Höllviken, Sweden. E-mail cp.hollviken'at'swipnet.se

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AbstractThis paper deals with Dunlin bill-length distributions based onlarge samplesand measured on live material. There are indications that the constituent (male and female) distributions of an overall bill-length distribution of Dunlin are not normal; they may have different skewness and kurtosis patterns. In the present study a large Waddensea material (Rösner 1997) is combined with material from the Sound area, S. Sweden, in order to establish as far as possible the general character of these distributions. Both materials have identical mean values, they are bimodal, and the constituent distributions are not normal. From the Waddensea material an algorithm is created, allowing separation of the constituents of any distribution, provided that the overall distribution is reasonably congruent with the German Waddensea material (and provided that the original sexing of this material is reasonably accurate). The algorithm correctly returns the sex ratio of the morphologically sexed Falsterbo material from July - August: 56 : 44. It is argued that an intrinsic error will leave the general character of the constituent distributions fairly unaffected. In a following step the algorithm is applied to five different materials from the Falsterbo peninsula: among migrating adults (no catch before 15 July) the male : female ratio is 2 : 1 according to the algorithm, in autumn juveniles exactly 1 : 1, this also applies to the staging birds in October - November. In winter, a separation of sexes seems to take place; males lagging behind on the Falsterbo peninsula (10 ‰ salinity), females accumulating at Barsebäck (15 ‰ salinity) further north.

- 1. Introduction.
- 2. Material and results
- 2.1. Parameters of two modern bill-length materials
- 2.2. A provisional algorithm for separating bill length distributions of sexes in Dunlin
- 2.3. Separation of male and female bill-length distributions in five materials from the Falsterbo peninsula
- 3. Discussion

Table I lists sample sizes and mean values in male and female Dunlin from the above papers, in addition Mascher & Marcström 1976 and one material from Greenwood 1979 are quoted:

Reference; subsp., area | Mean value: male / female | 99 % interval: male / female | n: male / female | Age |

Soikkeli 1966; schinzii, Finland | 27.8 / 31.8 | 24.5-30 / 29-34.5 | 96 / 71 | 2y+ (live) |

Pienkowski & Dick 1975; schinzii+alpina, Morocco | 29.5 / 32.8 | 24.9-34.1 / 28.2-37.4 | total: c911 | 1y (live) |

Pienkowski & Dick 1975; schinzii+alpina, Morocco | 28.8 / 33.0 | 24.7-32.9 / 27.8-38.2 | total: c563 | 2y+ (live) |

Pienkowski & Dick 1975; schinzii, Mauritania | 29.0 / 33.4 | 25.4-32.6 / 29.8-37.0 | total: 1408 | 1y (live) |

Pienkowski & Dick 1975; schinzii, Mauritania | 29.6 / 33.0 | 25.5-33.7 / 28.9-37.1 | total: c675 | 2y+ (live) |

Griffiths 1970 (from Soikkeli 1966; Ottenby) | 31.3 / 35.2 | 27.5-35 / 31.5-39 | total: 2418 | 2y+ (live) |

Mascher & Marcström 1976; Ottenby | 31.2 / 34.5 | 27-37 / 32-37 | 46 / 33 | 1y (frozen) |

OAG Münster 1976; inland Germany | 29.9 / 34.3 | 26-34 / 30-39 | total: 306 | 1y (live) |

Greenwood 1979; alpina, Scand., Russia | 31.1 / 32.9 | 27-35 / 29-36 | 48 / 30 | 2y+ (skins) |

