Maud Poisbleau
email: maud.poisbleau@ua.ac.be

A E Noe

Guillon Ae

Herve ´fritz

F M Bairlein

Á N Poisbleau

Á H Guillon

Fritz

Preservation of winter social dominance status in Brent Geese Branta bernicla bernicla within and across winters

Keywords: Social dominance status conservation, Wintering, Age class, Social unit size, Flock dynamics

Dominant and subordinate individuals in a group may benefit from the stability of the social dominance organisation, avoiding excessive waste of time and energy in aggressive interactions and reducing injury risks. Nevertheless, the likely evolutionary incentive for individuals to become, and furthermore to stay, dominant may destabilise such dominance hierarchies. In this context, the relative importance of fixed (e.g. sex, morphological size) and fluctuating (e.g. body condition, mating status, reproductive success, social unit size) traits influencing the establishment and preservation of dominance relationships could play a key role in group structure. We investigated the relative role of fixed and fluctuating traits on social status in Dark-bellied Brent Geese Branta bernicla bernicla which form large fairly unstable groups both within and across winters. We compared individual dominance scores of ringed Brent Geese during four consecutive winters. Brent Geese conserved their dominance score within a given winter irrespective of their age but were generally unable to conserve it across consecutive winters. As winter dominance scores correlated best with social unit size, dominance status thus appeared to be mostly a by-product of a fluctuating trait: breeding success in the previous summer. When we considered only adults that had the same social unit size during two consecutive winters, we observed a significant preservation of dominance scores. This result suggests that a fixed trait such as sex or morphological size may still play a role in setting dominance status.

Introduction

Social dominance organisation occurs in most gregarious species (see [START_REF] Krause | Living in groups[END_REF] for examples), probably because the benefits of such an organisation outweigh the costs. Indeed, a social organisation based on dominance relationships allows the reduction of costs associated with gregariousness. It also provides social stability through a predictable social environment, mostly by structuring access to limited resources, such as food or a mate [START_REF] Rowell | The concept of social dominance[END_REF][START_REF] Bernstein | Dominance: the baby and the bathwater[END_REF][START_REF] Archer | The behavioural biology of aggression[END_REF]). This stable hierarchy avoids an excessive waste of time and energy in aggressive interactions and reduces injury risks, especially between individuals that differ considerably in dominance status. However, although both subordinate and dominant individuals in a group may benefit from a stable social organisation, there is probably an evolutionary incentive for individuals living in groups to access and retain dominant status [START_REF] Krause | Living in groups[END_REF].

When a social dominance hierarchy is established within a group, the outcome of subsequent aggressive interactions is often predictable from previous encounters in the short term [START_REF] Rowell | The concept of social dominance[END_REF]), but also possibly from one year to the next (Lamprecht 1986b). In the long term, the preservation of dominance status may only be possible within groups undergoing few changes in composition. If a group remains stable (without immigration or emigration), we could expect that its social organisation also remains stable because, once established, social dominance hierarchy between members could be conserved by social inertia [START_REF] Guhl | Social inertia and social stability in chickens[END_REF][START_REF] Archawaranon | Social inertia and hormonal control of aggression and dominance in white-throated sparrows[END_REF][START_REF] Wiley | Social inertia in white-throated sparrows results from recognition of opponents[END_REF] when no environmental change affects members' social behaviour (e.g. [START_REF] Sloman | The effects of increased flow rates on linear dominance hierarchies and physiological function in brown trout, Salmo trutta[END_REF]. On the other hand, stability of dominance hierarchies may depend on the phenotypic traits influencing individual social dominance status, such as fixed (e.g. sex, morphological size) or fluctuating characteristics (e.g. body condition, mating status, reproductive success, social unit size; see [START_REF] Piper | Social dominance in birds. Early findings and new horizons[END_REF] of the individuals constituting the group. In unstable groups and fluctuating environmental conditions, individuals may conserve their social status if fixed traits influence social dominance more than fluctuating ones. However, if dominance status depends mostly on fluctuating individual traits, individual dominance status in time could be constantly reconsidered and stability within the group may be impossible. In free-living animals, preservation of dominance status may thus depend on group structure, environmental conditions and individual phenotypic traits involved in the expression of dominance status. The relative importance of fixed and fluctuating traits determining dominance status and its preservation should therefore be a key issue when investigating gregarious species.

