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Keywords:

  • avian brood parasitism;
  • co-evolution;
  • egg phenotype;
  • frequency-dependent selection;
  • oscillation;
  • population genetics model

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

In avian brood parasitism, egg phenotype plays a key role for both host and parasite reproduction. Several parrotbill species of the genus Paradoxornis are parasitized by the common cuckoo Cuculus canorus, and clear polymorphism in egg phenotype is observed. In this article, we develop a population genetics model in order to identify the key parameters that control the maintenance of egg polymorphism. The model analyses show that egg polymorphism can be maintained either statically as an equilibrium or dynamically with frequency oscillations depending on the sensitivity of the host against unlike eggs and how the parasite targets host nests with specific egg phenotypes. On the basis of the model, we discuss egg polymorphism observed in parrotbills and other host species parasitized by the cuckoo. We suggest the possibility that frequencies of egg phenotypes oscillate and we appeal for monitoring of cuckoo–host interactions over a large spatiotemporal scale.


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Polymorphism in natural populations occurs as discontinuous, discrete assemblages of individuals with a shared phenotype. Such polymorphism may evolve and can be maintained as a consequence of frequency-dependent selection (Kettlewell, 1973; Majerus, 1998; Bond, 2007). Rare mutants for novel discrete phenotypes may spread if they enjoy a selective frequency-dependent advantage during interactions with any selective agents. Here, we focus on polymorphic phenotypes of the eggs that birds produce because they can be a crucial component for successful reproduction by both host and parasite in avian brood parasitism.

Accepting brood parasitism usually results in significant reduction in reproductive success for the host (Rothstein, 1990; Davies, 2000). This strong parasitism pressure constitutes a driving force for co-evolutionary interactions between the parasite and the host where the host evolves defences against parasitism like the ability to recognize and reject parasitic eggs that look dissimilar in appearance to its own eggs (Rothstein, 1975; Davies & Brooke, 1988; Moksnes et al., 1990) and the parasite evolves better egg mimicry to counter the host defence (Brooke & Davies, 1988). The host may further counter egg mimicry by the parasite by decreasing intraclutch variation and increasing interclutch variation in egg phenotype (Øien et al., 1995; Stokke et al., 2002, 2007).

The common cuckoo Cuculus canorus, one of the best studied avian brood parasites, parasitizes several parrotbill species of the genus Paradoxornis. Kim et al. (1995) showed that the vinous-throated parrotbill P. webbianus in Korea exhibits clear dimorphism in egg colour; a clutch contains either white or blue eggs. Lee & Yoo (2004) and Lee et al. (2005) demonstrated that the vinous-throated parrotbill has the ability to recognize and reject unlike eggs as do many other cuckoo hosts. Despite the dimorphism in egg colour of the parrotbill, however, only blue eggs are found in the cuckoo in Korea (Kim et al., 1995; Lee & Yoo, 2004; Lee et al., 2005).

Recently, Yang et al. (2010) showed that the ashy-throated parrotbill P. alphonisianus in southern China, a closely related species of P. webbianus, shows clear polymorphism in egg colour, with white, pale blue and blue eggs occurring in both the parrotbill and the cuckoo population. Yang et al. (2010) also demonstrated that the ashy-throated parrotbill has a fine-tuned ability to recognize and reject eggs that are dissimilar beyond a certain threshold. They suggested the possibility that egg polymorphism both in the parrotbill and in the cuckoo has evolved as a result of co-evolutionary interaction between them.

Apparent absence of white cuckoo eggs in Korea should favour parrotbills that produce white eggs because such ‘white parrotbills’ can reject blue cuckoo eggs better than ‘blue parrotbills’ (Lee et al., 2005). Increase in the frequency of white parrotbills may be followed by the emergence of ‘white cuckoos’ that can exploit white parrotbills more efficiently than blue cuckoos. This parasitic interaction naturally raises an intriguing question about the maintenance of such mutual egg dimorphism in both host and parasite under frequency-dependent selection. Presence of the three egg colours both in the parrotbill and in the cuckoo in China also poses the same question: How can egg polymorphism be maintained in both host and parasite in this co-evolutionary arms race?

Egg phenotype including background colour, patterns like spots, blotches and lines is likely genetically determined, and a female produces eggs of a constant phenotype throughout her lifetime (Collias, 1993; Gibbs et al., 2000; Gosler et al., 2000; Mahler et al., 2008; Moksnes et al., 2008; Morales et al., 2010). Several independent cases of evolution of egg polymorphism (Kilner, 2006) suggest that the underlying genetic mechanisms are simple and possibly only involve one or at most a few loci. A recent study on the common cuckoo indicates that the genes determining egg coloration are most likely found on autosomal loci, rather than the W-chromosome as previously assumed, and therefore are subject to Mendelian inheritance (Fossøy et al., 2011). Also, several empirical studies suggest that the background egg coloration is governed by at least two autosomal loci in birds (Wei et al., 1992; Collias, 1993; Ito et al., 1993). Because egg phenotype plays a key role for both the host and the parasite to successfully reproduce in avian brood parasitism, frequency-dependent selection is expected to work on egg phenotype to cause the egg polymorphism we observed in Korea and China.

