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doi: https://doi.org/10.1038/d41586-022-00888-9

ReferencesLeBrun, E. G., Jones, M., Plowes, R. M. & Gilbert, L. E. Proc. Natl Acad. Sci. USA 119, e2114558119 (2022).PubMed 
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Ethics oversightThis study was carried out within the frame of our housing and breeding permit (311.4-si) granted by the Landratsamt Starnberg, Germany. Attachment of backpacks was approved by the Regierung von Oberbayern, Germany (ROB-55-2-2532. Vet_02-17-211).Study populationsWe used four zebra finch populations that are genetically differentiated due to founder effects and selection (see Supplementary Fig. 1 & Fig. 2): two domesticated populations (D1 and D2) that have been maintained in captivity in Europe for about 150 years and two populations (W1 and W2) that have been taken from the wild about 10–30 years ago (see Supplementary Fig. 1). We ran all experiments in two independent replicates. We used individuals from populations D1 and W1 for replicate 1 and individuals from D2 and W2 for replicate 2.Breeding experiment Generation 1We created four groups of 36 individuals (9 males and 9 females from both a domesticated and a wild-derived population, two groups within each replicate) and put each group separately in an indoor aviary (5 m × 2.0 m × 2.5 m). All individuals had been reared normally by their genetic parents in similar breeding aviaries, were inexperienced (never mated before) and unfamiliar to all opposite-sex individuals. In replicate 1 (W1 – D1, starting December 2016), birds were 142 ± 32 days old at the start of the experiment (range: 101–191 days); in replicate 2 (W2 – D2, starting March 2017), birds were 241 ± 47 days old (range: 151–306 days). In each aviary, we provided nest material and nest boxes to stimulate breeding and observed pair-bonding behaviour for ca. 60 h spread over 14 days. Two observers recorded all instances of allopreening, sitting in bodily contact, and visiting a nest box together, which reflects pair bonding64.In total, we observed 3166 instances of heterosexual association among the 4 × 36 individuals (Supplementary Table 3). We defined a pair-bond between two opposite-sex individuals if they were recorded in pair-bonding behaviour at least five times (mean: 22 ± 14 SD, range: 5 – 73). This cut-off was chosen (blind to the outcome of data analysis) based on the frequency distribution showing a clear deviation from a random, zero-truncated Poisson distribution (Supplementary Fig. 8). Using this definition, we identified a total of 60 pairs (30 in each replicate). Of all females, 48 and 6 had a pair-bond with one and two males, respectively (18 females remained unpaired). Conversely, 34, 10, and 2 males had a pair-bond with one, two, and three females, respectively (26 males remained unpaired).Cross-fostering for Generation 2 experimentsAfter the breeding experiment of Generation 1, in 2017, we established two different cultural lineages within each genetic population by cross-fostering eggs, either within or between populations (Fig. 3). For this purpose, we used 16 aviaries (four per population), each containing 8 males and 8 females of the same population (Generation 1). Individuals were allowed to freely form pairs and breed. We reciprocally exchanged eggs shortly after laying between two aviaries per population (within-population cross-fostering) and between pairs of aviaries from different populations (between-population cross-fostering). This resulted in four cultural lineages per replicate (DD, DW, WD, and WW; Fig. 3). Each lineage was maintained in two separate breeding aviaries to ensure the availability of unfamiliar opposite-sex Generation 2 individuals from the same line. Offspring remained with their foster parents until they reached sexual maturity, when the following experiment started.Social experiment Generation 2Between December 2017 and March 2018, we put four groups of individuals (two groups for each replicate) in indoor aviaries (same as in Generation 1 experiment). Each group consisted of 10 males and 10 females from each of the cross-fostered groups DD, WW, DW and WD, i.e., a total of 80 birds per aviary, except that one aviary of replicate 2 only consisted of 63 individuals (7DD, 8WW, 8DW and 8WD) due to a shortage of birds. In replicate 1 (W1 – D1, starting December 2017), birds were 170 ± 25 days old at the start of the experiment (range: 105–199 days); in replicate 2 (W2 – D2, starting January 2018), birds were 200 ± 29 days old (range: 120–241 days). We recorded the position of individuals using an automated barcode-based tracking system31. We fitted each individual with a unique machine-readable barcode (Supplementary Fig. 4a) and placed eight cameras (8-megapixel Camera Module V2; RS Components Ltd and Allied Electronics Inc.), each connected to a Raspberry Pi (Raspberry Pi 3 Model Bs; Raspberry Pi Foundation) in each aviary. For 30 consecutive days, the cameras recorded individuals at six perches and at two feeders (Supplementary Fig. 4b, c). Between 05:30 and 20:00, when lights were switched on, each camera took a picture every two seconds.Each day, pictures stored on the Raspberry Pis were downloaded to a central server and processed using customised scripts. The customised software used the PinPoint library in Python65 to identify each barcode in each picture, allowing us to simultaneously track the position and orientation of each individual (Supplementary Fig. 4b) for the duration of the experiment. The tracking system generated 118 million observations across all four aviaries (Supplementary Fig. 4c). From these data, we extracted the average distance between the male and the female (in mm) for each male-female dyad, either daily or across the entire 30-day period (for comparison, such distance data were also extracted for all male-male