Thomas K Gottschalk
email: thomas.gottschalk@allzool.bio.uni-giessen.de

Falk Huettmann

Justus Liebig

Comparison of Distance Sampling and Territory Mapping

Keywords: bird census, detection probability, number of registrations, habitat type, survey design zwischen Distance Sampling und Revierkartierung zur Erfassung von Vogelbeständen Rahmen eines wissenschaftlichen Experiments beide Methoden parallel durchgeführt und die zwischen den Lebensräumen und zwischen den Vogelarten. Die Dichten, die mit Hilfe Während die Anzahl der ermittelten Arten bei beiden Methoden in etwa gleich war, zeigten die Abundanzen von 15 Arten signifikante Unsere Ergebnisse zeigen, das zur Wertung eines Reviers im Durchschnitt vier Registrierungen eines Vogels bei acht Begehungen notwendig sind, um realistischere jeweiligen Fehler liefert

Distance sampling (DS) and territory mapping (TM) are globally applied bird survey techniques. However, specifically designed studies comparing results of both methods in different habitats in the framework of a scientific experiment have rarely been conducted. To provide a more generalized guidance for the field surveyor, here we evaluated estimates of bird abundances and number of bird species in four different habitats (broad-leaved forest, coniferous forest, open woodland and farmland) in central Germany. Abundances were estimated in parallel by TM and DS in 2006 and 2008, following standard protocols.

Detection probability differed significantly among habitats and species. Density estimates by DS were in total 24% lower than those estimated by standardized TM. While the number of bird species detected with both methods was approximately the same, the estimated abundances of 15 bird species showed significant differences. Increasing the number from two to four and five registrations to count a territory by using TM decreased the density on average about 28% and 42%, respectively. Using standardized TM resulted in an overestimation of abundances of species showing a high detection probability. In contrast, DS estimated very high densities for species that had a very low detection probability. In fact, a highly negative correlation was found between the density estimated by DS and the detection probability. Using standardized TM and setting a fixed number of registrations before a location qualifies for a bird territory cannot compensate for the large differences in species detectability. Instead, the number of registrations required to count a territory should be adjusted to differences in detection probabilities and seasonal activity. From our results we can recommend a mean of four registrations if eight visits were conducted to count a territory.

However, the lack of any statistically-based quality assessment reduces the serious usability of TM for estimating densities for science-based management applications. Whereas, the clear advantage of DS is that it provides error estimates and considers differences in species detectability.

Gottschalk -5

Distance sampling (DS) and territory mapping (TM, or spot-mapping) are survey techniques for estimating bird abundance [START_REF] Bibby | Bird census techniques[END_REF][START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF]. The TM method is based on counting territories of all species within a defined study plot. Locations of all birds, particularly singing males, are mapped on paper replicas of the plot during visits (usually eight or more) during the breeding season. Data for each visit are transcribed to speciesspecific sheets, and territory boundaries are identified for clustered multiple registrations at the end of the census.

The DS method is based on counting birds detected as heard or seen from a point or transect [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF], and takes into account the fact that some birds are detectable at greater distances than others, that a species may be more easily detected in one habitat than another, and that detectability can change with time of day. Therefore, for each bird detected, the distance between observer and bird must be estimated accurately. A detection function is estimated from these distance data, and is then used to compute the probability of detection.

Crucial to DS is the estimated detection function that compensates for the fact that detectability decreases with increasing distance from the observer. Studies have shown that DS delivers reliable results and is efficient for sampling large areas [START_REF] Norvell | A seven-year comparison of relative-abundance and distance-sampling methods[END_REF][START_REF] Somershoe | Combining breeding bird survey and distance sampling to estimate density of migrant and breeding birds[END_REF][START_REF] Newson | Use of distance sampling to improve estimates of national population sizes for common and widespread breeding birds in the UK[END_REF][START_REF] Ronconi | Estimating seabird densities from vessel transects: distance sampling and implications for strip transects[END_REF]. To create detection functions for each bird species, a minimum number of observations in each main habitat is required. Alternatively, detection probability functions of biologically similar species can sometimes be used [START_REF] Buckland | Estimating bird abundance: making methods work[END_REF]). However, the assumption of a similar, constant and transferable detection probability can be difficult, especially for rare species and when survey conditions vary. Experience shows that detection is highly dynamic and can vary with time of day, between seasons, years and other factors [START_REF] Norvell | A seven-year comparison of relative-abundance and distance-sampling methods[END_REF][START_REF] Robbins | Estimating Numbers of Terrestrial Birds[END_REF].

