Z theor BioL (1986) 118, 253-258

The Evolution of Empty Flowers

GRAHAM BELL

Biology Department, McGill University, 1205 Avenue Dr. Penfield, Montreal,

Quebec, Canada H3A 1B1
(Received 3 January 1985)
This paper describes a simple model intended as a first step towards a quantitative
theory of the flower. It divides both flowers and their visitors into two categories:
the flowers may either produce a substantial volume of nectar ("secretors") or none
("cheaters"), while the insects either attempt to discriminate between them and
enter only secretors ("selectors") or enter any flower encountered ("neglectors").
If the insects can learn the state of the plant population and behave accordingly,
then they should all either select or all neglect exclusively; the plants adopt a mixed
strategy with each plant producing a similar proportion of cheating flowers. The
proportion increases as the time required by the selectors to discriminate between
cheaters and secretors increases relative to the time required to extract the nectar.
The argument is of general interest because it represents a new theory of nectar
concealment. Cheating is known to occur in some plants, but it is not known how
widespread the strategy is.

The main function of flowers is to attract the insects which import and export pollen
grains. The way in which the architecture of the flower permits or enforces the
correct reception of the pollen onto the stigma and its deposition from the anthers
has been the subject of a large literature over the last century (see Knuth (1906)
for a wealth o f detail), especially in relation to the means for securing cross-
fertilization. Despite the volume of this descriptive work, however, there is virtually
no quantitative theory of pollination from the plant's point of view. An increased
number of insect visits must generally result in a greater proportion of the ovules
being fertilized, and more particularly in a greater proportion of the pollen being
dispersed. At the same time, further increases in the number of visits must sooner
or later yield diminishing returns: if ten insect visits are sufficient to remove
three-quarters of the pollen or to fertilize three-quarters of the ovules then twice as
m a n y visits cannot double the benefit to the plant. Plants should therefore evolve
so as to optimize their expenditure on flowers, by maximizing the number of
fertilizations obtained per unit of secondary allocation to floral attractants and
rewards.
The purpose of this note is not to develop a general quantitative theory of flowers,
but rather to describe one special case, in which a dichotomy in floral allocation
seems to arise naturally from the interaction between flowers and their visitors, and
whose validity can be investigated by simple measurements of flowers in natural
populations.
Suppose that some plants ("secretors") have expensive nectar-bearing flowers
whilst others ("cheaters") have empty flowers which are cheaper to make. The
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254 o. B E L L

insects which visit these flowers are also of two kinds: "selectors", which visit only
secretors but can distinguish them from cheaters only at the cost of an increase in
the length o f time needed for each visit, and "neglectors", which enter all the flowers
they encounter. In their simplest form, these rules describe a cyclical game (Fig. 1)

PLANT
Cheoter Secrelor

Selector

I NSECT

Neglecter

FIG. 1. Cheating as a strategy which leads to a cyclical game. The payoff to the plant is indicated
before the diagonal in each cell and the payoff to the insect after: " + " indicates that relevant strategy
is better than the alternative a n d " ' - " that it is worse, when the populations of plants a n d insects are
nearly fixed for the two relevant pure strategies.