Previous theoretical studies have demonstrated that host–parasite co-evolution can promote polymorphism in the levels of host resistance and parasite virulence with their levels fluctuating cyclically and that such co-evolutionary cycles are likely to occur in antagonistic interactions in general (Sasaki, 2000; Nuismer & Thompson, 2006; Nuismer et al., 2007; Tellier & Brown, 2007a,b). Yoder & Nuismer (2010) have analysed the effect of ecological interactions on phenotypic diversification and shown that antagonism generally promotes phenotypic diversification both in hosts and in parasites. Thus, the polymorphism, an extreme phenotypic diversification, observed in the cuckoo and parrotbill interactions could be conceptually understood by these previous models; egg polymorphism may be maintained with oscillating frequencies of each egg phenotype. However, these models are based on simplified assumptions such that hosts and parasites encounter each other completely randomly and that the trait in focus is asexually inherited in haploid organisms. We consider these models too simplistic to provide any quantitative and empirically testable predictions on frequency changes of egg phenotypes observed in avian brood parasitism.

In order to understand how egg polymorphism can be maintained in avian brood parasitism and to provide quantitative predictions that can be empirically tested, we here construct a population genetics model with biologically plausible genetic and ecological assumptions. On the basis of the model analysis, we identify the key parameters that control the maintenance of egg polymorphism in avian brood parasitism. We suggest the possibility that the parrotbill–cuckoo interactions exhibit oscillations in frequencies of distinct egg colours and that egg polymorphism can be statically or dynamically maintained in the host and the parasite population depending on the sensitivity of the host when recognizing unlike eggs and how the parasite targets to parasitize host nests with a certain egg phenotype. We discuss egg polymorphism observed in other brood parasitic interactions with implications for the importance of studying the behaviour by the parasite when utilizing hosts with polymorphic eggs as well as the genetic mechanism of egg phenotype. Finally, we emphasize that our findings have general applications outside avian brood parasitism to include antagonistic interactions in general.

The model

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

We assume sexual and diploid population for both the host and the parasite. We assume three distinct egg colours, white, pale blue and blue, as observed in the parrotbill–cuckoo interaction (Yang et al., 2010).

We assume the following genetics for the expression of egg phenotype. Egg colour is controlled by two autosomal loci. The first locus controls the presence or absence of blueness with two alleles, b and B. The allele B expresses blueness and is assumed to be dominant over b. The second locus controls the expression of the blueness with two alleles, m and M. The allele M modifies the expression of blueness and makes eggs pale blue and is assumed to be dominant over m. This diallelic two-locus assumption is based on the sexual inheritance of egg colour observed in chickens Gallus gallus, village weavers Ploceus cucullatus and Japanese quails Coturnix japonica (Wei et al., 1992; Collias, 1993; Ito et al., 1993). Recombination rate of the two loci is r (0 ≤  1/2).

As a notation, we hereafter denote egg colour as 0 (white), 1/2 (pale blue) and 1 (blue). Ten genotypes are possible in a population, each indexed as i and having a phenotypic value as follows (= 1, 2,..., 10): (Genotype i, its phenotype): (bm/bm, 0), (bm/bM, 0), (bM/bM, 0), (bm/Bm, 1), (bm/BM, 1/2), (bM/Bm, 1/2), (bM/BM, 1/2), (Bm/Bm, 1), (Bm/BM, 1/2) and (BM/BM, 1/2). Let hi and pi be the frequency of genotype i in the host and the parasite population, respectively. The phenotypic frequency of egg colour 0, 1/2, 1, respectively, is denoted as fh0 = h1 + h2 + h3, fh1/2 = h5 + h6 + h7 + h9 + h10, fh1 =  h4 + h8 for the host and fp0 = p1 + p2 + p3, fp1/2 =  p5 + p6 + p7 + p9 + p10, fp1 = p4 + p8 for the parasite (fh0 + fh1/2 + fh1 = 1, fp0 + fp1/2 + fp1 = 1). We assume infinitely large population, random mating and nonoverlapping generations.

We next assume the following ecological situations. Each host female builds a nest and completes a clutch of egg colour determined by her genotype. Each nest is parasitized by parasitism rate P (0 < < 1), and the probability that a nest of host female genotype i is parasitized by a parasite genotype j conditional on the nest is parasitized is assumed to be φipj where φi is the probability that a parasite targets the nest of a host genotype i. We here assumed that a host nest is parasitized once at most. Multiple parasitism is ignored in our model, which is a close approximation of reality because the probability of multiple parasitism is low in parrotbills and most other hosts (but see Moskát & Honza (2002), Takasu & Moskát (2011) for high parasitism rate that remained constant among years).

