James J H St Clair

Philippherrmann

Robin W Woods

Tamás Székely

Diel sex-roles in Two-banded Plovers Female-biased incubation and strong diel sex-roles in the Two-banded Plover Charadrius falklandicus

Keywords: Incubation, Nest camera, Nest Predation, Nocturnal, Sex-role, Temperature datalogger

The relative contributions of males and females to incubation, and the diel schedules by which incubation is shared, are important breeding system traits. We used infra-red sensitive cameras to record incubation patterns at 13 nests of the Two-banded Plover Charadrius falklandicus in the Falkland Islands during both day and night. Because predation risk can affect incubation behaviour, we also recorded the diel pattern of nest predation in the wider study population. We found high nest attendance, femalebiased incubation, and strong diel sex-roles, with females incubating during the day and males at night. We also found that incubation intermissions tended to be short but frequent, and were correlated strongly with the diel pattern of nest predations which occurred exclusively in the daylight hours (probably due to the absence of terrestrial mammals from the study site). Our results suggest that sex-roles are unusually strict in the Two-banded Plover, and that these strict sex-roles lead to inequality in incubation sharing and the level of exposure to sources of energetic cost such as disturbance by nest predators.

Introduction

Shorebirds (sandpipers, plovers and allies) exhibit a wide spectrum of sex-roles in incubation, with some species showing uniparental (female-only or male-only) incubation and a majority of species in which the male and female always contribute [START_REF] Reynolds | The evolution of parental care in shorebirds: Life histories, ecology, and sexual selection[END_REF]. It is increasingly realised that incubation is a highly demanding component of parental care, being both energetically expensive and risky [START_REF] Amat | Predation risk on incubating adults constrains the choice of thermally favourable nest sites in a plover[END_REF][START_REF] Reid | Energetic consequences of clutch temperature and clutch size for a uniparental intermittent incubator, the starling[END_REF]. It is thus a prime candidate for sexual conflict, the divergent interests of males and females over reproduction [START_REF] Poyser | Sexual conflict[END_REF]: by transferring the burden of incubation to its partner, an individual could increase the time and energy available for breeding attempts with other partners and potentially increase its fitness [START_REF] Brunton | Energy expenditure in reproductive effort of male and female Killdeer (Charadrius vociferus)[END_REF][START_REF] Kosztolányi | Negotiation between parents over care: reversible compensation during incubation[END_REF]). Variation in the relative contributions of males and females to incubation can thus be a sensitive indicator of sexual conflict and an important trait with respect to natural and sexual selectionas [START_REF] Trivers | Parental investment and sexual selection[END_REF] noted, heritable variations in parental investment can have major evolutionary consequences. However, the relative contributions of males and females have only been recorded in a small proportion of biparentally incubating bird species [START_REF] Wallander | Sex roles during incubation in the Common Ringed Plover[END_REF][START_REF] Whittingham | Male parental care and paternity in birds[END_REF].

Studies of parental care in wild populations often seek to investigate patterns including the relative contributions of males and females to care based on data collected exclusively in the daylight hours (e.g. [START_REF] Bergstrom | Daylight incubation sex-roles in Wilson's plover[END_REF]). While the approach may be acceptable for mainly diurnal behaviours such as the provisioning of young by diurnal altricial birds, it can lead to biased estimates of parental roles whenever the sexes have a tendency to divide care according to a diel schedule. Methods which account for parental activity during both the day and the night are therefore preferable: such methods include the use of infra-red cameras [START_REF] Thibault | Day-and night-time parental investment by incubating Wilson's Plovers in a tropical environment[END_REF] passive transponders [START_REF] Kosztolányi | Using a transponder system to monitor incubation routines of Snowy Plovers[END_REF] and triangulation of radio transmitters [START_REF] Warnock | Nocturnal nest attendance of killdeers: More than meets the eye[END_REF].

We present nest-camera data on diel patterns in incubation in the Two-banded Plover Charadrius falklandicus, a small biparentally-incubating shorebird (body mass 70 g) that is widely distributed in the southern Neotropical region (del Hoyo et al. 1996). We investigated this species in the Falkland Islands, where it is abundant [START_REF] Nelson | Atlas of breeding birds of the Falkland Islands[END_REF]. We quantify the relative contributions of males and females to incubation, determine whether incubation is shared according to a diel schedule, and explore patterns in nest attentiveness over the course of the day.

Because nest predation risk has the potential to explain much of the variation in incubation patterns [START_REF] Ghalambor | Comparative manipulation of predation risk in incubating birds reveals variability in the plasticity of responses[END_REF] we also recorded the diel distribution of nest predations in the wider study population using temperature dataloggers [START_REF] Hartman | An inexpensive method for remotely monitoring nest activity[END_REF].

Methods

Fieldwork

Field data were collected between October and December 2008 on Sea Lion Island at the south of the Falkland Islands archipelago in the South Atlantic (52° 26′ 0″ S, 59° 5′ 0″ W); the island has an area of 905 ha, and lacks mammalian nest predators such as cats, foxes and rats. Two-banded Plover nests were located by searching suitable habitat on foot within the study area. Motion-sensitive cameras were then used to monitor incubation routines at 13 nests at which at least one parent had been captured, sexed and colour-ringed, allowing the sexes to be distinguished. Twobanded Plovers can be reliably sexed in the field, as males typically have stronger black breast-bands and a rich chestnut plumage on the nape [START_REF] Woods | Guide to birds of the Falkland Islands[END_REF]. The camera systems are composed of an Infra-Red LED array and Infra-Red sensitive digital camera mounted on a short pole, and a data-logger and power supply which were buried up to 4 m from the nest scrape [START_REF] Bolton | Remote monitoring of nests using digital camera technology[END_REF]. A software trigger was used to record a series of images whenever movement occurred in the centre of the camera's frame (ie, whenever the sitting bird moved its head, shifted position on the eggs, or left or arrived at the nest). An average of 1381 images were taken per hour of recording. The images were written to SD-type data cards with a unique electronic time stamp. The data cards were changed approximately every 10 hours (mean ± SD: 10.3 ± 1.8 hrs, n = 13 nests).

In the nest-camera images, individuals were identified by recording the presence or absence of colour rings on both the outgoing and incoming bird. Additionally, differences between individual males and females in the width and completeness of the upper breast-band allowed identifications to be confirmed while the birds were incubating.

Temperature data-loggers were installed at 50 further nests using methods described by [START_REF] Hartman | An inexpensive method for remotely monitoring nest activity[END_REF] to record the diel timing of predation events.

Briefly, a temperature data-logger was buried in the centre of each nest-scrape, under approximately 5 mm of lining material; a control data-logger was similarly buried approximately 30 cm from the nest scrape. Of the 50 nests, 27 predation events were recorded and the time of predation was clearly discernable (from a rapid decrease to ambient temperature) in 24 of these.

Data analysis

To compare the relative contributions of males and females over the full day using nest-camera images, 24 on-the-hour imagesone for each hour, from 00:00 h to 23:00 h -were taken from each nest, recording the presence of 'male', 'female' or 'neither'. Because some records on a given day were often missing (for example, due to a lack of camera memory) or potentially compromised (for example, due to an observer visit to the nest during the proceeding hour) the 24 on-the-hour records were sampled randomly from the remaining 'good' records across all recorded days for that nest. If a nest lacked a record for a particular hour on every day of recording, the nest was excluded from this analysis (leaving n = 8 nests in total). For each nest, the number of records (of 24) contributed by the male and female were then expressed as a proportion; male and female per-day contributions were compared by subtracting male from female proportions, and testing for a difference from zero (paired t-test).

Nests which provided fewer than 24 on-the-hour records were not excluded from graph of diel sex-roles because the graph shows the pattern of nest attendance in the population, and not within individual nests.

A 'shift' was considered as the period from the time a parent settled on the eggs to the time it left to be replaced by its partner. Change-over times (when one partner replaced the other) were also recorded. Because birds often took breaks (henceforth 'intermissions') from incubation during their shift, incubation was broken up into shorter periods (henceforth 'bouts'). For each nest, the median durations of bouts and intermissions were recorded separately for males and females. Shifts, bouts and intermissions were excluded from analysis if they were compromised by observer activity (for example, the bouts preceding and following observer visits to the nest were excluded).

Rates of intermission in incubation were estimated by counting the number of intermissions (excluding observer-caused intermissions and change-overs between the sexes) that occurred in each two-hour period during the day. Linear quadratic regressions were used to test the relationships between time-of-day and predation events, and time-of-day and frequency of intermissions in incubation; for this test, a single (mean) frequency of intermission was calculated for each time period. Finally, we tested for correlation between the response variables (nest predation events and mean frequency of intermission).

Data are presented as mean values ± 1 Standard Deviation when normally distributed, and differences between males and females were analysed using paired ttests. Intermission and bout durations and change-over times were not normally distributed and are thus presented as median values with inter-quartile range (1 st to 3 rd quartile, in parentheses), and analysed using Mann-Whitney U-tests. To avoid pseudoreplication, when more than one datum was recorded for a given individual, nest or time-period, an average was calculated for analysis.

Results

Diel patterns of incubation

Incubation was biparental, with males tending to incubate at night and females tending to incubate during the day, and change-overs between the sexes occurring around sunrise and sunset (Fig. 1); female shifts started at a median of 69.0 minutes after sunrise (1.4 -119.0 min, n = 12 nest medians), and male shifts started at a median of 15.0 minutes before sunset (64.4 --2.5 min, n = 13 nest medians).

Including only those nests for which attendance was recorded at least once for every hour of the day (n = 8), total nest attendance was 92%, and males were on the nest for a significantly lower proportion of records than females both over a full 24 hours and during the daylight hours only, although they attended the nest on the majority (85%) of the hours of darkness (Table 1).

Consistent with the period of daylight exceeding the dark period, female incubation shifts were significantly longer than those of males, although there was no significant difference between the sexes in the duration of intermissions or bouts of incubation (Table 2).

Diel sex-roles in Two-banded Plovers Change-overs between the sexes were quick, lasting a median of 0.1 minutes (0.1 -0.2 min, n = 5 nest medians). Observer-caused intermissions were relatively infrequent, occurring only when data cards were exchanged (2 -3 per day). The parents returned quickly to the nest following these visits: from the moment the observer walked away from the nest to the return of the incubating parent took a median 1.7 minutes (1.0 -3.4 min, n = 13 nest medians).

