Evolutionary Ecology Research, 2003, 5: 345–362

Co-evolutionary dynamics of egg appearance
in avian brood parasitism
Fugo Takasu*
Department of Information and Computer Sciences, Nara Women’s University,
Kita-Uoya Nishimachi, Nara 630-8506, Japan

ABSTRACT
In avian brood parasitism, interactions between parasites and their hosts lead to aco-evolutionary process called an arms race. Field studies have shown that many host species
have evolved an ability to recognize and reject parasite eggs that look unlike their own andthat some parasites lay eggs of sophisticated mimicry. Egg appearance is a crucial factor that
determines the reproductive success both of the parasites and their hosts in the arms race. The
appearance of eggs, however, is a quantitative trait and the variation within a population could
critically affect the dynamics of the arms race; parasite individuals that lay eggs more similar to
those of the host are more likely to reproduce successfully, whereas host individuals that lay
eggs more unlike those of the parasite have a better chance of rejecting parasitism and reproducing, which would de-stabilize the parasites’ good mimicry. To explore the arms race for egg
appearance, I constructed a model described by an integro-difference equation. The temporalchange of the ‘distributions’ of egg appearance along a continuous spectrum was analysed bothfor the parasite and host populations. The model analyses show that (1) a conventional view ofan arms race – ‘parasites chasing their hosts escaping from the parasites’ – is realized in theearly stage of the dynamics, but (2) the distributions of egg appearance finally converge todiscrete point-distributions (polymorphism) even when any continuous distributions are usedas the initial state. The latter might be relevant to the maintenance of polymorphism of egg
appearance. Based on the model analysis, I discuss the implications of this model for general(co)evolutionary processes for quantitative traits and suggest a new modelling framework to
account for such processes.
Keywords: arms race, avian brood parasitism, co-evolution, egg appearance, ideal freeevolution, quantitative trait.

INTRODUCTION
Brood parasites exploit parental care of their hosts and usually reduce the hosts’ reproductive success. It is therefore expected that the hosts will evolve some adaptation to reduce
the reproductive damage caused by parasitism, such as an ability to recognize and reject* e-mail: takasu@ics.nara-wu.ac.jp
Consult the copyright statement on the inside front cover for non-commercial copying policies.
© 2003 Fugo Takasu

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Takasuparasite eggs, which, in turn, should select for the parasites’ counter-adaptation, such as
egg mimicry. Field studies have shown that some host species have evolved an ability torecognize and reject parasite eggs that look unlike their own (Rothstein, 1975; Davies andBrooke, 1989; Moksnes and Røskaft, 1989; Brown et al., 1990; Soler, 1990; Soler and
Møller, 1990; Moksnes et al., 1993) and that there are various degrees of egg mimicry by
the Common Cuckoo Cuculus canorus, one of the most extensively studied avian broodparasites (Brooke and Davies, 1988; Davies and Brooke, 1988; Higuchi, 1989, 1998;
Moksnes, 1992). Avian brood parasitism is an excellent system for studying the coevolutionary arms race, where parasites are selected to exploit their hosts more efficiently,
while the hosts evolve to escape the reproductive loss caused by the parasites (Dawkins andKrebs, 1979; Rothstein, 1990; Rothstein and Robinson, 1998; Davies, 2000).
In the co-evolutionary process, the appearance of eggs seems to be crucial because
acceptance or rejection of parasitism by the host generally depends on differences inappearance of the host and parasite eggs (Victoria, 1972; Brooke and Davies, 1988;
Higuchi, 1989, 1998; Moksnes et al., 1993). Recent field studies have demonstrated that thepattern of the distribution of egg appearance within a cuckoo host population is correlated
to the history of parasitism (Øien et al., 1995; Soler and Møller, 1996; Stokke et al., 1999).
This suggests that parasitic interactions act as selective agents for the co-evolution of egg
appearance.
Egg appearance is a quantitative phenotypic trait composed of many characteristics,
including background colour and coverage of spots, and it usually exhibits a continuous
spectrum within a population. Assuming within-population variation in egg appearance,
the consequence of the arms race is open to question because, even if a parasite has
evolved very good mimicry, host individuals that lay eggs more dissimilar to those of
the parasite can reproduce more and hence the parasite’s good mimicry is easily
destabilized.
Kelly (1987) presented a mathematical model to determine the rate of spread of
hypothetical genes responsible for egg mimicry by parasites and rejection behaviour by
hosts. But the classification of parasitic eggs as simply a ‘good’ or ‘poor’ match to those of
the host misses the essence of the co-evolutionary process, because egg appearance is aquantitative trait and this binary classification is unlikely to be justified. Takasu (1998b)
presented a model to explore the arms race between a parasite and its host population,
where the degree of egg mimicry was more realistically given by a parameter measuring theprobability of acceptance. The model, however, ignores the within-population variation ofegg appearance and it cannot fully explore the co-evolutionary process of egg appearance asa quantitative trait. Rodríguez-Gironés and Lotem (1999) presented a model for recognitionand learning based on signal-detection theory, but did not analyse co-evolution of egg
appearance.
The aim of this study was to examine the co-evolutionary dynamics of a parasite and its
host population focusing on egg appearance as a quantitative trait. I constructed a model
described by an integro-difference equation to analyse the co-evolutionary change of the‘distributions’ of egg appearance. Although the model itself targets avian brood parasitism,the modelling framework can easily be applied to general co-evolutionary processes ofhost–parasite interactions and others. Previous studies have adopted integro-difference
equations for the dynamics of quantitative traits (cf. Slatkin, 1980; Taper and Case, 1985,
1992; Maynard Smith and Brown, 1986; Doebeli, 1996a,b, 1997; Sasaki and Godfray,1999). Because of the difficulty of an analytic treatment, however, most of these wereCo-evolution in avian brood parasitism