During winter, goose populations live in large groups for several months. Even if site fidelity provides a relatively stable group composition [START_REF] Spaans | Inland pastures are an appropriate alternative for salt-marshes as a feeding area for spring-fattening dark-bellied brent geese Branta bernicla[END_REF]and authors' unpublished data), groups are not always stable both within and across winters, due for instance to constant exchanges of individuals between wintering sites and processes related to local recruitment, emigration and immigration. In geese, sex, age, mating status (paired or single) and reproductive success (number of juveniles) have been found to be variously related to dominance status (e.g. Lamprecht 1986b;[START_REF] Black | Determinants of social rank in goose flocks: acquisition of social rank in young geese[END_REF][START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF]Poisbleau et al. 2006a[START_REF] Poisbleau | Dominance relationships in dark-bellied brent geese Branta bernicla bernicla at spring staging areas[END_REF]. [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF] found that social dominance increased with age for female Barnacle Geese Branta leucopsis whereas it was less obvious for males in a group of non-breeding moulting individuals. By contrast, Lamprecht (1986b) noted a better correlation between dominance and male age in winter flocks of Barheaded Geese Anser indicus. These results pointed out that individual dominance status may normally increase progressively through life. However, both sex and age are good integrators of various parameters potentially determining social dominance status within a class of social unit size. For instance, sex is strongly related to morphological size and body mass, while age is correlated with experience, mating status, reproductive success, morphological size and body mass. Pairs of Bar-headed Geese improved their dominance status after successfully completing the breeding season (Lamprecht 1986b). Dominant White-fronted Geese Anser albifrons lost their hierarchical position after losing their partner, and decreased their dominance score to the level of unpaired individuals [START_REF] Boyd | On encounters between wild white-fronted geese in winter-flocks[END_REF]. When the brood size of Barnacle Geese changed as a result of natural events or experimental manipulation, social dominance status changed too [START_REF] Loonen | The benefit of large broods in barnacle geese: a study using natural and experimental manipulations[END_REF]. Moreover, Barnacle Geese that remained with young for the longest period bred more successfully the next year than those with shorter periods of parental care, and females returning to the winter area with a brood were heavier than those that had nested but had no young in winter [START_REF] Owen | Factors affecting the survival of barnacle geese on migration from the breeding grounds[END_REF]Black and Owen 1989b). For wintering Dark-bellied Brent Geese Branta bernicla bernicla, social dominance for a given winter was significantly correlated with body mass, morphological size and body condition during the previous winter (Poisbleau et al. 2006a). Hence free-living goose populations are a pertinent model to test dominance preservation and explore the relative role of fixed and fluctuating factors in setting social status within and across winters.

We investigated dominance relationships in Dark-bellied Brent Geese Branta bernicla bernicla during four consecutive winters. For the first time, we tested preservation of social dominance status within and across wintering periods using ringed individuals. Previous investigations on geese showed that the best predictor of dominance status in geese was social unit size (Lamprecht 1986a, b;[START_REF] Gregoire | Agonistic behavior and dominance relationships among lesser snow geese during winter and spring migration[END_REF][START_REF] Loonen | The benefit of large broods in barnacle geese: a study using natural and experimental manipulations[END_REF][START_REF] Siriwardena | Parent and gosling strategies in wintering barnacle geese[END_REF][START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF]Poisbleau et al. 2006a). However, the individual reproductive success of Brent Geese seems to vary significantly across years, mostly due to changes in predation pressure and climatic conditions at the Arctic grounds [START_REF] Ebbinge | A multifactorial explanation for variation in breeding performance of brent geese Branta bernicla[END_REF][START_REF] Underhill | Breeding of waders (Charadrii) and Brent geese Branta bernicla bernicla at Pronchishcheva Lake, northeastern Taimyr, Russian, in a peak and decreasing lemming year[END_REF][START_REF] Spaans | Dark-bellied brent geese Branta bernicla bernicla forego breeding when Artic foxes Alopex lagopus are present during nest initiation[END_REF][START_REF] ˆty | Shared predators and indirect trophic interactions: lemming cycles and arcticnesting geese[END_REF]. We therefore expect a strong influence of these stochastic events on individual dominance status. We predict that dominance preservation should be high within a given winter but may vary across winters. Whenever environmental constraints are of prime importance, we predict that the preservation of dominance status should be poor between successive winters. If individual characteristics partly override the effects of fluctuations in environmental constraints, we then expect dominance status to be conserved between winters. We therefore investigated the effect of fixed (sex and morphological size) and fluctuating (social unit size) individual characteristics on dominance status.

Methods

Birds and study site

During winters between 2001 and 2005, we captured 127 Dark-bellied Brent Geese in the wintering quarter of Ole ´ron Island (175 km 2 , 45°56 0 N, 1°21 0 W, France) using a cannon net [START_REF] Bub | Bird trapping and bird banding. A handbook for trapping methods all over the world[END_REF] but no bait or decoy [START_REF] Weatherhead | Comment: a critical assumption of band-recovery models may often be violated[END_REF]. We caught birds on the beach, close to watering areas where Brent Geese came in small flocks. We ringed birds immediately following capture and released them together as soon as possible (less than 4 h after capture). We assumed that family integrity was not affected by capture and manipulations [START_REF] Bowler | Family cohesion in greylag goose Anser anser broods following catching and ringing[END_REF].