Diel pattern of nest predation

We recorded the exact times of 24 nest predation events in our study population; these events followed a diurnal pattern, with no events occurring during the hours of darkness and a peak during daylight hours (Fig 2 ); thus time-of-day accounted for most of the variation in nest predation frequency (F 2,9 = 10.23, P = 0.005, r 2 = 0.70, n = 12). Intermission rate and nest predation frequency followed the same diel pattern, and were thus highly correlated (Pearson correlation, r 2 = 0.68, P = 0.014, n = 12).

Discussion

The diel sex-roles in incubation are apparently stronger in the Two-banded Plover than in most other Charadrius species [START_REF] Wallander | Sex roles during incubation in the Common Ringed Plover[END_REF] with males attending the nest for almost all of the night, and females for almost all of the day (Table 1; Fig. 1). The pattern of female incubation during the day and male incubation at night is consistent with other members of the genus [START_REF] Wallander | Sex roles during incubation in the Common Ringed Plover[END_REF], although the sympatric Rufouschested Dotterel Charadrius modestus shows a reversal of this pattern (St Clair et al. 2010). A comparison among the biparentally incubating shorebirds demonstrates that the tendency for male incubation at night and female incubation during the day is not a rule: for example, the tundra plovers (Pluvialis spp) and several lapwings (Vanellus spp) exhibit more female incubation during the night [START_REF] Byrkjedal | Tundra plovers : the Eurasian, Pacific and American golden plovers and grey plover[END_REF][START_REF] Ward | Incubation temperatures and behaviour of crowned, black-winged and lesser black-winged plovers[END_REF]). These patterns suggest that among closely related species the same diel Diel sex-roles in Two-banded Plovers sex-roles tend to prevail, indicating that the roles are often evolutionarily conserved.

The other striking pattern is that diel sex-roles in incubation are apparently the norm, with weak or nonexistent diel sex-roles in biparental incubation seldom reportedweak diel sex-roles were reported for three sandpipers (Calidris spp) by [START_REF] Norton | Incubation schedules of four species of calidridine sandpipers at Barrow, Alaska[END_REF] and [START_REF] Pierce | Sex roles in the monogamous Purple Sandpiper Calidris maritima in Svalbard[END_REF] although their high-latitude study sites were in perpetual daylight.

Most explanations for diel sex-roles in incubation assume that incubation at different times of day has divergent costs and benefits that males and females are differently equipped to cope with. For example, the daylight-incubating sex may face greater risk from visual predators and be required to sit for more time (as daylight hours exceed night-time hours), whereas the night-time incubating sex must cope with a greater risk from olfactory predators, and possibly a higher energetic cost per unit of time (due to lower temperatures). Added to these factors, temporal variations in prey abundance and feeding opportunities may drastically change the costs of incubation during the day or night, as one sex is constrained to incubate while the other can feed freely. The relative magnitudes of these different costs and benefits, and their differential effects on male, female and offspring fitness, are still largely addressed speculatively rather than empirically. We found that females experience higher rates of intermission in incubation than males, an effect that appears to be associated with time-of-day rather than sexspecific behaviour because male intermission rate increases through the course of the morning and decreases during the evening (Fig 2). Our results also highlight a candidate mechanism for this difference in intermission rate: a strong diel pattern in nest predation risk which is highly correlated with frequency of incubation intermissions through the day. Because many factors besides predation risk vary according to diel patterns, it is not possible to distinguish causes from consequences in this casefor example, it may be that higher daytime temperatures allow females to leave the nest to feed, while the costs of doing so at night are prohibitive due to lower ambient temperatures. However, it seems probable that the frequent very short intermissions during the day are at least partly the result of anti-predator behaviour (flushing) in response to avian predators such as Striated Caracaras Phalcoboenus australis, which are abundant on Sea Lion Island [START_REF] Nelson | Atlas of breeding birds of the Falkland Islands[END_REF]. There are likely to be costs associated with such frequent anti-predator behaviour, including increased energy expenditure [START_REF] Biebach | Energetics of rewarming a clutch in starlings (Sturnus vulgaris)[END_REF]) and chronically elevated levels of stress hormones such as corticosteroids [START_REF] Clinchy | Balancing food and predator pressure induces chronic stress in songbirds[END_REF]; the lack of nocturnal nest predators (mammals) at our study site means that these potential costs are faced largely by the female. Furthermore, frequent intermissions are also likely to be costly to the developing embryo, potentially resulting in egg temperature fluctuations away from the developmental optimum, lower mean egg temperatures and consequently long incubation times [START_REF] Martin | Geographic variation in avian incubation periods and parental influences on embryonic temperature[END_REF][START_REF] Olson | Periodic cooling of bird eggs reduces embryonic growth efficiency[END_REF]. Consistent with the latter idea, Two-banded Plovers exhibits a remarkably long incubation period of 29.6 days (St Clair et al. in prep.).

Once specialised diel sex-roles in incubation arise, there are intriguing potential consequences for the evolution of breeding systems. For example, in biparental species the adaptation of the sexes to particular incubation roles may be reinforced by positive feedback: adaptation promotes role specialisation and role specialisation favours adaptation. The resulting reinforced specialisation of sex-roles may strongly favour continued cooperation and biparental care (Barta et al. in prep), as specialised parents may experience higher fitness than unspecialised 'jack-of-alltrades' parents through improved survival of themselves and their offspring. In turn, their specialisation may mean that they are unable to effectively assume the role of the other parent, ensuring that desertion by one parent severely reduces the success probability of the reproductive attempt. Such feedback adds a further dimension to the complexity of breeding system evolution, and it is increasingly realised that mate choice, mating systems and parental care are strongly interrelated (Alonzo 2009;[START_REF] Székely | Mating patterns, sexual selection and parental care: an integrative approach[END_REF]. If specialisation in parental roles reaches a threshold level, it could represent a powerful selection pressure for social monogamy, and a mechanism for the contingent irreversibility (sensu Szathmáry and Maynard Smith 1995) of biparental cooperation.

A further possibility is that if diel sex-roles are 'hard-wired' in that changeover behaviours are triggered by changing light levels at dawn and dusk, latitudinal shifts in breeding range towards the poles (where hours of darkness are reduced) would lead to changes in the relative amounts of incubation contributed by males and females; this idea may be assessed in the Two-banded Plover by testing the prediction that incubation will be more evenly shared at lower latitudes in continental South America where there is greater parity between the hours of daylight and darkness.

While biparental care may be most evolutionarily stable when day and night are not of very different lengths, changes in day-length would effectively reduce the time and energy available to partners of one sex for extra-pair courtship or polygamy, while increasing the variance in extra-pair possibilities for the other. In consequence, among taxa with an evolutionary history of strict diel sex-roles we might expect a higher incidence of uniparental care at very high latitudes, a trend which has been shown to exist among shorebirds in general with respect to post-hatching care [START_REF] Garcia-Peña | Breeding systems, climate, and the evolution of migration in shorebirds[END_REF]) but which has not yet been addressed with respect to care during incubation.

In summary, our findings show that Two-banded Plover pairs share incubation according to a diel schedule, and that males tend to incubate at night and females during the day. This diel schedule is associated with unequal sharing of incubation over 24 hours (because day-length exceeds night-length, females spend more time at the nest) and with other inequalities, such as a higher rate of intermissions in incubation for females. The diel pattern of intermissions is consistent with frequent flushing during the day in response to elevated predation risk, although there are other possible causes such as the need to forage or express territorial behaviour. These patterns suggest several directions for future research, both into the proximate causes of variation in incubation behaviour (factors such as photoperiod and predation risk), and the potential interaction between sex-role specialisation and the evolution of breeding systems. Table 1: Incubation sharing by Two-banded Plover pairs in the Falkland Islands.

Zusammenfassung

Figures & Tables

Proportion of on-the-hour records for which nests was attended by the male, female or no bird, for full days (24 h), daytime only (05:00 to 20:00 inclusive) and night-time only (21:00 to 04:00 inclusive).

Proportion of records (mean ± SD)

'

  Natural' intermissions in incubation were frequent and varied over the course of the day, with a low rate of intermission at night increasing to a peak rate of intermission during the middle of the day (Fig 2): time-of-day was thus a powerful predictor of intermission rate (F 2,9 = 19.69, P = 0.001, r 2 = 0.81, n = 12 (mean) intermission rates).

  Die vornehmlich weibliche Bebrütung und stark tageszeitabhängigen Geschlechterrollen des Falklandregenpfeiffers Charadrius falklandicus Die relative Beiträge von Männchen und Weibchen zur Bebrütung und der tägliche Rhythmus, in dem die Bebrütungswechsel erfolgen, sind wichtige Merkmale von Fortpflanzungsstrategien. An Hand von Infrarotkameras wurden Bebrütungswechsel an 13 Nestern des Falklandregenpfeiffers Charadrius falklandicus auf den Falklandinseln bei Tag und Nacht aufgezeichnet. Weil Bedrohung durch Fressfeinde das Bebrütungsverhalten verändern kann, wurden außerdem tägliche Erscheinungsmuster von Nesträubern in der gesamten Studienpopulation erfasst. Wir beobachteten hohe Nestpräsenz, Bebrütung vermehrt durch das Weibchen und ausgeprägte, tageszeitabhängige Geschlechterrollen, wobei Weibchen tagsüber brüteten und Männchen nachts. Wir fanden eine starke Korrelation zwischen Bebrütungsaussetzern, die meist kurz waren, aber häufig vorkamen, und dem Auftreten von Nesträubern, die nur tagsüber erschienen (wahrscheinlich, weil am Studienort keine terrestrischen Säugetiere vorkamen). Unsere Ergebnisse legen nahe, dass Geschlechterrollen bei Falklandregenpfeiffern ungewöhnlich genau festgelegt sind und dass diese strengen Geschlechterrollen zu einer Ungleichheit führen, sowohl in der Aufteilung der Bebrütung als auch in der Belastung durch Ereignisse mit hohen energetischen Kosten, wie die Beeinträchtigung durch Fressfeinde.