347

based on numerical analyses and there remains room to advance theoretical exploration ofthis type of modelling. In this paper, I show that the dynamics in the early stage can beanalytically traced under some circumstances and that the rate of the co-evolutionarychange of egg appearance can be estimated from ecological parameters, such as survivaland breeding rate. I also show that the distributions of egg appearance finally converge todiscrete point-distributions (polymorphism) both for the parasite and host populations,even if continuous initial distributions are used. Based on the model analysis, I discuss theimplications of my results.
THE MODEL
Consider a parasite population that parasitizes a host population. Let P(x) be the density ofparasite females that lay eggs of appearance x, and let H(x) be that of host females laying
eggs of x. Egg appearance in general is a multidimensional trait. In this model, for the sakeof simplicity, the appearance is assumed to be a one-dimensional trait on an arbitrary scale
within a certain range L (x ∈ [− L/2, L/2]) that both the parasite and host populations sharein common. Extension to a multidimensional trait is straightforward but is not dealt withhere.
The mechanism of genetic inheritance of egg appearance in birds is unclear, but it has
been suggested that independent female lines exist in a bird population, each producing eggs
of a distinct appearance directly inherited by daughters (Gibbs et al., 2000). It is possiblethat a sex-linked gene on the W chromosome controls the appearance (Brooke and Davies,
1988; Marchetti et al., 1998; but see Gibbs et al., 1996; Davies, 2000; Gosler et al., 2000).
I assume that the appearance is transmitted asexually to offspring with no effect on mating.
I also assume that the sex ratio is 1:1 and that all females share the same ecological characteristics except egg appearance. Relaxation of this assumption, such that the reproductive
output is maximized at some particular egg appearance, is addressed later.
Let A(x, y) be the probability that a parasite egg of appearance x, laid in the nest of ahost female that lays eggs of appearance y, is accepted. Hereafter, A(x, y) is referred toas the acceptance probability function. Field experiments have demonstrated that hostindividuals able to reject unlike parasitic eggs tend to accept if the parasitic eggs are mimetic
to their own (Victoria, 1972; Brooke and Davies, 1988; Higuchi, 1989, 1998; Moksnes et al.,1993). Thus, the acceptance probability can reasonably be assumed to be a decreasing
function of the absolute difference in appearance, A(x, y) = A(| x − y | ). If there is no difference in appearance (x = y), the parasite egg will look identical to those of the host and it will
always be accepted, A(0) = 1. In the following analysis, I adopt the following function as theacceptance probability:
A(x, y) = exp[−γ(x − y)2]
where the parameter γ measures the sensitivity of the hosts to discriminate against unlike
eggs. The greater the value of γ, the less tolerant hosts are to unlike parasitic eggs. The–
amount of tolerance is measured by the width of the function, defined as w = 1/√γ.
The probability of a host nest being parasitized by any parasite is given as follows, using
the zero-th term of the Poisson distribution (May and Robinson, 1985; Takasu et al., 1993;
Takasu, 1998a,b) with the mean amount of parasitism being proportional to the total
parasite density:
1 − exp[−a ∫ P(x)dx]

348

Takasuwhere a is the search efficiency of the parasites. In all cases, the domain of the integrationis set to x ∈ [− L/2, L/2] unless otherwise stated. Here I assume that the conditional
probability of a host nest being parasitized by a specific parasite of egg type x is
proportional to its density P(x). The density of the parasites of egg type x in the following
year, P⬘(x), is then obtained as the sum of the adult survivors to the following year and the
newly recruited yearlings that fledged from the nests of host y with probability A(x, y),summed over all possible y, asP⬘(x) = sPP(x) + (1 − exp[−a ∫ P(x)dx])

