i. Zitting cisticolas.

Differences between host and experimental eggs in three traits predicted rejection in zitting cisticolas: color (Estimate = 0.80 ± 0.22SE, Z₈₆ = 3.60, P < 0.001), marking luminance (Estimate = 0.60 ± 0.19SE, Z₈₆ = 3.16, P = 0.002), and mean marking size (Estimate = 2.22 ± 0.85SE, Z₈₆ = 2.60, P = 0.009; Nagelkerke’s R² = 0.70; Table 1). To the human eye, each of these three traits is categorically distributed in this species, with six observed trait combinations (Figs 1a and 2a). Almost all (51/52) foreign eggs that differed from the host clutch in one or more human-assigned category were rejected; almost all (35/38) eggs that did not differ in any category were accepted. Partitioning Around Medioids (PAM) cluster analyses (n = 119 for color, n = 400 for pattern traits) showed that indeed each trait was best described as two clusters, with the number of clusters differing significantly from one (all P < 0.01; Table 1), and with each cluster corresponding well to the human categorization of phenotypes (Fig 2a). These results were supported by direct tests for multimodality (S1 Text §d). In summary, zitting cisticolas showed categorical variation in traits predicting rejection.

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Table 1. Summary of results from egg rejection experiments and cluster analyses.

https://doi.org/10.1371/journal.pbio.3003667.t001

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Fig 2. Distributions of the traits predicting rejection in the four hosts studied.

Histograms illustrating categorical variation (all traits in (a); one trait in (b)) include dotted lines indicating the cutoffs between the two clusters, and bars colored according to the proportion of each bar corresponding to visual categorization of each trait. These illustrate that the clusters corresponded well to human categorization of phenotypes. Histograms illustrating continuous variation (two traits in (b); all traits in (c) and (d)) include gray bars. Insets show representative egg phenotypes from near the ends of each distribution. The data underlying this figure are available at https://doi.org/10.17863/CAM.116928.2.

https://doi.org/10.1371/journal.pbio.3003667.g002

ii. Croaking cisticolas.

Differences between host and experimental eggs in three traits predicted rejection in croaking cisticolas: color (Estimate = 0.79 ± 0.25SE, Z₄₇ = 3.18, P = 0.002), pattern coverage (Estimate = 30.83 ± 14.46SE, Z₄₇ = 2.13, P = 0.03; Nagelkerke’s R² = 0.43 for this model), and marking luminance (which, alongside color, predicted rejection when eggs were not immaculate: Estimate = 0.38 ± 0.16SE, Z₃₅ = 2.38, P = 0.02; Nagelkerke’s R² = 0.76 for this model excluding immaculate eggs; see S1 Text §b for details; also see Table 1). A PAM cluster analysis (n = 166 for marking luminance; n = 194 for pattern coverage) showed that for both traits the number of clusters did not differ significantly from one (marking luminance: P = 0.16; pattern coverage: P = 0.51; Table 1). However, color was best described as two clusters (n = 73; P = 0.03; Table 1) which corresponded well to human categorization of phenotypes (Fig 2b). These results were largely supported by tests for multimodality (S1 Text §d). Thus, croaking cisticolas exhibited both continuous and categorical variation in traits predicting rejection (Fig 2b).

iii. Tawny-flanked prinias (hereafter, prinias).

Differences between host and experimental eggs in three traits predicted egg rejection in prinias: color (Estimate = 0.27 ± 0.06SE, Z₁₅₃ = 4.2, P < 0.001), pattern coverage (Estimate = 15.46 ± 4.56SE, Z₁₅₃ = 3.39, P < 0.001), and pattern dispersion (Estimate = 3.38 ± 2.12SE, Z₁₅₃ = 1.59, P = 0.11; Nagelkerke’s R² = 0.36; Table 1). A PAM cluster analysis (n = 254 for egg color, n = 511 for pattern traits) showed no evidence that the number of clusters for any of the three traits differed from one (egg color: P = 0.06, pattern coverage: P = 0.18; pattern dispersion: P = 0.89; Table 1; see S1 Text §c and reference [27] for further evidence against egg color being categorically distributed in prinias). These results were supported by tests for multimodality (S1 Text §d). Thus, prinia eggs showed continuous variation in traits predicting rejection (Fig 2c).

iv. Red-faced cisticolas.

