Karel Lemr

Preparation of Knill-Laflamme-Milburn states using tunable controlled phase gate

Keywords: numbers: 42, 50, Dv, 03, 67, a

. Although the usage of such states is widely investigated, considerably less effort has been invested into experimentally accessible preparation schemes. This paper discusses the possibility to employ a tunable controlled phase gate to generate an arbitrary Knill-Laflamme-Milburn state. In the first part, the idea of using the controlled phase gate is explained on the case of two-qubit KLM states. Optimization of the proposed scheme is then discussed for the framework of linear optics. Subsequent generalization of the scheme to arbitrary n-qubit KLM state is derived in the second part of this paper.

Introduction

Important developments have been demonstrated in quantum information processing (QIP) in the past few decades [START_REF] Alber | Quantum Information[END_REF][START_REF] Bowmeester | The Physics of Quantum Information[END_REF][START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. Several outcomes of this scientific field such as quantum cryptography [START_REF] Benett | IEEE International Conference on Computers, Systems, and Signal Processing[END_REF][START_REF] Benett | [END_REF]7,8] or random number generation [9,10,11,12,13] have already found their industrial applications. In other cases a lot of effort has yet to be invested into the research. Mainly the lack of some experimental tools (e.g. strong optical nonlinearity [14]) prevents from developing efficient quantum devices. An important discovery has been achieved by Knill, Laflamme and Milburn [1], when they have derived that a specific class of partially entangled states (so called Knill-Laflamme-Milburn states, or simply KLM states) can be used to significantly improve the efficiency of quantum computing. They have proposed a nearly deterministic teleportation based protocol for quantum computation using the KLM states as ancillas. In this protocol the overall success probability of quantum computation goes asymptotically to unity with growing number of photons in the ancillary KLM state. Their work has been followed by several other related proposals and experiments [15,16,17]. Franson et al. [15] have generalized the original KLM scheme so that the success probability of quantum computing scales better with growing number of photons, but at the expense of lower fidelity of the output states. Several schemes for preparation of KLM states have also already been proposed. The general preparation idea has been mentioned in the original KLM paper [1] though there was no specific recipe. The first explicit scheme for preparation of the KLM states was proposed by Franson et al. and it uses non-deterministic controlled sign gates and single photon interference to generate arbitrary photon-number KLM states [18]. Another scheme limited only to 2-photon KLM states, but not requiring any post-selection, was also proposed [19] and subsequently experimentally implemented [20].

This paper investigates yet another approach for experimentally accessible preparation of KLM states using the controlled phase gate (c-phase gate). The advantage in using this gate is the fact that the c-phase gate is considered an important part of the QIP toolbox [21,22]. The Franson et al. scheme also employs the controlled phase gates (or in their case controlled sign gates) but with constant phase shift set to π. In this paper a fully tunable controlled phase gate is considered and a scheme for it's usage as a resource for KLM state generation is developed. By this strategy the overall success probability of the KLM state preparation can be increased considerably for some KLM states as it is shown in this paper. The presented scheme is fully general and allows to prepare KLM states of arbitrary number of qubits. Also no previous entanglement between the input qubits is required as the entangling capability of the gate itself is sufficient. The fully tunable controlled phase gate capable of imposing any phase shift in the range from 0 to π has already been both proposed theoretically [23] and implemented experimentally [24] on the platform of linear optics and thus can be considered experimentally accessible.

Basic 2-qubit scheme

Using the qubit representation, one can express then-qubit KLM state in the form of

|ψ KLM = n j=0 α j |1 j |0 n-j . (1)
The original definition by Knill, Laflamme and Milburn setsα j = 1 √ n+1 for j = 0, ..., n, but the subsequent research carried out by Franson et al. [15] indicates, that additional benefits can be found in using general amplitudes α j . Their research revealed that one can increase the efficiency of teleporation based quantum computing for instance by choosing triangular shaped amplitudes α j (that is α 0 = α n = 0 and alpha linearly growing towards maximum at α n/2 and then decreasing). This improvement is obtained at the expense of lower fidelity of the output state. (For more details please consult [15]).

In the first part of this paper let us consider the preparation of two-qubit KLM states (see figure 1). The generalization to an arbitrary number of qubits would be presented later. Using the general definition for the KLM states (1) one can find that the two-qubit KLM states are in the form of

|ψ 2-QUBIT KLM = α 0 |00 + α 1 |10 + α 2 |11 , (2)
where α j (for j = 0, 1, 2) are arbitrary complex amplitudes following the normalization condition

2 j=0 |α j | 2 = 1.
Having the target state well defined let us now inspect the properties of the c-phase gate.

The c-phase gate is a two-qubit quantum gate whose action in the gate's computational basis reads

|00 → |00 |01 → |01 |10 → |10 |11 → e iϕ |11 (3)
with numbers in brackets denoting first and second qubit state. General c-phase gate can be set to impose an arbitrary phase shift ϕ to the two-qubit state |11 . Any signal and control qubit can be expressed in terms of the gate's computational basis

|ψ c,s = cos θ c,s |0 c,s + e iφc,s sin θ c,s |1 c,s , (4)
where indexes c and s denote the control and signal qubit. Please note that this state can always be prepared with high fidelity using only single qubit transformations (e.g. wave-plates in the case of photon polarization encoding). The separable input state|ψ c ψ s is transformed by the gate yielding

|ψ OUT = cos θ c cos θ s |00 + e iφs cos θ c sin θ s |01 + + e iφc sin θ c cos θ s |10 + + e i(φc+φs+ϕ) sin θ c sin θ s |11 . (5)
Using the expression for signal qubit (4), the output state can be rewritten to the following form

|ψ OUT = cos θ c |0ψ s + e iφc sin θ c τ |1ψ s + |1ψ ⊥ s , (6)
where |ψ ⊥ s is the orthogonal state to |ψ s so that ψ ⊥ s |ψ s = 0 and the parameters τ and are defined as