It remains unknown how a parasite decides to parasitize a host nest having a certain colour of eggs in the presence of egg polymorphism in the host population. Because parasites often visit host nests without laying parasitic egg presumably to check nest content (Moksnes et al., 2000), the way a parasite utilizes a host nest may not be random (Avilés et al., 2006; Cherry et al., 2007), that is, it may be influenced by frequencies of egg phenotypes in the host population and φi might not necessarily be the same as the frequency hi of host genotype i. Such nonrandom search has been demonstrated as frequency-dependent switching when birds use search images to look for prey of a particular colour or pattern (Bond, 1983). In order to consider nonrandomness of parasitic behaviour, we assume that φi is given as follows using the host phenotype frequencies, fh0, fh1/2, fh1.

  • image
  • image
  • image

where the exponent n controls the propensity that the parasite is attracted to parasitize hosts having a certain egg colour; the parasite utilizes hosts just randomly according to the actual frequencies (= 1 so φi = hi), the parasite is more likely to utilize a host with rare egg colour than would be expected by chance (< 1), or the parasite is more likely to utilize a host with common egg colour than would be expected by chance (> 1) (Fig. 1).

image

Figure 1.  The way the parasite targets to parasitize a host nest. The inner disc represents host phenotypic frequency of the three colours, 0, 1/2 and 1, each being 16.7%, 33.3% and 50.0% (1/6, 2/6, 3/6), respectively, as an example. The outer disc represents relative frequencies of host nests being actually parasitized, φ1 + φ2 + φ3, φ5 + φ6 + φ7 + φ9 + φ10 and φ4 + φ8. If parasites randomly search for host nests to parasitize (= 1), the relative frequencies become identical to the actual frequencies of the three colours (Left). If parasites utilize hosts with an egg colour in minority more than actual frequencies (< 1), hosts with colour 0 are more likely parasitized (Middle). If parasites utilize hosts with an egg colour in majority more than actual frequencies (> 1), hosts with colour 1 are more likely parasitized (Right).

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We assume that all host males and females have the same ability to recognize and reject unlike eggs and this recognition ability is innate, not learnt. The probability that a host with egg colour CH accepts parasitic egg CP in the nest is denoted as A(CH, CP) and it is assumed to be a decreasing function of the absolute difference in colour |CH − CP| (CH, CP = 0, 1/2, 1); the greater the contrast in colour, the smaller the acceptance probability as has been demonstrated in many host species and modelled (Higuchi, 1998; Takasu, 2003; Stokke et al., 2007; Yang et al., 2010). To simplify the notation, we denote a0 = A(0, 0) = A(1/2, 1/2) = A(1, 1), a1/2 = A(0, 1/2) = A(1/2, 0) = A(1, 1/2) = A(1/2, 1), a1 = A(0, 1) = A(1, 0) (1 ≥ a0 > a1/2 > a1 ≥ 0). A parasitized host breeding pair can produce own offspring only when it rejects parasitism successfully. Otherwise, a parasite chick fledges from the parasitized host nest. Sex ratio is fixed 1:1, and the genotype frequencies are the same in males and females both in the host and in the parasite. Fitness is defined as the reproductive success from a pair of two genotypes. Table 1 summarizes these ecological assumptions.

Table 1.   Combinations of host and parasite genotypes conditional on parasitism having occurred with parasitism rate P. The upper is host fitness as the probability of rejecting parasitism. The lower is parasite fitness as the probability of parasitism being accepted.
 Host
ParasitesPhenotype: 0 Genotype freq.: h1, h2, h3 Prob.: φ1, φ2, φ3Phenotype: 1/2 Genotype freq.: h5, h6, h7, h9, h10 Prob.: φ5, φ6, φ7, φ9, φ10Phenotype: 1 Genotype freq.: h4, h8 Prob.: φ4, φ8
Phenotype: 01 − a01 − a1/21 − a1
Genotype freq.: p1, p2, p3a0a1/2a1
Phenotype: 1/21 − a1/21 − a01 − a1/2
Genotype freq.: p5, p6, p7, p9, p10a1/2a0a1/2
Phenotype: 11 − a11 − a1/21 − a0
Genotype freq.: p4, p8a1a1/2a0

Using vector notation = (h1, h2,..., h10) and = (p1, p2,..., p10), the genotype frequencies at the next generation h′ and p′ are given as follows:

  • image(1)
  • image(2)

Here, inline image and inline image refer to the average fitness of the host and the parasite, respectively. TH and TP refer to a 10 by 102 transmission matrix that describes the distribution of offspring genotypes produced from a pair of two genotypes as a male and a female for the host and the parasite, respectively. WH and WP refer to a 102 by 102 diagonal matrix with coefficients of fitness as the reproductive success from a pair of two genotypes as a male and a female for the host and the parasite, respectively. ⊗ is Kronecker product, and hh and pp refer to the frequencies of mating pairs for the host and the parasite, respectively. See Appendix (A1 through A9) for the derivation.

The coupled dynamics of eqns (1) and (2) describes temporal change in the genotype frequencies hi and pi (= 1, 2,..., 10) and hence the phenotype frequencies fh0, fh1/2, fh1, fp0, fp1/2, fp1 under the genetic and ecological assumptions explained above. In the next section, we analyse the frequency dynamics of egg colours, mainly focusing on the maintenance of egg polymorphism, firstly for asexual and secondly for sexual inheritance of egg colour.