Figure legends Figure 1 :

 legends1 Figure legendsFigure 1: Diel sex-roles in nest attendance in the Two-banded Plover in the Falkland Islands. On-the-hour nest attendance records (male, female or neither) were sampled from nest camera footage of 13 nests (maximum one record per nest per hour) and expressed as a proportion of the total number of records for each hour. Median sunrise occurred at 04:43h and sunset at 20:52h.

Figure 2 :

 2 Figure 2: Rate of intermission in the Two-banded Plover. Each point represents the number of intermissions at a single nest during a two hour period; jitter is added for clarity. Change-overs and observer-caused intermissions are not counted here. Black symbols for male-only periods, Grey symbols for female-only periods, and Empty symbols for periods during which a change-over occurred (ie both sexes present). Bars show frequency of actual nest predations recorded by temperature data-loggers over the course of the study, in two-hour bins.

  Figure 1

Sex comparison (paired t-tests)

		Male	Female	Neither	n nests	t	P
	Full day	0.34 ± 0.08 0.58 ± 0.06 0.08 ± 0.04	8	4.83 0.002
	Daytime only 0.06 ± 0.08 0.85 ± 0.09 0.09 ± 0.07	8	14.60 <0.001
	Night-time only 0.85 ± 0.14 0.05 ± 0.07 0.09 ± 0.13	8	12.38 <0.001

Acknowledgements

The study was funded by a University of Bath studentship (JS), the Falkland Islands Government (JS) and the European Union's Leonardo da Vinci program (PH). We would like to thank the RSPB for loaning us the nest cameras, the Falkland Islands Development Corporation and the owners and management of Sea Lion Lodge for allowing research to proceed, various assistants for their help in the field, and Thomas W. P. Friedl and two anonymous referees for their comments. Research was conducted under licence from the Falkland Islands Government (R08/2007).

Diel sex-roles in Two-banded Plovers


Rafał Martyka
email: rafal.martyka@uj.edu.pl

Joanna Rutkowska

Anna Dybek-Karpiuk

Mariusz Cichoń

Kazimierz Walasz

Sexual dimorphism of egg size in the European blackbird Turdus merula

Keywords: egg volume, hatching asynchrony, laying sequence, maternal effects, sex allocation

Recently, a number of studies have shown that female birds are able to control the sex of their progeny at the stage of the gamete. There is also some evidence that females adjust their investment in offspring depending on the sex of the embryo during egg formation.

Differential maternal investment to the eggs depending on their sex is usually interpreted as an adaptive strategy, by which females can increase competitive abilities of the smaller sex, or preferentially invest towards the sex with the potentially higher fitness returns. Here, we studied variation in egg size in relation to embryo sex and laying order in the European blackbird Turdus merula. We found male and female eggs to differ in size, with larger eggs containing male embryos, as well as a significant interaction between embryo sex and laying order. This interaction resulted from the fact that egg size increased with the laying sequence among eggs bearing females but did not change with laying order among eggs bearing males.

There was no relationship between offspring sex and the laying sequence within a clutch. We suggest that sexual dimorphism in egg size recorded in the European blackbird may reflect favoritism of the sex which may give higher fitness returns.

Introduction

Sex allocation theory predicts that if a social or ecological circumstance differentially affects the reproductive value of male and female offspring, parents should skew their investment towards the sex with higher fitness returns. Differential sex allocation can be potentially achieved in two ways, by the production of different numbers of sons and daughters, or by unequal allocation of resources to the members of each sex [START_REF] Trivers | Natural selection of parental ability to vary the sex ratio of offspring[END_REF]Willard 1973, Charnov 1982). In birds, females are the heterogametic sex, and thus have the opportunity to vary their investment in sons and daughters both during egg production and after hatching by differential provisioning of the offspring. In most bird species, the male is usually the sex with higher variance in reproductive success [START_REF] Griffith | Extra pair paternity in birds: a review of interspecific variation and adaptive function[END_REF]. Thus, according to sex allocation theory mothers may be expected to allocate more resources to male offspring [START_REF] Charnov | The theory of sex allocation[END_REF]. The first evidence showing disparities in size between eggs bearing male and female embryos comes from studies on the White-crowned sparrow Zonotrichia leucophrys oriantha, in which authors detected that male offspring hatch from larger eggs [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF]. The same pattern of egg sexual dimorphism where males (the larger sex) come from larger eggs has also been found in other bird species (e.g. [START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF][START_REF] Cunningham | Reply to Petrie et al[END_REF][START_REF] Leitner | Male song quality, egg size and offspring sex in Captive canaries (Serinus canaria)[END_REF]. If male offspring benefit more from increased maternal investment early in life compared to female offspring in terms of lifetime reproductive success, the observed pattern of egg provisioning may be considered as an adaptive favoritism of the sex with potentially higher fitness returns (e.g. [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF][START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF].

Alternatively, it is suggested that females may differentiate investment to eggs to boost the competitive ability of the smaller sex. Two studies on sexually size dimorphic birds, the American kestrel Falco sparverius and the Brown songlark Cinclorhamphus cruralis, show that eggs containing the smaller, less competitive sex are larger [START_REF] Anderson | Sexually dimorphic eggs, nestling growth and sibling competition in American kestrels Falco sparverius[END_REF][START_REF] Magrath | Egg size and laying order in relation to offspring sex in the extreme sexually size dimorphic brown songlark, Cinclorhamphus cruralis[END_REF]. Females may also affect performance of sons and daughters by placing male and female eggs early or late in the laying sequence (e.g. [START_REF] Cichoń | Laying order and offspring sex in blue tits Parus caeruleus[END_REF][START_REF] Lezalova | Should males come first? The relationship between offspring hatching order and sex in the black-headed gull Larus ridibundus[END_REF]. Such differential allocation to sons and daughters according to the laying order might be an important strategy of compensating negative effects of asynchronous hatching (e.g. [START_REF] Rutkowska | Egg size, offspring sex and hatching asynchrony in zebra finches Taeniopygia guttata[END_REF].

Here, we examined whether eggs containing male and female embryos differ in size and whether sexes are non-randomly ordered in the laying sequence in the European blackbird Turdus merula. Sexual dimorphism in egg size might be expected in this species as sexes differ in size (adult males are ca. 5% larger than adult females; Cramp1988, Wysocki 2002, Walasz, unpublished data), and fitness returns resulting from additional maternal investment to sons and daughters may differ, due to the high frequency of extra-pair paternity (in the studied population up to 18% of offspring originate from extra-pair copulation, Rutkowska et al, unpublished data) and possibility of facultative male polygyny (Wysocki 2004). As the European blackbird shows substantial hatching asynchrony (Magrath 1989, Walasz, unpublished data), strategies of allocation of offspring sex to early or late eggs in the laying order might be expected. Thus, we predicted that males would be placed in larger and earlier eggs.

Methods

The study was carried out in an urban blackbird population inhabiting the Dr Jordan Park in Kraków (southern Poland). The European blackbird is a medium-sized, sexually dichromatic, strongly territorial, socially monogamous, and open cup nesting passerine bird [START_REF] Cramp | Birds of the Western Palearctic[END_REF]).

The park is located close to the city center and covers 21.3 ha; density of the breeding blackbirds averaged 1.9 pairs per 1 ha (ca. 40 pairs every year). Most males occupy their territories from March and April and some of them stay on their territories throughout the whole year. Females start laying eggs at the end of March and continue into July. In the study population, females usually lay 3-5 eggs (mean ± SD: 4.3 ± 0.7, N = 99 clutches). As a result of high nest predation, about 95% of females repeat breeding attempts. Most nests are destroyed during egg laying (very often after laying first or second egg) and incubation. All territories were monitored daily and nests were found by following nest-building females and searching bushes and trees throughout the park. Almost all breeding birds in the park were individually marked with an aluminum band and a combination of colour bands. In years 2000-2003 and 2005-2006 laying date of the first egg, clutch size, hatching date and number of nestlings were recorded in most of nests. Additionally, data on laying and hatching order were recorded in a sample of nests. In most cases, data on egg size and sex were obtained from clutches deserted during incubation (N = 45 eggs from 25 clutches). In this case, samples of tissue from embryos were collected. In some nests nestlings hatched successfully (N = 33 hatchlings from 27 clutches), and blood was sampled for further sex determination within 1-2 days after hatching of the first nestling. In the case of 2 clutches, genetic material was collected from both unhatched eggs (N = 2) and alive nestlings (N = 2). Laying order was determined by inspection of nests every day during egg laying. On the day of laying, eggs were numbered with a permanent marker, measured with an electronic calliper (length and width to the nearest 0.1 mm) and weighed with an electronic balance (to the nearest 0.01 g).

Starting from the day before expected hatching, nests were visited daily to assign each marked egg to its corresponding hatchling. Nestlings were then marked by nail clipping. In some nests, we were not able to assign hatchlings to the eggs from which they hatched and so they were excluded from the analyses. To calculate egg volume, we used the formula, V = 0.51 × LW 2 , where L = egg length and W = egg width, which can be applied to most bird eggs [START_REF] Hoyt | Practical methods of estimating volume and fresh weight of bird eggs[END_REF]. Egg volume calculated according to this formula is highly correlated with the egg mass (r = 0.85, P < 0.0001, N = 87). Overall, data on laying order and sex were collected for 82 eggs originating from 54 different females (clutches). 93% of these clutches were incomplete because of egg infertility (N = 4), unknown hatching order of nestlings (N = 4), partial predation (N = 17) and discovery of the nest after egg-laying initiation (N = 25).

Blood and tissue samples were kept in 0.5 ml of 96% ethanol and stored at room temperature. DNA was extracted using Nucleo Spin Blood kit (Macherey-Nagel) following the manufacturer's protocol. Molecular sexing was performed by PCR amplification of two homologous genes (CHD1-W and CHD1-Z) following the protocol of [START_REF] Griffiths | A DNA test to sex most birds[END_REF].

PCR products were separated by electrophoresis on 3% agarose gels, which were stained with ethidium bromide and visualized under UV trans-illumination. The sex was assessed according to the presence of one band for a male or two bands for a female. This method was positively validated by blindly testing blood samples originating from 10 adult birds of known sex.