P(x)∫ H(y)A(x, y)dyΓ
∫P(x)dx(1)

where sP and Γ are the survival rates of adult parasites and of parasitic eggs after acceptanceof parasitism, respectively. Here I have assumed implicitly that only one parasite egg
survives in a host nest even if the nest was parasitized more than once (as in the case of
cuckoo parasitism) and that no particular parasite trait is preferred at this stage.
The density of the hosts of egg type x in the following year, H⬘(x), is derived in the sameway. The number of the host yearlings of egg type x that fledged from an unparasitized nest,
WNP, is given as
WNP = f exp[−a ∫ P(x)dx]
where f is the reproductive success of the host in the absence of parasitism and exp[*] is theprobability of a host nest escaping any parasitism. The number of yearlings of egg type x
from a parasitized nest, WP, is given as the product of the reproductive success of the hostin the absence of parasitism, the probability of the nest being parasitized by a parasitey, and the probability of rejecting the parasitism 1 − A(x, y), summed over all possible y,
to give
WP = f(1 − exp[−a ∫ P(x)dx])∫ P(y){1 − A(x, y)}dy∫ P(x)dx

The total density of the host population is generally regulated by a density-dependent
effect in the absence of parasitism. Then, the density of the hosts x in the following year is
obtained as follows:H⬘(x) =k[sH + WNP + WP]H(x)k + ∫ H(x)dx(2)

where the first term on the right outside the square brackets represents the densitydependent effect (1/k measures the intensity of the density-dependent effect) andsH is the survival rate of adult hosts. Here I also assume that the density-dependent
effect is independent of host egg appearance. I discuss relaxation of this assumption
later.
The integro-difference equations (1) and (2) constitute the model that describes the
temporal change of the distributions of egg appearance, P(x) and H(x), for x ∈ [−L/2, L/2]
starting from some initial distributions. We can reduce the number of parameters by
rescaling x → x/L, a L P(x) → P(x), a L ΓH(x) → H(x), a k Γ → k, and γ L2 → γ. Thismeans that we can set L = 1, a = 1 and Γ = 1 without loss of generality to examine thequalitative behaviour of the model.

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349

ANALYSES AND RESULTS
Monomorphic caseFirst, I analyse the special case when both the parasite and host populations aremonomorphic. In this case, within-population variation is zero and both populations
are homogeneous. The distributions P(x) and H(x) are given as
P(x) = Pδ(x − mP)

H(x) = Hδ(x − mH)(3)

where P and H are the total densities and mP and mH are the egg appearances of the parasite
and host populations, respectively. δ(x) is the delta function that represents a monomorphic
point distribution. Without loss of generality, I assume mP < mH.
By substituting (3) into (1) and (2) and integrating with respect to x, we obtain thefollowing conventional population dynamics that relates the total densities of parasites andhosts (P and H) to those in the following year (P⬘ and H⬘):P⬘ = sPP + (1 − e
H⬘ =

−P

)He − γd2(4)k[sH + fe − P + f (1 − e − P)(1 − e − γd )]H
k+H
2

(5)

where d is the distance between the egg appearances (d = mH − mP > 0).
The behaviour of the monomorphic dynamics (4) and (5) can be readily analysed. Thereexists a critical threshold dC, defined asdC =—————————
1k( f + sH − 1)log
γ
1 − sP冪

冤

冥

which is proportional to the width of the acceptance probability function w. If d ≥ dC, thepopulation dynamics converges to the equilibrium state in which the parasites go extinct
and only the hosts survive: P → 0, H → k ( f + sH − 1). In contrast, if d < dC, the dynamics
converges to the equilibrium state in which both the parasites and hosts co-exist: P → P*,
H → H*, where (P*, H*) is the unique internal equilibrium of (4) and (5). A monomorphic
parasite population of poor mimicry beyond the threshold (d ≥ dC) cannot reproduce
enough and goes extinct. On the other hand, a parasite population of good mimicry (d < dC)
can reproduce enough to sustain the population. This monomorphic case corresponds toa conventional population dynamics in which the (co)evolutionary change of a trait infocus is controlled by a parameter and the condition of a new trait to invade and replace
the existing population is analysed (e.g. Takasu, 1998b). In the next sub-section, I relax themonomorphic constraint and introduce within-population variation to examine how the
distributions P(x) and H(x) shift with time.
Co-evolution of egg appearance
Let P(x) and H(x) be distributed around some population means with certain variances.
Note that each trait x is assumed to reproduce asexually and no genetic structure isconsidered in this model. Co-evolution of egg appearance is defined in terms of thetemporal change of the distributions P(x) and H(x).