Results

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Asexual inheritance of egg colour

When egg colour is asexually inherited to daughters, the coupled dynamics (1) and (2) is reduced to the simpler dynamics of the six phenotype frequencies fh0, fh1/2, fh1, fp0, fp1/2 and fp1 (see Appendix). When the parasite utilizes host nests randomly according to their frequencies (= 1), the reduced dynamics is analytically tractable.

We first look for equilibria at which the six phenotype frequencies temporarily remain unchanged, that is,

  • image

(* denotes equilibrium). Under the ecological assumptions that parasites randomly parasitize (= 1) and that unlike eggs are more likely to be rejected (1 ≥ a0 > a1/2 > a1 ≥ 0), there exists a unique trimorphic equilibrium where all colours are present in both the host and the parasite population as

  • image(4)

if and only if the condition

  • image(5)

is satisfied. The condition (5) requires that the host accepts moderately mimetic eggs with probability a1/2 being less than the arithmetic average of a0 and a1, which biologically means that the host has high sensitivity to discriminate unlike eggs (Fig. 2).

image

Figure 2.  The acceptance probabilities of parasitic eggs by the host, a0, a1/2 and a1, plotted against the difference in egg colour. The host does not tolerate moderately mimetic eggs (thick lines) and host sensitivity is high (a0 − 2a1/2 + a1 > 0). The host tolerates the moderately mimetic eggs (grey lines) and sensitivity is low (a0 − 2a1/2 + a1 < 0). The unique trimorphic equilibrium (4) is possible if and only if the condition (5), a0 − 2a1/2 + a1 > 0, is satisfied.

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Besides the trimorphic equilibrium (4), there are fourteen equilibria where hosts and parasites are either monomorphic or dimorphic: nine equilibria where both hosts and parasites are monomorphic with one of the three phenotypes; one equilibrium where hosts are dimorphic with 0 and 1 and parasites are monomorphic with 1/2; one equilibrium where hosts are monomorphic with 1/2 and parasites are dimorphic with 0 and 1; and three equilibria where both hosts and parasites are dimorphic with 0 and 1/2, 0 and 1, and 1/2 and 1, respectively (Fig. 3). These fourteen equilibria are possible irrespective of the condition (5).

image

Figure 3.  Possible equilibria of the frequency dynamics of (1) and (2) when egg colour is asexually inherited and the parasite parasitizes randomly (= 1). The horizontal axis represents the egg colour of the host and the parasite. The vertical axis represents the frequency (scale is arbitrary except for monomorphic equilibria). (a) Both the host and the parasite are monomorphic in egg colour. (b) The host is dimorphic with 0 and 1 and the parasite monomorphic with 1/2. Or the host is monomorphic with 1/2 and the parasite dimorphic with 0 and 1. (c) Both the host and the parasite are dimorphic. (d) All colours are present both in the host and in the parasite. This trimorphic equilibrium is possible if and only if condition (5) is satisfied.

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Local stability of these equilibria can be analysed by linearizing the dynamics (1) and (2) (Murray, 2007). It turns out that all of the equilibria derived above are unstable and that the linearized dynamics when at least two phenotypes are present both in the host and in the parasite population shows oscillations with a period dependent on a0, a1/2, a1 and P. See Appendix for the derivation of these results.

Instability of equilibria where both the host and the parasite are monomorphic can be readily shown as follows (see Fig. 3a). When both are monomorphic with the same phenotype, rare host mutants having different phenotype can always invade the host population because these have higher chance to reject parasitism and increase in frequency. In the same logic, when both are monomorphic but with different phenotype, rare parasite mutants having phenotype more mimetic to that of hosts can always invade the parasite population because these have higher chance of parasitism acceptance and increase in frequency. Therefore, monomorphic hosts and monomorphic parasites cannot be maintained stably. The same logic applies to equilibria where either hosts or parasites, or both, are dimorphic lacking a particular egg colour (Fig. 3b, c).

Figure 4 shows typical frequency dynamics for asexual inheritance of egg colour. For dimorphic hosts and dimorphic parasites with white and blue eggs (pale blue egg is absent in both populations) and the parasite utilizes hosts randomly (= 1), the frequencies of the two phenotypes continue to oscillate. Amplitude of oscillation is larger in the parasite than in the host and parasite frequencies apparently converge to a heteroclinic cycle where one phenotype dominates for a longer and longer time but is eventually replaced by another phenotype (Fig. 4a) (Seger, 1988). When the parasite utilizes hosts with egg colour in minority more than its frequency (< 1), the frequency dynamics can be stabilized and the two phenotypes coexist but dynamically (Fig. 4b). In contrast, when the parasite utilizes hosts with egg colour in majority more than the frequency (> 1), the dynamics is more destabilized and shows a heteroclinic cycle similar to the case = 1 (not shown).