and all female-female dyads). We used this dataset to identify the nearest opposite-sex individual for each of 151 males and females (55% of these 151 associations were reciprocal). Out of 151 nearest males to females, 74 (49%) paired with that female in the following breeding experiment (see below) and this proportion strongly increased as the average distance between partners decreased (Supplementary Fig. 9).Breeding experiment Generation 2Immediately after the social experiment, we moved each group into a separate semi-outdoor aviary (5 m × 2.5 m × 2.5 m) and provided nest material and nest boxes. During the next 2 months, three observers scored heterosexual associations to identify pair bonds as described for ‘breeding experiment Generation 1’ (ca 300 h per replicate). In total, we observed 6072 associations involving 284 individuals (Supplementary Table 3). Consistent with the previous experiment, we defined a pair-bond when a male-female dyad was observed in pair-bonding behaviour at least five times during the entire experiment (mean: 18 ± 13 SD range: 5 – 61; Supplementary Fig. 8). Using this definition, we identified 147 pairs (79 pairs in replicate 1 and 68 in replicate 2). Of all males, 97, 22 and 2 had a pair-bond with 1, 2 and 3 females, respectively (27 males remained unpaired). Conversely, 99, 21 and 2 females had a pair-bond with 1, 2 and 3 males (26 females remained unpaired).Breeding experiment Generation 3Between April and December 2018, we housed the four cultural lineages (DD, WW, DW and WD) separately again. We placed 8 males and 8 females in each of 16 breeding aviaries (four per lineage) and allowed them to freely form pairs and breed. The offspring belong to four lineages (Fig. 3): two lineages with individuals that were raised by parents that had not been cross-fostered between the domestic and wild-derived population (DDD and WWW) and two lineages with individuals from the same genetic background, but where their parents had been cross-fostered and raised by the other population (DDW and WWD).Between December 2018 and February 2019, we put four groups of 36 birds (two per replicate, i.e., 2 with 18 DDD and 18 DDW individuals and 2 with 18 WWW and 18 WWD individuals; 9 males and 9 females per lineage; Supplementary Table 3) in an outdoor aviary (same as above). In replicate 1 (W1 – D1, starting December 2018), birds were 172 ± 44 days old at the start of the experiment (range: 131–195 days); in replicate 2 (W2 – D2, starting January 2019), birds were 191 ± 40 days old (range: 122–230 days). During 14 days, two observers recorded all pair-bond behaviours as described under ‘breeding experiment Generation 1’. In total, we observed 3378 instances of pair-bond behaviour involving 137 individuals (Supplementary Table 3). As above, we defined a pair-bond when a male-female dyad was observed in pair-bonding behaviour at least five times during the entire experiment (mean: 18 ± 11 SD, range: 5 – 47; Supplementary Fig. 8). We identified 82 pair bonds (37 in replicate 1 and 45 in replicate 2). Of all males, 34, 16, 4 and 1 had a pair-bond with 1, 2, 3 and 4 females, respectively (17 males remained unpaired). Conversely, 42, 16, 1 and 1 females had a pair-bond with 1, 2, 3 and 5 males, respectively (12 females remained unpaired).Morphological measurementsAfter birds had reached sexual maturity ( >100 days of age), we measured body mass (to the nearest 0.1 g), tarsus length (to the nearest 0.1 mm), and wing length (to the nearest 0.5 mm) of all individuals (all measured by WF). We included these three variables in a principal component analysis (PCA) and used the first principal component (PC1, 67% of variation explained) as a measure of body size.Song recording and analysis approachWe recorded the songs of the parental males from Generation 1 (16 aviaries x 8 males = 128 males, of which 122 were successfully recorded between November and December in 2017) and of their offspring (Generation 2; 146 out of 152 males were successfully recorded between March and May 2018). To elicit courtship song, each male was placed together with an unfamiliar female in a metal wire cage (50 cm × 30 cm × 40 cm) equipped with three perches and containing food and water. The cage was placed within one of two identical sound-attenuated chambers. We mounted a Behringer condenser microphone (TC20, Earthworks, USA) at a 45° angle between the ceiling and the side wall of the chamber, such that the distance to each perch was approximately 35 cm. The microphone was connected to a PR8E amplifier (SM Pro Audio, Melbourne, Australia) from which we recorded directly through a M-Audio Delta 44 sound card (AVID Technology GmbH, Hallbergmoos, Germany) onto the hard drive of a computer.Previous studies that quantified differentiation of songs between zebra finch populations using specific song parameters (e.g., duration and frequency measures) largely failed to detect prominent differences12,49,50. We, therefore, used the following two approaches (Sound Analysis Pro and Machine Learning) to quantify the extent to which a given male’s song resembled the songs of other males.Song similarity analysis with SAPUsing Sound Analysis Pro (SAP) version 2011.10427, we quantified song similarity (ranging from 0 to 100) by direct pairwise comparison of song motifs (the main part of a male’s song that is stereotypically repeated and about 0.8 s long, excluding introductory syllables). Pair-wise comparisons of two males (based on one representative motif recording per male) revealed higher within-population similarity than between-population similarity (Supplementary Table 2, data from Generation 1). Further, for offspring that were cross-fostered between populations (N = 73 males from Generation 2) song similarity to their foster father was higher than song similarity to their genetic father (80 versus 68, paired t-test: p  More

