Behavioral Ecology Vol. 13 No. 3: 321-327
© 2002 International Society for Behavioral Ecology
A game theoretical approach to conspecific brood parasitism
a Centre for Statistics and Stochastic Modelling, School of Mathematical Sciences, University of Sussex, Brighton, UK b Division of Environmental and Evolutionary Biology, Institute of Biomedical and Life Sciences, University of Glasgow, Glasgow, UK
Address correspondence to G.D. Ruxton, Division of Environmental and Evolutionary Biology, Institute of Biomedical and Life Sciences, Graham Kerr Building, University of Glasgow, University Avenue, Glasgow G12 8QQ, UK. E-mail: g.ruxton{at}bio.gla.ac.uk . M. Broom is also a member of the Centre for the Study of Evolution at the University of Sussex.
Received 3 October 2000; revised 21 April 2001; accepted 2 July 2001.
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ABSTRACT |
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We constructed a game theoretical model to predict optimal patterns of egg laying in systems where individuals lay in the nests of others as well as in their own nests. We show that decreasing the effect of position within an egg-laying sequence on the worth of an egg should lead to reduced parasitism. Indeed, parasitism can only flourish if the worth of an egg to its biological parent declines with the total number of eggs laid in that nest. Further, we found that increasing the intrinsic costs of egg production should lead to an increased propensity for conspecific brood parasitism. The model also predicts that variation in hosts' ability to reject parasitic eggs has little effect on parasitism until this ability is well developed.
Key words: conspecific brood parasitism, egg dumping, host—parasite systems, intraspecific parasitism, parental care.
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONREFERENCES |
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Yom-Tov (1980
) defines
conspecific brood parasitism as the laying of eggs in the nest of another
individual of the same species without taking part in the subsequent processes
of incubation and caring for the hatchlings. At the latest count
(Eadie et al., 1998), such
behavior had been recorded in 185 species. The recent advent of molecular
techniques such as DNA fingerprinting has greatly aided field study of this
behavior. This may explain why, unusually for behavioral ecology, empirical
study greatly dominates theoretical underpinning of this subject. Hence we
begin to redress this imbalance by using a game theoretical approach to
explain the observation of parasitism by individuals that also raise a brood
themselves.
Davies (2000) described
three different kinds of conspecific brood parasitism. The first type involves
individuals that attempt to nest normally but whose nest is destroyed, say, by
weather or predators. If such a female has begun the process of egg laying,
then she may not have time to rebuild the nest, and she may turn to parasitism
to "make the best of a bad job." In the second type, some
individuals make no attempt to nest themselves but instead choose pure
parasitism. Clearly, the success of such a strategy depends on the number of
individuals adopting it. The more parasites there are, the more competition
there is between them for fewer nests. Such a situation can best be understood
using game theory, as applied to this problem by Andersson
(1984) and Eadie and Fryxell
(1992). The final type of brood
parasitism occurs when parasitic individuals build nests that are not
destroyed and lay eggs in their own nests, but also lay some of their eggs
parasitically. This is the type that concerns us here, and it was first
considered theoretically by Lyon
(1998).
Lyon (1998) argued that the
worth of an egg to its parent can be thought of in terms of a "fitness
increment," defined as survival of offspring from that egg minus the
costs to producing it and any negative impact that the egg or its hatchling
has on the survival of siblings because of competition for limited parental
care. This constraint on the investment that parents can make means that every
egg laid in the home nest yields a lower fitness increment than the last. In
the absence of the option to parasitize, the optimal number of eggs to lay is
n, where the n + 1 egg would be the first to yield a
negative fitness increment. However, if the average fitness increment that a
parent can obtain from a parasitic egg is some positive value (P),
then the optimal number of eggs to lay in an individual's own nest changes.
Now an individual should lay n1 eggs in its own nest, when
the n1 + 1 egg is the first to provide a fitness increment
below P, all subsequent eggs should be laid parasitically. Lyon's key
prediction was that n1 would be less than n; in
other words, the opportunity to parasitize would force a reduction in the
optimal clutch size laid in an individual's own nest.
One important simplification in this argument is the assumption that the benefit gained from a parasitic egg (P in Lyon's model) is a constant. In practice, the worth of a parasitic egg will depend on both the number of eggs that individuals lay in their own nest and the amount of parasitism. However, both of these will be influenced by the worth of parasitism. To cope with this interdependence, a game theoretical model is required. The aim of this study was to develop such a model. This model should predict the optimum numbers of eggs laid in an individual's own nest and laid parasitically and predict how these numbers are influenced by ecological variables, such as the costs of egg production and strength of competition between nest mates.