We determined sex by cloacal examination [START_REF] Hochbaum | Sex and age determination of waterfowl by cloacal examination[END_REF]. Juveniles (first-winter) were distinguished from birds in adult plumage (second-winter sub-adults and older) by the white edges of the wing coverts (Lambeck 1990a). We measured tarsus and culmen lengths to the nearest 0.01 mm using an electronic calliper and wing length to the nearest millimetre with a ruler. One observer (M.P.) carried out all measurements to minimise observer biases. We used wing ? culmen ? tarsus lengths as a body size index (Poisbleau et al. 2006a). We marked each bird individually with coded Darvic leg-rings, which we could identify from a distance of up to 250 m with a 20 9 60 telescope. Using the same colour combination for each bird, we avoided potential inter-individual differences in the colour-band effect within the studied birds (see, for examples, [START_REF] Burley | Comparison of the band-colour preferences of two species of estrildid finches[END_REF][START_REF] Cuthill | Colour bands, dominance, and body mass regulation in male zebra finches (Taeniopygia guttata)[END_REF]. The European Brent Geese network have used Darvic rings for the last 14 years and reported no significant effects on the ecology and survival of this species.

Behavioural observations

Each winter, we conducted behavioural observations from arrival to departure of Brent Geese on Ole ´ron Island (i.e. from late September to late March). We determined the social dominance organisation of the flock by observing interactions (see also Poisbleau et al. 2006a, b). For a given ringed individual, we considered any aggressive interactions with any opponent, marked or not, and noted the outcome, while the geese were foraging or drinking on Zostera spp. and Enteromorpha spp. beds along coastlines. We defined an interaction as a direct confrontation between two birds, ranging from threats with lowered head and neck to active chases with flapping wings [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF]. We considered that an individual won an aggressive interaction when its opponent turned and walked or ran away [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF]. The social dominance score was defined as the percentage of interactions won by a focal bird divided by the total number of interactions in which the bird participated (for method, see [START_REF] Ens | Interference among oystercatchers, Haematopus ostralegus, feeding on mussels, Mytilus edulis, on the Exe Estuary[END_REF]Lamprecht 1986b). We calculated a dominance score for each marked individual with a minimum total number of 26 interactions in order to obtain an error of less than ±10% for the estimated score (Poisbleau et al. 2006b). The more dominant a bird is, the closer to 100% is its dominance score. Thus, we assigned an individual winter dominance score for each observed ringed Brent Goose if involved in at least 26 interactions. During the winter 2003-2004, we assigned a monthly individual dominance score between October 16 and March 16 in order to test social status preservation across months and finally used all interactions to calculate the global winter dominance score.

Because parents and their offspring normally stay together until spring (Lambeck 1990b), we determined the size of ringed birds' social units using behavioural observations. Family members coordinate their behaviour and stay in proximity, i.e. family members move as a unit, and conspecifics coming too close to the family are pecked at or are chased away [START_REF] Gregoire | Agonistic behavior and dominance relationships among lesser snow geese during winter and spring migration[END_REF]. For adults, social unit size was the combination of mating status and breeding success where 1 meant unpaired bird, 2 meant paired bird without juveniles, 3 meant paired bird with one juvenile, etc. In this dataset, no juvenile was observed with only one parent or only with siblings but some were seen alone. Therefore, juveniles' social unit size could be 1 (juvenile alone) and otherwise ranged from 3 (juvenile without siblings) to 7 (juvenile with four siblings).

Statistical analysis

Dominance scores were percentages, hence were arcsinetransformed to conform to the normal distribution requested for parametric tests [START_REF] Sokal | Biometry[END_REF]. To test the correlation of individual dominance scores between different periods and the general trend within the studied population, we analysed the dominance score preservation between consecutive periods (months or winters) using a linear regression between each pair of consecutive periods. If the linear regression was significant, we first tested that the slope did not differ from one, and then that the intercept did not differ from zero, using a hypothesis test for intercept [START_REF] Scherrer B ; ) Biostatistique | Environmental influence on life-history traits: growth, survival, and fecundity in black brant (Branta bernicla)[END_REF]. For instance, if the slopes are the same, but the intercept increases winter after winter, it could be an indication of an increase in score with age, irrespective of the initial position of the individual in the hierarchy.

To explore the dominance score correlates for juveniles, we used a General Linear Model with sex as a fixed factor and social unit size and morphological size as covariates. Because 5 of the 16 juveniles came from the same two families, we randomly included only 1 juvenile from each family in this analysis. To investigate the relative contribution of fixed (between-individual components: sex and morphological size) and fluctuating (within-individual component: social unit size) traits in the variance of adults' dominance scores, we performed two General Linear Mixed Models with social unit size as a random variable, sex as a fixed factor and morphological size as a covariate. Because sex and morphological size were not independent for adults (F 1,118 = 95.852, P \ 0.001), we used sex in a first model and morphological size in a separate, second one. We performed all statistical analyses using SPSS 10.0 software (SPSS 1999), and present values as means ± standard error.

Results

Dominance score preservation within winter

The individual dominance score appeared to be conserved within a given winter, as all the linear regressions on social dominance scores between consecutive months were significant (Table 1). No slope was significantly different from one, and no intercept was significantly different from zero. The correlation between the first month (just after the arrival on winter areas) and the second was slightly less significant than the correlations between other months.