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BioassaysSalmon-louse bioassays were performed by the BC Centre for Aquatic Health Sciences (CAHS) as described in Saksida et al.10. Briefly, motile (i.e., pre-adult and adult) L. salmonis were collected from 11 salmon farms in the Broughton Archipelago (BA) between 2010 and 2021 and transported to CAHS in Campbell River, BC. Within 18 h of collection, healthy lice were separated by sex and randomly placed into petri dishes each containing approximately 10 lice (mean ± SD = 9.6 ± 1.1) and subjected to one of six EMB concentrations (either 0, 31.3, 62.5, 125, 250, and 500 ppb or 0, 62.5, 125, 250, 500, and 1000 ppb, depending on suspected variation in EMB sensitivity11). Each collection corresponded to one bioassay, and each bioassay contained roughly four replicates for each sex (4.0 ± 1.3 for females and 3.6 ± 0.9 for males). After 24 h of EMB exposure, lice were classified as alive if they could swim and attach to the petri dish, or moribund/dead otherwise. Lice were kept at 10 °C throughout the process. In total, 34 bioassays were conducted from 11 farms between October 2010 and November 2021.We analysed the proportion of lice that survived exposure to EMB, using standard statistical descriptions that accounted for within-assay dependencies (generalized linear mixed models (GLMMs) with logit link functions, fitted separately to the data from each bioassay). The models included fixed effects for EMB concentration, sex, and the interaction between the two, as well as a random intercept for petri dish. For each analysis, we centered concentration values and scaled them by one standard deviation. We used the GLMM fits to calculate the effective concentrations at which 50% of the lice survived (EC50) in each bioassay. The GLMM for one bioassay produced a singular fit because there was not enough variation in the female survival data to warrant the random-effects structure. We retained the EC50 values resulting from this singular fit because re-fitting without the random intercept yielded identical EC50 values, and removing the entire bioassay from the overall dataset did not qualitatively affect the subsequent analysis.To assess whether the sensitivity of salmon lice to EMB has decreased over time, we fitted a set of five standard GLMs with gamma error distributions and log link functions to the maximum-likelihood EC50 estimates. Each of these five models included binary effects for sex and for whether the farm’s stock had previously been treated, since both affect EMB sensitivity in lice10. The first model included only these two effects and served as a null model that assumed lice did not evolve EMB resistance over time. The second model added a fixed effect for time (i.e., the number of days since January 1, 2010), while the third model included an interaction between time and sex. The fourth and fifth models were identical to the second and third, but with a quadratic effect for time, to account for possible first-order nonlinearity. We were unable to add an effect for farm due to small sample sizes. We performed model selection using the Akaike Information Criterion penalized for small sample sizes AICc25, treating AICc differences of less than two as being indistinguishable in terms of statistical support and selecting the least complex model when that was the case26. The ΔAICc values for the EC50 models were 48.1, 6.1, 4.9, 0, 1.75, respectively.Field efficacyWe used relative salmon-louse counts after EMB treatment (i.e., the post-treatment count divided by the pre-treatment count) as our measure of EMB field resistance between 2010 and 2021 (higher relative counts imply lower treatment efficacy). We defined “pre-treatment” as one month prior to treatment and “post-treatment” as three months after treatment (roughly when one would expect to find the lowest counts in louse populations previously unexposed to EMB), as in Saksida et al.10. We excluded EMB treatments for which an additional, non-EMB treatment was performed within the following three months. In total, there were 73 EMB treatments for which we were able to calculate relative post-treatment counts.Salmon-louse counts were performed by farm staff as described by Godwin et al.27. In short, salmon-louse counts were usually performed at least one per month by capturing 20 stocked fish in each of three net pens using a box seine net, then placing the fish in an anesthetic bath of tricaine methanesulfonate (TMS, or MS-222) and assessing the fish for motile (i.e., pre-adult and adult) L. salmonis by eye.The treatment dataset included the date and type of every treatment that has been performed on a BA farm (i.e., not just the 11 farms with bioassay data). In total, 88 EMB treatments were conducted between 2010 and 2021, of which we were able to calculate relative post-treatment counts for 73 because some months lacked counts or had a non-EMB treatment performed within the following three months. An additional 22 non-EMB treatments (e.g., freshwater and hydrogen baths) were performed, all since the beginning of 2019, but we excluded these data from our analysis.To determine whether field efficacy of EMB treatments has decreased over time, we used GLM-based “hurdle models”—standard statistical descriptions used to accommodate an over-abundance of zeroes in data being analysed. A hurdle model uses two components—one model for whether a count is nonzero and another for the value of the nonzero count—to predict overall mean count. To this end, we fitted three binomial GLMs paired with three gamma GLMs to the relative-count data, each of the paired models being structurally identical in terms of predictors. All of these submodels included a binary fixed effect for previous treatment, as in the EC50 models. The null pair of submodels included no additional terms, the second pair of submodels included a fixed effect for time (i.e., the number of days since January 1, 2010), and the third pair of submodels included a quadratic effect of time (again, to account for possible first-order deviations nonlinearity). We were unable to add an effect for farm due to small sample sizes. We performed model selection of the hurdle models, again using the Akaike Information Criterion penalized for small sample sizes. The ΔAICc values for the three hurdle models were 39.6, 18.3, and 0, respectively. We performed our analyses in R 3.6.028, using the lme4 package29. More