image

Figure 4.  Frequency dynamics of the three phenotypes of the host and the parasite when egg colour is asexually inherited. Black curve represents frequency of white 0, grey curve for blue 1 and dotted curve for pale blue 1/2. (a) Pale blue eggs (1/2) are absent in both the host and the parasite. Parasites search for host nests randomly (= 1). (b) Same as (a) but parasites utilize host nests with an egg colour in minority more often than its frequency (= 0.5). (c) All colours are present and parasites search for host nests randomly (= 1). (d) Same as (c) but parasites are attracted to hosts with an egg colour in minority (= 0.5). Initial frequencies of the three colours in the host and the parasite (fh0, fh1/2, fh1, fp0, fp1/2, fp1) are set equal to those observed in Yang et al. (2010), (313/547, 0, 234/547, 10/21, 0, 11/21) for a) and b), (313/555, 8/555, 234/555, 10/24, 3/24, 11/24) for (c) and (d), respectively. Parameters used in common are a0 = 0.8, a1/2 = 0.3, a1 = 0.1 and = 0.05 based on Yang et al. (2010).

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For trimorphic hosts and trimorphic parasites when condition (5) is satisfied and the parasite utilizes hosts randomly (= 1), the frequencies of the three phenotypes exhibit complex dynamics, eventually converging to a heteroclinic cycle (Fig. 4c). For n being sufficiently small (< 1), an equilibrium can be reached where all three phenotypes are stably maintained both in the host and in the parasite (Fig. 4d). When the parasite utilizes hosts with egg colour in majority more than the frequency (> 1), the frequencies exhibit complex behaviours and eventually show a heteroclinic cycle similar to the case = 1 (not shown). When condition (5) is not met, the dynamics starting from all phenotypes present converges to an equilibrium where hosts are dimorphic with white and blue and parasites are monomorphic with pale blue (not shown).

Figure 5 shows the dependency of the period of oscillation in the early dynamics on the parasitism rate P. The phenotype frequencies oscillate roughly with a period proportional to the inverse of the square root of the parasitism rate P as predicted by local stability analysis (see Appendix).

image

Figure 5.  Dependency of oscillation period T on the parasitism rate P. Dot and rectangle show period T for asexual and sexual inheritance, respectively. Dotted and thick curve represent least-square fit of the inverse of square root of the parasitism rate P, 20.0/√P (asexual) and 41.9/√P (sexual). Oscillation period T was calculated by Fourier analysis from time series of the phenotypic frequencies where pale blue was absent and parasites utilize hosts randomly (= 1). For asexual inheritance, time series data in the early 300 generations were used to avoid the effect caused by heteroclinic cycle.

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Sexual inheritance of egg colour

We next focus on the case that egg colour is sexually inherited, a likely case in the cuckoo–parrotbill interactions. The dynamics (1) and (2) are intractable and we numerically analyse the behaviour.

Figure 6 shows typical frequency dynamics for sexual inheritance of egg colour. For dimorphic hosts and dimorphic parasites where there are no pale blue eggs (the allele M is absent) and the parasite utilizes hosts randomly (= 1), the phenotypic frequencies oscillate but the oscillation lasts more stably with a longer period compared with the asexual case (Fig. 6a, cf. Fig. 4a). When the parasite utilizes hosts with egg colour in minority more than its frequency (< 1), the dynamics is stabilized where both the two phenotypes coexist nearly equally (Fig. 6b). In contrast, when the parasite utilizes hosts with egg colour in majority more than the frequency (> 1), the dynamics is more destabilized and shows a heteroclinic cycle similar to the asexual case = 1 (not shown).

image

Figure 6.  Frequency dynamics of the three phenotypes of the host and the parasite when egg colour is sexually inherited. Black curve represents the frequency of white 0, grey curve for blue 1 and dotted curve for pale blue 1/2. (a) Pale blue eggs (1/2) are absent in both the host and the parasite (the allele M is absent). Parasites search for host nests randomly (= 1). (b) Same as (a) but parasites utilize host nests with an egg colour in minority more often than its frequency (= 0.5). (c) All colours are present and parasites search for host nests randomly (= 1). (d) Same as (c) but parasites are attracted to hosts with an egg colour in minority (= 0.5). Note that the dynamics (1) and (2) are calculated up to 4000 generations for (c) and (d). Initial frequencies of ten genotypes in the host and the parasite are set equal to the Hardy–Weinberg equilibrium whose phenotypic frequencies match those observed in Yang et al. (2010), (fh0, fh1/2, fh1, fp0, fp1/2, fp1) = (313/547, 0, 234/547, 10/21, 0, 11/21) for (a) and (b), (313/555, 8/555, 234/555, 10/24, 3/24, 11/24) for (c) and (d), respectively. Parameters used in common are a0 = 0.8, a1/2 = 0.3, a1 = 0.1, = 0.05 and recombination rate rH = rP = 0.5 (the two loci are not linked).