Adult dominance score preservation across winters

For adults, linear regressions on dominance score were not statistically significant for one of the three comparisons of consecutive winters, and of borderline significance for another (Table 2). Slopes were significantly different from one and intercepts were all significantly different from zero. Even when individual dominance status was approximately maintained across winters, subordinate birds had increased and dominant birds had decreased their dominance score.

Dominance score correlates for juveniles

Morphological size of juveniles did not significantly affect their dominance scores (F 1,9 = 0.143, P = 0.714). Juvenile social dominance scores were mainly explained by social unit size (F 1,10 = 92.071, P \ 0.001; Fig. 1). Juveniles in larger families were dominant over others, and isolated juveniles were subordinate. Accounting for social unit size, sex tended to be significant, males being almost always dominant over females (F 1,10 = 3.934, P = 0.075; Fig. 1). All interactions between age class, sex and morphological size were statistically non-significant (all P [ 0.500).

Relative contribution of fixed and fluctuating traits on dominance score for adults Adult males had significantly higher social dominance scores than adult females (F 1,149 = 31.750, P \ 0.001; Fig. 2). Sex explained 18.5% of the variance. Morphological size was significantly correlated with dominance score (F 1,113 = 4.822, P = 0.030) but explained only 4.1% of the variance. In both models, social unit size was positively correlated with the social dominance score and

Discussion

Dominance preservation within and across winters

According to both our predictions and Lamprecht's results (1986b), Brent Geese conserved their individual social dominance score within the same wintering period. Consequently, all observations made during winter could be summed to assess the overall winter dominance score. Nevertheless, because dominance scores for the first 2 months were more weakly related to each other than were those for subsequent pairs of months, dominance relationships seemed to be slightly less stable just after Brent Geese arrival in the wintering areas. Just after arrival on the wintering grounds, birds may encounter unfamiliar individuals for the first time and learn and/or test their respective dominance status. Perhaps it is also a period of intense migration, potentially increasing group instability [START_REF] Summers | Phenology of migration and use of wintering sites by the increasing population of dark-bellied brent geese Branta bernicla bernicla[END_REF], where interactions with transient birds could potentially influence the dominance score of residents. Consistent with our predictions, the individual dominance scores fluctuated significantly between winters. Several factors may explain these results. The first is the major influence of social unit size, as it appears to drive individual dominance status in our studies, as in others. Reproductive success, however, seems to be partly disconnected from individual quality, probably through the major influence of environmental conditions in the breeding grounds [START_REF] Ebbinge | A multifactorial explanation for variation in breeding performance of brent geese Branta bernicla[END_REF][START_REF] Scherrer B ; ) Biostatistique | Environmental influence on life-history traits: growth, survival, and fecundity in black brant (Branta bernicla)[END_REF][START_REF] Spaans | Dark-bellied brent geese Branta bernicla bernicla forego breeding when Artic foxes Alopex lagopus are present during nest initiation[END_REF][START_REF] Sedinger | Early environment and recruitment of black brant (Branta bernicla nigricans) into the breeding population[END_REF]. Similarly, some birds may have skipped breeding in a given year instead of having a failed breeding attempt. Nevertheless, dominance order was preserved once the effect of social unit size was accounted for. This additionally suggests, as our results showed, that some fixed individual characteristics such as sex or morphological size may also contribute to the dominance status of a given bird. Some of the fluctuations in dominance status and score preservation may on the other hand come from the fact that Brent Geese group composition fluctuates considerably across winters, and individual recognition between two or more subsequent winters is complicated. In this context, geese need to establish new dominance relationships each winter according to individual fighting quality and status signalling.

Dominance status as a by-product of breeding success in free-living goose populations For both juveniles and adults, dominance score was mainly and positively correlated with social unit size. As parents and their offspring normally stay together until spring, with no or few temporary separations between family members (Lambeck 1990b), this result is coherent with the high dominance score preservation within winters. Indeed, like social unit size, dominance score is a fluctuating trait across winters but a stable trait within a winter. While the fixed traits tested in this study had a much lower contribution to dominance score, sex (and secondary morphological size) also had a significant impact on the individual dominance score with adult males being dominant over adult females (and larger birds dominating smaller). In Bar-headed and Barnacle Geese, male characteristics and behaviour mainly or entirely determine the dominance status of a pair or family, and the bigger the family, the more motivated is the male (Lamprecht 1986b;Black and Owen 1989a). The same mechanism seems to act in our Dark-bellied Brent Geese study population.