Differences between host and experimental eggs in two traits predicted rejection in red-faced cisticolas: marking luminance (Estimate = 0.30 ± 0.12SE, Z₅₅ = 2.43, P = 0.02) and the number of SIFT features (Estimate = 0.03 ± 0.01SE, Z₅₅ = 2.56, P = 0.01; Nagelkerke’s R² = 0.34; Table 1). A PAM cluster analysis (n = 182) showed that for both traits the number of clusters did not differ significantly from one (marking luminance: P = 0.44; number of SIFT features: P = 0.09; Table 1). These results were supported by tests for multimodality (S1 Text §d). Thus, red-faced cisticola eggs showed continuous variation in traits predicting rejection (Fig 2d).

Overall, for each species, the results of cluster analysis (and tests for multimodality; see S1 Text §d) corresponded well with human perception of their egg phenotypes. We also note that for each of the four species, none of the traits predicting rejection were highly correlated with each other (all R² < 0.23). Thus, each trait can be considered separately when testing for categorical variation.

i. Threshold-based rejection.

We first simulated continuous and categorical populations (Fig 4a) with threshold-based rejection, where eggs differing from the host phenotype by more than a specified threshold distance are rejected. In general, as the threshold increases, the rate of Type II errors (accepting parasitic eggs) will increase, since more parasitic eggs will fall within the threshold and thus be accepted. Similarly, as the threshold increases, the rate of Type I errors (rejecting own eggs) will decrease, since more of a host’s eggs will fall within the threshold and thus be accepted.

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Fig 4. Estimated rates of Type I errors (rejecting own eggs), Type II errors (accepting parasitic eggs), and overall success rates in simulated populations with parasitism rates of 50%, such that the maximum Type II error rate is 50%.

In panels (b), (c), (e), and (f), “threshold distance” refers to the distance whereby eggs differing from the host phenotype by more than that distance are rejected. Above: Threshold-based rejection of categorical vs. continuous egg phenotypes: (a) Simulated categorical (red) and continuous (blue) populations, each with mean trait value 0.5 and standard deviation 0.25. (b) Across a range of rejection thresholds, the probability of egg survival in each of the populations. At intermediate thresholds (ca. 0.25–0.55) the success rate in the categorical population is greater than that of the continuous population, due to (c) a reduced rate of Type II errors at these thresholds, without a concurrent increase in Type I error rate. Note that the dashed red and blue lines are virtually identical because, under threshold-based rejection, Type I error rates are unaffected by distribution shape (S1 Text §h). Below: Category-based rejection of categorical egg phenotypes vs. threshold-based rejection of continuous egg phenotypes. (d) Simulated categorical (red) and continuous (blue) populations. Here, categorical trait variation is modeled as binary, with no intra-clutch variation, to simulate category-based rejection. (e) Category-based rejection occurs in the categorical population at thresholds below 0.5. At all such thresholds, the categorical population experiences greater (or equal) success than the continuous population, due to (f) zero Type I errors and a low frequency of Type II errors. Indeed, the Type II error rate of 0.25 in the categorical population is only achievable in the continuous population when thresholds are lower than ~0.25. The simulated data underlying this figure are available at https://doi.org/10.17863/CAM.116928.2.