Model assumptions
First, we assume that the worth (as defined by
Lyon, 1998) to the genetic
mother of the ith egg laid in a nest that has an eventual clutch size
of T is given by
 | (1) |
< ß < 1. The biological
basis for this assumption, and the meaning of the parameters, can be
understood as follows. Each egg laid is less valuable than the last; indeed,
its worth is always a constant fraction, ß, of that of the preceding egg:
 | (2) |
 | (3) |
is used to control the effect of final clutch size
on the worth of eggs in individual positions in the laying sequence. This can
be thought of in terms of finite resources leading to more intense competition
in larger clutches. There is a cost to parasitism implicit in this assumption:
parasitism leads to increased clutch sizes and so reduces the worth of all
eggs in the clutch because of increased competition. The extent of this effect
is controlled by the value of .
The total worth of a clutch is given by
 | (4) |
< ß) increases with T. The theoretical
maximum worth of a clutch is obtained by allowing the clutch size T
to tend to infinity:
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Second, we assume that the probability that the owner of a nest does not
reject a newly laid parasitic egg is a constant 
(0, 1).
Third, we assume that the fitness cost of laying an egg is also a constant
(C) and is greater than zero. The effect of this cost on the relative
payoffs of different strategies depends only on its size relative to the
available reward, and so we will work with a variable (R), which is
the maximum worth of a clutch divided by C:
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The fourth assumption is that all individuals begin laying on the same day.
The final assumption is that each individual lays a single egg each day. A
given individual's strategy is defined as {n1,
n2}, indicating that it lays its first
n1 eggs in its own nest, then lays another
n2 eggs. Each of these is placed in the nest of another
individual, chosen at random from the available population, independently for
each egg. Effectively, each laying sequence is split into two rounds: in the
first, all individuals lay in their own nests; in the second, any remaining
eggs are laid parasitically. It is generally true that parasites lay their
parasitic eggs before laying eggs in their own nests
(Davies, 2000), but the key
biological feature that we need to capture is that parasitic eggs are not the
first to be laid in a nest. Females reject alien eggs placed in their nests
before they have started their own laying
(Davies, 2000). Hence, in our
model, we assume that parasitic eggs are always laid after all of the host's
eggs. This slightly underestimates the effectiveness of real parasitism, but
we believe it is an acceptable compromise between analytic tractability and
biological realism.
The evolutionarily stable strategy
We now find the evolutionarily stable strategy (ESS) of our model. The
concept of an ESS was introduced by Maynard Smith and Price
(1973; see also
Maynard Smith, 1982). If a
system possesses a unique ESS, then (usually) the population should settle on
playing that strategy by natural selection; if there are multiple ESSs, then
the one that the population chooses depends on the initial conditions of the
system and chance (see Hofbauer and Sigmund,
1988,
1998, for a detailed discussion
of the dynamics of biological systems). It has not been possible to prove that
our game always yields a unique ESS. However, in every case that we consider,
we have been able to find only one ESS, despite often considering several
potential candidates.
Suppose all N individuals in the population play
{n1, n2}. Every nest contains
n1 of the owner's eggs and a number of parasitic eggs. If
N is large, because parasitized nests are chosen at random, then the
number of parasitic eggs in any given nest will be closely approximated by a
Poisson distribution; in other words, the probability of a given nest having
j parasitic eggs is given by
 | (5) |
n2. To find the ESS (or more properly, as discussed above, ESSs), we now need to consider two pair of strategies in competition: {n1, n2} versus {n1, n2 + 1} and {n1 + 1, n2} versus {n1, n2}. The way the model has been formulated, the choice of whether to lay one more (or less) parasitic egg and whether to lay one more (or less) egg in an individual's own nest are independent. In addition, because increasing n1 or n2 both reduce the worth of extra eggs, if laying more than one extra egg is beneficial, then laying exactly one extra certainly will be, and so these two competitions are the only ones we need consider.
Laying an extra egg parasitically costs an extra amount, C. It is
laid after all the other eggs and will be the last egg laid in a nest that
already contains n1 + j eggs, where j is
drawn from the Poisson distribution of Equation 5. Thus the worth of this egg
is
 | (6) |
 | (7) |
We now consider the costs and benefits to a single individual of switching
to an alternative strategy where it lays the same number of parasitic eggs as
the other individuals but lays one more egg in its own nest before switching
to parasitism. We assume that this mutant individual lays its
n1 + 1 egg in its own nest before any parasitic eggs are
placed in it, but that this has no effect on the eventual number of parasitic
eggs laid in this nest. In addition, we assume that this also has no
detrimental effect on the positioning of the individual's own parasitic eggs.