In stable flocks, the outcome of a fight between two ganders determines their future dominance relationships (Fischer 1965and Scott 1978cited in Lamprecht 1986b). Lamprecht (1986b) therefore suggested that birds in stable flocks know each other individually, and that the hierarchical order may also be partly stabilised by the birds' experiences obtained during earlier interactions. In our field conditions, with large and partially unstable group compositions, individual recognition [START_REF] Halpin | Individual odors and individual recognition: review and commentary[END_REF][START_REF] Dugatkin | Individual recognition, dominance hierarchies and winner and loser effects[END_REF] and thus preservation of dominance relationships by social inertia [START_REF] Guhl | Social inertia and social stability in chickens[END_REF][START_REF] Archawaranon | Social inertia and hormonal control of aggression and dominance in white-throated sparrows[END_REF][START_REF] Wiley | Social inertia in white-throated sparrows results from recognition of opponents[END_REF] are probably not sufficient for the establishment of a stable social relationship. Brent Geese therefore may use an alternative means of status signalling [START_REF] Ketterson | Status signaling in dark-eyed juncos[END_REF][START_REF] Brotons | Status signalling in the coal tit (Parus ater): the role of previous knowledge of individuals[END_REF] to reduce costs of interactions. As suggested in former studies (see also Poisbleau et al. 2006a), social unit size may be the first indicator of pair quality and status in larger groups, dominant geese potentially having a higher reproductive success (Lamprecht 1986a, b), but also being more motivated in encounters. As paired birds had higher hierarchical positions than singletons in a flock of non-breeding moulting Barnacle Geese, [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF] also advocated the hypothesis of mating status as a signal for dominance. The lack of a significant correlation between social unit size and dominance status in stable flocks (Lamprecht 1986b) may also support this hypothesis. The age effect on individual dominance scores in geese could thus be only an artefact of the age-related increase in the probability of being paired and having goslings. This probably explains why the dominance score of both sexes increased drastically between their second and third winter, the age when birds began to form pairs [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF]. Indeed, age and breeding status were positively correlated [START_REF] Stahl | Subordinates explore but dominants profit: resource competition in high Arctic barnacle goose flocks[END_REF].

Zusammenfassung

  Fig.2).Adult dominance score preservation across winters after controlling for social unit sizeTo test for the existence of a stable dominance hierarchy once social unit size was accounted for, we tested for the preservation of dominance score among individuals that had kept the same social unit size during two consecutive winters. The analysis was only possible for the most common social unit size: paired without juveniles (social unit size equal to 2). Between the first and second winter, only five birds met these criteria, preventing rigorous tests. Between the second and third winters, individual dominance scores for paired birds without juveniles were highly correlated, showing a preservation of dominance order (F 1,11 = 106.808, P \ 0.001), although the slope differed from one [a = 0.563, t(a) = -8.035, P(a) \ 0.001] and the intercept differed from zero [b = 15.332, t(b) = 6.709, P(b) \ 0.001]. This suggested that the score of more dominant birds tended to decrease slightly whereas those from subordinates increased. Between the last two winters, birds significantly conserved their dominance status as they rigorously conserved their dominance score [F 1,19 = 21.217, P \ 0.001, slope: a = 0.779, t(a) = -1.308, P(a) = 0.206, intercept: b = 5.531, t(b) = 0.812, P(b) = 0.427].

Fig. 1 Fig. 2

 12 Fig.1Mean (±standard error) of juvenile Dark-bellied Brent Goose Branta bernicla bernicla dominance scores (percentage) according to social unit size, from 1 for single juveniles to 7 for juveniles with their parents and four siblings. No juvenile was with only one parent or only with siblings. Inset mean (±standard error) of juvenile dominance scores according to sex. Numbers above and below error bars represent sample sizes

  Stabilita ¨t im winterlichen sozialen Dominanzstatus von Ringelgelga ¨nsen Branta bernicla bernicla innerhalb und zwischen Wintern Dominante wie untergeordnete Individuen ko ¨nnten von einer Stabilita ¨t der sozialen Dominanzstruktur profitieren, weil dadurch der hohe Zeit-und Energieaufwand aggressiver Interaktionen vermieden und das Verletzungsrisiko reduziert wird. Dennoch ko ¨nnte der wahrscheinlich vorhandene Evolutionsdruck auf Individuen dominant zu werden und weiterhin zu bleiben, solche festen Dominanzhirarchien destabilisieren. In diesem Zusammenhang ko ¨nnte der relativen Bedeutung der festen (z.B. Geschlecht, Ko ¨rpergro ¨ße) und vera ¨nderlichen Merkmale (z.B. Ko ¨rperkondition, Familienstand, Fortpflanzungserfolg, Gro ¨ße der sozialen Einheit) auf die Etablierung und den Erhalt der Dominanzbeziehungen eine Schlu ¨sselrolle in der Gruppenstruktur zukommen. Wir untersuchten die relative Rolle fester und vera ¨nderlicher Merkmale auf den sozialen Rang von Ringelga ¨nsen Branta bernicla bernicla, die u ¨ber und zwischen Wintern große, ziemlich unstabile Gruppen bilden. Wir verglichen die Dominanzwerte von beringten Ringelga ¨nsen wa ¨hrend vier auf einander folgenden Wintern. Ringelga ¨nse behielten ihre Dominanzwerte wa ¨hrend eines betrachteten Winters unabha ¨ngig von ihrem Alter, aber waren in der Regel nicht dazu in der Lage, sie u ¨ber zwei aufeinander folgende Winter aufrecht zu erhalten. Da die Winterdominanzwerte am besten mit der Gro ¨ße der sozialen Einheit korrelierten, erschien der Dominanzstatus u ¨berwiegend ein Nebenprodukt eines vera ¨nderlichen Merkmals zu sein, na ¨mlich des Bruterfolgs im vorherigen Sommer. Betrachteten wir nur Altvo ¨gel, deren soziale Gruppeneinheit in zwei aufeinander folgenden Wintern gleich groß war, beobachteten wir eine signifikante Aufrechterhaltung der Dominanzwerte. Dieses Ergebnis deutet darauf hin, dass auch ein festgelegtes Merkmal wie Geschlecht oder Ko ¨rpergro ¨ße eine Rolle bei der Festlegung von Dominanzra ¨ngen spielt.