Soikkeli and Griffiths measured to the nearest 0.5 mm, OAG Münster to the nearest 0.1 mm, Mascher & Marcström and Greenwood obviously to the nearest mm. Soikkeli measured on live breeding birds in the field, the Griffiths material is based on measurements of live, migrating adults (Ottenby material: P. Martin-Löf), Mascher & Marcström measured frozen carcasses, Pienkowski & Dick as well as the OAG Münster measured live, migrating birds, and Greenwood dry skins of breeding adults. The widest ranges were noted at Münster (13 mm) and Ottenby (11.5 mm). Turning to the mean values, the Greenwood value for females is substantially lower than the values given by Griffiths, Mascher & Marcström and OAG Münster; there must be some kind of bias here, the OAG male value in turn may be influenced by some type of error. One obvious conclusion from table I is, that different race mixes and different types of treatment of the material obstruct the evaluation of and comparison between material from different sites; this cannot be done with older materials. In the Baltic, an assumed increasing occurrence of large-sized Siberian birds (subspecies "centralis") from September onwards (Martin-Löf 1958) adds to the biometric confusion; in the past experienced workers like Nörrevang 1955, Brenning 1987 and Meltofte 1993 - again judging from biometrical data - have even suggested that the East Siberian race *sakhalina* could be a rare, but regular visitor to the Baltic.

Bill length distributions from Fig.3 in "Phenology and biometry..." and Rösner 1997 suggest, that the assumption of normality in Dunlin bill-length distributions may be incorrect, however. In order to investigate these distributions further, a "hermeneutic", step-by-step approach will be adopted; the whole thing starting with the observation of some sort of anomaly, next this anomaly will be quantified as far as possible, finally a new algorithm will be constructed and tested on bimodal distributions. The larger Waddensea material will serve as starting-point and reference, the smaller Sound materials as guinea pigs. I hope to be able to repeat the whole process with better materials in a near future.

Rösner's bill-length material was collected in the German part of the Waddensea, 10 - 31 May 1979 - 1994, bills measured to feathers with 0.1 mm accuracy, birds sexed according to morphological characters given by Ferns & Green 1979 and Stiefel & Scheufler 1989. I have extracted the material from the somewhat coarse original diagram, the approximate sample size is 3,408: 1,519 males and 1,889 females. There is some small error due to the extraction, most certainly negligible in this kind of estimate. Rösner (personally measuring the lion's share) points out, that there is some degree of overlap in the sexual characters, but the total material from May numbers 3,688, so 8.5 % of all birds (more males than females) were left unsexed. His reservation should be borne in mind; but my own impression is, that the material is consistent. The suspicion that skewed distributions could be artefacts (e.g. caused by forbidden glances at biometry) is reduced by my parallel experience with morphological sexing in July - August (Fig.3); the mean values of sexes are identical (Table II) and the sex ratio is returned by the algorithm. In addition I think that some degree of wrong identification throughout the materials will not affect the general character of the distributions. So, at any rate the materials - in particular the large Waddensea one - can be used to establish a general pattern, even with an unknown, intrinsic error. Plain histograms of both sexes from Rösner 1997 are brought together in Fig.1, parameters related to the material given in Table II together with parameters of the Falsterbo material.

Statistic | W: Male (n=c1509) | W: Female (n=c1889) | S: Male (n=182) | S: Female (n=144) |

Range (mm) | 27-40.5 | 27.5-40 | 27.0-38.2 | 29.0-41.2 |

Median (mm) | 31.6 | 34.6 | 31.4 | 34.4 |

Mean ± 1 s.e. (mm) | 31.8 ± 0.05 | 34.4 ± 0.05 | 31.9 ± 0.2 | 34.5 ± 0.2 |

S.d. (mm) | 2.0 | 2.2 | 2.7 | 2.9 |

Skewness | 0.66 | -0.42 | 0.60 | -0.25 |

Kurtosis | 0.55 | -0.26 | 0.02 | -0.31 |

In a perfect normal distribution both skewness and kurtosis are zero. A positive value indicates skewness to the right (males), a negative value skewness to the left (females). The kurtosis describes the ratio (center + tail) : (shoulders) of the distribution, if the value is positive (center + tail) are in majority (leptokurtic distribution, here: males), if the value is negative the shoulders are in majority (platykurtic distribution, here: females). So, in spite of apparent symmetry relative to a vertical central axis, the male distributions seem to be more skewed than the female distributions, and the two distributions have different kurtosis patterns; both female distributions are platykurtic. The female mean values do not differ much from values calculated from Ottenby material by Griffiths 1970 and Mascher & Marcström 1976, the male values are slightly higher.