analogous to Dawkins' (1976) "battle of the sexes". Suppose that the plant popula-
tion at some point in time contains a large proportion of cheaters, while the insects
are mostly selectors. The selector strategy benefits the insects, since they can avoid
the many cheaters, but most of the plants are never visited. The plants should
therefore switch to a secretor strategy; they then receive more visits, but the insects
are now paying an unnecessary cost in terms of the time required to discriminate
between cheaters and secretors, which lowers their net rate of energy acquisition.
The insects should then become neglectors, lowering the payoff to the plants from
expensive secretor flowers and so favouring cheaters. When the cheaters again
prevail in the plant population the insects do better by being selectors, and the cycle
is complete.
We can easily formalize this game. Let the average number of visits received by
a flower be V; a visit may or may not result in pollination, depending on whether
or not the insect enters the flower--neglectors always enter, but selectors enter only
secretors. Each entry results in the removal o f a quantity G of pollen. In practice
the quantity of pollen removed may be a decreasing function of the quantity
remaining from previous visits, and visits serve to fertilize ovules as well as to
remove pollen. So long as the number o f fertilizations obtained (through pollen or
ovules) is an increasing function of the number of visits, however, the argument
given here is qualitatively valid. Each secretor flower produces a quantity N of
nectar, which is related to the total cost of the flower by a constant k: a cheater
costs kN units of resource to make, while the cost of a secretor is (1 + k)N. Among
the insects, some variables take the same value for neglectors and selectors: the rate
of encounter with flowers is h, the handling time per flower is H, and the nectar
reward from secretors is R. The selectors, however, spend a certain length of time
D in ascertaining whether each flower encountered is a cheater or a secretor.
 E V O L U T I O N OF E M P T Y F L O W E R S 255
When each plant produces only one kind of flower, the payoff from each pure
strategy is
cheater: V(1 - p ) G / k N
secretor: VG/(I+k)N
where p is the frequency of selectors among the insects, and the payoff is in terms
of the total number of pollen grains exported per plant. If each insect likewise plays
only one of the two pure strategies, the payoffs are
selector: (1-q)R/[A-~+H(1-q)+D]
neglector: (1-q)R/[A-~ + H]
where q is the frequency of cheaters among the plants, and the payoff is in terms
of the quantity of energy harvested per unit time; the form of the expressions is
standard in the theory of optimal diets (see Charnov, 1972).
These four payoffs describe a cyclical game. If the plant and insect strategies are
both determined by alleles at a single haploid locus, then it is easy to show that
gene frequencies in both populations cycle through time, usually with long period
and large amplitude, as the result of the time-lagged frequency-dependence built
into the model. However, such a situation is quite unrealistic. In the first place, if
plants play a pure strategy then the insects could quickly learn to leave a plant after
encountering a single empty flower. Secondly, it is assumed that plants and insects
play alternate moves; but in practice insects can quickly learn to respond appropri-
ately to the reward schedule offered by the plants (see Heinrich, 1978), whereas
only a genetic response is available to the plants. A more realistic model, therefore,
would incorporate a mixed strategy for the plants and learning by the insects.
To calculate the payoff for a plant using strategy (q), suppose that it possesses
a total of S units of resource, and produces C cheating flowers. It can then produce
( S - CkN)/(1 + k)N secreting flowers, so that the total payoff to the plant is
E(q) = CV(1 - p ) G + V G ( S - CkN)/(1 + k)N.
Since the frequency of cheating flowers on the plant is by definition q = C/[C+
( S - CkN)/(1 + k)N] = (1 + kN)/(1 + S~ C), we have C = qS/(1 + k - q)N to sub-
stitute into the expression for E(q), yielding
E(q) = constant x (1 -pq)/(1 + k - q).
The payoff to an insect playing strategy (p) is
F(p) = p(1 - q)R/[A-~+ H(1 - q) + D] + (1 -p)(1 - q)R/(A -~ + H),
so that aF(p)/Op < 0 if q > D/H. Since the insects can learn very quickly relative
to the generation time of the plants, the strategy used by the insects should be
p=O ifq<D/H;
p=l ifq>D/H.
I have simulated this interaction by allowing each of the 101 equidistant phenotypes
256 G. BELL
in q ~ (0, 1) to be determined by a different allele at a haploid locus. Both plant
and insect populations change through time, the insect population flipping from
p = 0 to p = 1 or vice versa according to the average state of the plant population,
while the average state of the plants responds in a continuous fashion to the strategy
of the insects. However, the fluctuations of q quickly settle down to an irregular
oscillation of very small amplitude centred around q = D / H . Moreover, extreme
phenotypes (q very small or very large) are rapidly eliminated, and the plant
population eventually comes to comprise only a narrow range of phenotypes close
to D / H . In similar models with diploid genetics, or in which the insects adopt an
indeterminate mixed strategy when q is very close to D / H , plant strategy is even
more stable.
The major conclusion from this argument is that we expect plants to adopt a
mixed strategy q ~ D / H , with all individuals producing a similar substantial fraction
of cheap flowers virtually devoid of nectar if the time required by the insects to
discriminate between cheating and secreting flowers is substantial relative to the
time required to extract the nectar from a secretor. It is predicted, therefore, that
plants whose nectar is deeply concealed within the flower will often bear cheating
flowers; and that the frequency of cheats will rise as the discrimination time increases
relative to the handling time.
The standing crop of nectar in flowers is often measured during foraging studies,
and in many cases a standard deviation is cited, though the frequency distribution
of nectar volume is rarely given. If cheating flowers are common, we expect that
many flowers will contain an abundance of nectar whilst others bear none. Unfortu-
nately, such standing-crop measurements cannot be used to test the theory since a
similar pattern will be created by the activity of insects even in the absence of
cheaters (Brink, 1982; Pleasants & Zimmerman, 1983). If insects visit flowers at
random then the arrival of bees at flowers is a Poisson process, and for any given
flower the waiting time until the next visit will be exponentially distributed. Since
nectar volume usually increases more or less linearly after depletion (Cruden et al.,
1983), the standing crop of nectar will also become exponentially distributed once
insect visits have got under way. The frequency distribution of standing crops will
therefore have a mode at very small nectar volumes, even if all flowers secrete nectar.
This simple hypothesis is readily tested using the large data set supplied by Cruden
et al. (1983), since for an exponential distribution the standard deviation is equal
to the mean. It turns out to be false (Fig. 2), but the variance of the standing crop
is actually less than that predicted on the basis of random visits to flowers. I suggest
that this effect may be due to a preference of visitors for flowers whose nectar
volume is great enough to yield a rate of reward equal to or exceeding that of the
average of its neighbours, as predicted by the marginal value theorem of Charnov
(1976); but at all events the measurement of standing crop gives no support to the
argument developed here, and perhaps cannot supply a decisive test of the existence
of cheating flowers.
The crucial measurement is therefore the frequency distribution of nectar produc-
tion rates, got by measuring nectar volume in bagged flowers. The only such data
I have found in the literature are the frequency distributions given by Feinsinger
 EVOLUTION OF EMPTY FLOWERS 257
/
/ /o
/
/
2
g // ¢
,/
E'/ %,e
FF//~ o%
o a o