https://doi.org/10.1371/journal.pbio.3003667.g004

At intermediate thresholds, categorical populations experienced greater success than continuous populations (Fig 4b and Fig C in S1 Text). This difference arose because, in categorical populations, rates of Type II errors increase in a stepwise manner with increasing threshold, since the threshold encompasses first one and then both of the peaks of the distribution (solid lines in Fig 4c). To visualize this, consider a host whose eggs lie in the left-hand peak of the bimodal distribution (the following explanation applies similarly to a host whose eggs lie in the right-hand peak). With a low threshold, this host only accepts eggs close to that peak, so all eggs in the right-hand peak are rejected. Increasing the threshold to intermediate values mainly adds eggs from the low-density “valley” between the two peaks, so relatively few additional parasitic eggs fall within the threshold, and the Type II error rate therefore rises only slightly. By contrast, when the threshold is large enough that eggs from the right-hand peak also fall within it, a substantial additional number of parasitic eggs now lies within the threshold, and the Type II error rate increases sharply. Thus, increasing the threshold from low to intermediate values admits few extra parasitic eggs, whereas increasing it from intermediate to high values admits many, producing the stepwise increase in Type II error rate with threshold in categorical populations (solid red line in Fig 4c). This contrasts with continuous populations, where the increase in Type II error rate with threshold is smooth (blue solid line in Fig 4c). As a result, categorical variation allows lower Type II error rates at intermediate thresholds than continuous variation. Meanwhile, Type I error rates remain effectively identical between continuous and categorical populations (dashed lines in Fig 4c; S1 Text §h).

Overall, these simulations demonstrate that distribution shape can influence success rate under threshold-based rejection due to its effect on the proportion of foreign eggs that lie outside a given host’s threshold.

ii. Category-based rejection.

A second potential mechanism for why categorical variation can reduce error rates involves rejection that is not threshold-based, but is instead category-based. A categorical distribution allows for category-based rejection: hosts lay eggs of one category, always accept eggs of that category, and reject all eggs of different categories. We simulated this process for a trait with two categories by generating a truly binary distribution (Fig 4d) with no within-clutch variation, since under category-based rejection, bimodal phenotypes are perceived in a binary manner. We compared error rates between a categorical population exhibiting category-based rejection, and a continuous population (modeled in an identical manner to the continuous population described in (i); see Fig 4a and 4c), exhibiting threshold-based rejection. Category-based rejection does not involve well-defined thresholds, but can be thought of as occurring when the “threshold” is between zero and the distance between the peaks. Under these circumstances, hosts accept eggs of their own category and reject eggs of the other category – i.e., rejection is category-based. When the “threshold” is above the distance between the peaks, all eggs are accepted, and rejection is therefore no longer category-based.

This simulation showed that when eggs are rejected according to their category (i.e., 0 < threshold distance < 0.5), categorical variation can result in greater success rate than continuous variation (Fig 4e). This is because under category-based rejection, a host’s own eggs are always accepted (they always share the host’s category), so no Type I errors are made (Fig 4f). Category-based rejection also affects Type II error rates. If there are n equally common egg categories, then the parasitic egg in a parasitized nest matches the host’s category with probability 1/n, and is otherwise rejected. Thus, the probability of a Type II error in a parasitized nest is 1/n. Across the whole host population, the overall frequency of Type II errors across all nests (both parasitized and unparasitized) is P/n, where P is the parasitism rate (Fig 4f). In Fig 4d–4f, for example, P = 0.5 and n = 2, so the Type II error rate under category-based rejection is 0.25 (i.e., 25% of host nests will involve a Type II error; Fig 4f). Whenever P/n is lower than the Type II error rate in the continuous case (i.e., for sufficiently many categories), the categorical population will exhibit fewer Type I and Type II errors than the continuous population, thus escaping the Type I:Type II error trade-off.

Overall, category-based rejection alongside categorical variation can effectively eliminate Type I errors, and concurrently reduce Type II errors relative to continuous variation.

i. Estimating Type II error rates for each species.

Our experiments provided estimates of Type II error rates (accepting parasitic eggs) for each species – these “observed” rates were simply the proportion of experiments for which each host species accepted the conspecific egg placed in their nest to simulate parasitism. However, we also estimated rejection rates based on the rejection model of each species to control for the possibility that, by chance, different species were given differently-well-matched eggs across experiments (c.f. [28,44]). While we aimed to randomly choose experimental eggs with respect to host clutch phenotypes, our choice of experimental eggs was dependent on the host eggs present at the time of experimental manipulation that could be used as experimental eggs. Thus, as with all egg rejection experiments using conspecific or parasitic eggs, we could not ensure that all hosts were given equally difficult tasks [28,44]. If we had inadvertently given one species more difficult tasks than another, this would artificially increase the observed Type II error rate of the first species relative to the second, without necessarily indicating that the first is actually poorer at discriminating its eggs from those of its parasite. The solution to this problem is to estimate rejection rates based on all phenotypic data available [28,44]. Note that we did not use statistical analyses here (following previous work [44]) due to sample sizes being arbitrary—results can be visually compared in Fig 3.