To examine variation in egg volume in relation to embryo sex and laying sequence, we used the linear mixed model (PROC MIXED in SAS). In this model, the sex of the embryo was introduced as a categorical variable and laying order as a covariate. To control for possible non-linear effects, the laying order was introduced to the model as a linear and quadratic term and their interactions with sex were also tested. In addition, year as a categorical variable and laying date and clutch size as covariates were introduced to the initial model. To distinguish between within and among clutch effects, clutch sex ratio was entered to the initial model and although non-significant it remained in the final model. This clutch sex ratio was derived only of eggs with known sex and laying order. The level of significance was set at P < 0.05 and non-significant terms were systematically removed from the initial model to increase the power of tests. Female identity was entered in each model as a random factor. To analyze the relationship between offspring sex and laying order, we used the generalized linear mixed model using GLIMMIX macro in SAS with logit link function and binomial error variance. To check possible non-linear effects of the position of the egg in the laying sequence on offspring sex we introduced to the model laying order as a linear and quadratic term. Female identity was introduced to the model as a random factor. Because we did not find any non-linear effects, laying order squared was removed from the final model.

All analyses were performed in SAS version 9.1 and denominator degrees of freedom were approximated by the Satterthwaite method.

Results

We found that within a clutch, eggs containing male and female embryos to differ in size (Table 1). Male eggs were significantly larger (least sq mean ± SE; egg volume: 6988.7 ± 112.9 mm 3 ) in comparison to female eggs (least sq mean ± SE; egg volume: 6842.6 ± 110.9 mm 3 ). There was also a significant interaction between embryo sex and laying order (Table 1, Figure 1). Separate analysis performed for each sex showed that egg volume significantly increased with the laying sequence among eggs bearing female embryos (F 1, 25.4 = 8.46, P = 0.007; Figure 1). The effect of laying order was not significant among eggs bearing male embryos (F 1, 23.4 = 1.44, P = 0.24; Figure 1). We also found a non-linear effect of position of the egg in the laying sequence on egg size (Table 1). This results from the fact that initial eggs were larger than middle eggs and final eggs were the largest within a clutch. There were no between clutch effects on egg volume (Table 1), thus the observed within-clutch effects were independent from differences in sex ratio. The sex ratio in the studied sample of eggs was not significantly different from parity (49% males; Binomial test, P = 0.90). Offspring sex did not vary with the position of the egg in the laying sequence analyzed both as a linear (laying order: F 1, 80 = 1.62, P = 0.21) and quadratic term (laying order: F 1, 79 = 0.26, P = 0.61; laying order laying order: F 1, 79 = 0.08, P = 0.78).

Discussion

In this study, we report within-clutch differences in eggs size between sexes but do not find evidence that offspring sex is differently allocated in early and late laid eggs in the European blackbird. Sexual egg size dimorphism has been reported in at least six passerine species [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF][START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF], 2001[START_REF] Arnold | Sex-specific hatching order, growth rates and fledging success in jackdaws Corvus monedula[END_REF][START_REF] Magrath | Egg size and laying order in relation to offspring sex in the extreme sexually size dimorphic brown songlark, Cinclorhamphus cruralis[END_REF][START_REF] Leitner | Male song quality, egg size and offspring sex in Captive canaries (Serinus canaria)[END_REF]. There are also similar findings from non-passerine species [START_REF] Anderson | Sexually dimorphic eggs, nestling growth and sibling competition in American kestrels Falco sparverius[END_REF], Petrie et al. 2001; see also [START_REF] Cunningham | Reply to Petrie et al[END_REF][START_REF] Blanco | Mass provisioning to different-sex eggs within the laying sequence: consequences for adjustment of reproductive effort in a sexually dimorphic bird[END_REF][START_REF] Müller | Within clutch co-variation of egg mass and sex in the black-headed gull[END_REF]. However, the observed patterns of egg sexual size dimorphism are inconsistent and multiple hypotheses have been suggested to explain the possible adaptive value of disparities in egg size between male and female eggs. Among passerines, female offspring of the Spotless starling Sturnus unicolor and the Brown songlark hatched from larger eggs [START_REF] Cordero | Seasonal variation in sex ratio and sexual egg dimorphism favouring daughters in first clutches of the spotless starling[END_REF][START_REF] Magrath | Egg size and laying order in relation to offspring sex in the extreme sexually size dimorphic brown songlark, Cinclorhamphus cruralis[END_REF]. This pattern of egg provisioning was interpreted as an adaptive strategy of support for the more vulnerable smaller sex [START_REF] Cordero | Seasonal variation in sex ratio and sexual egg dimorphism favouring daughters in first clutches of the spotless starling[END_REF][START_REF] Magrath | Egg size and laying order in relation to offspring sex in the extreme sexually size dimorphic brown songlark, Cinclorhamphus cruralis[END_REF]. In contrast, in the White-crowned sparrow, the House sparrow Passer domesticus and the captive canaries Serinus canaria, males were found in larger eggs, despite them being larger than females [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF][START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF][START_REF] Arnold | Sex-specific hatching order, growth rates and fledging success in jackdaws Corvus monedula[END_REF][START_REF] Leitner | Male song quality, egg size and offspring sex in Captive canaries (Serinus canaria)[END_REF]. [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF] and [START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF] explained their results in light of the sex allocation theory, arguing that better provisioning of eggs bearing male embryos may be a adaptive strategy of increased investment to the sex with the potential to give higher fitness returns. In fact, males of those species exhibit higher variation in reproductive success compared to females resulting from extra-pair copulation, very common in these species. Therefore, the increased maternal investment towards male offspring might be expected. Support for this hypothesis comes from studies on mammals, which indicate that the breeding success of a male is related to maternal investment at early stages of life, however, only mothers in good condition can produce more successful males (e.g. [START_REF] Clutton-Brock | Maternal dominance, breeding success and birth sex ratios in red deer[END_REF][START_REF] Clutton-Brock | Great expectations: dominance, breeding success and offspring sex ratios in red deer[END_REF][START_REF] Sheldon | Maternal dominance, maternal condition, and maternal offspring sex ratio in ungulate mammals[END_REF]. Thus, extra investment in eggs containing male embryos may be considered as an adaptive strategy by which females could increase reproductive success of their sons (e.g. [START_REF] Mead | Sexual dimorphism in egg size and implications regarding facultative manipulation of sex in mountain white-crowned sparrows[END_REF][START_REF] Cordero | Sexual dimorphism in house sparrow eggs[END_REF]. Such scenario may also contribute to explain sexual egg size dimorphism in the European blackbird, in which extra-pair paternity is common and generate a substantial variation in male reproductive success (Creighton 2000, Rutkowska et al, unpublished data). However, rigorous test of the adaptive role of sex-specific variation in egg size is missing. The observed pattern of sexual dimorphism in egg size may alternatively be explained without invoking adaptive explanations. For instance, if the same maternal hormones regulate oocyte growth and simultaneously affect sex-determining chromosome segregation, sexual egg size dimorphism may arise as a not-necessarily adaptive consequence of this hormonal mechanism (e.g. [START_REF] Young | Evolution of sex-biased maternal effects in birds: I. Sexspecific resource allocation among simultaneously growing oocytes[END_REF]).

In the current study we found a significant interaction between sex and laying order resulting from the fact that egg size increased with laying order among female but not male eggs. An increase in egg size is often explained in terms of compensatory mechanism of negative effect of hatching asynchrony (e.g. [START_REF] Clark | Avian breeding adaptations: hatching asynchrony, brood reduction and nest failure[END_REF]. Such compensation could be expected in the blackbird, as eggs hatch asynchronously in this species (Magrath 1989, Walasz, unpublished data). Interestingly, such compensation would occur only among eggs bearing daughters. Since eggs bearing daughters are smaller, they may pay larger cost of hatching asynchronously and the increase in egg size in late laid eggs may offset negative effects of hatching asynchrony. Such sex-specific effect has already been reported in asynchronously hatching zebra finches Taeniopygia guttata, in which egg mass and egg laying sequence were significant predictors of differential survival of male and female offspring [START_REF] Rutkowska | Egg size, offspring sex and hatching asynchrony in zebra finches Taeniopygia guttata[END_REF]. sex ratio, clutch size and laying date as covariates. Laying order was entered to the model both as a linear and quadratic term and all possible interaction between sex and laying order were tested. Non-significant terms were systematically removed from the initial model. In

Zusammenfassung

Sexueller

  Dimorphismus der Eigröße in Amseln Turdus merula In letzter Zeit haben mehrere Studien zeigen können, dass weibliche Vögel in der Lage sind, das Geschlecht ihrer Nachkommen während des Keimzellenstadiums zu kontrollieren. Ferner gibt es Hinweise darauf, dass Weibchen ihre Investitionen in ihre Nachkommen von dem Geschlecht des Embryos während der Eiformation abhängig machen können. Die unterschiedliche mütterliche Investition in Eier in Abhängigkeit des Geschlechts wird normalerweise als adaptive Strategie interpretiert, wobei Weibchen die Wettbewerbsfähigkeiten des kleineren Geschlechts verbessern können, oder vorzugsweise in das Geschlecht mit dem potenziell höheren Fitnessgewinn investieren. In der vorliegenden Arbeit haben wir Variation in Eigröße relativ zum Geschlecht der Embryos und Legereihenfolge in Amseln Turdus merula untersucht. Männliche und weibliche Eier waren unterschiedlich großgrößere Eier enthielten männliche Embryos. Wir fanden auch eine signifikante Interaktion zwischen dem Geschlecht von Embryos und der Legereihenfolge. Diese Interaktion entstand durch eine Zunahme von Eigröße mit Legereihenfolge innerhalb von Eiern die weibliche Embryos enthielten; die Eigröße von männlichen Embryonen veränderte sich dagegen nicht mit der Legereihenfolge. Wir fanden keinen Zusammenhang zwischen dem Geschlecht der Nachkommen und der Legereihenfolge innerhalb einzelner Gelege. Wir schlagen vor, dass der sexuelle Dimorphismus der Eigröße bei Amseln möglicherweise die Begünstigung des Geschlechts mit dem höheren Fitnessgewinnversprechens reflektiert.

  table F and P values, degrees of freedoms (d.f.) and estimates with SEs presents values for the final model, whereas in the case of rejected terms on the time when they were dropped from the model. Clutch sex ratio was entered to the initial model and remained in the final model to control for between clutch effects. Female identity was included as a random factor (Z = 3.75, P < 0.0001; the final model).

Figure 1 .

