in the population and p is a linear function of
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trait attractiveness ranging from 0 to 1. For each i the distributions of d (unpollinated flowers
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or intact, unpreyed fruits) and 1 – d (pollinated flowers or preyed fruits, respectively),
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selection gradients and mean interaction intensities can be calculated.
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Given that flowers and fruits are pollinated or attacked one by one, and that attractiveness is
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not affected by such events, saturation and depletion are slower for many-flowered plants.
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Assuming that all plants produce the same number of flowers, interaction intensity and
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selection strength will depend on separate functions for each class with a specific number of
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unpollinated flowers or intact, unpreyed fruits, respectively. With increasing number of
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pollinator or predator visits, each class will receive plants from the class with more
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unpollinated flowers or more intact, unpreyed fruits and lose plants to the class with fewer
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unpollinated flowers or fewer intact, unpreyed fruits according to
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𝑖
dk = ( ) × d0 × p k × (1 – p) i-k
𝑘
(18)
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where dk is the distribution of the class with k pollinated and n – k unpollinated flowers or
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preyed and unpreyed fruits, respectively, for i > k (for i ≤ k, dk = 0). However, from the class
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where n = k, i.e., with all flowers pollinated or all fruits preyed, no plants can be lost to a class
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with fewer unpollinated flowers or fewer intact, unpreyed fruits (the saturation and depletion
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effect). This class can be calculated by subtracting the other dk distributions from d0. For each
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i it is now possible to calculate distributions for plants with each number of intact fruits, and
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hence a selection gradient and a mean interaction intensity.
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Pollinators and seed predators selecting on the same trait
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If both pollinators and seed predators are attracted to the same trait, but the interactions occur
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in sequence (a fruit has to be initiated before it can be attacked), the above models can first be
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used to estimate a pollinator-produced distribution of plants with fruits, dpoll. This dpoll can
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then be used as d0 for a model of seed predator mediated selection (Fig. 2).
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