Estimation of rejection rates involved parametrizing the rejection model with data from real eggs to estimate the likelihood of rejection based on random laying by the parasite. Parasites appear to lay at random with respect to host phenotype in this system [26,28,44].

For prinias and zitting cisticolas, data on parasitic eggs from 2012 to 2023 were available such that rejection rates of real parasitic phenotypes could be estimated. Each parasitic egg analyzed was laid in a different host nest (where more than one parasitic egg was laid in the same host nest, which is common [65], we randomly selected one for further analysis). This resulted in a dataset of n = 129 eggs laid in prinia nests (published in reference [29]) and n = 34 eggs laid in zitting cisticola nests (see S1 Text §f). We randomly selected one host egg from a clutch to represent the phenotype of the clutch. We then randomly selected a parasitic egg, and calculated the probability that it would be accepted according to the rejection model for that species. We generated 1,000 such combinations, with resulting estimates of Type II error rate shown in Fig 3.

The estimated Type II error rate using parasitic eggs is not necessarily comparable between species, since different hosts may be subjected to differing degrees of mimetic accuracy and phenotype distributions by the parasite lineages (known as “gentes”) that specialize on each host. We therefore also estimated parasitism events using host eggs as a proxy for parasitic eggs. This assumes that the distribution of parasitic phenotypes is identical to the distribution of host phenotypes. Using host eggs as proxies for parasitic eggs does not necessarily correspond to true biological reality (since in at least two of the host species studied here, there are consistent differences between parasitic and host eggs [19,28,29,34]), but it allows us to compare host species’ propensity for Type II errors controlling for potential differences in the accuracy of parasitic mimicry for each host. As with the estimated parasitism events described above, we randomly selected one host egg from a clutch. We then randomly selected another host egg from a different clutch, as a proxy for a parasitic egg, and calculated the probability of rejection of this egg based on the rejection model for that species. We generated 1,000 such combinations, with resulting estimates of Type II error rate shown in Fig 3.

Estimates of Type II error rates calculated as above are provided in the main text and summarized in Fig 3. These estimated Type II error rates were lower than observed in experiments (Fig 3). This is likely due to us inadvertently providing birds with slightly more difficult decisions (i.e., more closely matched eggs) than the intended random assignment. As discussed above, the assignment of “experimental” eggs to host clutches depended on the host eggs available to us at the time, and therefore it was not possible to make this assignment truly random. Furthermore, the bias could result because, in order to test effects of pattern differences, we needed to provide good color matches (and vice versa); this likely resulted in providing a greater proportion of good color and pattern matches than a truly random assignment would generate [18].

ii. Estimating Type I error rates for each species.

We estimated Type I error rates by generating combinations of eggs, each from the same clutch, quantifying the differences between them and estimating the probability that one of the eggs would be rejected according to that host’s rejection model. Only a subset of host clutches had reflectance spectra measured for multiple eggs within the same clutch (n = 73, 28, 21, and 30 for prinias, zitting cisticolas, red-faced cisticolas, and croaking cisticolas, respectively) because reflectance spectra were collected indoors at the field site base, not in the field. We avoided removing multiple eggs for spectrophotometry because this might influence rejection behavior. To incorporate color differences in our error rate estimations, we therefore randomly assigned to each pair of eggs one of the values for intra-clutch color difference from the subset of nests for which multiple eggs were measured. Because the JNDs were randomly assigned to comparisons, and because pattern and color traits are uncorrelated in hosts [19,44] (also see Results), this should not introduce any bias into these estimations.