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For trimorphic hosts and trimorphic parasites when condition (5) is satisfied and the parasite parasitizes the host randomly (= 1), the frequencies of the three phenotypes exhibit complex dynamics, eventually converging to a heteroclinic cycle (Fig. 6c). For n being sufficiently small, a stable equilibrium can be reached where all three phenotypes are stably maintained both in the host and in the parasite (Fig. 6d). When condition (5) is not met, the dynamics starting from all phenotypes present converges to an equilibrium where hosts are dimorphic with white 0 and blue 1 and parasites are monomorphic with pale blue 1/2 (not shown).

Dependency of the period of oscillation on the parasitism rate P for the sexual case is shown in Fig. 5. Oscillation period is proportional to the inverse of the square root of P but it is nearly two times larger than that of the asexual case.

We also investigated the effect of recombination rate of the two loci, rH and rP, on the frequency dynamics. The dynamics are quantitatively similar unless the recombination rate is nearly zero, which is consistent with linkage disequilibrium decreasing exponentially whereby all genotypes are eventually created (Crow & Kimura, 1970).

Discussion

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

We have shown that the frequency dynamics of genotypes/phenotypes critically depends on ecological factors, the sensitivity of hosts when recognizing unlike eggs (acceptance probabilities a0, a1/2 and a1), and how the parasite utilizes the host (the exponent n).

Egg polymorphism with three phenotypes present can be maintained only when the host has a high sensitivity to reject moderately mimetic eggs (a0 − 2a1/2 + a1 > 0). Otherwise, an equilibrium is reached where the parasite shows monomorphism with the intermediate pale blue colour and the host showing dimorphism with two extremes of white and blue.

The way the parasite targets hosts, the exponent n, critically affects if polymorphism is statically or dynamically maintained; only when n is sufficiently small so that hosts with an egg colour in minority are more parasitized, the frequency dynamics converges to a stationary polymorphism. With smaller n, a negative feedback operates on the risk of a host being parasitized and this contributes to stabilize the frequency dynamics. Otherwise, frequencies oscillate with a period roughly proportional to the inverse square root of the parasitism rate and polymorphism is dynamically maintained. In this case, the amplitude of oscillation is always larger in the parasite than in the host. This is because selection operates stronger in the parasite than in the host; all parasites experience judgment of a host’s accepting or rejecting parasitism, whereas not all hosts are parasitized. If the amplitude is large enough, a certain phenotype may be lost by chance in the parasite population when the frequency becomes extremely low. This demographic stochasticity has been ignored in our model but it could be significant as we argue in the real system below.

These results remain qualitatively the same, irrespective of the asexual or sexual inheritance of egg colour, although it quantitatively affects the period with which frequencies oscillate; sexual inheritance results in nearly two times longer period of frequency oscillation compared with asexual case.

We assumed that the acceptance probabilities A(CH, CP) are a decreasing function of the difference in colour |CH − CP| (CH, CP = 0, 1/2, 1) and that these can be represented by three parameters, a0, a1/2 and a1 (1 ≥ a0 > a1/2 > a1 ≥ 0), that is, the dynamical systems (1) and (2) are structurally symmetric. We have carried out numerical analyses where each of the acceptance probabilities A(CH, CP) is randomly perturbed around a0, a1/2, a1 with a certain range to incorporate asymmetry and found that all the results remain qualitatively similar. Thus, we conclude that our results are robust and not artefacts caused by the symmetry of the model structure.

We assumed diallelic two-locus genetics where both the blue allele B and the modifier allele M are dominant. We confirmed that modification of this genetic assumption of dominance does not change the results greatly. Simple genetics as we assumed might be justified by independent evolution of polymorphism in egg colour in avian brood parasitism (Kilner, 2006). Further empirical study to elucidate the detailed genetic mechanisms underlying egg phenotype expression is certainly needed.

The importance of host sensitivity against unlike egg has been suggested in theoretical studies where egg phenotype is assumed to be a continuous trait that is asexually inherited; the higher the host sensitivity to discriminate unlike eggs, the more discrete egg phenotypes can coexist both in the host and in the parasite population (Takasu, 2003, 2005). We have obtained qualitatively the same result in our model. Our results also corroborate previous theoretical studies that co-evolutionary dynamics of adaptive traits in antagonistic interactions between prey/host and predator/parasite tend to exhibit oscillation in the level of adaptive traits of hosts and prey (resistance to parasitism or predation) and parasites and predators (virulence or attack rate) (Seger, 1988; Gavrilets & Hastings, 1998; Gandon, 2002; Nuismer et al., 2005; Kopp & Gavrilets, 2006; Nuismer & Thompson, 2006; Tellier & Brown, 2007a,b). In these previous models, however, simpler assumptions like haploid populations, asexual inheritance of phenotypes and random encounters of antagonistic organisms (= 1 in our model) are assumed. Random encounters that many of the previous models have assumed may be justified for microparasites like viruses that passively contact with target hosts. In contrast, avian brood parasites do not necessarily parasitize hosts randomly (Avilés et al., 2006; Cherry et al., 2007) as complex cognitive mechanisms may be involved in search for target hosts (Bond, 1983). We have shown that relaxing the random encounter rule results in the novel finding that polymorphism can be statically maintained if a rarer host egg phenotype attracts more parasites. The way the parasite utilizes hosts is likely to evolve, although it has not been considered in our model. We stress the need for theoretical study that focuses on the evolution of parasite behaviours coupled with the evolution of egg polymorphism.