Table 1

 1 Results of linear regressions and slope and intercept tests on arcsine transformed dominance scores of Dark-bellied Brent Geese Branta bernicla bernicla between each five consecutive monthly periods during winter 2003-2004 Number of ringed Brent Geese recorded in both periods, R 2 and P results of the linear regression, a slope of the linear regression, b intercept of the linear regression, t(a) and P(a) results of the test of comparison of an estimated slope with a theoretical slope (here equal to 1), t(b) and P(b) results of the hypothesis test for intercept (here theoretically equal to 0)

	Months	n	R 2	P	a	b	t (a)	P(a)	t(b)	P(b)
	1 vs 2	13	0.646	0.001	0.948	2.401	-0.246	0.810	0.216	0.833
	2 vs 3	14	0.703	\0.001	0.903	8.095	-0.573	0.577	0.954	0.359
	3 vs 4	21	0.702	\0.001	0.866	9.580	-1.035	0.314	1.579	0.131
	4 vs 5	24	0.580	\0.001	0.789	1.701	-1.473	0.155	0.245	0.809
	n

Table 2

 2 Results of linear regressions and slope and intercept tests on arcsine transformed dominance scores across consecutive winters for adult

	Brent Geese
	Winters	n	R 2	P	a	b	t (a)	P(a)	t(b)	P(b)
	2001-2002 vs 2002-2003	14	0.286	0.049	0.385	34.198	-3.506	0.004	3.558	0.004
	2002-2003 vs 2003-2004	24	0.011	0.624	0.093	35.822
	2003-2004 vs 2004-2005	37	0.195	0.006	0.516	17.723	-2.730	0.010	2.224	0.033

n Number of ringed Brent Geese recorded in both winters, R 2 and P results of the linear regression, a slope of the linear regression, b intercept of the linear regression, t(a) and P(a) results of the test of comparison of an estimated slope with a theoretical slope (here equal to 1), t(b) and P(b) results of the hypothesis test for intercept (here theoretically equal to 0) explained most of the variance (F 6,144 = 36.894, P \ 0.001, 60.6% of the variance in the model with sex and F 5,113 = 28.629, P \ 0.001, and 55.9% of the variance in the model with morphological size;

Acknowledgments We are grateful to Pierre-Yves Perroi, Le ´opold Denonfoux, Coraline Moreau, Thomas Biteau, Se ´bastien Dalloyau, Alain Kim, David Pinaud, Alexandre Millon, Diane Desmonts, Sandra Blais, Daphne ´Durant and more particularly Bart Ebbinge, Gerard Mu ¨sken and Janien van der Greft for help with Brent Goose manipulations and observations. We also greatly appreciated the statistical assistance of Loı ¨c Hardouin. The manuscript benefited greatly from critical comments by Marcel M. Lambrechts, Christophe Bonenfant, Matthieu Guillemain, David Carslake, Laurent Demongin and anonymous reviewers. Thank you to Philippe Delaporte and Fre ´de ´ric Corre for letting us work in the Re ´serve Naturelle of Moe ¨ze-Ole ´ron. The programme was funded by the CNRS, the Office National de la Chasse et de la Faune Sauvage and the EU ''Coast Bird Diversity '' program EVK2-2000-00612.


Benjamin Samson

Frédéric Verpillat

Michel Gross

Michael Atlan

Video-rate laser Doppler vibrometry by heterodyne holography

We report a demonstration video-rate heterodyne holography in off-axis configuration. Reconstruction and display of 1 Megapixel holograms is achieved at 24 frames per second, with a graphics processing unit. Our claims are validated with real-time screening of steady-state vibration amplitudes in a wide-field, non-contact vibrometry experiment.

The laser Doppler method is the most common optical interferometry technique used for non-contact measurements of mechanical vibrations. Though highly effective for single-point vibration analysis, this technique is much less adapted to wide-field imaging than holography. Homodyne [1][2][3] and heterodyne [4][5][6] holographic recordings in off-axis configuration enabled reliable measurements of mechanical vibrations, but none of them allowed real-time monitoring, which is an essential feature. Matching the display rate of opticallymeasured Megapixel digital holograms with real-time imaging standards is demanding in terms of computational power. Holographic measurements are performed in a diffraction plane. Hence, image formation requires to simulate the back-propagation of an optical field. Such propagation involves turning the data measured in the plane of an array detector into a reciprocal plane with at least one bi-dimensional numerical Fourier transformation, typically a Fast-Fourier transform (FFT). Recently, real-time display of digital holograms with Graphics Processing Units (GPUs) [7,8] has alleviated the issue of the high computational workload needed for such image reconstruction. Parallel computations on GPU consistently increase the throughput with respect to CPU for computer-generated holograms, which demonstrated the performance of GPUs in streamline image processing [9], [10].