Estimates of Type I error rates are provided in the main text and summarized in Fig 3. These estimates predicted higher rates of Type I errors than observed during field experiments, except for croaking cisticolas (see main text; also see S1 Text §g for a potential explanation for this relating to croaking cisticolas possessing large bills [66]). It may therefore be that hosts use other (unmeasured) cues to recognize their own eggs and thus avoid rejecting them more than predicted by rejection models. Another possible (and not mutually exclusive) explanation is that a foreign egg placed in a nest draws attention away from any of the host’s own eggs that are somewhat dissimilar and might otherwise be rejected.

We calculated standard errors (SEs) for all observations and estimates of Type I and Type II errors (Fig 3). For observations, SEs were estimated using the formula

where p is the frequency of errors and n is the sample size. For estimates of Type II error rates, we arbitrarily chose a sample size of 1,000 combinations of eggs, and calculated the mean error rate. We estimated SEs of these means using bootstrapping. To do this, we randomly sampled (with replacement) 100 combinations from the 1,000, and calculated the mean error rate. We repeated this 100 times and calculated the SE of the 100 estimated error rates. For estimates of Type I error rates, which were not based on 1,000 replicates, we calculated SEs directly without bootstrapping.

i. General model description.

Each individual-based simulation involves a host and a brood parasite population, where, for simplicity, each population is haploid. Because these are not evolutionary simulations, the ploidy of the population does not influence results. In each replicate, all nests have a nesting attempt, a potential parasitism event, and a series of host decision processes for all eggs within the nest. There are nesting sites and each of these is occupied by a host parent, who lays a clutch size of eggs (constant across all nests). Each nest has a probability of being parasitized, , and if the nest is parasitized, the parasite lays eggs in it. For all simulations and . Although cuckoo finches often, though not always, lay more than one egg in each host nest (perhaps because increasing the proportion of parasitic eggs in a host nest reduces the likelihood of the host rejecting the parasitic egg or eggs [65]), we keep constant to ensure all parasitized host nests are comparable, and because these simulations are intended to test the role of trait distribution shape in a generalizable way, rather than encapsulating all aspects of natural history in the cuckoo finch system.

Each simulated host and parasite population possesses one egg trait, distributed on a one-dimensional space. The host uses only this single trait in deciding whether to reject eggs. Egg traits are products of the parental genotype.

The survival of host eggs (“success”) is entirely based on rejection decisions (i.e., food availability, predation, offspring quality, and other potential complications are excluded for simplicity), and is hierarchical based on whether a parasitic egg is accepted. This means that, if a parasitic egg is accepted (i.e., a Type II error occurs), no host offspring are produced, as is nearly always the case in the cuckoo finch system [44]. By contrast, if a parasitic egg is rejected or the nest is not parasitized, the host then inspects each host egg. If a host egg is rejected, a Type I error occurs; if a host egg is accepted, the resulting host offspring survives.

Hosts and parasites are defined as having the same genotype distribution, which underlies the egg trait involved in rejection. The genotypic values corresponding to the trait are distributed either continuously or categorically. Specific details for how these are generated depend on the particular distribution being created, and are therefore detailed below.

Host egg phenotypes are generated from the genotype values using a layer of normal distribution to represent the within-clutch variation, centered at the genotype of the focal host,

(1)

where is the phenotype of the focal host egg, is the phenotype of the focal parasitic egg, is the index of the host, , and is the index of the host egg, . and are the genotypes of the host and parasite individual at the focal nest, and is the degree of host within-clutch variation, which is assumed to be a constant for the entire host population. The parasite population does not require within-clutch variation because in all models, parasites only lay one egg (i.e., ).

The decision to accept or reject is based on the threshold distance of the host parent (). Specifically, two thresholds are created based on the genotype of the host and the threshold distance: an upper threshold, , and a lower threshold, . If an egg in the host nest (either laid by the host itself or laid by a parasite) has a trait value between both thresholds, then it is accepted by the host parent. The parameters and take different values in each model (see S1 Text §h for details). We note that while, for simplicity, we have used strict thresholds (i.e., where all eggs falling outside the threshold will be rejected, and all eggs falling within the threshold will be accepted), thresholds are likely to be less strict in nature, such that some eggs falling outside thresholds will be accepted and vice versa.