 1 Figure 1. Predicted egg volume (least square means ± SEs) in relation to embryo sex and

Table 1 .

 1

	Final model	Rejected terms	Estimate	SE	d.f.	F	P
	Sex		-720.03	281.67	1, 37.3	6.53	0.015
	Laying order		-390.81	202.28	1, 36.7	2.47	0.124
	Laying order laying order		73.95	30.57	1, 35.4	5.85	0.021
	Sex laying order		187.51	78.52	1, 42.8	5.70	0.021
	Clutch sex ratio		398.13	245.61	1, 74	2.63	0.109
		Laying date	6.90	3.73	1,45.5	3.43	0.071
		Sex laying order laying order	-59.61	67.10	1, 38.3	0.79	0.380
		Year			5, 47.2	0.70	0.627
		Clutch size	41.69	168.11	1, 45.4	0.06	0.805

Table 1 .

 1 Results of the Linear Mixed Model (PROC MIXED in SAS) analyzing variation in egg volume in relation to embryo sex and year as categorical variables, laying order, clutch

Acknowledgements

We thank S. Verhulst, D. Acevedo Seaman and anonymous reviewers for comments. This research was founded by Polish Ministry of Science and Higher Education in years 2003-2006, partly by DS/WBiNoZ/INoŚ/757/06 and EC Center of Excellence IBAES. J.

Rutkowska is supported by The Foundation for Polish Science. This study was conducted in agreement with the current laws of Poland.


Anja Gladbach
email: anja.gladbach@gmx.de

David Joachim Gladbach

Petra Quillfeldt

Seasonal clutch size decline and individual variation in the timing of breeding are related to female body condition in a non-migratory species, the Upland Goose Chloephaga picta leucoptera

Keywords: clutch size decline, egg size, waterfowl, Tadornini, Upland Goose, Chloephaga picta leucoptera, reproductive investment, body condition

Several studies have shown that waterfowl females in good condition lay larger clutches and start egg-laying earlier in a breeding season. However, most of these studies lack corresponding data on individual condition and timing of breeding in different years. We analysed data on clutch size, egg size, hatching date and female body condition of a non-migratory species, the Upland Goose Chloephaga picta leucoptera, recorded in three seasons on New Island, Falkand Islands. We found a strong seasonal decline in both egg and clutch size. Egg size increased with clutch size. The mean egg volume and total clutch volume increased with female body condition and hatching date was earlier for females in higher body condition. Chicks hatched from nests with higher mean egg volume had higher early body condition. We also compared individual females between different study years and found that individual body condition showed a significant repeatability between years, whereas hatching date did not. While female aging did not systematically affect hatching date, females produced clutches that hatched earlier in years of higher body condition. We could thus show for the first time that the frequently reported negative relationship between female condition and hatching date also holds true on an individual basis. To our knowledge, this is the first study on seasonal clutch size decline in sheldgeese and one of a few in waterfowl comparing condition and timing of breeding within individuals between different years.

INTRODUCTION

There are numerous studies on seasonal variation in reproductive success in birds, one of the most common patterns being a seasonal decline in clutch size [START_REF] Klomp | Determination of clutch-size in birds[END_REF]. Early-nesting birds often lay bigger clutches; their offspring grow more rapidly and have a higher chance of survival and recruitment than late-nesting birds [START_REF] Blums | Patterns of reproductive effort and success in birds: path analyses of long-term data from European ducks[END_REF][START_REF] Drent | The prudent parent -energetic adjustments in avian breeding[END_REF][START_REF] Hochachka | Seasonal decline in reproductive performance of song sparrows[END_REF]; [START_REF] Sockman | Orchestration of avian reproductive effort: an integration of the ultimate and proximate bases for flexibility in clutch size, incubation behaviour, and yolk androgen deposition[END_REF], which results in a decrease in offspring value within a season. In precocial birds such as waterfowl young receive parental care in the form of vigilance, social support and defence against predators (e.g. [START_REF] Lamprecht | Dominance and reproductive success in Bar-headed Geese (Anser indicus): a multiple-regression analysis[END_REF][START_REF] Owen | Waterfowl Ecology[END_REF][START_REF] Schindler | Increase of parental effort with brood size in a nidifugous bird[END_REF][START_REF] Winkler | The determination of clutch size in precocial birds[END_REF]. Parents also lead their offspring to good feeding sites and enable chicks to gain access to these sites when competitors are present [START_REF] Lepage | Seasonal decline of growth and fledging success in Snow geese, Anser caerulescens: an effect of date or parental quality?[END_REF][START_REF] Winkler | The determination of clutch size in precocial birds[END_REF]. However, chicks do not depend on parental feeding, a component that is correlated with clutch size in altricial species [START_REF] Lack | Ecological adaptations for breeding in birds[END_REF]). Instead, the investment into clutch formation plays an important role for precocial chick survival. Most hypotheses concerning clutch size in waterfowl have implied that the condition of the female prior to laying is a determinant for reproductive success [START_REF] Bengston | Variations in clutch-size in ducks in relation to the food supply[END_REF][START_REF] Johnsgard | Proximate and ultimate determinants of clutch size in Anatidae[END_REF][START_REF] Lack | Ecological adaptations for breeding in birds[END_REF][START_REF] Ryder | A possible factor in the evolution of clutch size in Ross' Goose[END_REF][START_REF] Winkler | The determination of clutch size in precocial birds[END_REF]. The body condition of individuals can vary according to individual quality and experience, leading to increased foraging efficiency, as well as the food resources available in the territory. Thus, high-quality individuals in high-quality territories may gain condition more rapidly after the winter and would therefore be the first to start laying in the breeding season, while low quality individuals in lower quality territories or individuals facing less favourable feeding conditions would lay later [START_REF] Rowe | Time, condition, and the seasonal decline of avian clutch size[END_REF][START_REF] Sockman | Orchestration of avian reproductive effort: an integration of the ultimate and proximate bases for flexibility in clutch size, incubation behaviour, and yolk androgen deposition[END_REF]. [START_REF] Drent | The prudent parent -energetic adjustments in avian breeding[END_REF] suggested that the conflict between early breeding (high offspring value) and late breeding (higher accumulated condition and potential clutch size) results in the observed timing of laying and clutch size. From a graphical model [START_REF] Drent | The prudent parent -energetic adjustments in avian breeding[END_REF] concluded that birds in poorer condition might have more to gain from a delay than birds in good condition. Field studies have shown that in most species females in good condition lay larger clutches and earlier in the season [START_REF] Klomp | Determination of clutch-size in birds[END_REF][START_REF] Sockman | Orchestration of avian reproductive effort: an integration of the ultimate and proximate bases for flexibility in clutch size, incubation behaviour, and yolk androgen deposition[END_REF]. Experiments manipulating parental condition by supplementary feeding prior to egg laying resulted in earlier laying and increased clutch sizes [START_REF] Karell | Maternal effects on offspring Igs and egg size in relation to natural and experimentally improved food supply[END_REF][START_REF] Meijer | Female condition and reproduction -effects of food manipulation in free-living and captive Kestrels[END_REF][START_REF] Meijer | Family-planning in the Kestrel (Falco tinnunculus) -the proximate control of covariation of laying date and clutch size[END_REF][START_REF] Rowe | Time, condition, and the seasonal decline of avian clutch size[END_REF].

However, most of the studies on the relationship between body condition and hatching date have been conducted either using data on population-level (e.g. [START_REF] Bety | Body condition, migration, and timing of reproduction in snow geese: A test of the condition-dependent model of optimal clutch size[END_REF][START_REF] Dalhaug | Seasonal decline in clutch size of the Barnacle Goose in Svalbard[END_REF]e.g. Devries et al. 2008;[START_REF] Sjoberg | Early breeding leads to intra-seasonal clutch size decline in Canada Geese[END_REF] or, if conducted on an individual basis, then only on repeated observations of clutch size and hatching date without collecting corresponding data on individual condition (e.g. [START_REF] Hamann | Age effects on clutch size and laying dates of individual female Lesser Snow Geese Anser caerulescens[END_REF][START_REF] Petersen | Reproductive ecology of Emperor Geese -annual and individual variation in nesting[END_REF]). To our knowledge there is but one study that compares individual females between different years, finding a positive relation between Common Eider Somateria mollissima clutch size and female mass, but no relation with hatching date [START_REF] Erikstad | Clutch-size determination in precocial birds -a study of the Common Eider[END_REF]).

Egg size is another important component of parental effort in birds and several studies have shown a positive relationship between egg size and offspring fitness, in altricial as well as precocial birds. Larger eggs may enhance offspring fitness by increasing survival prospects in the first days after hatching and competitive power through a larger offspring size and the possession of more resources to survive adverse conditions [START_REF] Amundsen | Egg size and parental quality influence nestling growth in the shag[END_REF][START_REF] Anderson | Composition and growth of King eider ducklings in relation to egg size[END_REF][START_REF] Ankney | Egg weight, survival, and growth of Lesser Snow Goose goslings[END_REF][START_REF] Christians | Avian egg size: variation within species and inflexibility within individuals[END_REF][START_REF] Dawson | Effects of hatching date and egg size on growth, recruitment, and adult size of Lesser Scaup[END_REF][START_REF] Goth | Egg size predicts motor performance and postnatal weight gain of Australian Brush-turkey (Alectura lathami) hatchlings[END_REF][START_REF] Magrath | The effect of egg mass on the growth and survival of blackbirds: a field experiment[END_REF][START_REF] Potti | Maternal effects and the pervasive impact of nestling history on egg size in a passerine bird[END_REF][START_REF] Rutkowska | Egg size, offspring sex and hatching asynchrony in zebra finches Taeniopygia guttata[END_REF]. Increased egg size through food supplementation could be linked to improved hatchability of the egg, increased size at hatching, early growth and survival during the nestling stage (reviewed in Williams 1994 ;[START_REF] Christians | Avian egg size: variation within species and inflexibility within individuals[END_REF], see also Grindstaff, Demas & Ketterson 2005).