In this letter, we report an experimental demonstration of an image acquisition scheme designed to perform video-rate image reconstruction and display from heterodyne holographic measurements on a 1 Megapixel sensor array. Image reconstruction of steady-state vibration modes up to 100 kHz at a rate of 24 images per second is achieved. GPU processing is shown to enable holographic reconstruction and display with 3 FFT calculations per recorded frame, which covers the processing throughput needs of three reconstruction approaches : the convolution, angular spectrum, and Fresnel transform methods [11].

The acquisition setup consists of an off-axis, frequency-shifting holographic scheme, used to perform a multipixel heterodyne detection of optical modulation sidebands. Optical heterodyning is a process for placing information at frequencies of interest (e.g. the mechanical vibration of an object under investigation) into a useful frequency range by mixing the frequency content of the probe beam with a reference (or local oscillator, LO) beam. The optical frequency of the reference beam is shifted to generate a beat frequency of the interference pattern within the sensor bandwidth, which carries the information at the original frequency of interest. The Mach-Zehnder heterodyne interferometer used for the detection of an object field E in reflective geometry, beating against a LO field E LO , is sketched in fig. 1. The main optical radiation field is provided by a 100 mW, single-mode, doubled Nd:YAG laser (Oxxius SLIM 532) at wavelength λ = 532 nm, and optical frequency ω L /(2π) = 5.6 × 10 14 Hz. The optical frequency of the LO beam is shifted by an arbitrary quantity ∆ω in the low radiofrequency (RF) range (0 Hz ≤ ∆ω/(2π) ≤ 100 kHz) by two acousto-optic modulators (AA-electronics, MT80-A1.5-VIS). The LO field has the form

E LO = E LO e i(ωL+∆ω)t
, where E LO is its complex amplitude. The object studied is a sheet of paper, whose lateral dimensions are 9 × 26 mm, shined over 9 × 17 mm.

It is attached to a piezo-electric transducer (PZT, Thor-labs AE0505D08F), vibrating sinusoidally, driven at 10 V. Its local vibration is z(t) = z max sin(ω M t), where z max is the vibration amplitude, provokes a modulation of the optical path length of the object field. It induces a local optical phase modulation of the backscattered field at the mechanical vibration frequency φ(t) = φ 0 sin(ω M t), where φ 0 = 4πz max /λ is the modulation depth of the optical phase. Holographic analysis of objects in sinusoidal vibration with a frequency-shifted LO beam tuned to the n-th optical modulation sideband was carried-out extensively in [4]. The frequency filtering properties of time-averaged holography were introduced in [1]. The same filtering properties were described in digital holography [12,13]. It was shown that holograms of optical sidebands could be recorded selectively by matching the frequency shift of the local oscillator beam with the frequency of the band of interest. In the case of sinusoidal phase modulation, the hologram amplitude is of the form J n (φ 0 ), where J n is n-th order Bessel function of first kind. This modulation yields fringes that correspond to local extrema of J n . It was also shown that imaging at harmonics of the vibration frequency could enable robust assessment of vibration amplitudes which are much greater than the optical wavelength [6]. The interference pattern I is measured with a Pike F421-B camera on a Kodak KAI-04022 interline, progressive-scan CCD sensor (2048 × 2048 pixels, pixel size d px = 7.4 µm). The camera is run in binning mode (an effective pixel is made of 4 adjacent pixels); 16 bit, 1024 × 1024 pixels images are sampled at ω S /(2π) = 24 Hz throughout the experiments described hereafter. The RF command signals at frequencies ∆ω, ω M , and ω S are phase-locked. The temporal part of object field undergoing sinusoidal phase modulation can be decomposed in a basis of Bessel functions using the Jacobi-Anger identity

E = Ee iωLt+iφ(t) = ∞ n=-∞ EJ n (φ 0 ) e i(ωL+nωM)t (1)
where EJ n (φ 0 ) = E n is the weight of the optical modulation sideband of order n, and where E is the complex amplitude of the field. If the frequency detuning ∆ω is set close to the n-th modulation harmonic, i.e. |∆ωnω M | < ω S , and if the modulation frequency is much greater than the sampling frequency, i.e. ω M ≫ ω S , the time-averaging-induced bandpass filter of the detection process will isolate the term of order n in eq. 1 and reject all other optical sidebands. In the sensor plane, the interference pattern of E and E LO takes the form

I(t) = |E + E LO | 2 = |E| 2 + |E LO | 2 + EE * LO + E * E LO
, where * denotes the complex conjugate. The frame I(t) acquired at time t by the framegrabber is moved to a frame buffer in the GPU RAM by a CPU thread (fig. 2).