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TakasuThe integro-difference equations (1) and (2) allow only a limited analytical treatment, andthe following analyses in part are carried out by numerical calculation. As will become clear,
however, we can explore the dynamics analytically by focusing on the population means andvariances of egg appearance if initial distributions are given proportional to normal.
Assume that the initial distributions P0(x) and H0(x) are given proportional to normal
distributions with mean mP and mH and variance σ 2P and σ 2H, respectively, as follows:
P0(x) = PN(mP, σ 2P)

H0(x) = HN(mH, σ 2H)

where N(m, σ2) is a normal distribution with mean m and variance σ2, and P and H are theinitial total densities of the parasite and host populations, respectively. This would bereasonable if new traits are created by random mutation from an existing monomorphic
trait. Without loss of generality, we set mP = 0 and mH = d > 0 as the initial state. If the initialvariances, σ 2P and σ 2H, are sufficiently small, we can approximate that the range of thetrait space L is infinite and, by integrating (1) and (2) for x ∈ [− ∞, ∞], we obtain the total
densities, P⬘ and H⬘, and the population means of egg appearance, m⬘P and m⬘H, in thefollowing year as functions of those in the previous year (P, H, mP, mH, σ 2P and σ 2H) as
follows (see Appendix):P⬘ = ∫ P⬘(x)dx = sPP + (1 − e−P)HCH⬘ = ∫ H⬘(x)dx =m⬘P = ∆mP =(6)

kH[ f + sH − f(1 − e−P)C]
k+H

2(1 − e−P)Hγdσ 2PC{1 + 2γ(σ 2P + σ 2H)} {sPP + (1 − e−P)HC}m⬘H = d + ∆mH = d +2(1 − e−P)fγdσ 2HC{1 + 2γ(σ 2P + σ 2H)}{ f + sH − f(1 − e−P)C}

(7)
(8)
(9)

where
1
− γd 2
C = ———————exp
1 + 2γ(σ 2P + σ 2H)√ 1 + 2γ(σ 2P + σ 2H)

冤

冥and d is the difference between the means of the parasite and host populations in theprevious year (d = mH − mP). In the same way, we can derive the variances of egg appearance
in the following year, σ⬘2P and σ⬘2H, as functions of P, H, mP, mH, σ 2P and σ 2H (the expressions
are too lengthy to be listed here; see the Appendix for their full expression):σ⬘ 2P = FP (P, H, mP, mH, σ 2P, σ 2H)(10)σ⬘ = FH (P, H, mP, mH, σ , σ )

(11)

2
H

2
P

2
H

In the strictest sense, the distributions in the following year, P⬘(x) and H⬘(x) given byequations (1) and (2), are no more normally distributed if the initial distributions areproportional to normal, because the reproductive successes of the parasites and hosts are
not uniformly constant depending on trait x. Numerical analyses, however, show that P⬘(x)
and H⬘(x) can be approximately represented as normal again if the initial variances are
sufficiently small (σ 2P, σ 2H Ⰶ 1). This provides an analytical tool to explore the full dynamics(1) and (2) by an approximated method, which I refer to as normal distribution approximation. In this approximation, the set of equations (6)–(11) describes an approximatedCo-evolution in avian brood parasitism351

dynamics of the full dynamics under the restriction that the distributions P(x) and H(x) are
fixed proportional to normal. Note that the full dynamics (1) and (2) itself does not require
the normality of the distributions. With the approximated dynamics, however, we can
explore the temporal change of the distributions of egg appearance in more detail, as shown
below.
Figure 1 illustrates a typical example of the temporal change of P(x) and H(x) of thefull dynamics (1) and (2), starting from initial distributions set proportional to normal.
We see that the distributions P(x) and H(x) retain almost the same shape and theparasites run after the hosts. Figure 2 shows the corresponding temporal change of thetotal densities, P and H, the population means of egg appearance, mP and mH, andthe variances, σ 2P and σ 2H, calculated from the full dynamics (1) and (2), together withthose calculated from the normal distribution approximation (6)–(11). As shown, theapproximated dynamics provides a very good fit to the original full dynamics in the earlystage of the dynamics.
We can see the early dynamics in more detail by focusing on the distance between themean egg appearances of the parasite and host populations, d = mH − mP. When the change
in the distance per year is positive (∆d = ∆mH − ∆mP > 0), the hosts run away from theparasites. Otherwise (∆d < 0), the parasites catch up with the hosts. It is not straightforward
to see the sign of ∆d from (8) and (9). But if the variances are effectively small (σ 2P, σ 2H Ⰶ 1)
and we assume that co-evolution starts from the non-trivial monomorphic equilibrium [thetotal densities P and H are given equal to the non-trivial equilibrium P* and H* of themonomorphic dynamics (4) and (5)], ∆d is reduced as follows by ignoring the higher ordersof the variances (see Appendix):∆d = ∆mH − ∆mP = 2

冤k( f + sH − 1) − H*
H* + k冥σ 2H − (1 − sP)σ 2P γd(12)Fig. 1. Temporal change in the density distributions P(x) and H(x), when the sensitivity of the hosts
against unlike eggs is not high, γ = 2. Initial distributions are set proportional to the normal distribution N(m, σ2). Black lines = parasite, grey lines = host. Parameters used are f = 0.7, sH = 0.5, k = 100,sP = 0.5, a = 1.0, Γ = 1.0, γ = 2. Initial distributions are P0(x) = P0N(mP, σ 2P), H0(x) = HN(mH, σ 2H),
where mP = 0.1, mH = 0.3 and σP = 0.02, σH = 0.02. The total densities, P0 and H0, are set to the nontrivial equilibrium value of the corresponding monomorphic case d = 0.2; that is, P0 = 0.3560,
H0 = 0.6438.