τ = ψ s | cos θ s |0 + e i(φs+ϕ) sin θ s |1 = = cos 2 θ s + e iϕ sin 2 θ s , = ψ ⊥ s | cos θ s |0 + e i(φs+ϕ) sin θ s |1 = = e iφs sin θ s cos θ s 1 -e iϕ . (7)
After performing the single qubit transformation

|ψ s → |0 , |ψ ⊥ s → |1 (8)
in the signal mode, one can clearly recognize the two-qubit KLM state in the output state of the gate

|ψ OUT = cos θ c |00 + e iφc τ sin θ c |10 + e iφc sin θ c |11 , (9)
The remaining task is to map the complex amplitudes in (9) to the original amplitudesα j and to show that any two-qubit KLM state is achievable. First let us consider the relative amplitude ratio and phase betweenα 0 and (α 1 + α 2 ). Any amplitude ratio can easily be set just by the choice of theθ c parameter of the input control state

|α 1 | 2 + |α 2 | 2 |α 0 | 2 = tan 2 θ c . (10)
As for the phase, the freedom in setting any value of φ c assures that any phase shift between α 0 on one side and α 1 and α 2 on other side is achievable. The relation between α 1 and α 2 is also simple. For instance setting the phase shiftϕ = π simplifies the amplitude ratio to

|α 2 | |α 1 | = | | |τ | = tan 2θ s (11)
and an arbitrary phase shift between α 1 and α 2 can be set by the choice of φ s . Please note that setting ϕ = π allows to cover the whole class of KLM states. This fact will be used for the discussion in section 5. The equations (10 and 11) manifest that any amplitude ratio between α 0 , α 1 and α 2 is achievable since tan goes from 0 to ∞.

Success probability optimization

One may conclude that the tunability of the gate in the phase shiftϕ is a redundant feature. However this parameter can be used for optimization of the procedure. One of the most promising platforms for QIP is linear optics [25,26,27,28,29]. For this reason let us now focus on the optimization of the proposed procedure for linear optics. Recently Kielinget al. [23] have identified the maximum success probability of a c-phase in the framework of linear optics as

P C (ϕ) = 1 + 2 sin ϕ 2 + 2 3/2 sin π -ϕ 4 sin ϕ 2 1/2 -2 , ( 12
)
which does not depend on the input state. The optimization of the proposed scheme seeks to maximize the success probability of the c-phase gate used for KLM state preparation. With respect to that a numerical simulation (or optimization) has been carried out to reveal the maximum achievable success probability for several KLM states. The target KLM state of presented numerical simulation is the mono-parametric class of two-qubit KLM state motivated by Franson's et al. definition [15] (triangular-shaped amplitude function)

|ψ KLM = α 0 |00 + α 1 |10 + α 0 |11 . (13)
The amplitudes α 0 and α 1 are now considered to be real numbers as it has been shown above that the phase can always be set by the choice of φ c and φ s . These phases are independent of the gate phase shift ϕ and therefore have no effect on the success probability. The presented optimization will focus on the amplitude ratio |α 0 /α 1 | and investigate the corresponding success probability. First numerical simulation has been performed to determine the maximum achievable |α 0 /α 1 | ratio for a given phase shift. Results of this simulation are presented in figure 2. One can observe that maximum achievable|α 0 /α 1 | ratio grows monotonously with the phase shift ϕ. For reference the success probability (12) as a function of the phase shift ϕ is also depicted along with the reference ratio |α 0 /α 1 | = 1 corresponding to the original KLM state definition.

The second numerical simulation has been carried out to determine the maximum achievable success probability for a given |α 0 /α 1 | ratio (see figure 3). Also the setting of the phase shift ϕ and the parameter of the signal qubit θ s are depicted to illustrate the optimal strategy. This strategy is different in two regions separated by the amplitude ratio |α 0 /α 1 | ≈ 0.54. In the first region (|α 0 /α 1 | ≤ 0.54) setting θ s = π 4 and the phase shift ϕ accordingly is the optimal way. One tries to minimize the phase shift used for the KLM state preparation, because the success probability is a decreasing function of the phase shift. To keep the phase shift minimal, one has to set θ s = π 4 , because for a given phase shift ϕ the setting θ s = π 4 maximizes the |α 0 /α 1 | ratio. On the other hand, in the second region (|α 0 /α 1 | > 0.54) the previously mentioned strategy will not yield optimal results. This is because of the success probability not being monotonous in this region. Setting ϕ = π and adjusting the θ s instead is the optimal way here. Both this and the original Franson et al. scheme requires n -1 times using the c-phase gate in order to generate n-qubit KLM state. This leads to the overall success probability for n-qubit KLM state

P KLM = n-1 i=1 P C (ϕ i ), (14)
where n denotes the number of qubits and P C (ϕ i ) is the success probability of the controlled phase gate set for the phase shift ϕ i used in the i th repetition of the c-phase gate. The Franson et al. proposal considers only ϕ i = π for all values of i. So for example in the 2-qubit case, the success probability of Franson scheme would yield a constant value of0.11 (based on the optimal linear optical controlled phase gate). To emphasize the improvement achieved by the tunability of the phase gate, let us consider an example of|α 0 /α 1 | = 0.25.

For this particular choice the success probability of the scheme proposed in this paper would be 0.18, which is a 60% improvement. This improvement in success probability varies with the particular choice of the target KLM state (see figure 3).

Generalization to n-qubit KLM states

The proposed two-qubit scheme can be generalized to prepare KLM states of an arbitrary number of qubits. For simplicity let us now presume all complex amplitudes of the nqubit KLM state being equal (original KLM state definition). To illustrate the generalization procedure the step from two-qubit to three-qubit KLM state is explained and also illustrated in figure 4). Going from two to three qubit KLM state means to perform the following transformation where indexes 1 and 2 denote the first and second original qubits of the two-qubit KLM state and the index 3 denotes the newly added qubit. This transformation can be implemented by addition of a new qubit initially in the state |0 . This new qubit is firstly subjected to the Hadamard gate

|0 1 0 2 → |0 1 0 2 0 3 (15) |1 1 0 2 → |1 1 0 2 0 3 (16) |1 1 1 2 → |1 1 1 2 0 3 + |1 1 1 2 1 3 , (17)
|0 → |0 + |1 . (18)
After that it is propagated through the c-phase gate set to phase ϕ = π 2 along with the last of the original KLM qubits. At the end an inverse Hadamard gate is placed in the new qubit mode. One can see that in the case of the last original qubit being|0 , the phase shift imposed to the new qubit is zero and the new qubit leaves the scheme in the state|0 . On the other hand if the last original qubit is in the state |1 the new qubit gets a π 2 phase shift and yields |0 + |1 after leaving the inverse Hadamard gate.