ModelsConsider an IPD game with misperception such as implementation errors and observation errors22,23,31. Due to the misperception, the parameter in the real game changes from (omega _1=[T_1,R_1,P_1,S_1]) to (omega _2=[T_2,R_2,P_2,S_2]), and only player X notices the change. Thus, player Y’s cognition of the parameter is (omega _1), while player X’s cognition of the parameter is (omega _2). In each round, player X chooses a strategy from its strategy set (Omega _X={{mathbf {p}}=[p_{cc},p_{cd},p_{dc},p_{dd}]^T|p_{xy} in [0,1],xyin {cc,cd,dc,dd}}), e.g., (p_{xy}) is player X’s probability for cooperating with given previous outcome (xyin {cc,cd,dc,dd}). Similar to (Omega _X), player Y’s strategy set is (Omega _Y={{mathbf {q}}=[q_{cc},q_{dc},q_{cd},q_{dd}]^T|q_{xy} in [0,1],xyin {cc,dc,cd,dd}}). According to Press and Dyson7, this game can be characterized by a Markov chain with a state transition matrix (M=[M_{jk}]_{4times 4}) (see “Notations” for details). Denote ({mathbf {v}}=[v_{cc},v_{cd},v_{dc},v_{dd}]^T) as a probability vector such that ({mathbf {v}}^T M={mathbf {v}}^T) and (v_{cc}+v_{cd}+v_{dc}+v_{dd}=1). Let ({mathbf {S}}^{omega _i}_{X}=[R_i,S_i,T_i,P_i]^T), and ({mathbf {S}}^{omega _i}_{Y}=[R_i,T_i,S_i,P_i]^T,) (iin {1,2}). The expected utility functions of players are as follows:$$begin{aligned} begin{aligned} u_X^{omega _i}({mathbf {p}},{mathbf {q}})={mathbf {v}} cdot {mathbf {S}}^{omega _i}_{X}, u_Y^{omega _i}({mathbf {p}},{mathbf {q}})={mathbf {v}} cdot {mathbf {S}}^{omega _i}_{Y},iin {1,2}. end{aligned} end{aligned}$$Denote (G_1 = {{mathbf {P}}, {varvec{Omega }}, {mathbf {u}}, omega _1}), and (G_2={{mathbf {P}},{varvec{Omega }},{mathbf {u}},omega _2}), where ({mathbf {P}}={X,Y}), ({varvec{Omega }}=Omega _Xtimes Omega _Y), and ({mathbf {u}}={u_X^{omega _i},u_Y^{omega _i}}, iin {1,2}). Thus, the actual utilities of players are obtained through (G_2), and in the view of player Y, they are playing game (G_1). In the view of player X, they are playing game (G_2) but player X knows that player Y’s cognition is (G_1). (G_1) and (G_2) are shown in Table 2.Table 2 Utility matrices in IPD games with misperception.Full size tableLet ({mathbf {p}}_0=[1,1,0,0]^T). For (iin {1,2}), ({mathbf {p}}=alpha {mathbf {S}}^{omega _i}_{X} +beta {mathbf {S}}^{omega _i}_Y +gamma {mathbf {1}}+{mathbf {p}}_0), where (alpha ,beta ,gamma in {mathbb {R}}), is called a ZD strategy7 of player X in (G_i) since the strategy makes the two players’ expected utilities subjected to a linear relation:$$begin{aligned} alpha u_X^{omega _i}({mathbf {p}},{mathbf {q}})+beta u_Y^{omega _i}({mathbf {p}},{mathbf {q}})+gamma =0, end{aligned}$$for any player Y’s strategy ({mathbf {q}}). All available ZD strategies for player X in G can be expressed as (Xi (omega _i)={{mathbf {p}}in Omega _X|{mathbf {p}}=alpha {mathbf {S}}^{omega _i}_{X} +beta {mathbf {S}}^{omega _i}_Y +gamma {mathbf {1}}+{mathbf {p}}_0,alpha ,beta ,gamma in {mathbb {R}} }.) Also, the three special ZD strategies are denoted as:

(1)

equalizer strategy7,12: ({mathbf {p}}=beta {mathbf {S}}^{omega _i}_{Y}+gamma {mathbf {1}}+{mathbf {p}}_0);

(2)

extortion strategy7,13: ({mathbf {p}}=phi [({mathbf {S}}^{omega _i}_X-P_i{mathbf {1}})-chi ({mathbf {S}}^{omega _i}_Y-P_i{mathbf {1}})]+{mathbf {p}}_0,chi geqslant 1);

(3)

generous strategy14,15: ({mathbf {p}}=phi [({mathbf {S}}^{omega _i}_X-R_i{mathbf {1}})-chi ({mathbf {S}}^{omega _i}_Y-R_i{mathbf {1}})] +{mathbf {p}}_0,chi geqslant 1).

Based on the past experience, player Y knows that player X prefers ZD strategies, which has been widely considered in many IPD games7,9. To avoid that player Y notices the change, which may result in potential decrease of player X’s utility21 or collapse of the model28, player X keeps choosing ZD strategies according to (G_1), such that the strategy sequence matches player Y’s anticipation. To sum up, in our formulation,

the real game is (G_2);

player Y thinks that they are playing game (G_1), and player X thinks that they are playing game (G_2);

player X knows that player Y’s cognition is (G_1);

player Y believes that player X chooses ZD strategies;

player X tends to choose a ZD strategy according to (G_1) to avoid player Y’s suspicion of misperception.

In fact, player X can benefit from the misperception through the ZD strategy. For example, player X can adopt a generous strategy in (G_1) to not only promote player Y’s cooperation behavior, but also make player X’s utility higher than that of player Y, if the generous strategy is an extortion strategy in (G_2). A beneficial strategy for player X is able to maintain a linear relationship between players’ utilities or improve the supremum or the infimum of its utility in its own cognition. In the following, we aim to analyze player X’s implementation of a ZD strategy in IPD with misperception, and proofs are given in the Supplementary Information.Invariance of ZD strategyPlayer X’s ZD strategies may be kept in IPD games with misperception from implementation errors or observation errors. In particular, player X keeps choosing a ZD strategy ({mathbf {p}}) in (G_1) to avoid player Y’s suspicion about possible misperception. In the view of player X, it can also enforce players’ expected utilities subjected to a linear relationship if ({mathbf {p}}) is also a ZD strategy in (G_2). The following theorem provides a necessary and sufficient condition for the invariance of the linear relationship between players’ utilities.Theorem 1
Any ZD strategy ({mathbf {p}}) of player X in (G_1) is also a ZD strategy in (G_2) if and only if$$begin{aligned} frac{R_1-P_1}{2R_1-S_1-T_1}=frac{R_2-P_2}{2R_2-S_2-T_2}. end{aligned}$$
(1)

If (1) holds, player X can ignore the misperception and choose an arbitrary ZD strategy based on its opponent’s anticipation since it also leads to a linear relationship between players’ utilities, as shown in Fig. 1; otherwise, player X can not unscrupulously choose ZD strategies based on player Y’s cognition. There is a player X’s ZD strategy in player Y’s cognition which is not the ZD strategy in player X’s cognition. Further, because of the symmetry of (omega _1) and (omega _2), player X’s any available ZD strategy ({mathbf {p}}) in (G_2) is also a ZD strategy in (G_1) if and only if (1) holds. It indicates that (Xi (omega _1)=Xi (omega _2)) and player X can choose any ZD strategy based on its own cognition, which does not cause suspicion of the opponent since it is also consistent with player Y’s anticipation. Additionally, the slopes of linear relations between players’ utilities may be different, as also shown in Fig. 1, and player X can benefit from the misperception by choosing a ZD strategy to improve the corresponding slope.In fact, (1) covers the following two cases:

(1)

(2P_i=T_i+S_i), (iin {1,2}), is a sufficient condition of (1). Thus, when (2P_i=T_i+S_i), (iin {1,2}), player X’s any ZD strategy ({mathbf {p}}) in (G_1) is also a ZD strategy in (G_2). Actually, (2P_i=T_i+S_i), (iin {1,2}), means that the sum of players’ utilities when players mutual defect is equal to that when only one player chooses defective strategies.