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TakasuFig. 2. Corresponding temporal change of the population means and variances of egg appearance,
and the total densities of the parasite and host populations calculated from the full dynamics (solidlines). The normal approximation dynamics is shown by dashed lines.

As the distance between the means, d = mH − mP, increases up to the threshold dC given in
the previous sub-section, the monomorphic equilibrium density H* approaches the stableequilibrium density of the hosts in the absence of parasites, k( f + sH − 1), and the first termin the bracket becomes negligible to make ∆d negative. This shows that the further the
population mean of the parasite egg appearance is from that of the host, the fasterthe parasites chase the hosts. This is due to the in-built inequality that all parasites are at themercy of the host to accept or reject, and that not all hosts are parasitized. We also see from
(12) that the sensitivity γ and the distance d scale the rates of the co-evolutionary change ofthe population means of egg appearance and that a smaller parasite survival rate facilitates
the change in the parasite’s mean egg appearance.
In this sub-section, I have shown that the normal distribution approximation preciselyreflects the early transient stage of the full dynamics if initial distributions are given
proportional to normal. In this case, we can evaluate the rates of the co-evolutionarychange in mean egg appearance using ecological parameters. The approximated dynamics
eventually fails to apply because the distributions can no longer be assumed to be normal astime passes. Note again that the full dynamics (1) and (2) does not require the normality of
distributions. Without assuming the normality, however, it would not be possible to deriveCo-evolution in avian brood parasitism353

the approximated dynamics. Extensive numerical analyses, however, show that qualitatively
the same transient behaviours are obtained starting from a broad set of initial unimodal
distributions; a conventional view of an arms race – ‘parasites chasing their hosts escaping
from the parasites’ – holds again as long as the initial distributions P0(x) and H0(x) are
unimodal.
Equilibrium distributions of egg appearance
Even when unimodal distributions are used as initial distributions, their unimodality is
eventually lost and complex behaviour arises. Figure 3 illustrates the full dynamics after thenormal approximation fails to apply. Some host traits behind the parasite distribution
emerge. This ‘jumping emergence’ is essentially the same as in Doebeli (1997), where theclassical Nicholson-Bailey host–parasitoid model was extended to cover quantitative traits.
Jumping is always possible for any initial distribution, the support of which is equal to thetrait space – that is, ∀P0(x), H0(x) > 0 for ∀x ∈ [−L/2, L/2] (the normal distribution satisfies
this condition). If all traits are available initially in a population, they never disappear
within a finite time, although some of them might go extinct as time goes to infinity. Thus,
as the population mean of the parasite egg appearance shifts rightward, host traits near the
opposite side of the trait space can reproduce more successfully and begin to increase. In
response, parasite traits near the newly emerged host traits begin to increase and theparasite’s population mean is pulled back leftward.
As time passes further, P(x) is distributed around the midpoint of the trait space (x = 0.5
in Fig. 3) and H(x) is distributed at both the edges of the trait space (x = 0 and x = 1).
Numerical calculation shows that P(x) and H(x) converge to a stable equilibrium where theparasite population is monomorphic at x = 0.5, but the host exhibits dimorphism with two
discrete traits at x = 0 and x = 1.
It can be shown that non-trivial equilibrium of the full dynamics (1) and (2), if any, isgiven as a combination of the delta functions (point distributions) – that is, only discrete
traits are allowed to survive in an equilibrium state. It is, however, extremely difficult
to predict stable equilibrium (the positions of the discrete traits and their equilibrium

Fig. 3. Temporal change of P(x) and H(x) after 5000 years calculated from the full dynamics.
Parameters and initial distributions are the same as in Fig. 1. Both the parasite and host populations
begin to show spike-like distributions, but have not yet reached stable equilibrium.

354

Takasudensities) for general parameter values. Extensive numerical analyses have shown that (1)the initial distributions P0(x), H0(x) do not affect which discrete traits survive, as long as
their supports coincide with the trait space (P0(x), H0(x) > 0 for ∀x ∈ [−L/2, L/2]), and (2)the sensitivity of host discrimination γ is key for determining the numbers and the positionsof the discrete traits. Figure 4 shows the full dynamics for a larger γ, in which three discrete
traits survive both in the parasite and host populations. Although the dynamics does notreach equilibrium and shows a chaotic-like oscillation after 10,000 iterations, the discrete
traits have been fixed in the trait space.
The dependence of the surviving discrete traits on γ is illustrated in Fig. 5. As γ isincreased, the discrete traits bifurcate to give rise to new ones and greater numbers surviveboth in the parasite and host populations; both exhibit polymorphism of egg appearance.
The interval between these discrete traits is roughly proportional to the width of theacceptance function w.