The generalization to an arbitrary number of qubits is straightforward. To generate an (n + 1)-qubit KLM state from an n-qubit KLM state (n ≤ 2) a new qubit is added at the end of the original qubits and subjected to the procedure described in previous paragraph. The general scheme is depicted in figure 4.

Optimization of the generalized scheme

The previous section is just a proof of the scalability of the scheme, but does not give optimal setting with respect to the success probability. A similar optimization as for the two-qubit KLM states can be considered to maximize the yield of the scheme. Hadamard gates can be replaced by more general single qubit transformations and together with the tunability of the phase shift imposed by every controlled phase gate the overall success probability can be optimized with respect to the selected target KLM state.

One can use the iterative procedure starting from n-qubit KLM state with amplitudes α

[n] j , j = 0...n and going to (n + 1)-qubit KLM state with amplitudes α

[n+1] j , j = 0...n + 1.

Here the upper index denotes the n-qubit starting KLM state and (n + 1)-qubit target KLM state. Note that in this case the c-phase gate is applied to the last of the original qubits ( n th qubit) and a newly added (n+1) th qubit. This new qubit can be expressed in the form of |ψ s as defined by ( 4) and the last original qubit takes effectively the form similar to |ψ c with

cos θ c = n-1 j=0 |α [n] j | 2 (corresponding to the |0 state) sin θ c = |α [n] n | (corresponding to the |1 state) φ c = arg α [n] n (19)
also following the original definition [START_REF] Nielsen | Quantum Computation and Quantum Information[END_REF]. With this mapping one can proceed in the similar way as explicitly described in the second section. The resulting amplitudes α

[n+1] j are then in the form

α [n+1] j = α [n] j , for j = 0...n -1 α [n+1] n = |α [n] n |e iφc τ α [n+1] n+1 = |α [n] n |e iφc , (20)
where φ c is defined by (19) and τ and by (7). The equations become increasingly complicated with the growing number of qubits. For this reason one can seek the solution numerically.

As a result of such a numerical optimization, one can for example prepare a 4-qubit KLM state of the triangular-shaped amplitudes in the form of

|ψ KLM = 1 N 4 j=0 α j |1 j |0 n-j (21)
α 0 = α 4 = 1, α 1 = α 3 = 3, α 2 = 6 (22)
(N = n j=0 |α j | 2 ) with the success probability of 0.19% while the original proposal would give only 0.14% success probability (40% improvement). Note that this improved success probability would allow almost 1.5 times higher rate of preparation of KLM states for the "nearly deterministic" protocol proposed by Knill, Laflamme and Milburn [1]. The reason for the improvement in the success probability is the fact that using a tunable phase shift, one can operate the controlled phase gate at optimal phase shift. Because one can always set the gate to operate at the phase π and set single qubit operations accordingly, the proposed scheme would never give lower success probability as the one proposed by Franson et al. The optimal strategy for setting the phase shift imposed by the gate in every step of the generalized procedure is similar to the strategy discussed in the Sec. III for the 2-qubit case. This can be summarized by an inequality P C (ϕ i ), (23) where the left-hand side corresponds to the success probability of the Franson et al. proposal and the right-hand side corresponds to the success probability of the scheme described in this paper. In the worst case scenario hereby proposed scheme allows to set ϕ = π to generate any KLM state and in this case the inequality would be saturated.

Conclusions

The scheme presented in this paper shows how a tunable controlled phase gate can be used to generate arbitrary n-qubit KLM states. In comparison with the Franson et al. proposal, this scheme gives higher success probability depending of the requested KLM state. It can offer a significant improvement in generation of ancillary states for efficient quantum computing. Please note that this paper discusses the improved generation success probability (rate) for the KLM ancillary states. It should not be confused with the success probability of the teleportation based KLM scheme that employs these ancillary states and considers them as already prepared. Several specific KLM states are discussed in this paper and their preparation success probabilities shown to demonstrate this improvement.

Figure 1 .

 1 Figure 1. Scheme of the proposed procedure for generation of two-qubit KLM states. The signal and control input qubit undergo a c-phase gate with tunable phase shiftϕ yielding the two-qubit KLM state.

Figure 2 .

 2 Figure 2.Maximum achievable |α 0 /α 1 | ratio for a given phase shift of the c-phase gate. The success probability of the optimal linear optical c-phase gate as a function of its phase shift is also depicted for reference.

Figure 3 .

 3 Figure 3. Maximum achievable success probability and corresponding optimal θs and ϕ parameters are plotted as a function of |α 0 /α 1 | ratio. Please note that the optimal setting of ϕ for |α 0 /α 1 | > 0.54 is ϕ = π (this explains the step of ϕ at |α 0 /α 1 | = 0.54).

Figure 4 .

 4 Figure 4. Generalization of the two-qubit scheme to an arbitrary number of qubits. Inputnqubit KLM state is combined with a new qubit initially in|0 state. H denotes the Hadamard gate and C denotes the c-phase gate (this time set to impose the phase shiftϕ = π 2 ).

P

  C (π) = P C (π) n-1 ≤ P KLMnew = n-1 i=1

Acknowledgement

The author would like to thank his colleagues Jan Soubusta and Antoní n Černoch for fruitful discussion on the subject of this paper.

The author gratefully acknowledge the support by the Operational Program Research and Development for Innovations -European Regional Development Fund (project CZ.1.05/2.1.00/03.0058 and the Operational Program Education for Competitiveness -European Social Fund (project CZ.1.07/2.3.00/20.0017 of the Ministry of Education, Youth and Sports of the Czech Republic, by Palacky University (internal grant PrF-2011-009).