We studied seasonal egg and clutch size decline and their relation to female body condition in a non-migratory waterfowl species, the Upland Goose Chloephaga picta leucoptera. Upland Geese belong to the order of the sheldgeese (Tadornini), a group that physically resembles true geese (Anserinae) and shows similar habits but is more closely related to shelducks and ducks. The smaller (migratory) sub-species C. picta picta breeds on the South American mainland, whereas the slightly larger one (C. picta leucoptera) is restricted to the Falkland Islands. Their basic breeding biology and life-cycle has been studied in the Falklands from 1977 to 1980 [START_REF] Summers | The life-cycle of the Upland goose (Chloephaga picta) in the Falkland-Islands[END_REF]. Upland geese are highly territorial and socially monogamous, usually returning to the same territory with the same mate every year. Territories are taken up from August onwards and eggs laid between late September and end of October. The mean clutch comprises 6 eggs [START_REF] Summers | The Upland Goose[END_REF]. Hatching mainly takes place between mid-October and mid-November and fledging starts with about 70 days of age.

Family groups then stay together for the austral summer and winter [START_REF] Summers | The life-cycle of the Upland goose (Chloephaga picta) in the Falkland-Islands[END_REF]) and during the breeding season no creching occurs.

Additionally to the general analysis of the relationship of clutch size, hatching date and body condition in this non-migratory waterfowl species, we were able to compare data from individual females captured in subsequent years in the same territories. To our knowledge, this is the first study on seasonal clutch size decline in sheldgeese and one of a few comparing condition and timing of breeding within individuals between different years in waterfowl.

We tested the following hypotheses:

1. Egg and clutch size decrease with increasing hatching date.

2. Egg size, clutch size and hatching date is related to female body condition.

3. Individual variation in hatching dates is related to individual variation in body condition. 4. Chicks hatched from larger eggs are in a better condition than chicks from smaller eggs.

MATERIALS AND METHODS

The study was carried out in the New Island Nature Reserve, Falkland Islands (51°43'S, 61°17'W) from October to December in 2005, 2007 and 2008. New Island has an estimated 84km of coastline embracing some 2,300 ha. The island has been established as a nature reserve in 1970 when all livestock was removed from the island. This led to an increase in the density of Upland Geese, which is now with approximately 0.8 pairs ha -1 one of the highest in the Falkland Islands [START_REF] Quillfeldt | Escape decisions of incubating females and sex ratio of juveniles in the Upland Goose Chloephaga picta[END_REF].

For the study of seasonal decline, a total of 84 nests were included in the analysis (29 from 2005, 25 from 2007 and 30 from 2008). At the start of each field season we mapped nests using GPS.

Females left their nests at varying distances when approached, eggs were covered with down after measurements and all females returned to their nests afterwards. For each nest, we determined clutch size, measured length (L, expressed in cm) and width (B, expressed in cm) of each egg to the nearest 0.1 mm using callipers and weighed each egg to the nearest 0.1 g using a digital balance. Egg volume (V; in cm 3 ) was calculated as V = (L x B 2 x 0.507) following [START_REF] Furness | A technique for estimating the hatching dates of eggs of unknown laying date[END_REF]. We defined total clutch volume as the sum of the volumes of each egg in the clutch. Avian eggs decrease in density with increasing length of incubation mainly due to water loss. In 2005, a subsample of eggs was reweighed at a later date to estimate the density loss. Eggs lost an average of 0.0063 ± 0.0005 SE g/cm 3 per day. We determined expected hatching date as follows: (1) we calculated egg density (D) from egg mass M (expressed in g) and V. (2) Our 2005 data showed that hatching occurred at a mean egg density of 0.89g/cm 3 ± 0.01g/cm 3 SE (n=14 eggs). (3) Thus, the number of days to hatching T was estimated as T = (D -0.89g/cm 3 ) / 0.0063g/cm 3 . We then visited nests again at least once a day, starting at the estimated hatching date. All eggs hatched within 0-2 days of the estimated hatch date.

We caught adults during the period when they attended their brood (mean chick age: 14.8 ± 4.1 days) using a 3 x 5 m whoosh net. One person herded the family of geese slowly to the catching area, and when they arrived directly in front of the furled net, the other researcher pulled the trigger. Adults were marked with an individual metal ring on one leg and a plastic ring readable from a distance on the other and weighed to the nearest 10g using a digital spring balance. Head length, culmen length, and tarsus length were measured to the nearest 0.1 mm using callipers; wing length (maximum flattened chord) was measured to the nearest 1 mm using a foot rule. Chicks were marked individually using web-tags placed in the footwebs. We left enough space for the foot web to grow and did not observe any adverse effects on gosling growth. Chicks were weighed to the nearest 1g using a spring balance (<300g) or to the nearest 10g using a digital spring balance (> 300g). We measured head length, culmen length, wing length and tarsus length (±0.1 mm) using callipers. Chick age was determined from a growth curve for head and tarsus lengths established from chicks of known age in 2005. For each clutch, the mean chick age and thereby mean hatching date were calculated. As hatching dates could not be determined from egg density for all individual females, we used this estimated hatching date for further analyses. Estimated hatching dates from egg measures and chick measures were highly correlated (R = 0.973, P < 0.001, n = 41).

The body condition of females then was determined accounting for structural size and chick age, as we did not catch females during the prelaying period and captured females had goslings of different age. Females lose weight during incubation, and regain weight after hatching of the chicks [START_REF] Summers | The Upland Goose[END_REF]. We estimated an expected body mass for each individual based on a multiple linear regression of body mass on the first principal component (PC1) of measurements of wing, head, bill and tarsus and chick age. Body condition was calculated as the ratio of the observed body mass to the derived expected body mass, according to female size and chick age. For the analysis of the effect of mean egg volume on chick body condition we only included clutches with chicks ≤20 days of age as until then no sex-specific growth was observed (A.Gladbach, unpublished data). Chick body condition was expressed as the ratio of observed to expected body mass. To avoid using the same data to determine chick body condition and chick age (see above), in this case we used the estimated hatching date determined from egg density data. Expected body mass was the population mean of chicks of the same age (based on body mass data of a total of 553 chicks), following [START_REF] Quillfeldt | Begging in the absence of sibling competition in Wilson's storm-petrels, Oceanites oceanicus[END_REF].

Using this method, age is included indirectly into the model and hence controlled for. As a result, body condition is by definition uncorrelated (orthogonal) to age. For each clutch, we calculated a mean body condition of chicks.

Statistical analysis

Statistical tests were performed in SPSS 11.0 and R 2.9.1 (R Development Core Team 2009, www.r-project.org). Normality was tested with Kolmogorov-Smirnov tests and data were transformed, if necessary. Means are given with standard errors. We assessed significance using Pearson correlations or F-statistics in General Linear Models (GLM), based on Type III Sum of Squares. To indicate the direction (positive or negative) of the correlation to the covariable, we included t values, and as a measure of effect sizes, we included partial eta-squared values (η 2 ) (i.e. the proportion of the effect + error variance that is attributable to the effect). The sums of the partial eta-squared values are not additive (e.g. http://web.uccs.edu/lbecker/SPSS/glm_effectsize.htm). To analyse the dependency of mean egg volume on clutch size, we used a non-parametric Spearman correlation.

For the analysis of individual data from different years we used a linear mixed effects model (lme) in R 2.9.1. with hatching date as the response variable, body condition and year (as ordered fixed factor) as explanatory variables and female identity as a random effect. Thereby we could simultaneously test whether hatching date on an individual basis depended on body condition or if there was a relation with increasing age (year). In the final model, "year" was excluded based on the stepwise AIC function, an information-theoretical approach [START_REF] Burnham | Trade-offs between egg size and number in waterfowl: an interspecific test of the van Noordwijk and de Jong model[END_REF]. We here present results of this final model.

Statistical repeatabilities of variables in different years were calculated by the method of [START_REF] Lessells | Unrepeatable repeatabilities -a common mistake[END_REF]. Significance level was set to p<0.05. Some territories were not visited before chicks hatched and hence some data on clutch size and egg volumes were missing, resulting in different sample sizes for the respective tests.

RESULTS

Egg and clutch size decline

Clutch size and mean egg volume were significantly higher in clutches with earlier hatching date (Fig. 1, Table 1). Mean egg volume increased significantly with increasing clutch size (Pearson correlation, R = 0.377, P < 0.001). This correlation remained significant when controlling for female structural size (R = 0.373, P = 0.018), but not when controlling for female mass (R = 0.221, P = 0.171) or body condition (R = 0.238, P = 0.139).

Female body condition and hatching date, clutch size and mean egg volume

Hatching date and female post-hatching body condition were positively related, with females in a better body condition having earlier clutches (Fig. 2,Table xy).

We compared data of 28 recaptured individuals (5 captured in all three study years, 4 in 2005 and 2008, 4 in 2005 and 2007 and 15 individuals in 2007 and 2008). Body condition data showed significant repeatability (r = 0.308, F 27,60 = 1.969, P = 0.032), whereas repeatability of hatching date was non-significant (r = -0.171, F 27,60 = 0.683, P = 0.844). Hatching date was significantly negatively related to body condition within individuals (lme in R; body condition: F 1,32 = 18.80, t = -4.336, P = <0.001), while changes in year (i.e. female age) did not systematically affect hatching date as it was dropped from the full model (see "Statistical analysis").

Clutch size and mean egg volume were higher in females in better post-hatching body condition and this relationship was independent of the sample year (Fig. 3, Table xy).

Egg volume and chick body condition

Chick body condition was positively related to mean egg volume and negatively to hatching date, i.e. chicks hatched from nests with higher mean egg volume and earlier clutches had significantly higher early body condition (Table 3, Fig. 4). Clutch size had no effect on chick early body condition (Table 3). Mean body condition of chicks from females with high body condition was higher compared to chicks of low body condition females (Table 3).

DISCUSSION

In our present study on Upland Geese we found that:

1. Clutch size and egg size decreased with later hatching date.

2. Clutch size variables and hatching date were related to female post-hatching body condition.

3. Individual variation in hatching dates was related to individual variation in body condition. 4. Egg size had a direct and positive effect on chick body condition in the first weeks after hatching.

We found the observed seasonal decline of clutch and egg size on the population level in Upland Geese to be mainly caused by the early onset of reproduction by females in good body condition. We could show, to our knowledge for the first time, that the commonly observed positive relation between female condition and hatching date also holds true on an individual basis. Annual variation in individual body condition had a direct effect on the onset of reproduction, with females in an individually better condition having earlier hatching dates. The significant repeatability of posthatching body condition indicates that differences in quality between individuals remain stable over several years, but that despite this stability small individual changes in body condition affect timing of reproduction. However, as we did not measure female pre-breeding condition we are aware that our results must be treated cautiously as it must not necessarily be linked to her post-hatching condition, an assumption that still needs to be tested in Upland Geese.