These assumptions are likely not to be quite true in real systems. In reality,
the number of parasitic eggs laid in the individual's nest may be less because
other individuals may prefer nests with fewer eggs to parasitize. On the other
hand, the individual will start laying its own parasitic eggs a day later, so
that their worth will, on average, be less. Hence, these two assumptions have
opposite effects on the payoff to the mutant. These simplifying assumptions
are adopted because the effects are small and not additive, and they buy
significant tractability to the analysis. Thus, from its own nest, the
individual gains an extra amount given by
 | (8) |
However, the mutant individual's cost will be increased because it lays one
more egg. For this strategy not to be advantageous, the benefit of laying the
extra egg must be at most equal to C:
 | (9) |
 | (10) |
, ß, , and R. These occur at equality for the two
equations when the ESS values of n1 and
n2 are both positive. In general, these will not be
integer valued. If we find that the ESS value of n1 is
6.7, then this should be interpreted as follows. If the whole population lays
six eggs in their own nests, then a mutant that lays seven would do better;
conversely, if the population all lays seven eggs in their own nests, then a
mutant laying six would do better. Hence, in the population at equilibrium, we
would expect to find 70% of individuals laying seven eggs and 30% laying six.
If Equation 10 is satisfied when n2 = 0, then the expected
reward for laying a parasitic egg is less than its cost, even when no others
are laying parasitically, so that the optimal strategy is
n2 = 0, and no parasitic eggs should be laid. If Equation
10 is not satisfied, then (n1, n2) is
not an ESS; if Equation 7 is not satisfied, then a parasitic level greater
than n2 is favored, so that again (n1,
n2) is not an ESS. Parasitism makes no positive value of
n1 viable, so no nest building is optimal. Due to the complexity of Equations 7 and 10, the ESSs can generally only be found numerically. Before we do this, we explore four limiting cases, where analytical methods are effective.
Case 1: the worth of an egg does not decrease with clutch size
If we make the assumption that = 0, so that the worth of an egg
depends only on the position of that egg in the nest and is independent of the
total number of eggs in the nest, then considerable simplification occurs.
Equation 7 becomes
 | (11) |
 | (12) |
Because the number of parasitic eggs has no effect on the payoff to hosts,
it is no surprise that this expression for n1 is
independent of n2. Unless ß and are both
equal to 1, the left-hand side of Equation 11 is always less than that of
Equation 12, so that the only solution is to satisfy Equation 12 with equality
and Equation 11 with inequality, so that the optimal value of
n2 is zero, and parasitism should not take place. This
makes intuitive sense because laying further eggs in your own nest does not
decrease the worth of previously laid eggs, so (in this case) there is no
advantage to parasitism, and individuals should lay all their eggs in their
own nest.
Equation 12 can be rearranged to give the optimal number of eggs laid in an
individual's own nest, namely
 | (13) |
Case 2: parasitic eggs are never rejected
If = 1, then for n2 = 0 to be evolutionarily
stable, from Equations 7 and 10, we require that
 | (14) |
Case 3: the total clutch worth is independent of the number of eggs
it contains
In this case = ß, and so there is a fixed worth, R,
to be divided between all members of the clutch, no matter how many there are.
It is clear that if n2 = 0, then the left side of Equation
10 reduces to 0 because adding an extra egg to the nest does not increase the
overall worth at all, so that there is no value of n1 that
generates such an ESS solution. Thus, in this case also, some parasitism is
always favored. This can be explained as follows. Laying extra eggs in your
own nest is especially detrimental to the original eggs because any worth
obtained by the new egg corresponds with an identical drop in worth from the
others, so that relatively few eggs are laid in an individual's own nest.
Indeed, if there was no parasitism, a single egg would be optimal. Thus
parasites will take advantage of this fact because they do not mind devaluing
existing eggs.
Case 4: an egg's worth is independent of how early in the sequence it
was laid
Here ß takes its other extreme value, namely 1. Note that Equation 10
is no longer valid, as it required ß < 1. Through similar working, we
can obtain
 | (15) |
. For there
to be any parasitism, it is easy to show from Equation 7 that
 | (16) |
 | (17) |
 | (18) |
Numerical results: the general case
Equations 7 and 10 can be used to find the ESS combinations of
n1 and n2 for specified values of
, ß, , and R. In order to advance, we must now
postulate values for these parameters. The variable is the probability
that a host does not reject a parasitic egg. Obviously, when has a low
value, then parasitism is greatly disfavored, so we will concentrate on the
more interesting case, especially evolutionarily, where rejection is
relatively unlikely, and assume that lies somewhere between 0.75 and
1.0. Each egg in a laying sequence is worth a fraction, ß, of the last
laid one, and we postulate that ß is likely to lie in the range 0.7-1.0.