c
o
~ 0
/ =
_.J
//l •
/
-1

I I I I
-L2 -1 o 1 z 3
Log meon volume
FXG. 2. The relationship between the mean and the standard deviation of nectar standing crop. Plotted
points are data from Cnsden et aL (]983), different symbols indicating different pollinators: • bat; O
hawkmoth; A hummingbird; A sunhird; [ ] "oriole-starling" guild; bees, butterflies and small moths.
The broken line has unit slope and represents the expected outcome of an exponential probability
distribution of nectar volume. The "F" symbols are Feinsinger's data for bagged flowers of five tropical
plants, included on the diagram for comparison with the standing crop data.

(1978) for five tropical plants visited by hummingbirds. In all five cases his figures
show a large proportion of the flowers producing little or no nectar, whilst a few
produce very large quantities. Feinsinger interprets this pattern as a device which
forces the hummingbirds to visit more flowers in order to obtain an adequate diet,
and the model developed in this p a p e r can be taken as an extension and a confirma-
tion of his reasoning.
Feinsinger's data establishes the existence of cheating flowers, but their generality
remains unknown, and other data are less conclusive. Brink & deWet (1980) found
that nectar production was very variable within and between populations of a species
of Aconitum, and some of their samples included substantial numbers of non-
secreting flowers. However, this variance was largely accounted for by differences
in flower age within populations and by differences in mean nectary depth between
populations; moreover, populations with more deeply concealed nectar had lower
frequencies of empty flowers, the reverse o f the pattern predicted by the model.
I f the nectar is so effectively concealed that discrimination time exceeds handling
time, then plants should produce cheating flowers exclusively. The argument then
becomes too complex for a simple model to represent, since the pollinators could
learn to avoid such plants entirely and forage on a different species. A pure cheating
strategy can be sustained only if the time required to discriminate between species
is prohibitively long. I suggest that this extension of the basic model provides an
interpretation of cases such as Cephalanthera rubra, an orchid whose large and
conspicuous flowers secrete no nectar but closely resemble those of nectariferous
Campanula (Nilsson, 1983), or the similar relationship between Calopogon pulchellus
and Pogonia ophioglossoides (Krebs & Davies, 1981).
The traditional interpretation of nectar concealment has been that it enforces
pollinator constancy and therefore the correct reception and deposition of pollen
258 G. BELL

in m o r p h o l o g i c a l l y c o m p l e x flowers, an i d e a d a t i n g b a c k at least to M u l l e r (1883).

A n a r r o w r a n g e o f p o l l i n a t o r s p e c i e s is c e r t a i n l y a c o n s e q u e n c e o f n e c t a r c o n c e a l -
ment, a n d l e a d s in t u r n to t h e e v o l u t i o n o f c o m p l e x floral a r c h i t e c t u r e . T h e h y p o t h e s i s
I have d e v e l o p e d a b o v e , h o w e v e r , suggests t h a t n e c t a r c o n c e a l m e n t is m o r e f u n d a -
m e n t a l l y a d e v i c e w h i c h p e r m i t s e c o n o m y in a l l o c a t i o n to flowers b y facilitating
the e v o l u t i o n o f cheating. F u r t h e r studies o f the v a r i a b i l i t y o f n e c t a r p r o d u c t i o n in
b a g g e d flowers w o u l d b e o f g r e a t interest.

This work was supported in part by an Operating Grant from the National Science and
Engineering Research Council of Canada.

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