Differences in body condition between individuals may arise due to differences in the inherent quality of individual birds (e.g. [START_REF] Jensen | Sexual variation in heritability and genetic correlations of morphological traits in house sparrow (Passer domesticus)[END_REF]e.g. Merila et al. 2001), which is supported by the significant repeatability of body condition in our study. As Upland Geese return to the same territories every year, also differences in territory quality may affect female body condition. Differences in individual body condition could have more immediate causes such as differences in weather, food availability, health state etc.. That small differences in condition affect the timing of the onset of reproduction supports the idea that a) egg laying starts when a certain individual threshold body condition is reached and that b) the investment into egg size and number decreases as the season advances, as expected offspring value is assumed to decline during a breeding season. Alternatively, if female-post-hatching condition is closely linked to her pre-laying condition, the investment into more and heavier eggs could also be explained by this better body condition. In contrast to previous studies that found an age effect with earlier hatching dates in older age classes to be the main cause for interindividual variation in hatching dates [START_REF] Blums | Patterns of reproductive effort and success in birds: path analyses of long-term data from European ducks[END_REF][START_REF] Finney | Reproductive habits in Snow Goose -influence of female age[END_REF][START_REF] Hamann | Age effects on clutch size and laying dates of individual female Lesser Snow Geese Anser caerulescens[END_REF][START_REF] Hamann | Intra-seasonal decline of clutch size in Lesser Snow Geese[END_REF] we did not observe a decline in hatching date within individual females over the study years. Upland Geese have an average adult life expectancy of about five years [START_REF] Summers | The Upland Goose[END_REF], our study period thus covered a considerable amount of this expected lifespan. However, the small sample size of individuals sampled in three different years (N=5) may also explain why we could not detect an age effect in our study. One hint for the possibility of an age effect at the very end of an individual life span is one female found dead at the end of the last study year which had hatched chicks 30 days earlier in 2008 than in 2005 and 2007.

In precocial species like waterfowl, where the young feed for themselves after hatching, the role of the parents and their effect on offspring performance is harder to detect than in altricial species.

Parental quality may not only be reflected in the effect of social factors like predator avoidance/defence and social competence for variation of reproductive success. High quality parents may also be able to produce larger clutches and to accumulate more resources and more rapidly than low quality individuals which would allow an earlier nesting [START_REF] Drent | The prudent parent -energetic adjustments in avian breeding[END_REF] and thereby providing better growing conditions for the offspring [START_REF] Hoekman | Factors affecting survival of Mallard ducklings in southern Ontario[END_REF][START_REF] Lepage | Seasonal decline of growth and fledging success in Snow geese, Anser caerulescens: an effect of date or parental quality?[END_REF][START_REF] Lepage | Seasonal variation in growth of greater snow goose goslings: the role of food supply[END_REF]. Female egg investment may play an important role in determining survival prospects of offspring [START_REF] Amat | Intraclutch egg-size variation and offspring survival in the Kentish Plover Charadrius alexandrinus[END_REF]) and thereby increasing individual fitness. We here show that early chick body condition increased with increasing egg size. This is line with other studies on waterfowl, where chick body size was also found to be related to egg size (e.g. [START_REF] Christians | Avian egg size: variation within species and inflexibility within individuals[END_REF][START_REF] Ankney | Egg weight, survival, and growth of Lesser Snow Goose goslings[END_REF][START_REF] Anderson | Composition and growth of King eider ducklings in relation to egg size[END_REF].

Interestingly, we found that egg size increased with increasing clutch size. Other studies on waterfowl already have shown that the theoretical trade-off between size and number of offspring as proposed by [START_REF] Lack | Ecological adaptations for breeding in birds[END_REF] can be observed on an interspecific level [START_REF] Figuerola | A comparative study of egg mass and clutch size in the Anseriformes[END_REF], but is not always present on an intraspecific level (Christians 2000; see Christians 2002 for a review). The intraspecific relationship does not show a consistent pattern, as there was no correlation between egg size and clutch size in some species but a positive correlation in others (see Christians 2002 for a review). Hõrak et al. (2008) pointed out that the absence of a trade-off between egg size and number may be explained by the lower quality of young hatched from larger clutches, i.e. the trade-off occurring between egg number and hatchling quality. We found no evidence that this is true for Upland Geese, as clutch size had no influence on chick body condition. [START_REF] Flint | Allocation of limited reserves to a clutch: A model explaining the lack of a relationship between clutch size and egg size[END_REF] suggested that in waterfowl which rely on endogenous reserves for egg production, large-bodied birds may tend to lay larger c1utches of larger eggs than do small-bodied birds. Our results show, that in Upland Geese body size is no proxy for egg size as we failed to detect a relationship between structural size and either egg or clutch size. In contrast, the availability of endogenous reserves (reflected in body mass and body condition) was an important source for variation in egg and clutch size between female Upland Geese. This supports the idea that high-quality females are able to produce more eggs and larger eggs than low-quality females, although we could not distinguish whether the higher body condition of these females was caused by higher foraging

FIGURESFigure 1 .

 1 FIGURES

Figure 2 .

 2 Figure 2. Negative relationship between female body condition and hatching date in Upland Geese.

Figure 3 .

 3 Figure 3. Positive relationship between female body condition and clutch size (a), and mean egg

Figure 4 .

 4 Figure 4. Relationship between chick body condition, hatching date and mean egg volume in Upland

  Figure 1.

Table 3 Figure 5 .

 35 Figure 5. Relation of female body condition, hatching date and clutch size (left) and egg size (right) in

Table 1

 1 Results of General Linear Models (GLM) showing the effects of year and hatching date on clutch size and mean egg volume in adult female Upland Geese. Clutch size or mean egg volume were used as dependent, year as a fixed factor and hatching date as covariate. η 2 was included to indicate the effect size and t-values to indicate the direction (positive or negative) of the correlation to the covariable.

	Parameter	year	hatching date (hd)	interaction (year*hd)
	clutch size	F 2,84 = 0.469, P = 0.628, η 2 = 0.012	F 1,84 = 13.213, P < 0.001, η 2 = 0.145, t = -1.300	F 2,84 = 0.434, P = 0.649, η 2 = 0.011
	mean egg volume	F 2,84 = 1.505, P = 0.228, η 2 = 0.037	F 1,84 = 20.183, P < 0.001, η 2 = 0.206 t = -1.273	F 2,84 = 1.528, P = 0.223, η 2 = 0.038

Table 2

 2 Results of General Linear Models (GLM) showing the effects of year and body condition on hatching date, clutch size and mean egg volume in adult female Upland Geese. Hatching date, clutch size or mean egg volume were used as dependent, year as a fixed factor and hatching date as covariate.

Acknowledgments

We are grateful to the New Island Conservation Trust for the possibility to carry out this study on the island and for providing accommodation and transport. We would like to thank Riek van Noordwijk, Rafael Matias and Andreas Michalik for their help in capturing "difficult" goose families. The manuscript benefited from the comments of two anonymous referees and Yann Clough checked the English. This work would not have been possible without the support of Ian, Maria and Georgina Strange and Dan Birch. A.G. received financial support by the Bayerische Eliteförderung, the Arthurvon-Gwinner-Foundation, the German Academic Exchange Service (DAAD) and the German Ornithological Society (DO-G). P.Q. was funded by DFG, Germany (Emmy Noether Programme, Qu148/1-3).


Jules Sadefo Kamdem
email: sadefo@lameta.univ-montp1.fr

Gini Index and Polynomial Pen's Parade

Keywords: Gini, Income inequality, Polynomial pen's parade, Ranks. JEL Classication: D63, D31, C15

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Interest in the link between income and its rank is known in the income distribution literature as Pen's parade following [START_REF] Pen | Income Distribution[END_REF][START_REF] Pen | A parade of dwarfs (and a few giants)[END_REF]. The precede has motivated research on the relationship between Pen's parade and the Gini index that is a very important inequality measure. However, this research has so far focused on the linear Pen's parade for which income increases by a constant amount as its rank increases by one unit (see [START_REF] Milanovic | A simple way to calculate the Gini coecient, and some implications[END_REF]. However, a linear parade does not closely t many real world income distributions, Pen's parade, which is convex in the absence of negative incomes (i.e., incomes that increase by a greater amount as its rank increases by each additional unit). [START_REF] Mussard | Quadratic Pen's parade and the computation of Gini Index[END_REF] has recently introduced the computation of Gini index with a convex quadratic Pen's parade (or second degree polynomial) parade for which income is a quadratic function of its rank.

In this paper, we extend the computation of gini index by using a more general and empirically more realistic case in which Pen's parade is a convex polynomial of nite order k ∈ N * . Hence, the Gini indexes for a linear Pen's parade and for quadratic pen's parade becomes a special case of that for an higher degree polynomial Pen's parade under some constraints on parameters to keep convexity.

The rest of the paper is organized as follows: In Section 2, the specication for a higher degree polynomial Pen's parade is provided. In Section 3, the problem of tting a higher degree polynomial Pen's parade to real world data sets of [START_REF] Milanovic | A simple way to calculate the Gini coecient, and some implications[END_REF] is discussed. Concluding is in Section 4.