Fig. 4. Temporal change of P(x) and H(x) after 10,000 years calculated from the full dynamics fora larger host sensitivity γ = 9. Parameters and initial distributions are the same as in Fig. 1 except γ.

Fig. 5. Dependency of the surviving traits on the sensitivity γ. Density plots of P(x) and H(x)calculated from the full dynamics, averaged over 10,000 years after 1 million iterations, are shownfor the host sensitivities γ = 0.22 = 0.04, 0.42 = 0.16, 0.62 = 0.36, . . . , 152 = 225. High density is shownin white.

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355

Fig. 6. Dependence of the surviving traits on the sensitivity γ when a particular egg type is preferredin the host population in the absence of parasitism, k(x) = kmaxN(m, σ2). kmax = 100, m = 0.5, σ = 0.2.
Density plots of P(x) and H(x) calculated from the full dynamics, averaged over 10,000 years after
1 million iterations, are shown for the same range of the host sensitivity γ. High density is shownin white.

So far, I have assumed that ecological parameters are independent of egg appearance– that is, they are neutral in the absence of parasitism. Without parasitic interaction,
however, a certain type of egg appearance might be preferred for some reason, such as
anti-predator camouflage. This effect can easily be embedded in the model by assuming thatthe parameter k corresponding to the carrying capacity of the host population is a functionof x with a maximum at some appearance. The drawback is that this makes the normal
distribution approximation derived above virtually impossible. However, numerical analyses
have provided qualitatively the same results as shown in Fig. 6. Again, only discrete traits
survive both in the parasite and host populations. The innate preference confers anadvantage to a particular egg type, so that only one discrete trait survives in the hostpopulation when the sensitivity γ is small.
The discreteness of the equilibrium distributions remains qualitatively the same even if
we introduce mutation (e.g. females of one egg type produce a few offspring of slightly
different types). With such a mutation, the distributions after a long run again have several
peaks, each corresponding to the position of the discrete trait in the absence of mutation(the sharpness of the peaks depends on the mutation rate). Thus I conclude that thediscreteness (polymorphism) observed in this model is robust, not an artifact of the model
framework.
DISCUSSION
Co-evolution of egg appearance as a quantitative trait
In the model presented here, co-evolution of egg appearance is viewed in terms of the
dynamics of density distributions along a continuous spectrum. As shown in the analyses,
we observe a conventional view of the arms race as ‘parasites chasing their hosts escaping
from the parasites’ in the early stage of the dynamics provided that the initial distributions

356

TakasuP0(x) and H0(x) are unimodal. If P0(x) and H0(x) are given proportional to normal, we
can evaluate the rate of co-evolutionary change of total densities, population means andvariances of egg appearance using ecological parameters by an approximated dynamics.
Technically, the normal distribution approximation developed here appears to correspond to the quantitative genetic recursion (QGR) in Taper and Case’s notation (Taper andCase, 1992), although the present model ignores genetic structure. Quantitative genetic
recursion describes the temporal changes of total density, population mean and varianceassuming that the distribution of a phenotypic trait is always kept normal. Taper and Case’s(1992) discussion is based on a classical Lotka-Volterra competition model, but as theynote, it could be applied to general host–parasite or prey–predator interactions. Most
theoretical studies of quantitative traits follow the conventional schemes of quantitative
genetics that also assume the normality of the trait distribution (Lande, 1980). However, thenormality of distribution is not always guaranteed, as egg appearance is probably inheritedby a daughter from her mother (Gibbs et al., 2000; Gosler et al., 2000). It would be interesting to investigate the link between the present model and QGR or conventional quantitative
genetics models.
Theoretical viewpoints aside, we have few data to estimate the temporal change of egg
appearance of avian brood parasites and their hosts. Also, little evidence is available
showing that egg patterns have been changing co-evolutionarily. However, recent fieldstudies have shown the possibility that parasitic interactions affect egg appearance of hosts(Øien et al., 1995; Soler and Møller, 1996; Stokke et al., 1999). Nakamura (1990) reported
that the Common Cuckoo had started parasitizing a new host, Azure-winged Magpie
Cyanopica cyana, in central Japan in the previous two decades. The magpie population
suffered severe parasitism so that multiple parasitism was commonly observed. But it began
to show a rapid increase in egg discrimination (Nakamura et al., 1998). This should impose
strong selective pressure for a co-evolutionary change in egg appearance of both the magpie
and the cuckoo populations. Long-term monitoring and quantitative evaluation of egg
appearance is needed.
Discreteness of surviving traits
The model analyses have shown that the distributions P(x) and H(x) are eventually reduced
to point distributions; discrete traits are winnowed both in the parasite and host populations despite the initial continuous distributions. Although the model exhibits a chaotic-likebehaviour that depends on parameter values, it does not take much time for the surviving
discrete traits to be fixed in the trait space. It is surprising that the arms race for egg
appearance settles down to such a state of polymorphism. It would appear that thediscreteness is a general consequence of the arms race for a quantitative trait, wherematching differences in the trait is of primary importance as in avian brood parasitism. On
the other hand, in cases where a graded difference is of primary importance, escalation of
the trait is possible and the system might exhibit cyclic behaviour if the exaggerated trait
entails a cost (Sasaki and Godfray, 1999).
Brown and Vincent (1987, 1992) developed a game theoretic approach to model theco-evolution of predator–prey communities. In light of the evolutionarily stable strategy(ESS), they formulated an analytical procedure to calculate ESS assemblages of multiple
predator and prey species, where communities of one prey and one predator, two preys andone predator, two preys and two predators, and so on, are realized depending on theCo-evolution in avian brood parasitism