TM attempts to account for imperfect detection by using a fixed ratio of registrations of a species to the number of effective visits for that species. This ratio, usually similar for all Comparison of Distance Sampling and Territory Mapping Gottschalk -6 species, is used to decide whether a territory will be assigned for counting. [START_REF] Bibby | Bird census techniques[END_REF] recommended at least two registrations for a species if there were eight or fewer visits, and at least three registrations when there are nine or more visits. This rule corresponds to a fixed detection rate of around 0.25-0.33. Despite the inability to assess this rule or its validity in a scientific way, using a constant number in that range will result in uncertainties if used for estimating abundances for a whole study area. Furthermore, territories of birds can be highly dynamic within the season [START_REF] Knapton | Settlement patterns, territory size, and breeding density in the Song Sparrow (Melospiza melodia)[END_REF][START_REF] Finck | Seasonal variation of territory size with the Little Owl (Athene noctua)[END_REF][START_REF] Pasinelli | Oaks (Quercus sp.) and only oaks? Relations between habitat structure and home range size of the Middle Spotted Woodpecker (Dendrocopos medius)[END_REF], making the territory a questionable metric for abundance estimation. One assumed strength of TM is that it provides finer spatial detail and, therefore, can be better used to depict the spatial distribution pattern of birds in an area and, additionally, can be correlated with habitat distribution. Therefore, it is often applied in environmental assessment studies because areas important to birds can be identified.

Despite the popularity of both methods, few investigators have compared the actual results of bird abundance estimation using DS and TM in the same study area. Such studies are very helpful for assessing and interpreting the accuracy and possible biases of each method. To our knowledge, the only studies where birds were estimated by both DS and TM by using a standardized approach and the two methods compared were those of [START_REF] Gillings | Avian community composition and patterns of bird distribution within birch-heath mosaics in north-east Scotland[END_REF], [START_REF] Raman | Assessment of census techniques for interspecific comparisons of tropical rainforest bird densities: a field evaluation in the Western Ghats, India[END_REF] and [START_REF] Buckland | Point transect surveys for songbirds: robust methodologies[END_REF]. The results of these studies did not show a clear pattern, [START_REF] Buckland | Point transect surveys for songbirds: robust methodologies[END_REF] and [START_REF] Gillings | Avian community composition and patterns of bird distribution within birch-heath mosaics in north-east Scotland[END_REF] estimated for three species a lower, for three other species a higher and for two species a similar density using TM compared to DS. [START_REF] Raman | Assessment of census techniques for interspecific comparisons of tropical rainforest bird densities: a field evaluation in the Western Ghats, India[END_REF] estimated a higher density using TM compared to DS for two out of 13 species in a tropical rainforest. Although [START_REF] Bibby | Bird census techniques[END_REF] has shown that territory maps are not easy to analyze and can be interpreted differently, depending for instance on the number of registrations used to set a territory [START_REF] Gerß | Automatische Revierabgrenzung bei Siedlungsdichteuntersuchungen[END_REF], none of these studies reported how territories were detected, and with the exception of [START_REF] Gillings | Avian community composition and patterns of bird distribution within birch-heath mosaics in north-east Scotland[END_REF] the minimum number of registrations used to set a territory was not reported. None of the studies comparing DS and Comparison of Distance Sampling and Territory Mapping Gottschalk -7 TM analyzed the influence of the minimum number of registrations on the estimated densities. However, this number is important as it indirectly reports the assumed detection rate, which is crucial to reduce over-or underestimation of species density. Further, detection rates differ between habitats [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF][START_REF] Pacifici | Effects of vegetation and background noise on the detection process in auditory avian point-count surveys[END_REF]) but none of the previous studies have compared the densities estimated in different habitats and using both, TM and DS. Therefore, we used a standardized sampling design and conducted a field study in four different habitats which were selected for their differences in vegetation structure (Fig 1). Our objective was to provide guidance to the field surveyor. Therefore, we determined if