In order to investigate the symmetry further, the female distribution is inverted and brought together with the male distribution on ln scale in Fig. 2; the platykurtic and leptokurtic characters are indicated by different inclinations of the curves. Note that the distance from 50 to 95 % is c13 % of the mean value in females and c14 % in males, the distance from 5 to 50 % c8 % of the mean value in females, c9 % in males, i.e. both distributions have similar basic characters when measured as ratios of the mean value. Fig. 3 shows the relative balance between sexes along the bill-length axis. These two figures indicate a rather simple, *dynamic* morphogenesis when the bill-length of an individual is "produced"; what I see makes me doubt that bill-lengths are determined by heredity to the last digit. Some sort of catastrophic dynamics seems to be involved - maybe with an external "inlet" providing a governing parameter. But where: before the egg, in the egg, or after hatching?

**2.2. A provisional algorithm for separating bill length distributions of sexes in Dunlin.**

If the method of Griffiths 1968 is applied to the material (male and female pooled) of Rösner 1997, the value of the point of inflexion - and hence the sex ratio of the material - will be miscalculated by appr. 10 %. No conceivable transformation will change this state; the pooled distribution cannot be normalized due to different skewness of male and female component distributions. The problem must be attacked in a different way. From Fig. 3 it can be seen, that the male and female shares of a pooled distribution are relatively constant over two intervals: 30 - 32.9 mm (male side) and 34.5 - 37.4 mm (female side) resp. The ratio between these two intervals may be used as a measure of the sex ratio. Due to different skewness and kurtosis of the male and female distributions it will change in a non-linear way as the sex ratio of the pooled population changes. This is shown in Fig. 4, with the male share changing from 20 to 80 %.

With the sex ratio at hand, two model distributions can be created from the Waddensea material, and with these as point of departure the component distributions (male and female) of any pooled material can be reconstructed. In the following section, this method will be applied to materials from the Sound area. The morphologically sexed material from (Fig.3) in "Phenology and biometry..." provides an initial test of the accuracy. Here the ratio between intervals 30.0 - 32.9 and 34.5 - 37.4 is 110 : 68 = 1.62. In Fig. 4 the curve value of ratio 55 : 45 is 1.57 and of ratio 60 : 40 1.76. Interpolation gives the value 56.3 % males - and the true value is 55.8 %. This is quite acceptable, decimals are not significant with the crudeness of the whole procedure. 56 % males, and the algorithm gives 56 % males. With materials order of magnitude 1,000 birds I estimate the total error at 2 %, with 100 - 200 birds it will be at least 4 - 5 %. In the following section the algorithm will be applied to five unseparated materials from the Falsterbo peninsula: the overall adult material from July - August, the overall material of juveniles, fat juveniles in October / November and two winter materials. The histograms showing male and female distributions are not particularly interesting, they are shown here in order to accustom the reader to the whole approach. The really interesting return is the sex ratios from different situations.

**2.3. Separation of male and female bill-length distributions in five materials from the Falsterbo peninsula.**

**2.3.1. Summer adults.** Between 12 July and 31 August 1995 - 2002 the bill length from feathers was measured on 802 adults (283 2c, 42 2c+, 477 3c+) Dunlin on the Falsterbo peninsula. Three extreme measurements were disregarded: 25.7, 39.2 and 41.2 mm, bringing the material down to 799. The ratio between intervals 30.0 - 32.9 mm and 34.5 - 37.4 mm is 349 : 182 = 1.92, giving a sex ratio 64 : 36 (± standard error order of magn. 2 %) according to Fig. 4. In the Sound area there are two males to every female among migrating adult Dunlin between mid-July and the end of August. With this ratio at hand, the male and female distributions may be recreated with the distributions of Fig.1 as point of departure. Reduction to digits changes the sex ratio a little; the recreated distribution contains 518 males and 281 females, i.e. 65 % males, both distributions are brought together in Fig.5. Mean values reflect the values of the model distribution and are of little interest.