357

predator’s niche width. The width of the acceptance probability function in this model
appears to correspond to that of the predator in their models. Although I could find noanalytical procedure to predict the numbers and the positions of stable equilibrium discrete
traits in this model, there seems to be an important link with the models of Brown andVincent. A critical difference is that they assumed the dynamics always led to a stableequilibrium and did not consider explicit population dynamics. The game theoretic
approach by Brown and Vincent, however, might be helpful in deriving a stable equilibrium
state for this model. On the other hand, Kisdi (1999) studied the branching of a quantitative
trait under asymmetric competition. The branching process of the equilibrium discrete
traits, as shown in Figs 5 and 6, might be analysed following this approach.
The discreteness of phenotypic distributions, as shown in the present model, has been
demonstrated in other contexts. Sasaki and Ellner (1995) analysed the evolutionarily stablephenotype distribution in a population under a randomly fluctuating environment. They
showed that distinct phenotypic classes rather than continuous ones could be realized as an
ESS and that more distinct traits survived as the degree of environmental fluctuation wasincreased, as in Figs 5 and 6. In a rough comparison, the fluctuating environment in theirmodel would correspond to the host population as a background to support parasites in thepresent model. The discreteness also occurs if we read the continuous phenotypic spectrum
as space in which individuals interact. Sasaki (1997) analysed a model in which individuals
compete with neighbours in a continuous space and the effect of the competition
is described by an integral kernel as in the present model. Sasaki showed that a highly
clumped (discrete) distribution could occur due to the competition with neighbours. The
present results, together with these previous ones, suggest that the discreteness along acontinuous spectrum could occur in a wide range of contexts.
In the real world, it is unclear whether distinct egg types exist (polymorphism) in
sympatric parasite and host populations. It is clear that various egg types exist in a parasite
population; the Common Cuckoo lays eggs of a wide variety of types (Higuchi and Sato,
1984; Davies and Brooke, 1998; Higuchi, 1998; Nakamura et al., 1998) and the Red-chested
Cuckoo, Cuculus solitarius, in southern Africa lays eggs of at least three different types, noneof which matches those of the most commonly used host (Cherry et al., 1998; S.M. Kuiper,
personal communication). However, it is unclear whether such variations exist within a hostpopulation. The lack of polymorphism of host egg appearance might be attributed to: (1) aparticular egg type being favoured in the host population because of factors other thanparasitism, such as camouflage against predation; or (2) the range of possible appearances
is much narrower in the host population than in the parasite population (both the parasite
and host populations share the same trait space in this model) because selection operates
more harshly on the parasites due to the in-built inequality of parasitic interactions, so thatthe parasites have evolved various egg types but the hosts have yet to do so. Few data are
available to test the idea that the arms race for egg appearance results in polymorphism. A
comparative study focusing on quantitative evaluation of egg appearance is needed.
How to model co-evolution
The concept of a co-evolutionary arms race is widely accepted (Dawkins and Krebs, 1979).
The process could be modelled in various ways. Monomorphic dynamics would be thesimplest, in which the (co)evolutionary change of a particular ecological feature, such as theattack rate by a predator or the degree of egg mimicry, is reduced to the change in358