Throughout this article, we assume that brood parasitic pressure is a primary factor responsible for co-evolving egg phenotype. Egg phenotype, however, could be under different selective pressures such as selection by predators (crypsis or aposematism) and selection by mates (signal of female quality) under a certain nutritional and physiological constraints (Moreno & Osorno, 2003; Soler et al., 2005; Kilner, 2006; Moreno et al., 2008). We stress the need for comprehensive study focusing on the significance of egg phenotype that is subjected to a myriad of selection pressures.

The vinous-throated parrotbill in Korea shows the dimorphism in egg colour as white and blue, and the ratio of white to blue varies from 0.21:0.79 to 0.4:0.6, while only blue cuckoo eggs are found (Kim et al., 1995; Lee & Yoo, 2004). The two egg colours may coexist either statically or dynamically depending on how the cuckoo parasitizes the dimorphic parrotbill as our model has shown. Frequency of the cuckoo parasitism in blue or white parrotbills paralleled the egg colour ratio of the parrotbill population (Lee et al., 2005). This implies that the cuckoo parasitizes the parrotbill in Korea just randomly. In this case, we expect frequency oscillation with the period in the order of a few hundred generations because 5.3% of nests (10 of 190) were found parasitized in Korea (Lee et al., 2005), and actual parasitism rate would be higher as unlike cuckoo egg had been rejected before detection (Fig. 5). Then, the apparent absence of white cuckoo eggs in Korea may have occurred by chance; cuckoos producing white eggs were once too small in frequency and they were lost by chance. We speculate that white cuckoos, if emerging by some reason like immigration from other area, are likely to increase in frequency and frequency oscillation may last. Studying the way the cuckoo selects host nests is needed together with long-term monitoring of egg colour frequencies in Korea.

The ashy-throated parrotbill in southern China shows polymorphism in egg colour, white, pale blue and blue, as well as in the cuckoo population. The ratio of white, pale blue and blue is 0.564:0.014:0.422 in the parrotbill, whereas it is 0.417:0.125:0.453 in the cuckoo (Yang et al., 2010). The parrotbill in southern China recognizes and rejects unlike eggs sensitively (Yang et al., 2010), and it is likely that condition (5) is satisfied, so that all the three colours can be maintained either statically or dynamically. Temporal variation in the three egg colours does not clearly show a trend of oscillation in frequencies over the past 10 years (Yang et al., 2010). Most likely, 10-year monitoring is too short to detect the frequency changes because 4.3% of the parrotbill nests (24 of 555) were found to be parasitized (Yang et al., 2010) and this gives an oscillation period in the order of a few hundred generations (Fig. 5). We suggest that the frequencies will likely change in the next several decades if the cuckoo in China parasitizes the parrotbill nearly randomly, irrespective of the parrotbill egg colour. Further long-term monitoring and behavioural study about the way the cuckoo parasitizes the parrotbill is needed.

Discrete polymorphism in egg phenotype is rare but in avian brood parasitism, and egg polymorphism has likely evolved through co-evolutionary interactions between brood parasites and their hosts (Kilner, 2006). The common cuckoo as a species produces eggs with a variety of phenotypes but the cuckoo as a whole consists of several independent host races, each of which is specialized on a particular host species by producing eggs mimetic to those of the host (Moksnes & Røskaft, 1995; Davies, 2000). A similar system is found in the Red-chested cuckoo Cuculus solitarius, which has three distinct egg phenotypes as chocolate/coffee brown, green-blue with red/brown spots and coffee with brown freckling (Kuiper & Cherry, 2002; Honza et al., 2005). Egg polymorphism in avian brood parasitism like the cuckoo and the parrotbill interaction (Kim et al., 1995; Lee & Yoo, 2004; Lee et al., 2005; Yang et al., 2010) constitutes an ideal system for studying how polymorphism can be maintained in a co-evolutionary context (Rothstein, 1990). It is ideal also because the time scale of the life cycle for the parasite and the host is nearly equal in avian brood parasitism, so that the evolution of novel egg phenotypes would proceed at equal pace in the two parties. Although empirical data on phenotypic frequencies and their temporal changes are too limited to draw any conclusion, we suggest the possibility that egg polymorphism in avian brood parasitism is dynamically maintained with temporarily varying phenotypic frequencies.

In this model, we assumed that both the host and the parasite populations are closed in the sense that there is no gene flow to and from the outside. Extending our model to consider spatial structure enables us to study a ‘metapopulation’ genetics in which we might expect a geographical gradient in phenotypic frequencies (Thompson, 2005). The absence of white cuckoos in Korea and the presence of the three egg colours in southern China may be continuously linked at a larger spatial scale. We may further observe temporally and spatially fluctuating phenotypic frequencies where egg polymorphism is dynamically maintained. Study focused at a larger spatial scale is certainly needed in order to better understand how egg polymorphism is maintained spatially and to demonstrate co-evolution in action in avian brood parasitism, which will certainly contribute to enrich our general understanding of co-evolution in antagonistic interactions.