2

High degree Polynomial Pen's Parade Suppose that positive incomes, expressed as a vector y, depend on individuals' ranks r y in any given income distribution of size n. Suppose that incomes are ranked in ascending order and let r y = 1 for the poorest individual and r y = n for the richest one. Hence, following [START_REF] Lerman | A note on the calculation and interpretation of the Gini index[END_REF], the Gini index may be rewritten as follows:

G = 2 cov(y, r y ) n ȳ . (1)
the mean income. It is straightforward to rewrite (13) as:

G = 2 σ y σ ry ρ(y, r y ) nȳ , (2)
where ρ(y, r y ) is Pearson's correlation coecient between incomes y and individuals' ranks r y , where σ y is the standard deviation of y and where σ ry is the standard deviation of r y . Following [START_REF] Lerman | A note on the calculation and interpretation of the Gini index[END_REF] and under the assumption of a linear Pen's parade (i.e. y = a + b r y ), [START_REF] Milanovic | A simple way to calculate the Gini coecient, and some implications[END_REF] demonstrates that for a suciently large n, the Gini index can be further expressed as:

G = σ y √ 3y ρ(y, r y ) . (3)
Milanovic's result is very interesting since it yields a simple way to compute the Gini index. However, as mentioned by Milanovic (1997, page 48) himself, "in almost all real world cases, Pen's parade is convex: incomes rise very slowly at the beginning, then go up by their absolute increase, and nally increase even at the rate of acceleration". Thus, ρ(y, r y ) which measures linear correlation will be less than 1. Again, from Milanovic (1997), a convex Pen's parade may be derived from a linear Pen's parade throughout regressive transfers (poor-to-rich income transfers). Inspired by Milanovic's nding, we demonstrate in the sequel, without taking recourse to regressive transfers, that the Gini index can be computed with a general nonlinear polynomial function Pen's parade. The computation of the Gini Index using a polynomial function of order 2 (i.e. quadratic function) Pen's Parade has been introduced in [START_REF] Mussard | Quadratic Pen's parade and the computation of Gini Index[END_REF]. In this paper, we generalize the precedure to compute the Gini Index using a polynomial function of order k ∈ N * pen's parade. Note that a linear pen's parade corresponds to k = 1 and quadratic pen's parade corresponds to k = 2. Our result here will be available for each nite k ∈ N * .

Simple Gini Index with nonlinear power Pen's Parade

Consider a power function relation between incomes and ranks:

y = k-1 i=0 b i r i y + b k r k y . (4)
3 with k ∈ N * . The covariance between y and r j y for j ∈ N * is given by: cov(y,

r j y ) = k i=0 b i cov r i y , r j y (5) = k i=0   b i n r i+j+1 y - b i n 2 n ry=1 r j y n ry=1 r i y   = k i=0   b i n n ry=1 r i+j+1 y -b i n ry=1 r i y n ry=1 r j y  
The mean income ȳ is then:

y = k i=0 b i r i y = 1 n n r=1 k i=0 b i r i (6)

The coecient of variation

Since the incomes y are positive, we use (13) by assuming that b k > 0 and b j for j = 1, . . . , k -1 are chosen such that y > 0. For instance, if k = 2 then we can use b j for j = 1, . . . , k such that b 2 1 -4b 2 b 0 < 0. We are now able to compute the coecient of variation of incomes as follows:

σ y y = k i=0 k j=0 b i b j cov(r i y , r j y ) k i=0 b i r i y (7) = k i=0 b i k j=0 b j cov(r i y , r j y ) k i=0 b i r i y (8) = k j=0 b j cov(y, r j y ) k i=0 b i r i y (9) = k j=0 b j k i=0 b i n n ry=1 r i+j+1 y -b i n 2 n ry=1 r i y n ry=1 r j y 1 n n r=1 k i=0 b i r i .
where the variance of

r k y is cov(r k y , r k y ) = σ 2 r k y = 1 n n r=1 r 2k - 1 n n r=1 r k 2 , (10)
the covariance between r j y and r i y for 0 ≤ i, j ≤ k, is:

cov r i y , r j y = 1 n n r=1 r i+j - 1 n 2 n r=1 r i n r=1 r j . ( 11
)
After a double summation

cov k i=0 b i r i y , k j=0 b j r j y = k i=0 k i=0 b i b j 1 n n r=1 r i+j - 1 n 2 n r=1 r i n r=1 r j . ( 12
)
Lemma 2.1 When n → ∞, for q ∈ N * and r ∈ N * , we have that

n r=1 r q ≡ n q+1 q + 1 . (13)
Based on the preceding lemma, the variance of y when n → ∞ is equivalent to:

cov k i=0 r i y , k j=0 r j y ≡ b 2 k n n k+k+1 k + k + 1 - b 2 k n 2 n k+1 k + 1 n k+1 k + 1 ≡ (n k b k k) 2 (2k + 1)(k + 1) 2 .
(14) Therefore when n → ∞ the standard deviation of y is equivalent to

σ y ≡ (n k b k k) 2 (2k + 1)(k + 1) 2 ≡ k |b k | (k + 1) √ 2k + 1 n k . (15)
When n → ∞, the mean of y is equivalent to

y = 1 n n r=1 k i=0 b i r i ≡ b k n n k+1 k + 1 ≡ b k n k k + 1 (16)
Thereby, as n → ∞ the coecient of variation is is equivalent expressed as:

σ y y ≡ k |b k | (k+1) √ 2k+1 n k b k n k k+1
(17) then we have the following limit which depends on k and the sign of b k :

lim n→∞ σ y y = |b k | b k k √ 2k + 1 = sign(b k ) k √ 2k + 1 . ( 18
)
We have then proved the following theorem:

Theorem 2.1 Under the assumption of a nonlinear polynomial Pen's parade, i.e., y = k i=0 b i r i y , with b k = 0, when n → ∞, the coecient of variation of the revenue y has the following limit:

lim n→∞ σ y y = |b k | b k k √ 2k + 1 (19)
On the other hand, following [START_REF] Milanovic | A simple way to calculate the Gini coecient, and some implications[END_REF]:

lim 2 σ ry n = lim n→∞ n 2 -1 3n 2 = 1 √ 3 . ( 20
)
The product of (20), (18) and ρ(y, r y ) entails the following result:

Theorem 2.2 Under the assumption of a nonlinear polynomial Pen's parade, i.e., y = k i=0 b i r i y , with b k = 0, when n → ∞, the Gini index G k can be approximately compute as follows:

G k 1 √ 3 |b k | b k k √ 2k + 1 ρ(y, r y ), (21)
where ρ(y, r y ) is the correlation coecient between the revenue and the rank r y . (1993, annual), Romania (1994, monthly), Bulgaria(1994;annual). For estonia (1995; quarterly) and UK (1986; annual), we can also choose k = 1, but we consider k = 2 in the case where government policy prefers to overestimate inequality instead of underestimate inequality. For Russia (1993-4; quarterly), we consider k = 3 and for Kyrgyzstan (1993; quarterly) we choose Remark 3.2 In our polynomial pen's parade, if r y = 1, then y = y 1 = k i=1 b k . In practical applications, the parameters b k , estimated from the observed data using multiple regression, are chosen so that the revenue y > 0 (i.e. b k > 0). It's very important to note that, the correlation coecient between y and its rank r y depend on b k . Ignoring this fact will arbitrarily restrict the Parade to pass through the origin and may result in less accurate estimates of the Gini Index. [START_REF] Pen | A parade of dwarfs (and a few giants)[END_REF] Concluding Remarks Following Milanovic (1997), we have proposed another simple way to calculate the Gini coecient under the assumption of a general polynomial Pen's Parade of order k . By using the data in Table 2, we conclude that the computation of the Gini Index of each revenues data needs to nd a specic integer k, which is the order a specic polynomial Pen's Parade. Our Gini computation is useful in practical applications as soon as the limit expression obtained in ( 21)is a good approximation of the Gini Index for usual size n.

Two immediate and practical applications can be generated from this new Gini expression. First, the possibility to address a signicant test since our Gini Index (as well as Milanovic's) is based on Pearson's correlation coecient. Thereby, testing for the Gini Index signicance is equivalent to testing for the signicance of Pearson's correlation coecient (up to the constant k √ 2k+1 ). This test relies on the well-known student statistics based on the hyperbolic tangent transformation. Second, estimating the coecients b i for i = 1, . . . , k, e.g. with Yitzhaki's Gini regression analysis or a multiple regression, enables a parametric Gini Index to be obtained. That procedure depends on parameters reecting the curvature of Pen's Parade. This may be of interest when one compares the shape of two income distributions.

Remark 2 . 1 1 ρ(y,ry) 3 and for k = 2 , 15 .Remark 2 . 2

 211321522 Note that for k = 1, we have the result of Milanovic (1997), i.e. G we have the result of Sadefo Kamdem et al. (2010), i.e. G 2 2 ρ(y,ry) √ Remarks that the Gini Index G k approximation depends on k, the sign of b k and ρ(y, r y ). Since the Gini computation is independent of parameters b i for i = 0, . . . , k -1, we can compute the Pen's Parade as follows: y = b 0 + b k r k . Based on revenue data, it is simple to use a regression to estimate the parameters b 0 and b k . In relation with the paramater k and b k > 0, we have the following table:

Table 1 :

 1 Computation of some coecient of variation GV k . CV k denotes the coecient of variation for polynomial Pen's Parade of order k and GC k = CV k /

	k	1	2	3	4	5	6	7	8
	CV k 0,577 0,894 1,134 1,333 1,508 1,664 1,807 1,940
	GC k 0.333 0.516 0.655 0.770 0.870 0.961 1.043 1.120
	where √	3. Following Milanovic's data (1997), we obtain
	the following results:

Table 2 :

 2 Comparison the gini indexes of G k for k = 1, 2, 3, 6 with true G est In the preceding table, G est denotes the estimation of the true Gini index by using Milanovic data (1997). Remark 3.1 Based on the analysis of the preceding table, we propose to consider linear Pen's parade (k=1) to compute the Gini index of Poland

	Country (year)	n	ρ(y, r y )	G 1	G 2	G 3	G 6	G est
	Hungary (1993; annual) Poland (1993; annual) Romania (1994; monthly) Bulgaria (1994; annual) Estonia(1995; quarterly) UK (1986; annual) Germany (1889; annual) US (1991; annual) Russia (1993-4; quarterly)) 16356 0.812 0.271 0.419 0.532 0.780 0.502 22062 0.889 0.296 0.459 0.582 0.854 0.221 52190 0.892 0.297 0.461 0.584 0.857 0.288 8999 0.863 0.288 0.446 0.565 0.829 0.284 7195 0.889 0.296 0.459 0.582 0.854 0.308 8759 0.871 0.290 0.450 0.570 0.837 0.342 7178 0.815 0.272 0.421 0.534 0.783 0.320 3940 0.744 0.248 0.384 0.487 0.715 0.305 16052 0.892 0.297 0.461 0.589 0.857 0.391 Kyrgyzstan (1993; quarterly) 9547 0.586 0.195 0.303 0.384 0.563 0.551

k = 6.