**2.3.2. Juveniles between August and October.** From mid-August there is no overall tendency in bill-lengths, while July and early August juveniles have somewhat shorter bills than the rest of the autumn scene (Fig. 6 of "Phenology and biometry..."). Between 17 July and 19 November 1991 - 2002 there are 2117 measurements of bill-length from feathers in juveniles, the distribution is shown in Fig. 6. Here is an almost perfect bimodal distribution, composed by two very similar constituent distributions, one the reverse of the other. The ratio between the intervals 30.0 - 32.9 mm and 34.5 - 37.4 mm is 772 : 562 = 1.37, giving a sex ratio 49 : 51 (± standard error > 1.5 %) according to Fig. 4. The algorithm is not valid for the most short-billed birds (21 between 24.4 and 26.9 mm), furthermore there are relatively (and in absolute numbers) more birds in the intervals 27.0 - 29.4 mm than in the Waddensea in spring. My suggestion is, that the number of small males is reduced in Rösner's material because many of them leave the Waddensea area before 10 May, they belong to the "low fat-load" category of Goede, Nieboer & Zegers 1990. Exclusion of Falsterbo material between 17.7 and 15.8 (n=61) will not affect the overall distribution, there will still be too many short-billed birds. A slightly modified algorithm is called for here, one that can handle the left tail of the very regular and probably "correct" (more comprehensive than the Waddeensea material from 10.5 - 31.5) overall bill-length distribution. Experiments with the algorithm show that the overall sex ratio will be practically unaffected by manipulation; if there are more males to the left, there must be hidden, rather short-billed females in the centre as well. My best estimate is a ratio 51 : 49; there is one male to every female in the overall juvenile material from autumn, this is the original, "natural" sex ratio in Dunlin (cf. Clutton-Brock 1986).

**2.3.3. Birds with large fat deposits in late October/early November.** This material was presented in section 2.3. of "Phenology and biometry..." and shown in Figs. 10 and 11 of that paper. The severed part of Fig. 10 contains 155 birds, 4/5 of them between 21 October and 18 November, median 5 November, mean weight 65.8 ± 0.6 g, s.d. 7.0 g. The ratio between the intervals 30.0 - 32.9 mm and 34.5 - 37.4 mm is 56 : 42 = 1.33, giving a sex ratio 48 : 52 (± standard error > 4 %) according to Fig. 4. The extreme weights in autumn are not connected with large-sized females, they are characteristic of a particular *population* or group, with equal amounts of males and females.

**2.3.4. Sex ratios in the wintering population.** When it comes to wintering, the main interest is sex ratios, so, in order to obtain as large materials as possible, adults and juveniles were pooled. Bill-length distributions from three materials: Falsterbo till 31 December, Barsebäck by New Year and Falsterbo after 1 January are shown in Fig. 7, 8 and 9. The sites are shown in Fig. 1 of "Wintering and spring staging...". Parameters related to the materials are given in Table III:

Statistic | Falsterbo till 31.12 (n=234) | Barsebäck, New Year (n=122) | Falsterbo after 1.1 (n=96) | |

Skewness | -0.22 | -0.37 | -0.02 | |

Kurtosis | -0.35 | -0.50 | -0.64 | |

Interval ratio | 78 : 68 = 1.15 | 26 : 48 = 0.54 | 37 : 15 = 2.47 | |

Sex ratio; est. s.e. | 40 : 60; > 4 % | < 10 : 90; > 5 % | 75 : 25; > 10 % | |

Mean wing length ± 1 s.e., 1 s.d. (mm) | 119.4±0.2, 3.4 | 122.0±0.3, 3.3 | 116.9±0.3, 3.3 |

These figures show, what can be done from the study of bill-length distributions and sex ratios; here all parameters (skewness, sex ratio, wing-length means) indicate a segregation of sexes (or segregation according to size) in wintering birds by midwinter, females moving north to waters of higher salinity (15 ‰, larger prey), males lagging behind in an area with lower salinity (10 ‰, smaller prey). The skewness at Falsterbo before New Year and at Barsebäck indicate substantial female surplus: the Barsebäck wing length mean is 2 mm larger than the Nov/Dec-value from Falsterbo, while the Jan/Feb-value from this area has decreased by 2 mm. It could be added: out of seven birds caught on the Falsterbo peninsula in early March, six were by all likelihood males.