Steven J Portugal
email: s.portugal.1@bham.ac.uk

Matthieu Guillemain

VIGILANCE PATTERNS OF WINTERING WIGEON: FEMALE BENEFITS FROM MALE LOW-COST BEHAVIOUR

Keywords: female benefit, mate guarding, male constraint, male investment, vigilance, Anas penelope Wachsamkeitsmuster bei überwinternden Pfeifenten: Weibchen profitieren von "Kosten-effizientem" Verhalten der Männchen Besonders hohe Wachsamkeit männlicher Tiere wird in der Regel mit einer von drei Hypothesen erklärt: dem Bewachen ihrer Weibchen (male investment hypothesis), als sekundäres Geschlechtsmerkmal verbunden mit erhöhtem Risiko, erbeutet zu werden

Increased vigilance in male animals has been attributed to mate guarding (male investment hypothesis), secondary sexual characteristics increasing predation risk (male constraint hypothesis) or for benefit to the female (female benefits hypothesis). We studied Eurasian wigeon Anas penelope while they grazed on dry land, a "risky" foraging situation, at two points during the winter period (pre and post pair formation) to assess if their behaviour was more consistent with one of these three hypotheses. Males were always highly vigilant, and vigilance increased markedly in more risky situations (smaller groups, higher distance from water). Mean male vigilance only changed to a minor extent from the pre-to the post-mating periods. Conversely, female vigilance significantly decreased after pairing, and was then significantly lower than that of males. Prior to pairing, males and females had similar rates of vigilance bouts. Overall, our findings suggest that higher vigilance in male wigeons in this instance is best explained by the female benefits hypothesis. Because the wigeons were foraging on land and were highly vigilant, even when unpaired, the females could actually benefit from the males vigilance without males investing more time in vigilance. In such a situation, paired males rely on "low-cost vigilance" whereby vigilance serves as a safety mechanism while simultaneously benefiting the female.

(male constraint hypothesis), oder zum Nutzen der Weibchen (female benefits hypothesis). Wir untersuchten eurasische Pfeifenten (Anas penelope) beim Grasen auf trockenem Land, einer risikoreichen Art der Nahrungssuche. Die Beobachtungen fanden zu zwei Zeitpunkten während der Winterperiode (vor und nach der Paarbildung) statt, um zu prüfen, ob ihr Verhalten eine der drei Hypothesen speziell unterstützte. Die Männchen waren die ganze Zeit über sehr wachsam, und ihre Wachsamkeit stieg in risikoreichen Situationen (in kleineren Gruppen, oder weiter entfernt von Wasser) noch spürbar an. In der Zeit vor der Balz bis zu der Zeit danach stieg die mittlere Wachsamkeit der Männchen kaum an, während im Unterschied dazu die Wachsamkeit der Weibchen nach der Paarung deutlich nachließ und dann signifikant niedriger als die der Männchen war. Vor der Paarbildung zeigten Weibchen und Männchen ähnlich häufige Kurzphasen der Wachsamkeit. Unsere Untersuchungen legen nahe, dass im vorliegenden Fall die erhöhte Wachsamkeit männlicher Pfeifenten am besten von der "female benefits hypothesis" erklärt wird.

Weil die Pfeifenten an Land auf Nahrungssuche gehen und dabei besonders im unverpaarten Stadium außerordentlich wachsam sind, konnten die Weibchen von der Wachsamkeit der Männchen profitieren, ohne dass diese Extra-Zeit dafür aufwenden mußten. Die verpaarten Männchen begnügen sich mit dieser Kosten-effizienten Wachsamkeit (low-cost vigilance), wobei diese als ein Sicherheitssystem dient, das gleichzeitig den Weibchen nutzt.

Introduction

The role of vigilance and the factors affecting vigilance levels are among the central questions of behavioural ecology (e.g. [START_REF] Krebs | An introduction to Behavioural Ecology[END_REF]. While vigilance in animals can typically have an anti-predator role, it may also be directed towards conspecifics, for example to avoid kleptoparasitism or to copy the behaviour of others (reviewed in [START_REF] Elgar | Predator vigilance and group size in mammals and birds: a critical review of the empirical evidence[END_REF][START_REF] Lima | Predators and the breeding bird: behavioural and reproductive flexibility under the risk of predation[END_REF]. In paired males, vigilance may additionally be used to monitor the behaviour of competing other males (e.g., [START_REF] Dahlgren | Females choose vigilant males: an experiment with the monogamous grey partridge, Perdix perdix[END_REF]). In many species, especially birds, males show distinct and conspicuous secondary sexual characters. While these may have a crucial role in sexual selection processes, such characters may also make males more visible to predators than females of the same species. Paired males may then be more vigilant than females because they face a greater predation risk when in breeding plumage, not just because they guard their mate (e.g., [START_REF] Lendrem | Sleeping and vigilance in birds. l. Field observations of the mallard (Anas platyrhynchos)[END_REF][START_REF] Hart | Predator avoidance as a function of flocking in the sexually dichromatic Hawaii akepa[END_REF]. Lastly, males may use vigilance behaviours to enhance female survival and to allow females more foraging and resting time to enhance nutritional status and fecundity (e.g. Thompson and Popko 1981;[START_REF] Lamprecht | Mate guarding in geese: awaiting female receptivity, protection of paternity or support of female feeding?[END_REF][START_REF] Artis | Male vigilance in white-tailed ptarmigan, Lagopus leucurus: mate guarding or predator detection?[END_REF]Weibe and Martin 1988;Rodway 2007a), with many studies reporting increased foraging times for the female when accompanied by a mate (e.g. [START_REF] Gauthier | The adaptive significance of territorial behaviour in breeding buffleheads: a test of three hypotheses[END_REF]. Furthermore, evidence suggests that male vigilance is important to females in mate selection, as females will preferentially select vigilant males [START_REF] Dahlgren | Females choose vigilant males: an experiment with the monogamous grey partridge, Perdix perdix[END_REF][START_REF] Choudhury | Mate selection behaviour and sampling strategies in geese[END_REF]Pizzari 2003).