Acknowledgments

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

This work was supported by Program for New Century Excellent Talents in University (NCET-10-0111) to LW, by National Natural Science Foundation of China (No. 31071938) to LW, BGS and AA and by Centre for Advanced Study at the Norwegian Academy of Science and Letters (CAS) as a part of the project ‘Coevolutionary interactions and adaptations in a metapopulation context’. We thank the Forestry Department of Guizhou Province and Kuankuoshui National Nature Reserve for support and permission to carry out this study, and Y. Cai, J. Q. Wu, X. L. Guo, X. Xu, N. Wang and L. W. Wang for assistance with the field work. We also thank three anonymous reviewers for valuable comments and suggestions.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix

Appendix

  1. Top of page
  2. Abstract
  3. Introduction
  4. The model
  5. Results
  6. Discussion
  7. Acknowledgments
  8. References
  9. Appendix
Derivation of the model

In general, the frequency of genotype i in the next generation, inline image , is given as follows:

  • image((A 1))

where N is the number of genotypes (= 10 in our model), T (i[LEFTWARDS ARROW]j, k) is the transition probability that a breeding pair of genotype j as female and k as male produces offspring of genotype i, wj is the fitness as the reproductive success of the breeding pair in which the female has genotype j (males do not lay eggs and the male k does not influence the reproductive success in our model), and inline image is the average fitness defined by

  • image((A 2))

to normalize the genotype frequencies inline image (= 1, 2,..., 10) to sum up to 1. In vector and matrix notation, eqn (A1) is written as eqns (1) and (2) by replacing xi with hi and pi for the host and the parasite, respectively.

We here describe the details of fitness matrices WH, WP and the transmission matrices TH, TP.

Fitness matrix for the host WH is given as follows, using parasitism rate P:

  • image((A 3))

where the first term in the right-hand side is reproductive success when not parasitized that is assumed to be 1, and the second term is that when parasitized. Here, I is the 102 by 102 identity matrix with 1 along diagonal elements and 0 elsewhere. inline image consists of ten 10 by 10 diagonal matrices Wi defined as

  • image((A 4))

where wi is host reproductive success from host female genotype i (= 1, 2,..., 10). For the host to successfully reproduce, parasitism has to be rejected. Thus, from Table 1, wi is obtained as products of the probability of being parasitized by a parasite j and the probability of rejecting the parasitism

  • image((A 5))

where CH(i) and CP(i) refer to the egg colour of genotype i of the host and the parasite, respectively. We have assumed that the probability that a host genotype i is parasitized is given by φi in which the exponent n controls the way the parasite searches host nests. Thus, we have to divide the term by hi to derive fitness (when = 1, φi = hi).

For the parasite to successfully reproduce, parasitism has to be accepted. Thus, the fitness matrix for the parasite WP is given as follows:

  • image((A 6))

where the element wi of (A4) is replaced with

  • image((A 7))

as products of the probability of parasitism to host genotype j and the probability of the parasitism being accepted, summed over all possible host genotypes.

Transmission matrix T consists of ten 10 by 10 square matrices Ti (= 1, 2,..., 10) arranged horizontally

  • image((A 8))

The jth column of Ti represents the frequencies of offspring genotypes produced by a breeding pair (the female is genotype i and the male is j) and the column sum amounts to 1 (= 1, 2,..., 10). For example,

  • image((A9))

using recombination rate r of the two loci. In the same way, T2 through T10 are obtained. The host and the parasite have a recombination rate rH and rP, respectively.

Asexual inheritance

Asexual inheritance of egg phenotypes (no male contribution to offspring egg colour) can be implemented by setting the matrix Ti to have 0 in all elements except the ith row being 1 (= 1, 2,..., 10).

  • image((A 10))

This applies to both the host and the parasite. In asexual inheritance, the frequency dynamics of 10 genotypes in the host and the parasite populations (20 genotypes in total) can be reduced to the frequency dynamics of six phenotypes, = (fh0, fh1/2, fh1), = (fp0, fp1/2, fp1).

Local stability

Local stability of an equilibrium can be studied by examining eigenvalues of the linearized dynamics around the equilibrium.

For the equilibrium (4) when condition (5) is met and all three colours are present, we have two zero and four complex eigenvalues

  • image((A 11))
  • image((A 12))
  • image((A 13))

where

  • image((A 14))

is a positive real value when the condition (5) is satisfied and the parasitism rate P is small enough. The absolute value of the four complex eigenvalues is always larger than unity, and the equilibrium (4) is unstable; once perturbed, the six frequencies oscillate approximately with periods

  • image((A 15))

and

  • image((A 16))

both of which are proportional to the inverse of square root of parasitism rate P. A shorter period will dominate actual frequency oscillation.