(2)

(R_i+P_i=T_i+S_i), (iin {1,2}), is another sufficient condition of (1). Thus, when (R_i+P_i=T_i+S_i), (iin {1,2}), player X’s any ZD strategy ({mathbf {p}}) in (G_1) is also a ZD strategy in (G_2). Actually, (R_i+P_i=T_i+S_i), (iin {1,2}), means that the game has a balanced structure in utilities32. At this point, the relationship between cooperation rate and efficiency is monotonous, i.e., the higher the cooperation rate of both sides, the greater the efficiency (the sum of players’ utilities).

Furthermore, for the three special ZD strategies, player X can also maintain a linear relationship between players’ utilities in the IPD game with misperception.Figure 1Player X can also enforce a linear relationship between players’ utilities in its own cognition. Let (omega _1=[T,R_1,P_1,S]=[5,3,1,0]) and (omega _2=[T,R_2,P_2,S]=[5,frac{23}{7},frac{1}{7},0]), which satisfy (1). Consider that player X chooses two different ZD strategies in (a) and (b), respectively, and the red lines describe the relationships between players’ utilities in (G_1). We randomly generate 100 player Y’s strategies, and blue circles are ((u^{omega _2}_X,u^{omega _2}_Y)), correspondingly. Notice that blue circles are indeed on a cyan line in both (a) and (b).Full size imageEqualizer strategyBy choosing equalizer strategies according to player Y’s cognition, player X can unilaterally set player Y’s utilities, as shown in the following corollary.
Corollary 1
Player X’s any equalizer strategy ({mathbf {p}}) in (G_1) is also an equalizer strategy in (G_2) if and only if$$begin{aligned} frac{R_1-P_1}{R_2-P_2}=frac{R_1-T_1}{R_2-T_2}=frac{R_1-S_1}{R_2-S_2}. end{aligned}$$
(2)

(2) is also a sufficient condition of (1). If (2) holds, player X can unilaterally set player Y’s utility by choosing any equalizer strategy in (G_1) even though they have different cognitions; otherwise, player X can not unscrupulously choose an equalizer strategy based on player Y’s cognition since it may not be an equalizer strategy in player X’s cognition.Extortion strategyBy choosing extortion strategies according to player Y’s cognition, player X can get an extortionate share, as shown in the following corollary.
Corollary 2
For player X’s extortion strategy ({mathbf {p}}) with extortion factor (chi >1) in (G_1), ({mathbf {p}}) is also an extortion strategy in (G_2) if (1) and the following inequality hold:$$begin{aligned} begin{aligned} (S_1-P_1)(R_2-P_2)-(R_1-P_1)(T_2-P_2)-chi ((T_1-P_1)(R_2-P_2)-(R_1-P_1)(T_2-P_2))1) in (G_1), ({mathbf {p}}) is also a generous strategy in (G_2) if (1) and the following inequality hold:$$begin{aligned} begin{aligned}(S_1-R_1)(R_2-P_2)-(R_1-P_1)(T_2-R_2)-chi ((T_1-R_1)(R_2-P_2)-(R_1-P_1)(T_2-R_2))b^1_i, iin {1,2}, end{aligned} end{aligned}$$
(5)
where (a^1_i) and (b^1_i,iin {1,2}) are parameters shown in “Notations”.
Actually, when player Y chooses the always cooperate (ALLC) strategy35, i.e., ({mathbf {q}}=[1,1,1,1]^T), player X gets the supremum of the expected utility in (G_1) and player X’s utility is improved in the IPD game with misperception.Figure 4Player X can use either equalizer strategies and extortion strategies to raise the supremum of its expected utility or generous strategies to raise the infimum of its expected utility. (a) and (b) consider that (omega _1=[T,R_1,P,S]) and (omega _2=[T,R_2,P,S]), where (R_1ne R_2); (c) considers that (omega _1=[T,R,P_1,S]) and (omega _2=[T,R,P_2,S]), where (P_1ne P_2). The red lines in (a), (b), and (c) describe utilities’ relationships when player X chooses an equalizer strategy, an extortion strategy, and a generous strategy in (G_1), respectively; The yellow area contains all possible relationships between players’ utilities in (G_2) if player X does not change its strategy. In (a) and (b), r is the supremum of player X’s utility in (G_1), and (r’) is lower than the supremum of player X’s utility in (G_2); In (c), l is the infimum of player X’s utility in (G_1), and (l’) is lower than the infimum of player X’s utility in (G_2).Full size imageExtortion strategyBy choosing extortion strategies according to player Y’s cognition, player X can also improve the supremum of its expected utility.
Corollary 5
For player X’s extortion strategy ({mathbf {p}}) with extortion factor (chi >1) in (G_1), the supremum of player X’s expected utility in (G_2) is larger than that in (G_1) if$$begin{aligned} begin{aligned}a^2_ichi ^2+b^2_ichi +c^2_i1), the infimum of player X’s expected utility in (G_2) is larger than that in (G_1) if$$begin{aligned} begin{aligned}a^3_ichi ^2+b^3_ichi +c^3_i More

Comparison of T
ag calculated from publicly available data and actual measurement dataThe calculated Tag (m2/kg-FM) in each year is summarized for each species in Supplemental Table 1:

The geometric means (GMs) of Tag values calculated using the collected samples ranged from 8.1 × 10−6 to 2.5 × 10−2 m2/kg-FM; the minimum was for western bracken fern in 2019 and the maximum was for koshiabura in 2018 at Kawamata, Fukushima.

The GMs of Tag values calculated using the publicly available data ranged from 1.6 × 10−5 to 1.2 × 10−2 m2/kg-FM and thus were similar to the actual measurement data. The minimum GM was for udo in 2019 and the maximum was for koshiabura in 2019. The geometric standard deviation (GSD) range was 1.5–4.5.