Takasua parameter under focus (Abrams and Matsuda, 1997; Takasu, 1998b). Or we may model it
more realistically based on the assumption that phenotypic distributions are normal andco-evolutionary change of the means and variances is analysed (Slatkin, 1980; Taperand Case, 1985, 1992; Frank, 1993, 1994; Saloniemi, 1993; Dieckmann et al., 1995). In
either case, we look at co-evolution focusing on parameters that characterize the target
populations, such as population means and variances.
The assumption of monomorphic or normality of phenotypic traits, however, is notalways justified, as in the case of egg appearance in avian brood parasitism. Indeed, we needto describe and analyse the temporal change of the ‘distribution’ of such traits. This type ofmodelling is possible using integro-difference or integro-differential equations, an approachthat has recently been gaining support as a useful theoretical tool to model (co)evolution(Doebeli, 1996a,b, 1997; Sasaki and Godfray, 1999; Doebeli and Dieckmann, 2000). I
suggest calling this approach ‘ideal free evolution’. By ‘ideal free’, I mean that (1) all
possible traits are made available in the target populations along a continuous spectrum and(2) each trait reproduces asexually. The first point is not what occurs in nature; all evolution
is mutation-limited. As shown in the model analysis, however, surviving discrete traits are
independent of the initial distributions if all traits are provided initially. The second point
might also not be realistic. It ignores genetics but brings a merit of tractability to the model.
Most ESS models focus only on phenotypic traits and ignore the detailed genetic background only to seek a solution in terms of optimization, but they succeed in providing
insightful results. The ‘ideal free evolution’ approach appears to warrant further study.
This approach could be applied to general evolutionary topics that involve (co)evolutionary change in quantitative traits, such as predator–prey interactions, competition, symbiosis
and community dynamics. I believe that avian brood parasitism is an excellent system tostudy co-evolutionary processes using both empirical and theoretical approaches. Further
study will enrich our understanding of co-evolutionary processes in general.
ACKNOWLEDGEMENTS
I thank E. Røskaft and A. Sasaki for valuable comments on earlier drafts of the manuscript.
Discussion with N. Davies, M. Higashi, Y. Iwasa, A. Lindholm, H. Nakamura, S. Petrovskii, H. Seno,
N. Shigesada, B. Stokke and N. Yamamura improved the manuscript. This work was supported inpart by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (12740422).

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APPENDIX
Derivation of the normal distribution approximation
Given the population densities producing egg type x, P(x) and H(x), the population means of theparasite and host populations, mP and mH, respectively, are given as follows:mP =∫ xP(x)dx∫ P(x)dxmH =∫ xH(x)dx∫ H(x)dx

And the variances, σ 2P and σ 2H, are given as follows:
σ 2P =∫ x2P(x)dx− m 2P∫ P(x)dx

σ 2H =∫ x2H(x)dx− m 2H∫ H(x)dx

In the normal distribution approximation, the total densities, P⬘ and H⬘, the population means,m⬘P and m⬘H, and the variances, σ⬘2P and σ⬘2H, in the following year are given using the following
calculus:∞—

冮 exp[− (ax + bx + c)]dx = 冪 πa exp冤4ab − c冥
2

2

−∞
∞

−b —

冮 x exp[− (ax + bx + c)]dx = 2a 冪 πa exp冤4ab − c冥
2

2

−∞
∞

2a + b2

冮 x exp[− (ax + bx + c)]dx = 4a
2

2

2

−∞

—
π
b2
exp
−c
a
4a

冪

冤

冥

where a, b and c are real constants (a > 0).
The variance of parasite egg appearance in the following year, σ⬘2P, is given using the following
calculus:
E [x2] =∫ x2P⬘(x)dx sPP{1 + 2γ(σ2P + σ2H)}2 + (1 − e−P)H(1 + 2γσ2H)2 + 2γσ2H(1 + 2γd 2 + 2γσ2H)C=∫ P⬘(x)dx{1 + 2γ(σ2P + σ2H)}2 {sPP + (1 − e−P)HC}

as a function of P, H, d = mH − mP, σ 2P and σ 2H, where冤

1− γd 2
C = ———————exp
1 + 2γ(σ 2P + σ 2H)√ 1 + 2γ(σ 2P + σ 2H)

冥In the same way, we can derive the variance of the host, σ⬘2H, using
E [x2] =∫ x2H⬘(x)dx=∫ H⬘(x)dx(f + sH)(d 2 + σ2H){1 + 2γ(σ2P + σ2H)}2 − f (1 − e−P)(1 + 2γσ2P){d 2(1 + 2γσ2P) + σ2H{1 + 2γ(σ2P + σ2H)}}C{1 + 2γ(σ 2P + σ 2H)}2{f + sH − f (1 − e−P)C}362

TakasuChange in population means when the variances are small
If the variances, σ2P and σ2H, are small enough, the change in the population means per year, ∆mP and∆mH, can be Taylor expanded to give the following series:2(1 − e−P)He − γdγd σ2P + O(σ4P, σ4H)
sPP + (1 − e−P)He − γd
2

∆mP =

2

2 f (1 − e−P)e − γdγd σ2H + O(σ4P, σ4H)
f + sH − f (1 − e − P)e−γd
2

∆mH =

2

Substituting the internal equilibrium of the monomorphic dynamics (4) and (5) into these equations
gives (12).