To detect the heterodyne signal of interest EE * LO = E * LO E n e i(nωM-∆ω)t , a sliding 4-phase temporal demodulation is performed. The intermediate frequency nω M -∆ω is set within the camera bandwidth to be sampled efficiently. More precisely, for a LO detuning ∆ω = nω Mω S /4, the modulation sideband E n beats at the frequency ω S /4 (6 Hz) in I(t). To detect it, the following quantity is formed

H(t) = 3 p=0 I(t -2pπ/ω S ) e ipπ/2
(2)

H is a complex-valued array, proportional to the diffracted field E in the sensor plane. Its calculation requires the allocation of a stack of 4 arrays in the GPU memory, filled with I(t), I(t -2π/ω S ), I(t -4π/ω S ), and I(t -6π/ω S ). Each new frame grabbed at instant t yields a shift of the stack (fig. 2) : I(t) replaces the array

I(t -8π/ω S ).
In off-axis configuration, the spatial spectrum in the reciprocal plane (k x , k y ) of the term EE * LO in the expression of I(t) is shifted by the projection of the wave vector difference along a transverse direction x in the sensor plane ∆k x ∼ 2iπθ 0 /λ. Spatial filtering [14] of the time-demodulated signal H(t) in off-axis geometry is used to remove the remaining contributions of the zeroorder terms |E| 2 and |E LO | 2 the twin-image term E * E LO to enhance the detection sensitivity. It is made by multiplying the (k x , k y ) spectrum of H by a mask M , allowing only frequencies in the neighborhood of ∆k to pass.

F (t) = F -1 {M F {H(t)}} (3)
where F is a spatial FFT and F -1 is an inverse spatial FFT. This operation is handled by the GPU. Only the heterodyne contribution of interest F (t) = E * LO E n remains in the filtered frame.

Image rendering from F ∝ E n in the sensor plane involves a scalar diffraction calculation in the Fresnel approximation, performed with a discrete Fresnel transform [15]. The hologram f (t) back-propagated to the object plane is calculated by forming the FFT of the product of F with a quadratic phase map, depending on the relative curvature of the wavefronts of E and E LO in the sensor plane (x, y) via a distance parameter ∆z. This calculation is handled by the GPU.

f (t) = F {F (t) e iπ(x 2 +y 2 )/(λ∆z) } (4)
Finally, the GPU calculates the quantity |f

(t)| 2 ∝ |J n (4πz max /λ)| 2
, which is a map in the object plane of the composition of the local vibration amplitude field z max with the squared amplitude of the Bessel function of order n. Image brightness adjustment is also performed by the GPU. Those maps are displayed in fig3 for the modulation sidebands of order n = 0 (a), n = 1 (b), n = 3 (c), and n = 7 (d). Excited at ω M /(2π) = 10 kHz, the paper sheet builds up a steady-state vibrational mode with rectilinear nodes and bellies oriented along y, with a mechanical wavelength of ∼ 5 mm. In fig. 3 We have demonstrated that the detection and rendering of 1 Mega Pixel heterodyne holograms can be carried-out with a refreshment rate of 24 Hz with commodity computer graphics hardware. Video-rate optical monitoring of steady-state out-of-plane vibration amplitudes was reported. This demonstration opens the way to high throughput multipixel optical heterodyne sensing in real-time.

This work was funded by Fondation Pierre-Gilles de Gennes (FPGG014 grant), Agence Nationale de la Recherche (ANR-09-JCJC-0113 grant), And Région Îlede-France (C'Nano grant).

Fig. 1 .

 1 Fig. 1. Experimental image acquisition setup.

Fig. 2 .

 2 Fig. 2. Algorithmic layout of holographic rendering.

Fig. 3 .

 3 Fig. 3. Time-averaged holographic maps of the ω M /(2π) = 10 kHz vibrational mode of the paper sheet for the modulation sidebands of order n = 0 (a), n = 1 (b), n = 3 (c), and n = 7 (d). 1024 × 1024 pixels renderings. The detection sideband sweep is reported in media 1.The first modulation sideband is screened for excitation frequencies swept from 0 Hz to 20 kHz in media 2 and from 0 Hz to 100 kHz in media 3. The effect of eq. 3 filter is reported in media 4.

Table 1 .

 1 FFT calculations were made with the function cufftExecC2C() from the CUFFT 3.2 library on single precision floating point arrays. The program was compiled and run on Microsoft Windows 7 -64 bit. The computer hardware configuration was based on an ASUS P6T motherboard with a 2.67 GHz Intel core i7 920 CPU and a NVIDIA GeForce GTX 470 GPU. Image rendering calculations I → H → F → |f | 2 are performed sequentially, in the main GPU thread (fig.2); the whole processing time of one frame is reported in table 1. For benchmark purposes, the rendering performance of 2048×2048 pixels recordings read out at ω S /(2π) = 8 Hz is also reported in table 1. Benchmarks of image rendering time.

	(a), the