We conduct 1,000 replicates, each with 1,000 nests, so that in total independent nesting events are recorded for each combination of parameters. A full summary of parameter names, values, and description is given in Table C in S1 Text.

ii. Simulating threshold-based rejection for hosts with various shapes of continuous and categorical distributions.

This model (see Results Section d) requires a continuous and categorical population, that each reject eggs according to thresholds (in contrast to the category-based rejection model below, where the categorical population does not reject according to thresholds). We generated these populations from normal distributions.

Specifically,

(2)

for the continuous case, and

(3)

for the categorical case. We ensured that the overall standard deviation is identical for both continuous and categorical cases by applying the formula .

We varied the shape of the categorical distribution (from highly bimodal to almost unimodal) by changing the standard deviation of each of the two normal distributions making up the bimodal distribution (Fig Ca–e in S1 Text). By doing so, and by changing the half distance between the two peaks, we ensured that the overall standard deviation of the unimodal and bimodal distributions were always equal. We tested how a range of rejection thresholds impact the frequency of Type I and Type II errors for each distribution. One case is provided in the main text (Fig 4a–4c, duplicated in Figs Cd, Ci, Cn, and Cs in S1 Text); the remaining panels in Fig C in S1 Text illustrate how variation in the shape of the categorical distribution (from highly bimodal to almost unimodal) affects error rates compared to a comparable continuous distribution. Furthermore, panels k–t in Fig C within S1 Text illustrate how Type I and Type II error rates vary with threshold distance for each of the parameter combinations, and thus contribute to the egg success rate (Fig Cf–j in S1 Text). Within-clutch variation was set to 0.25 and the probability of parasitism was 0.5.

These individual-based simulations are highly simplified (e.g., simulating only one egg trait, keeping within-clutch variation constant, and keeping parasitism rate constant). All of these factors should affect success rate (e.g., more egg traits should increase the likelihood of mismatches in at least one trait between host and parasitic eggs; increased within-clutch variation should increase the frequency of Type I errors and thus favor higher rejection thresholds; increased parasitism rate should increase the probability of making Type II errors, and thus favor lower rejection thresholds); however, these impacts will be similar for both continuous and categorical distributions, so are not explored further here. Moreover, we only estimate error rates and not fitness consequences, since the latter depend on the relative cost to a host of being parasitized versus rejecting one or more of its own eggs (which in turn may depend on various ecological factors that are system-specific). More importantly, the individual-based simulations draw both host and parasitic eggs from the same distributions. This is justified by the empirical data and estimation of Type II errors reported in the main text, but nevertheless even highly mimetic parasites such as cuckoo finches are unlikely to have identical trait distributions to their hosts in nature). This may be because of evolutionary lag meaning that parasites are yet to “catch up” to hosts [19], weak selection on parasites due to imperfect host perception [29], chase-away selection driving hosts away from parasites [34,67], bet-hedging by parasites [68], the inability of parasites to produce certain phenotypes that hosts can produce [26], or even host perceptual biases such as Weber’s Law driving parasites to exhibit specific imperfections [6]. Such complexities mean that the individual-based simulations presented here do not necessarily reflect true biological reality. Instead, they illustrate in a generalizable way that trait distributions can influence error rates in ways not fully encompassed by the expectation of a trade-off between Type I and Type II errors.

iii. Simulating category-based rejection.

A categorical distribution allows for the possibility of category-based egg recognition, such that hosts lay eggs of one category, and always accept eggs of that category, but reject eggs of the other category (or categories). As discussed in the main text, this process can be simulated for a single egg trait with two categories by generating a truly binary distribution, since under categorical rejection egg phenotypes with two categories would be perceived in a binary manner. We therefore set standard deviation of each peak and within-clutch variation . We compared the Type I and Type II error frequency under this distribution with the error frequencies of a comparable continuous population (i.e., one with the same overall standard deviation as the categorical population). Threshold was varied as before; category-based rejection occurs in the categorical population when the threshold is set between zero and the distance between the peaks, . Results of this simulation are provided in Fig 4d–4f.