Annual GMs of Tag values calculated from publicly available data and actual measurement data are compared in Fig. 1. The values for individual years are represented by different points. The Tag values were distributed close to the 1:1 line, which suggested that Tag values calculated from the publicly available data generally agreed with those calculated from actual measurements. Hence, an obvious overestimation of Tag from the publicly available data described above was not observed in the present data. We confirmed that Tag calculated from the publicly available food monitoring data and the total deposition data from the airborne survey are reliable surrogates for actual measurement samples. We discuss Tag calculated from the publicly available data hereafter.Figure 1Comparison of annual geometric means of the aggregated transfer factor (Tag) calculated from publicly available data and actual measurement data. Circles, diamonds, and triangles indicate deciduous perennial spermatophytes, deciduous tree spermatophytes, and deciduous perennial pteridophytes, respectively. Values for individual years are represented by different points. Error bars indicate the geometric standard deviation in cases where more than three samples were available.Full size imageRelationship between soil deposition and radioactivity in edible wild plants from publicly available dataWe confirmed the relationship between deposition and concentration of 137Cs for the publicly available data for butterbur scape, fatsia sprout, and western bracken fern in a year (Fig. 2), as a representative deciduous perennial and tree spermatophyte, and deciduous perennial pteridophyte, respectively, in the year of the maximum number of detections. Butterbur scape, fatsia sprout, and western bracken fern showed positive significant, nonsignificant, and weak negative significant correlations, respectively (Spearman’s rank correlation, butterbur scape, p = 0.001, rs = 0.45; fatsia sprout, p = 0.85, rs = − 0.03; western bracken fern, p = 0.03, rs = − 0.21). Among 29 subdata with more than 20 detections for each species in a year, in addition to the data shown in Fig. 2, 13 showed statistically significant positive correlations (Butterbur scape in 2014 and 2016; bamboo shoot in 2012, and 2014 − 2019; fatsia sprout in 2013 and 2016; koshiabura in 2013; and ostrich fern in 2012), and western bracken fern in 2017 showed a significant negative correlation. These weak correlations may be affected by uncertainty in the deposition data. We used a representative deposition value for each municipality and the original deposition data grid was of low resolution (see the “Methods” section Radiocesium deposition data from airborne survey). Especially for the cases lacking a clear positive correlation, the degree of radiocesium absorption by edible wild plants was largely different even in the same deposition. Radiocesium uptake by plants in an environment is also affected by other factors (e.g., soil characteristics25,26). The edible wild plants targeted in the present study were not cultivated but were collected in a variety of environments, such as forests with high organic matter content in the soil and paddy field margins with poorly drained soil high in clay content, although we cannot precisely confirm the growth environment of each species included in the present study.Figure 2Correlation between deposition and concentration of 137Cs in three edible wild plants. Circles, diamonds, and triangles indicate butterbur scape, fatsia sprout, and western bracken fern, respectively. The three species are representative deciduous perennial and tree spermatophyte, and deciduous perennial pteridophyte, respectively, in the year of the maximum number of detections.Full size imageTemporal change in T
ag
The time-dependence of Tag for each species in the period 2012–2019 is shown in Fig. 3. The Tag values of deciduous perennial spermatophytes and pteridophytes showed a decreasing trend with time. Given that the bioavailability of 137Cs in the soil in the plant root zone decreased with time, as observed in previous studies27,28, we also observed a decrease in Tag. The Tag of deciduous trees did not show a decreasing trend with time.Figure 3Temporal change in the aggregated transfer factor (Tag) in the period 2012–2019. Circles, diamonds, and triangles indicate deciduous perennial spermatophytes, deciduous tree spermatophytes (including bamboo shoot), and deciduous perennial pteridophytes, respectively. Single exponential fitted lines are shown. Solid lines indicate statistically significant parameters (see Table 2).Full size imageAfter the Chernobyl nuclear accident, radiocesium concentrations in deciduous tree leaves decreased with time owing to the effect of direct deposition at an early stage and the following root uptake effect29, and the Tag of tree leaves decreased accordingly. In previous studies conducted in orchards after the Chernobyl and Fukushima accidents, radiocesium concentrations in deciduous tree leaves showed a decreasing trend30,31. The lack of a declining trend for woody edible wild plants Tag in the present study may be due to a smaller effect of direct deposition at the early stage resulting from interception by tall tree canopies in the vicinity. The height of trees with edible wild plants is usually at eye level. The samples collected soon after the accident were possibly affected by direct deposition, whereas in the latter study period, many of the data were from trees grown after the accident. If the effect of direct deposition was large, a declining trend in Tag might have been observed as observed in orchards. Thus, the absence of a declining trend in Tag indicates that the effect of direct deposition was relatively small.As an additional possibility for the absence of a declining trend in tree Tag, the continuous supply of bioavailable radiocesium from the organic layer on the forest floor may affect the temporal change in Tag. Compared with the managed conditions in orchards of previous studies30,31, an organic layer develops on the soil surface in a forest and, therefore, reabsorption of radiocesium from the organic layer via the roots may be more active. Imamura et al.17 also observed a similar trend to that in the present study, namely that radiocesium concentrations in leaves of the canopies of the deciduous tree konara oak (Quercus serrata) did not show a temporal change from 2011 to 2015 in two Fukushima forests. These authors’ results included the effect of direct deposition on the tree bodies at an early stage of the accident, although the emergence of leaves was after the deposition. Nevertheless, a clear decreasing trend in the radiocesium concentration was not observed, which implies that a deciduous tree actively absorbs radiocesium via the roots in Fukushima forests, and a sufficient amount of radiocesium is absorbed to conceal a decline at an early stage owing to the effect of direct deposition.Single exponential fitted lines for each species are shown in Fig. 3. The estimated parameters and the Teff (year) calculated with Eq. (2) in “Methods” section are presented in Table 2. The Teff for Tag values that showed a decreasing trend was approximately 2 years, except for bamboo shoot. Tagami and Uchida10 reported that the Teff of the slow loss component for three edible wild plants of deciduous perennial spermatophytes was 970–3830 days. The 137Cs decline in pteridophytes, and deciduous shrub and herbaceous species on the floor of European forests was reported to be 1.2–8 years for Teff excluding the rapid loss component after the Chernobyl nuclear accident32. The present results are thus within the range of previous studies.Table 2 Estimated parameters and standard errors for correlations of Tag (m2/kg-FM) in the period 2012–2019 with time (day) calculated using Eq. (3) and effective half-lives [Teff, (year)] calculated using Eq. (2) for 11 parts of 10 edible wild plant species. A0 is estimated initial Tag, and λ (/day) is the 137Cs loss rate in edible parts of the plants.Full size tableFor bamboo shoot, applying a single exponential function, a relatively long Teff of 8.3 years was estimated. The Tag decreased between 2012 and 2014, and thereafter no notable change was observed. This observation may reflect the effect of rapid and a slow loss components. Indeed, we applied a two-component exponential function for bamboo shoot, and observed Teff of 0.7 years and − 7.8 years for the rapid and slow loss components, respectively. For edible wild tree species, statistically significant single exponential fitted lines were not observed, which reflected the absence of change in Tag with time, as discussed above in this section.The Tag varied for all species, varying by 1–3 orders of magnitude within a year that included more than two detections (Fig. 3, Supplemental Table 1). As demonstrated in previous studies5, the present study also showed substantial variation in Tag values, which may be for several reasons. Recently, Tagami et al.12 calculated Tag using the radiocesium concentration in edible wild plants measured by local municipalities from higher-resolution publicly available data (accurate to district level) for giant butterbur, bamboo shoot, fatsia sprout, and koshiabura. The municipalities in these authors’ study are located within the present study area. These authors’ results differed in being one or two orders of magnitude smaller than the present results. The lower resolution of the present deposition data may be one of the causes of the greater Tag variation. The other source of variation is the site dependency of radiocesium absorption by edible wild plants from the soil as described above. Clarification of factors that contribute to the variation in Tag other than 137Cs deposition, and its trends consistent with species, is necessary, which will decrease uncertainty and lead to more accurate estimation of Tag of 137Cs with wild plants.Summary of T
ag for estimation of long-term ingestion dose to the publicTo estimate long-term potential ingestion dose to the public, Tag with small temporal variability excluding high values at the early stage after the accident is required. However, for the edible wild plant species in the present study, no Tag information in an equilibrium condition from before the Fukushima accident is available. Therefore, average values of Tag for the period after the decrease in Tag has weakened and a certain number of samples is available would be appropriate. The Teff for Tag showing a decreasing trend was approximately 2 years except for bamboo shoot, which has not shown any temporal variation since 2014. The Tag for the other species, udo, uwabamisou, momijigasa, fatsia sprout, koshiabura and Japanese royal fern, has not shown temporal variation throughout 2012–2019 (see the “Results and discussion” section Temporal change in Tag). Therefore, Tag values since 2014 are applicable for estimation of long-term potential ingestion dose to the public. The GMs and GSDs of the Tag values for 2014–2019 for each species are shown in Table 3 listed in order of decreasing GM.Table 3 Aggregated transfer factor (m2/kg-FM) calculated from publicly available data for 2014–2019 for 11 parts of 10 edible wild plant species.Full size tableSignificant differences in Tag were observed among the species (one-way ANOVA with Tukey’s post hoc test, p  More