The worth of an egg is also proportional to a factor Q which is a
function of both the total clutch size T and the parameter
according to
 |
: 0.5, 0.7, and 0.8. This
shows that = 0.5 represents a relatively small effect of total brood
size on the worth of an individual egg in a given position and = 0.8
represents a strong effect. We let vary in the range 0.5-0.8. For each
parameter, we set a default value that we consider to be a reasonable value.
We then set each parameter at its default value, and vary the chosen value
over a range of plausible values. The chosen default values are = 0.7,
ß = 0.9, and = 0.95. Further simulations (not shown) suggest that
our results are not qualitatively particular to these specific values. The
variable R is the maximum possible return from a breeding event
divided by the cost of laying a single egg. Because this is very difficult to
evaluate, we consider it over a very wide range of plausible values from 10 to
500 and always consider several values of R as we vary the other
parameters.
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Figure 2 explores the effect of the value of ß on the ESS strategy. Each egg in a laying sequence is worth a fraction, ß, of the last laid one. As we would expect, Figure 2a shows that increasing both ß and R increases the ESS number of eggs an individual lays in its own nest. It is initially surprising that for low values of R, this number decreases with ß at very high values. Perhaps even more surprising, because ß has no effect on the first-laid egg in a nest, n1 can fall below 1. This effect occurs because R and ß are not independent variables. Increasing ß and keeping R constant can only be achieved by increasing the cost of producing eggs (relative to their future worth). Hence, at very high ß, the cost of eggs has been raised so high that any egg laying is prohibitively expensive. Generally, in Figure 2b we find that the ESS number of parasitic eggs decreases with both ß and R, since increasing both of these factors make the cost of laying eggs in an individual's own nest smaller. Again, Figure 2b shows unusual behavior at high ß and low R because of the non-independence of these variables. There is a critical value, which we denote Rc and define as the highest value of R for which n2 is nonzero, and so parasitism occurs. This value decreases dramatically with increasing ß (see Figure 2c).
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Figure 3 explores the effect
of the value of on the ESS strategy. As
Figure 1 shows, at high brood
sizes, the value of has little effect. This can be seen in
Figure 3a, where at high
R values, the value of n1 is sufficiently big
that it is insensitive to . But at lower brood sizes, increasing
does have a significant effect, and this can be seen in the declining
brood sizes at high values for intermediate R values. But for
such intermediate R values, individuals compensate at high
values by switching to laying eggs parasitically
(Figure 3b). This situation is
carried to its extreme for low R values, where individuals lay more
eggs parasitically and practically no eggs in their own nest.
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Note that when final clutch size (T) is low, increasing
increases the worth of the eggs appreciably, so that n1
increases with . Under these circumstances, our assumption that all
individuals build a nest is likely to be false, as individuals that lay all
their eggs parasitically would not build a nest. Again, there is a threshold
value of R above which parasitism is not seen. As we would expect
from earlier discussion, Rc increases with
.
The variable is the probability that a host does not reject a
parasitic egg. We have already seen a pattern that at high values of
R, parasitism is not observed. Hence it is no surprise that
Figure 4a shows that at high
R values the value of n1 has a negligible
dependency on . Increasing promotes parasitism, but the effect
of this is less marked than might be expected, (see
Figure 4b,c). At lower values
of R, there is a tendency for n1 to increase
slightly with increasing . These two occurrences are linked and can be
explained as follows. When increases, the parasitic eggs are more
beneficial (to the parasite), so that it is better for the parasites to lay
more eggs. Parasitic eggs have a significant detrimental effect on a host's
eggs through increasing the clutch size and hence competition among nest mates
for resources. If the host lays more eggs in its own nest, this reduces the
advantage to parasites, thus reducing the number of parasitic eggs it is best
to lay, and so indirectly helping the host's own eggs. Thus, as
increases more host eggs are laid, and the rate of increase of parasitism is
less than might be expected.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONDISCUSSIONREFERENCES |
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One key assumption of our model is that the worth of the ith egg placed in a nest is a function not only of its position in the laying sequence (i.e., all the eggs placed in the nest before it), but also of the final number of eggs (including all the eggs that come after it). Without this assumption, parasitism is never evolutionarily stable in our model. In this case, extra eggs do not harm existing eggs, so there is no incentive to avoid laying in your own nest; indeed, the risk of another individual rejecting your egg makes it optimal not to do so. Lyon (1998
) did not make this
assumption; he implicitly assumed two classes of individuals, only one of
which is able to parasitize, and the other of which is only vulnerable to
parasitism. We predict that when egg production is relatively cheap (high
R), then brood sizes will be relatively insensitive to the strength
of this effect, and parasitism will not be favored. Conversely, when eggs are
relatively expensive (low R), the position effect is high (low
) and within-brood competition is strong (high ß), then parasitism
is highly favored. Indeed, under such conditions we predict that many
individuals would opt to lay all their eggs parasitically. In this instance,
our model needs some modification, as such individuals would not build nests
of their own. However, this result indicates that case 1 of Davies
(2000), discussed in the
Introduction, where birds become obligate parasites and do not build a nest,
need not be seen as a separate case to the one described here, but rather both
can be adopted within a more general framework. We hope that the methodology
presented here will be a useful foundation for that framework.