These possible causes of paired male vigilance have been commonly termed the "male investment", "male constraint" and "female benefits" hypotheses, respectively (e.g., [START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF][START_REF] Squires | Vigilance behaviour in the harlequin duck (Histrionicus histrionicus) during the preincubation period in Labrador: Are males vigilant for self or social partner?[END_REF]. The extent to which vigilance of males and females in a given species is more consistent with one or the other hypothesis may theoretically depend upon environmental conditions, though this has never been properly tested. Ducks are good model organisms to study vigilance behaviour patterns, because they are subject to frequent attacks by predators (including during daylight hours, Guillemain et al. 2007a), they are highly gregarious and form pair-bonds anew during each winter period that are maintained for a long time before the breeding season itself, during spring [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF]; see however [START_REF] Mitchell | Re-mating in migratory Wigeon Anas penelope[END_REF] for evidence of occasional re-mating over successive years in this species). In waterfowl, such pairing in late autumn or early winter is considered to be adaptive: paired birds indeed get dominant status within the flock which provides better access to food (e.g. Johnson and Rohwer 1988;[START_REF] Black | Ecology of social behaviour[END_REF], and for the male a early pairing should translate into higher eventual breeding probability if sexual competition for mates exists among individuals (e.g. [START_REF] Rohwer | Female-biased philopatry, monogamy, and the timing of pair formation in migratory waterfowl[END_REF][START_REF] Oring | The mating systems of waterfowl. Ecology and Management of breeding waterfowl[END_REF][START_REF] Bowler | Breeding strategies and biology[END_REF]. This in turn may allow paired females to improve body condition, sensu the female benefits hypothesis, which is likely to translate into higher breeding success [START_REF] Guillemain | The incomecapital breeding dichotomy revisited: late winter body condition is related to breeding success in an income breeder[END_REF]. For example, [START_REF] Lercel | Mate loss in winter affects reproduction of Mallards[END_REF] demonstrated that female ducks that lose their mate after pairing during winter experienced lower subsequent breeding success, hence demonstrating the fitness benefit of early pairing in such species. [START_REF] Squires | Vigilance behaviour in the harlequin duck (Histrionicus histrionicus) during the preincubation period in Labrador: Are males vigilant for self or social partner?[END_REF] determined for harlequin ducks (Histrionicus histrionicus) that male vigilance could be explained by the female benefits hypothesis, because unpaired males were less vigilant than paired males despite having the same plumage, although this does not rule out the male investment hypothesis. This is the opposite to what would be expected if the vigilance was a result of male constraint. In common eiders (Somateria mollisoma), conversely, it was concluded that male vigilance was primarily for paternity assurance, because widowed females foraged at the same rate as prior to the loss of their mate, and maintained body mass and clutch size which was comparable with previous breeding seasons when paired [START_REF] Hario | The role of mate-guarding in pre-laying common eiders (Somateria mollisima) in the northern Baltic Sea[END_REF]): females eiders had not been benefiting nutritionally from having a mate.

During winter, previous studies of Eurasian wigeon Anas penelope have demonstrated a significant difference in the length of time and frequency of vigilance behaviour between the sexes during winter [START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF]. At this time males are generally more vigilant than females (see also [START_REF] Goodburn | Mate guarding in the mallard, Anas platyrhynchos[END_REF][START_REF] Baldassarre | Waterfowl Ecology and Management[END_REF]). An earlier study in an acknowledged safe environment (mixed feeding flocks with Brent Geese Branta bernicla L on large and open Zoostera sp. beds of the Exe Estuary, UK, where predators were few) suggested that vigilance of male wigeons could be best explained by the male investment hypothesis [START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF]. This is because males showed an increase in vigilance behaviour after pairing, not when they acquired their brighter breeding plumage, and female vigilance rate did not differ between pre-and post-pairing. In the present study, we also aimed to compare the vigilance levels of male and female wigeons before and after the main period of pair formation, but in a less safe environment in which birds were grazing on dry land, away from water. In such situation, foraging wigeon use open water as a refuge in case of disturbance or predation event [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF] and consequently their feeding is restricted to strips of vegetation near to the water. Distance from water and flock size were measured as covariates indicating varying levels of risk (e.g. [START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF][START_REF] Elgar | Predator vigilance and group size in mammals and birds: a critical review of the empirical evidence[END_REF]. Vigilance in wigeons typically increases with distance foraging from water, and decreases with increasing flock size [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF]. Such a situation may alter the relative importance of the male investment, male constraint and female benefits hypotheses compared to other safety contexts.

We predicted that if increased vigilance in males is primarily related to predator detection (the male constraint hypothesis), then (1) the amount of vigilance behaviour exhibited by both sexes should be similar before and after pairing, (2) vigilance should decrease with increasing flock size at the same rate for both males and females, or at a higher rate in the more conspicuous males, and (3) vigilance should increase with increasing distance from water, again, at the same rate in males and females, or at a higher rate in the more conspicuous males.

If mate guarding and paternity assurance plays a significant additional role in vigilance (the male investment hypothesis), it was then predicted that (1) males should become more vigilant than females in late winter, or the difference between males and females should increase in late winter compared to early winter, (2) female vigilance should remain constant or decrease in late winter (because they are being guarded by their mate), and (3) male vigilance should increase with increasing flock size (because there are more competing males), or remain similar if the benefits in terms of safety from predators are counterbalanced by the costs in terms of sexual competition.

If the benefit to the female was the primary driving force of male vigilance (female benefits hypothesis), it was predicted that (1) females should become less vigilant in late winter compared to early winter as they are utilising their partners vigilance (2) male vigilance should remain the same or increase in late winter compared to early winter, and

(3) vigilance should not be affected by group size or distance from water in either males or females.