(1) the strength of differences between the results of both methods are habitat specific, (2) the number of registrations used to set a territory influence densities estimated by TM and (3) the differences in species detectability affect estimates of species densities by TM and DS, respectively. To do so, abundances and number of bird species were estimated by both methods.

We decided not to set one method as a benchmark a priori (e.g. [START_REF] Buckland | Point transect surveys for songbirds: robust methodologies[END_REF][START_REF] Desante | A field test of the Variable Circular-Plot Censusing Method in a Sierran Subalpine Forest habitat[END_REF][START_REF] Gale | A field test of distance sampling methods for a tropical forest bird community[END_REF]), as we do not assume that one of the methods provides greater precision per se. Further, intensive bird census techniques used in other studies [START_REF] Casagrande | Evaluation of four methods for estimating parrot population size[END_REF][START_REF] Desante | A field test of the Variable Circular-Plot Censusing Method in a Sierran Subalpine Forest habitat[END_REF][START_REF] Tarvin | A field evaluation of techniques used to estimate density of Blue Jays[END_REF]) like color-banding or nest-finding do not guarantee that individuals can be found or caught with equal ease and it is very difficult to be confident that all individuals have been found [START_REF] Bibby | Bird census techniques[END_REF].

Additionally, these techniques are likely to result in an unacceptable level of disturbance to birds.

Methods

All study sites were located in the Hoherodskopf, located 60 km northeast of Frankfurt am Main in Hessen, central Germany (9°21"E and 50°51"N). One study site was located in Six of eight surveys started 30 min before sunrise and finished between 08:00 and 11:00 (mean = 09:37). Two of the eight visits were in the evening to better sample species less active in the early morning, e.g., raptors and owls. All surveys were conducted by the same observer. To obtain comparable conditions and data, DS and TM were conducted on the same day; the second method was started after the first was completed. The order in which each method was used first was alternated. Before the field work was started, a route was established on a map that approached within 50 m of every point on the plot. In open woodland and farmland, where visibility was higher, this distance was set to a maximum of Gottschalk -9 observer (because of a priori knowledge from the first survey) and the birds (because of disturbances by the first survey), during the second survey of this day we considered these points with possible day-to-day differences in weather conditions [START_REF] Bibby | Bird census techniques[END_REF] if the census would have been conducted on two different days and differences in observers perception [START_REF] Diefenbach | Variability in grassland bird counts related to observer differences and species detection rates[END_REF] if the census would have been conducted from two or more observers.

Following [START_REF] Bibby | Bird census techniques[END_REF] for TM, the locations of all birds present in the plot were mapped on different days, and a territory was defined if at least two registrations were made of a bird singing or exhibiting breeding behavior (nest with eggs, young birds, or adults carrying nest material or food). Henceforth, we call this the "standardized TM" approach. To analyze the effect of the minimum number of registrations used to count a territory, the number of registrations used to set a territory was increased from two to five. Assuming that each territory was occupied by a pair, the number of territories was equivalent to the number of breeding pairs in the plot. DS was conducted using point counts where all birds heard or exhibiting breeding behavior within 5 min were mapped. We decided to use 5 min instead of 10 min as it reduces the chance of double counts and is widely used. Distance to each bird detected was estimated to the nearest 10 m-interval using binoculars (8 x 56) that included a laser range finder.

Sampling points were placed within the study sites where TM was conducted, and spaced at least 200 m apart to reduce spatial autocorrelation. Without overlap, six sampling points could be placed within each 25-ha study plot. All were marked for easy relocation on later visits.