**2.3.5.1. Two sexed materials from Gdansk and Yamal peninsula**. Field materials of sexed Dunlin, with good measurements made by experienced workers are rare. Soikkeli 1966 has published full material, but his population is marginal. J. Gromadzka and H. Behmann have given me access to two good materials: collected birds from Gdansk, sexed by dissection (n = 42 + 24) and breeding birds from the Yaibari area, Yamal (n = 52 + 52; H. Behmann, M. Gromadzki) sexed by means of morphological characters, often both contrahents of a pair simultaneously. Parameters of both materials are given in Table IV, the Yamal material and twenty-two mating patterns (H. Behmann in litt.) from the same area in Figs. 10 and 11.

Both materials were combined with the bill-lengths of 10 genetically/morphologically sexed individuals in Rösner 1997 and Wenink 1994, creating a sexed reference material of 180 birds (96 males, 84 females). The pooled distribution and its statistics are available under Bill-length account. I will continue this collecting of material till I have an adequate reference population for Baltic/Waddensea conditions.

Statistic | Gdansk, males n=42 | Gdansk, females n=24 | Yamal, males n=52 | Yamal, females n=52 |

Skewness | -0.34 | -0.38 | -0.17 | -0.28 |

Kurtosis | 0.14 | -0.65 | --0.06 | -0.43 |

Mean bill-length ± 1 s.e., 1 s.d. (mm) | 31.1 ± 0.2; 1.5 | 33.6 ± 0.4; 1.7 | 30.8 ± 0.2; 1.6 | 34.4 ± 0.2; 1.6 |

The skewness/kurtosis pattern of Rösner's Waddensea material (Fig.1) and the Sound material from summer (Fig.3) is not fully reproduced in the smaller Yamal material; the females are average, but the males remarkably short-billed (cf. Table II). Under such circumstances the algorithm is less useful; the ratio between intervals 30.0 - 32.9 and 34.5 - 37.4 mm is 31 : 28 (cf. Fig. 4), suggesting a sex ratio 4 : 6 - but the error of an estimate from this sample size is order of magnitude 10 %. A tailored, more compact model distribution would work better in this, and possibly in any particular case.

**2.3.5.2. Manipulation of bill-length-distributions, the possible gain.** Figs. 10 and 11 bring up the possibility of *local optimizations of bill length distributions* to the permafrost level at a given latitude and to different types of prey etc.; at 71° 40' N the Yamal males may have created a distinct image for themselves. The mean value of the 21 females mated with shorter-billed males of Fig. 11 is 34.4 ± 0.3 mm, s.d. 1.4 mm, i.e.: matching the overall population, while that of the males is 30.4 ± 0.4 mm, s.d. 1.8 mm, i.e.: below the population mean, but with larger variance. If there is a real tendency for skewness and kurtosis in male and female bill-length distributions in nature, its aim must be to get the most out of this disassortive mating (cf. Jönsson 1987). Fig. 12 shows the regression [difference between male and female bill-length] on male bill-length in the 21 cases where males had shorter bills than their female partners (material from Fig. 11). In Fig. 13 I have constructed two alternative model distributions, and in Fig. 14 the cells of these distributions have been "mated" in the same manner as in Fig. 12; the most short-billed available cell from the left distribution with the most long-billed available cell from the right distribution till each cell had a mate. Finally the differences between cell rankings were regressed. With increasing skewness, the regression coefficient decreases from -1.86 to -1.90 (all regression parameters are highly significant in these cases), i.e. by c2 %; the main gain seems to be, that *there will be extra room for favourable combinations short-billed females + long-billed males* - but can the existence of such combinations be demonstrated in nature? The single case of Fig. 11 stands as a reminder of this possibility, and a prompting for further field investigations. Summing up: the possible gain from "manipulation" of mating combinations by means of skewness and kurtosis of bill-length distributions is limited, the change in mean gap from case "A" to case "B" is negligible: from 2.75 to 2.81. Keeping within the scope of deviations from normality observed in nature, the possible overall gain (say: in terms of niche width) will be very limited relative to a disassortive mating of two normal distributions. But the *absolute* number of extreme combinations may be at least doubled, this outcome may be what counts.

First published 8.2.03, last major addition 12.3.03, last changed 27.3.03, links changed 10.2.07