Methods

Study site

Traeth Lafan (Lavan Sands) is a large intertidal area of sand and mud flats lying on the eastern edge of the Menai Straits, in Conwy Bay, North Wales, UK (53°17" N, 4°16" W). The area as a whole has a range of exposures and a diversity of conditions enhanced by freshwater streams that flow across the flats, and covers a total area of 2,643 ha. The area is well used by members of the public, and is bordered by a railway line and a series of public footpaths. The majority of the wigeon flock frequents a small intertidal area comprising Juncetum gerardii and Puccinellietum salt-marsh, approximately 1.5 km west of the village of Llanfairfechan (53°15" N, 4°1" W). The total number of wigeons present each day varied between 180 -440. Northern pintails Anas acuta regularly joined the wigeon flock during grazing periods, along with lesser numbers of redshanks Tringa totanus, Eurasian teals Anas crecca and mallards Anas platyrhynchos. There were no geese or swans present. The main footpath that runs along the edge of the marsh and shoreline experienced, on average, 37 human walkers along it per hour. Raptor flyovers were 2.9 and 2.1 per hour, for the November and March sampling sessions respectively. The main raptor species involved in this were peregrine falcons (Falco peregrinus) and common buzzards (Buteo buteo), and less frequently, merlins (Falco columbarius), common kestrels (Falco tinnunculus) and on one occasion, hen harriers (Circus cyaneus).

Vigilance recording

Wigeon were counted and observed using an Opticron telescope (30x) and Opticron binoculars (10 x 50). Observations were conducted from the 12 th -25 th November, 2003, and from the 4 th -16 th March, 2004. The individual pairing status of each bird could not be recorded (see below), so it is possible that some individuals from the November sample were already paired, and some late individuals from the March sample (especially juveniles) were still unpaired. However, November is when wigeons start to pair in significant numbers [START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF] and most of these are generally paired by the end of winter [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF], so that the two study periods were assumed to adequately represent the pre-and post-pairing periods. A total of 562 focal observations (sensu [START_REF] Altmann | Observational study of behaviour: sampling methods[END_REF] were used in the analyses, 244 in November and 318 in March.

Ducks were not individually marked, and it is therefore possible that some of them may occasionally have been sampled twice. However, the incidence of double sampling during the same day is likely to be low (see below), and thus should not cause an important problem for statistical analyses. The traditionally high turnover of individuals among wintering duck populations [START_REF] Pradel | Capture-recapture survival models taking into account of transients[END_REF][START_REF] Guillemain | What is the spatial unit for a wintering teal Anas crecca? Weekly day roost fidelity inferred from nasal saddles in the Camargue, southern France[END_REF]) also made the double sampling of the same birds over different days unlikely. Counts were made during daylight within 1.5 hours of low tide, for a total period of three hourstypically 1.5 hours either side of low tide. Observations were made from a bench overlooking the tidal salt marsh area, approximately 100 metres away from the region the wigeon most frequently grazed. Because the birds formed many small groups, it was possible to get a large sample number of many different flock sizes, and also limited the risk of sampling the same individual more than once.

Sampling protocol followed that of [START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF] and [START_REF] Guillemain | Feeding methods, visual fields and vigilance in dabbling ducks (Anatidae)[END_REF].

One focal bird was randomly selected from the approximate centre of the flock, sexed and observed for 300 seconds. So as to provide an approximate equal number of each sex, different sexes were observed alternatively, i.e. if the randomly selected individual was of the "wrong" sex, then the nearest bird of correct sex to this random one was observed [START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF].

Vigilance was measured as % time with head up in a grazing bird (i.e. grazing was its predominant activity). At the start and end of each 300 s observation period, the flock size was recorded and for each observation, a mean flock size allocated. The same principle was applied to flock distance from water. The distances were estimated by eye, in metres, but were considered to be reasonably accurate since the length of the area between the footpath where the observations took place and the low tide of the water was known [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF]. The wigeon typically fed in compact, closely-knit groups in the winter. However, at times there were a few individuals more widely dispersed. In order to restrict the observations to birds feeding in clearly defined groups, birds further than 20 metres from the centre of the flock were not included in the estimation of flock size, or selected for any vigilance recordings [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF]. As the focus was on one bird at a time (i.e. one bird typically filled the eyepiece of the telescope), it was not possible to accurately determine whether each bird was paired or not. The maximum flock size recorded during the sampling session was 350, and the maximum distance the wigeons foraged from the waters edge was 35 metres.

Statistical Analysis

We used General Linear Models (GLM) to test for differences in proportion of time spent vigilant by focal birds (arcsin-transformed during the statistical analyses, but presented as percentages in the figures to ease reading) between sexes and periods of the winter, taking into account the potential effects of distance from water and flock size.

Preliminary analyses suggested a non-linear relationship between vigilance rate and flock size, so the latter was log-transformed before inclusion in the models. Similarly, a significant correlation was initially found between flock size (log-transformed) and distance from water (P < 0.0001). However, the regression coefficient was relatively low (r² = 0.55), so that there was some variability in flock size that was not explained by variability in distance. In order to tease apart the relative role of each variable on wigeon vigilance (i.e., flock size and distance from water), both were therefore included at the beginning of the model selection procedure. The variables and factors initially entered into the GLM were, therefore, Sex, Month, Distance from water (hereafter "Distance"), Log(FlockSize), and all possible interactions with up to three terms (the biological meaning of more complicated models being difficult to interpret). A stepwise backwards model selection procedure was then used, where non significant terms at P = 0.05 were gradually removed, starting with the most complex interactions.

All values are presented as means ± SE. All tests are considered significant if P < 0.05.

Results

The final model of the stepwise procedure provided an excellent fit to the data (r² = 0.91; Table 1). It first highlighted the major role of danger in itself on individual vigilance rate, Log(FlockSize) alone being responsible of 75% of the explained variance (see ratio of partial F-values from Table 1). Males were generally more vigilant than females, and increased their vigilance rate from November to March, though to a limited extent (11.82% ± 0.54, n = 149 and 13.25% ± 0.58, n = 131), while that of females decreased twice as much (10.58% ± 0.41, n = 138 and 7.62% ± 0.42, n = 135; Fig. 1).

The two sexes fed in flocks of similar size both in November (ANOVA: F 1,285 = 0.02, r² = 0.00, P = 0.8848) and in March (ANOVA: F 1,264 = 0.77, r² = 0.00, P = 0.3819).