It is no surprise that our model predicts that increasing ability of hosts
to reject alien eggs (decreasing ) decreases the attractiveness of
parasitism. What is more interesting is that changing from a situation where
hosts reject no parasitic eggs to one where they reject 25% of them makes only
a very slight difference to the levels of parasitism that the model predicts.
This is due to the fact that in our model individuals must find optimal values
for the number of eggs laid in their own nest and laid parasitically and that
these values are linked. The detrimental effect of parasitic eggs can be
severe, so that as parasitism becomes more effective, the optimal strategy is
to lay more eggs in your own nest to discourage parasites. This effect,
combined with the risk of individuals mistakenly rejecting their own eggs
(Lotem, 1993), suggests that
the evolution of rejection by hosts is also worthy of further theoretical
effort.
Conspecific brood parasitism is not as well known to the general public as the parasitic behavior of cuckoos, but it contains many fascinating challenges for the evolutionary ecologist. Further developments of the theory must explore the consequences of intrinsic differences between individuals and host selection by parasites. However, there is still much need for empirical work if we are to fully explain the diversity of this mechanism shown by natural populations. We hope that others will challenge the predictions made here with empirical testing, either by experimental manipulation or (perhaps more amenably) by cross-species or cross-population comparisons. Some of the simplest of these to test are the following. Increasing the intrinsic costs of egg production should lead to an increased propensity for intraspecific brood parasitism. Decreasing the effect of position within a brood on the worth of an egg should lead to reduced parasitism. Variation in hosts' ability to reject parasitic eggs has little effect on parasitism until this ability is well developed.
Further, theoretical development may also be fruitful. To retain some
analytic tractability, we were required to remove any temporal component to
birds' strategies. Thus we imposed strict laying synchrony on all the birds.
This does not happen in the real world. Allowing birds to control the timing
of when to begin laying would be a very interesting development to this model.
Particularly, this would allow the host availability to parasites and parasite
pressure on hosts to vary over time and would naturally introduce variability
in host nest attractiveness (through differential clutch size) to parasites at
any given time. However, this added realism will necessarily incur costs in
increased model complexity. However, an added advantage is that it will also
allow relaxation of another assumption in our model, that an individual lays
parasitically after laying in its own nest. Generally the reverse is true in
nature. This assumption was forced on us, once we adopted the simplifying
assumption of complete synchrony of breeding, because the key biological
feature that we needed to capture is that parasitic eggs are not the first to
be laid in a nest. Females reject alien eggs placed in their nest before they
have started their own laying (Davies,
2000). Accepting the complexity produced by having a temporal
component to individual's strategies would allow more realistic ordering of
parasitism and laying in an individual's own nest. We are confident that the
work presented here will be a useful tool in aiding understanding of such more
complex models.
Another useful extension would be to explore the coevolution of
antiparasitism traits such as egg rejection along with parasitic traits.
Yamauchi (1993) described how
quantitative genetic modeling can be applied to such coevolution. Further work
(Yamauchi, 1995) described how
this framework can be extended to consider both interspecific and conspecific
brood parasitism simultaneous. Such a framework is vital if we are to
understand how the type of conspecific brood parasitism described here may
have provided an evolutionary stepping stone to the obligate interspecific
brood parasitism famously practiced by cuckoos and cowbirds.
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ACKNOWLEDGEMENTS |
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We thank three referees for perceptive comments.
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Lotem A, 1993. Learning to recognise nestlings is maladaptive in cuckoo Cuculus canorus hosts. Nature 362: 743-745.
Lyon BE, 1998. Optimal clutch size and conspecific brood parasitism. Nature 392: 380-383.
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