All point-count data were analyzed using the program DISTANCE (version 5.0, [START_REF] Thomas | Distance software: design and analysis of distance sampling surveys for estimating population size[END_REF]). We truncated point-count distances at 150 m. Therefore, an area of 42.4 ha in each habitat was analyzed. Detections for all visits were pooled for each of the four study sites. The survey effort parameter was set to eight based on the number of visits to each site. However, following [START_REF] Südbeck | Methodenstandards zur Erfassung der Brutvögel Deutschlands[END_REF], the survey effort parameter was set to seven

Comparison of Distance Sampling and Territory Mapping

Gottschalk -10 for two migrant species (Tree Pipits, Anthus trivialis, and Eurasian Blackcaps, Sylvia atricapilla) and, for Common Whitethroats (Sylvia communis), to six due to their later arrival on their breeding grounds. Abundance estimates of species showing a coefficient of variation (CV) higher than 40% were not analyzed. In our study, at least 20 registrations were needed for the CV to fall below this value. We did not pool data across habitats to facilitate comparisons of detection probabilities across individual habitats. An average of 66 detections per species of singing birds was used to analyze bird data with DISTANCE. For some species, the number of detections was lower than the 60 recommended by [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF], but reliable detection curves could still be fitted. According to the methodology and definition of both census methods used, the output of TM are territories and that of DS are birds. However, in fact in both methods singing birds, or birds showing clues of breeding behavior, were counted.

To examine how density is related to detectability, correlations between abundance values and the effective detection radius (EDR) were analyzed using Spearman rank order correlations. EDR represents the distance from the observer where the number of birds missed equals the number of birds observed farther away [START_REF] Gates | Line transect and related issues[END_REF]. EDR and its coefficient of variation for each species were calculated using the program DISTANCE. Densities determined for the two methods were compared using the Wilcoxon matched pairs test. To identify possible differences in detection probability among habitats, we used one-way ANOVA. All statistical analyses were conducted using the Statistica 7.1 software package (StatSoft Inc., Tulsa, OK).

Results

We detected 58 species with DS and 60 with TM. The small differences in number of species detected were caused by the varying number of non-breeding birds (overflying birds or migrants). Eight and six of these non-breeding species were found using TM and DS,

Comparison of Distance Sampling and Territory Mapping

Gottschalk -11 respectively. Most bird species were found in half-open woodland (39 species using DS and 38 using TM) and farmland (31 species using DS and 39 using TM) and lowest species diversity was observed in beech and coniferous forest (30 and 28 species using DS and 31 and 33 using TM, respectively). Thus, the number of bird species counted by DS and TM showed the highest difference in farmland. This difference is mainly caused by a relatively higher number of non-breeding birds species counted using TM (six species) than using DS (two species). In other habitats the same number of bird species or none non-breeding birds were recorded using TM and DS.

We calculated abundances for 15 of these species (species with CV below 40%) and compared all together 22 density values from the four habitats (Table 1). Densities estimated by DS were significantly lower (in total by 24%) than those estimated by the standardized TM approach (Wilcoxon Matched Pair Test, n=22, z = 2.68, p = 0.013). Abundances estimated from the standardized TM approach were up to 3.9 times higher (mean = 1.3) than those derived from DS. Only Chaffinches in beech forest and Firecrests and Goldcrests in coniferous forest showed significantly higher abundances using DS. The strength of differences between the estimated densities varied between habitats. On average differences increased by 1.0 territories / 10 ha in farmland, 2.0 territories / 10 ha in beech forest, 2.8 territories / 10 ha in open woodland and 5.5 territories / 10 ha in coniferous forest. Highest differences with more than four territories / 10 ha were estimated solely for species found in coniferous forest (Robin, Common Wood Pigeon, Common Chaffinch, Winter Wren, Common Blackbird, Firecrest and Goldcrest).