The differences in vigilance rates between the two time periods could not be explained by average flock size being different between November and March in either of the two sexes (ANOVA on average Log(FlockSize): Males: F 1, 278 = 0.58, r² < 0.01, P = 0.4485; Females F 1, 271 = 3.03, r² = 0.01, P = 0.0830). However, there was a significant effect of the Sex*Log(FlockSize), the Month*Log(FlockSize) and the Sex*Month*Log(FlockSize) interactions (Table 1) indicating that, despite average flock sizes being similar in all cases, i) vigilance decreased at a higher rate with increasing flock size in males than in females, ii) vigilance decreased at different rates in November and in March, and iii) the later difference was of different magnitude between males and females. In the later case, the negative slope of vigilance rate against flock size actually increased in magnitude from November to March in males, and decreased in females (Fig. 2).

Males increased their foraging distance from water between the two time periods (10.84m ± 0.54 and 13.79m ± 0.57 in November and March, respectively; ANOVA: F 1,278 = 13.99, r² = 0.05, P = 0.0002), while females showed an opposite trend (11.34m ± 0.58 and 7.93m ± 0.58, respectively; ANOVA: F 1,271 = 17.39, r² = 0.06, P <0.0001). While males and females fed at a similar distance from water in November (ANOVA: F 1,285 = 0.45, r² = 0.00, P = 0.5043), females thus fed at a shorter distance from water than males in March (7.93 m ± 0.60 and 13.79 m ± 0.61, respectively; ANOVA: F 1,264 = 46.70, r² = 0.15, P < 0.0001). In March, males spent, on average, 25% of their time being vigilant when foraging in small flocks (Fig. 2) compared to 15% of their time in November. Comparably, time spent being vigilant by females in November when foraging in small flocks, as with the males, was approximately 15%, reducing to 10% in March.

That no interaction with Distance was retained in the final model indicated that the benefit birds gained from foraging closer to the water in terms of decreased vigilance was similar in males and females, and did not differ between the two time periods, that is, the slopes remained similar.

Discussion

Our data provides little support for the male investment hypothesis [START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF], that males increase their vigilance to guard their mate when they become paired.

Even the slight increase in male mean vigilance from November to March could be simply explained by non-mating reasons since males, on average, fed at a greater distance from water in March. In both sexes, vigilance increased with foraging distance from water and decreased with increasing flock size. We have shown that wigeons on our wetland study site spent a c. 10% of their potential feeding time scanning their environment. This is similar to what wigeons have been recorded to do when foraging on dry land previously [START_REF] Guillemain | Feeding methods, visual fields and vigilance in dabbling ducks (Anatidae)[END_REF][START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF]. For comparison, female and unpaired male wigeons on average spent less than 5% of their time vigilant while foraging on the Exe Estuary, where males were apparently essentially vigilant to ensure paternity with their mate and for pair bond protection [START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF]. Compared to [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF], the wigeons in the current study foraged, on average, in larger flocks and closer to the water. For example, at Caerlaverock, Solway Firth [START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF], the wigeons maximum foraging distance from the water was 60 metres, compared to just 35 metres in the present study. This can be attributed to the smaller site in the present study when compared to Caerlaverock. The total distance from the high tide line of the sea to the footpath at the rear of the marsh was only approximately 80 metres.

Therefore, the wigeons were foraging at smaller distances from a water body, whether that is the sea or one of the many small pools that were present within the marsh.

Male vigilance increased markedly in more risky situations (smaller groups, longer distance from water), and only changed to a minor extent from the pre-to the post-mating periods. Moreover, male and female vigilance was similar in the November period. This suggests that the slightly higher vigilance rate of males during March compared to that of females is not a result of the male constraint hypothesis (it is not related to the male"s plumage). The vigilance patterns are consistent with the predicted commonly observed relationship between vigilance and distance from water or group size in wigeons [START_REF] Mayhew | Vigilance levels in European Wigeon, Anas penelope -sexual differences[END_REF][START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF]. The increase in foraging distance during late winter in the males supports the hypothesis that anti-predator vigilance plays a major role in the behaviour of wigeons when in foraging flocks. It is highly likely that males are forced to use these distant and more risky areas in late winter, when safer feeding patches closer to water have been depleted (following general central-place foraging theory, e.g. [START_REF] Stephens | Foraging theory[END_REF]. The significant decrease in female vigilance between the two periods and the lack of a similar increase in male mean vigilance (the difference in vigilance rates was twice larger in females than in males) lends support to the female benefits hypothesis.

Females fed closer to the water in March than in November, i.e. were less at risk when they were paired. Examination of Figure 2 shows that females decreased their vigilance from November to March when they were foraging in the most risky situations, i.e. when in smaller flocks. Male vigilance was maximum in such situations, and almost twice that of females: in flocks of 3-5 birds males spent 20.4% of their time vigilant (± 0.5, n = 53), while females spent 11.7 % ± 0.5 (n = 51). Paired females in March will thus benefit from the presence of their mate in such most risky situations, enabling them to continue foraging and build body stores.