Erebia pronoe exhibits highly structured and strongly differentiated mitochondrial lineages, which are consistent with the distribution of previously described morphotaxa and analyses of Dincă et al.10 These genetic lineages are also reflected to varying degrees in the nuclear markers. The observed mito-nuclear discordances can be explained by different evolutionary rates of genetic markers, the effects of Wolbachia infections, and introgression. These aspects are discussed in more detail in the following sections on the phylogeographic history of this species complex.Mito-nuclear discordance and the systematic status of Erebia melas
Based on genital morphology and nuclear markers, E. melas represents a distinct group to E. pronoe. The common area of origin of both species was probably located in the eastern Alps, which is supported by a RASP analysis based on the nuclear markers. However, E. melas acts as an ingroup of E. pronoe based on the mitochondrial markers, and a RASP analysis indicates a common origin for both taxa in the Carpathian region. Since most Erebia species in Europe have at least parts of their distribution in the Alps21 and are adapted to Alpine environments and habitats22,23, we consider an eastern Alpine origin of the ancestor of E. pronoe and E. melas more likely. This hypothesis subsumes the assumption that the genetic proximity on the mitochondrial level was probably caused by hybridisation and introgression events, which could have occurred as a result of several eastward advances of E. pronoe to the Balkan Peninsula (see below). This seems plausible, because the ability and tendency of E. pronoe to hybridise with other Erebia species have been demonstrated repeatedly12,24,25.The existence of Wolbachia strain 2 in both species, and its distribution from the Pyrenees (in E. pronoe) to the Balkan Peninsula (in E. melas) also speaks for a common origin of both species. Thus, Wolbachia strain 2 might represent the ancient strain present in the common ancestor of this species group, surviving today at the geographic margins (i.e. Pyrenees, western Alps, Balkan Peninsula), but which at some time was replaced in the centre of the butterfly’s range (i.e. the eastern and central Alps) by strain 1. The link between co-occurrence in a common area and prevalence of one Wolbachia strain was also recently demonstrated in other Erebia species26 and might facilitate mitochondrial introgression27.Intraspecific differentiation and glacial refugia of Erebia pronoe
The Pyrenean region is inhabited by one of the oldest and most differentiated intraspecific lineages of E. pronoe. The high genetic diversity in the Pyrenees speaks for large effective population sizes throughout time, enabled by mostly altitudinal shifts in response to climatic cycles, and a lack of major genetic bottlenecks. Compared to the Pyrenean group, the genetic diversity of the western Alpine populations, also well differentiated from all other groups, is lower. This lower diversity was probably the result of repeated cold stage retreat to a geographically more restricted refugium at the foot of the south-western Alps, a well-known refugial area for numerous species28.We cannot say conclusively whether the populations in the Pyrenean region or in the western Alps differentiated first, due to the contradictory genetical markers. The higher evolutionary rate of the mitochondrial markers, the allopatric distribution, and the hybridisation with diverse Erebia species may have led to a greater differentiation of the Pyrenees and/or a loss of the genetic link between the western Alps and the Pyrenees. Since a link between the western Alps and the Pyrenees is still well reflected in the nuclear data set and by the shared Wolbachia strain 2, we consider the most likely scenario to be an early Pleistocene or even Pliocene expansion from the western Alps to the Pyrenees, with subsequent isolation and differentiation. Thus, the Pyrenees-western Alps populations might first have separated as one group from an eastern Alps group s.l., as suggested by nuclear information, and not in two independent events, as suggested by mitochondrial genes.Simultaneously to the split between western Alps and Pyrenees, a separation of the eastern Alpine group s.l. into a southern Alpine subgroup and an eastern Alpine subgroup should have occurred. The southern Alpine subgroup displays a high genetic diversity in their nuclear markers, but a significantly lower diversity in the mtDNA. This might be explained by the existence of a cold-stage refugial area in the southern Alps or their margin, supporting the constant survival of large populations, but also a reshaping of the mtDNA patterns through introgression from the eastern Alpine subgroup during secondary contact when both subgroups expanded into formerly glaciated east-central Alpine areas. The isolated occurrence of Wolbachia strain 1 and mitochondrial haplotypes H29 and H30 (shared with the eastern Alps subgroup) in the southern Alps further support the hypothesis of gene flow from the eastern Alpine region into the southern Alpine populations and vice versa.The eastern Alpine subgroup probably survived glacial periods in a large, cohesive refugium at the eastern edge of the Alps, as has been demonstrated for numerous other species28. This area is also seen as a potential centre of origin of the entire taxon. From there, a recent (most likely postglacial) dispersal must have taken place, which should be responsible at least partly for the star-like pattern of this group in both mitochondrial and nuclear haplotype networks. However, further dispersal events out of the eastern Alps during previous interglacials and maybe even going back to the Pliocene have to be postulated to explain the entire range dynamics in E. pronoe.Apparently, multiple advances out of the eastern Alps into the Balkan mountain systems have taken place from several independent glacial refugia in the region, as indicated by the different mtDNA lineages in Slovenia, western Balkan mountains, and eastern Balkan mountains. A separation between the eastern and western Balkans, and hence also separate glacial refugia in both areas, was frequently observed for mountain taxa28,31. This pattern may have resulted from a succession of independent dispersal events from the eastern Alps throughout the younger Pleistocene, with subsequent regional extinction events and/or independent dispersal events across the Carpathians, as has been demonstrated for numerous other species29.A similar pattern of two independent colonisation events also applies to the Carpathians. Thus, the highly isolated populations in the south-eastern Carpathians must go back to an older expansion out of the eastern Alps. This probably took place during one of the last interglacial phases. The route most likely followed the Carpathian arc, but only a few populations survived at their south-eastern edge. This underlines the phylogeographic independence of this part of the Romanian Carpathians, which is also supported by studies on numerous other mountain species30,31,32. On the other hand, the Tatra mountains, as the northernmost part of the Carpathians, were colonised very recently, most likely postglacially, out of the eastern Alpine area. The strong and rather recent link between these two areas is also supported by phylogeographic studies on many taxa30,33,34.Because of the slower evolutionary rate of nuclear DNA and the resulting incomplete lineage sorting, nuclear markers can contribute little to the reconstruction of these presumably recent events. In line with that, the Valais lineage also has little nuclear differentiation but is clearly distinguished from the western and eastern Alpine lineages by the exclusive mtDNA haplotype H17 and Wolbachia strain 3. The presence of a single, highly differentiated mtDNA haplotype and an exclusive Wolbachia strain indicates a selective sweep. This lineage most likely represents a chronological relict of an interglacial expansion of the eastern Alpine subgroup to the western-central Alps surviving since then in this area, finding glacial refugia in nearby unglaciated areas and becoming infested by a Wolbachia strain not present in any other E. pronoe lineage, hence accelerating its differentiation.Another selective sweep was probably the cause of the mito-nuclear unconformity in the southern Alps lineage. The occurrence of the mtDNA haplotypes H29 and H30 and the Wolbachia strain 1 indicate mitochondrial hybridisation between the eastern and southern Alpine lineages during an expansive interglacial phase. As a result, Wolbachia infection probably occurred, which might have impoverished the mitochondrial diversity of the southern Alps lineage.Consequences for subspecific differentiation in Erebia pronoe
In general, the support given by our data for the so-far described subspecies decreases from west to east. Erebia pronoe glottis Fruhstorfer, 1920, distributed in the Pyrenees, represents the best-supported subspecies. Fixed mitochondrial amino acid changes emphasize the distinctness of this taxon, which might be well advanced in the process of speciation; we cannot even exclude the possibility that it has already reached full species rank. The genetic separation of the western Alps from the Valais, geographically separated along the main Alpine ridge, justifies the recognition of the taxa E. pronoe vergy (Ochsenheimer, 1807) and E. pronoe psathura Fruhstorfer, 1920, respectively, and is supported by both marker sets as well as by the existence of two different Wolbachia strains. The eastern Alpine subgroup resembles the nominotypical E. pronoe pronoe. The existence of at least one lineage in the southern Alpine area is supported by both marker sets. A finer separation based on the mitochondrial markers is not possible, because of recent introgression events affecting east Alpine haplotypes, as also indicated by the existence of Wolbachia strain 1. This population group could be assigned to the taxon E. pronoe gardeina Schawerda, 1924, or to E. pronoe tarcenta Fruhstorfer, 1920, considering their ranges. Nevertheless, a final decision requires further regional studies. Erebia pronoe fruhstorferi Warren, 1933 was accepted to be widely distributed in the Balkan mountain systems. However, our data suggest independent lineages in the western and eastern Balkan mountain systems of which only the eastern populations can be assigned to this taxon. The lineage of the Slovenian Alps is primarily based on mitochondrial markers and morphological characteristics7. The existence of an independent lineage for the highly isolated populations in the southern Carpathians, justifies the subspecies status of E. pronoe regalis Hormuzachi, 1937. Both marker sets display a differentiation, which was more pronounced in the nuclear than in the mitochondrial DNA. More