The female benefits hypothesis provides a satisfactory evolutionary framework for vigilance patterns in paired birds, as the effectiveness of mate guarding has been called into question [START_REF] Struchbury | Female control, breeding synchrony, and the evolution of extra-pair mating systems[END_REF], mainly because of time constraints on the male, and the tactics that females can adopt to circumvent male guarding efforts. However, it has been shown that mate guarding can be effective in partially assuring paternity, particularly when males are extremely attentive [START_REF] Chuang-Dobbs | The effectiveness of mate guarding by male black-throated blue warblers[END_REF]. For example, male common eiders spend 99% of their time mateguarding when the females are most fertile [START_REF] Christensen | Female pre-nesting foraging and male vigilance in Common Eider, Somateria mollissima[END_REF][START_REF] Hario | The role of mate-guarding in pre-laying common eiders (Somateria mollisima) in the northern Baltic Sea[END_REF]. However, male birds of other species continue to follow females after the fertile period has ended, suggesting that paternity assurance is not the primary reason for general mate-guarding, and increased vigilance [START_REF] Squires | Vigilance behaviour in the harlequin duck (Histrionicus histrionicus) during the preincubation period in Labrador: Are males vigilant for self or social partner?[END_REF]Rodway 2007a). Martin et al. (2003) proposed that predation risk was more important than partial loss of paternity, as a fatal attack on a female will result in a total loss of offspring. It is, therefore, better for the male to invest in the female through allowing her to feed at a higher rate than she would otherwise, and thus allowing the male to potentially increase his own fitness. [START_REF] Quan | Sexual differences of paired ruddy shelduck in winter[END_REF] suggested that male birds could also achieve this through maintenance of the pair bond and female preference. The question still remains, however, of whether females simply capitalise on male vigilance or stimulate it [START_REF] Artiss | Female foraging and male vigilance in white-tailed ptarmigan (Lagopus leucurus): opportunism or behavioural coordination ?[END_REF]. A recent theory encompasses benefits for both males and females. Rodway (2007aRodway ( , 2007b) ) studied the time unpaired male harlequin ducks spent engaged in display behaviour, which was mutually exclusive to foraging. When the males became paired, they significantly reduced displaying behaviour and increased foraging, thus increasing their energy intake. Simultaneously, the females also increased foraging (Rodway 2007a;Rodway 2007b). Therefore, both the male and female mutually benefit from pairing in terms of time-budget. However, it has been proposed that the female should only pair early if the bond is beneficial for her [START_REF] Artis | Male vigilance in white-tailed ptarmigan, Lagopus leucurus: mate guarding or predator detection?[END_REF]. If food is plentiful and predation risk low, the female should actually avoid the more conspicuous males [START_REF] Artis | Male vigilance in white-tailed ptarmigan, Lagopus leucurus: mate guarding or predator detection?[END_REF]Rodway 2007b). In wigeons, it is likely that as the wintering season progresses, food and resources will be further from the water (see above) and thus involves riskier foraging. Therefore, pairing up at this stage would be a good strategy for the female.

Both the male and female benefit from the female being in the best possible physiological condition.

That males responded to group size more than females (they decreased their vigilance to a greater extent with increasing group size), and did so to a higher extent in March than in November (while the reverse would have been expected if there was a fierce competition for females between males) suggests that the increase in male vigilance, particularly in small flocks, is for the benefit of the female. If this increase vigilance was anti-predator related, the relationships between male vigilance and flock size would be expected to remain unchanged between November and March. Because the present terrestrial feeding location is intrinsically dangerous [START_REF] Mayhew | The feeding ecology of wigeon. Wigeoncurrent trends and future needs[END_REF][START_REF] Mayhew | Feeding site selection by Eurasian wigeon, Anas penelope in relation to water[END_REF][START_REF] Larsen | Wigeon Anas penelope offsetting dependence on water by feeding in mixed-species flocks: a natural experiment[END_REF], vigilance levels in male wigeon are particularly high in Traeth Lafan, even in November. Once mated, the paired females are benefiting from this genuine high level of vigilance by their mate to decrease their own vigilance and dedicate more time to foraging (see also [START_REF] Jacobsen | Effects of waders on grazing and vigilance in breeding wigeon, Anas penelope[END_REF].

What is unusual is the lack of a large increase in male vigilance when paired (e.g., [START_REF] Martin | Reproductive defense priorities of male willow ptarmigan (Lagopus lagopus)enhancing mate survival or extending paternity options?[END_REF][START_REF] Guillemain | Increased vigilance of paired males in sexually dimorphic species: distinguishing between alternate explanations in wintering Eurasian wigeon[END_REF]. From our data set, it is not possible to determine whether males could increase their vigilance further at Traeth Lafan. It is likely male wigeons at Traeth Lafan are already close to the maximum vigilance level they can afford to be without compromising their daily energy intake. In this scenario, females are able to benefit from the males vigilance at now extra cost to the male, and the males only need provide minimal supplementary vigilance in late winter. This, coupled with already high levels of vigilant behaviour in males means the females are able reduce their vigilance and focus on foraging. This occurs with no detriment to the male"s daily energy intake. The only circumstance where this may alter is when foraging in small flocks, when males are both significantly more vigilant than females and more vigilant than when foraging at the same distance in early winter compared to late winter. If mate guarding and paternity assurance can be ruled out as a function of vigilance in the present study, paired males and females could potentially benefit from foraging in larger flocks to avoid the male having to increase vigilance.

The present pattern would thus be similar to the situation recorded in Eurasian teal foraging at deep depths, where males are already so vigilant because of their blind exposure to predators when up-ending to forage that they do not increase vigilance after acquiring a mate (Guillemain et al. 2007b). In both cases, this could explain how male birds with high energy demands can afford the cost of maintaining a pair-bond during such long periods in winter (see Rodway 2007b). Data are not available at present, but a comparison of forced copulation attempts on female wigeons at Traeth Lafan and on the Exe Estuary (where male wigeons are clearly vigilant for paternity assurance and/or pairbond protection) could allow testing the above hypothesis more specifically. This study, therefore, suggests that whether male and female vigilance in a given species is more likely to be driven by the male constraint, male investment or female benefits hypothesis may depend very much upon the environmental safety conditions, and is not a fixed speciesspecific life history trait. While they had long been considered as mutually exclusive activities, the traditional distinction between non-vigilant feeding bouts and non-feeding vigilance bouts in foragers was challenged a few years ago by [START_REF] Lima | Back to the basics of antipredator vigilance: can nonvigilant animals detect attack?[END_REF], showing that some individuals were able to maintain anti-predator vigilance while foraging with their head down. Similarly, the present study suggests that vigilance for several purposes may be combined by paired males in some situations.

Fig. 1 Fig. 2

 12 Fig. 1 Average percentage of time spent vigilant by male and female wigeon in November 550

Acknowledgements

We are grateful to Nigel Brown for assistance during the planning of the fieldwork, and to David Gardiner for logistical support. Graham Martin, Jon Green and an anonymous referee provided useful comments on an earlier version of the manuscript. We are also grateful to the numerous students, particularly Tom Richardson, who assisted with fieldwork, and to Claire Tyler and Craig White for useful discussions.

The authors declare that they have no conflict of interest.

Table 1 Best fitting models for the percentage of time spent vigilant by wigeon while foraging, testing for differences between males and females (Sex), November and March observation periods ("Month"), the effect of group size in which each focal bird foraged ("Log(FlockSize)") and the distance from water at which it did so ("Distance"), plus their interactions. Only the final model of the backwards stepwise model selection procedure is shown.

Variable