Densities of all bird species decreased with an increasing number of registrations used to count a territory. The mean density of the 15 bird species analyzed decreased about 34% from 8.8 territories/10 ha using two registrations to 5.1 territories/10 ha using five registrations. The number of registrations was negatively correlated with density (r s = -0.39, p = 0.000229) estimated by TM. Densities based on two and three registrations differed Firecrest EDR = 29 m) (Table 1). A negative correlation was identified between the density estimated by DS and the detection probability (EDR) (Spearman Rank Correlation: -0.66, p = 0.000853) (Fig. 2). All densities estimated by TM regardless the number of registrations used to count a territory did not correlate with the EDR.

The time required to conduct the bird survey varied between habitat and survey techniques (Table 2). Using TM, most time was spent in the coniferous forest, followed by beech forest, farmland, and open woodland. DS was the more efficient method: almost twice as much time was required to conduct the bird census using the TM method.

Discussion

We found that the number of species detected was similar using DS and TM.

Generally, the number of species detected was related mostly to habitat, the study area size (TM: 25 ha and DS: 42.4 ha), survey effort (TM: 137min and DS: 70min at mean), and the detection probability of each species (Fig. 1). Although the time spent on each study site was higher using TM and would allow more opportunity to detect bird species, the area surveyed using DS was larger and therefore potentially inhabited by a larger number of bird species.

Comparison of Distance Sampling and Territory Mapping

Gottschalk -13

The strength of difference between the estimated abundances of both methods was related to habitat. Largest differences between both results were found in coniferous forest and were lowest in farmland. This suggests that estimated densities from habitats like the coniferous forest where bird species show a low EDR are more sensitive to methods which take the detection probability into consideration.

One reason for the higher density estimates obtained in our study by the standardized TM approach when compared to DS was the static number of registrations. For our standardized TM approach, we followed [START_REF] Bibby | Bird census techniques[END_REF] who recommended at least two registrations if there were eight visits to the study area. This number essentially assumes a detection probability of 25% for all bird species and thus ignores crucial and dynamic differences in the detection probability between species. Densities estimated by TM based on four and five registrations (detection probability of 50%-62.5%) were lower and did not significantly differ from those estimated by DS. These results confirm the findings of [START_REF] Gerß | Automatische Revierabgrenzung bei Siedlungsdichteuntersuchungen[END_REF] who has shown in an experiment using an automated approach to demarcate territories that the number of territories is largely affected by the minimum number of observations used to count a territory. The minimum number of territories used in TM also explains the differences between our results and those of [START_REF] Gillings | Avian community composition and patterns of bird distribution within birch-heath mosaics in north-east Scotland[END_REF]. They compared bird density estimates using TM and DS in the UK, and conducted four visits and counted one territory if at least two registrations were made using TM and 0.5 territories if one registration was conducted. The chance to detect a bird during four visits was lower than in our study and thus, fewer birds were registered by using TM. Consequently, these densities were more similar to the lower densities estimated by DS. This example emphasizes that detected differences have to be analysed exactly by how territories have been estimated using TM.

The reasons in our study for the differences found between the four densities estimated by TM using a different minimum number of bird registrations are related to (a) edge clusters, when territories overlap the plot boundary [START_REF] Bibby | Bird census techniques[END_REF], and (b) the assumed minimum

Comparison of Distance Sampling and Territory Mapping

Gottschalk -14 Euclidian distance at which an observation will be included to a territory [START_REF] Scheffer | An automated method for estimating the numbers of bird-territories from an observation map[END_REF].

Following [START_REF] Dornbusch | Zur Methode der Ermittlung von Brutvogel-Siedlungsdichten auf Kontrollflächen[END_REF] we counted a territory as a half if more than 50% of the observations of an edge cluster were inside the plot. Thus, and confirming the finding of [START_REF] Gerß | Automatische Revierabgrenzung bei Siedlungsdichteuntersuchungen[END_REF], the reduction of the minimum number of registrations increases the chance that such a territory can be counted in. The second reason is related to the minimum distance at which an observation was assumed to belong to a territory. When increasing the minimum number of registrations used to count a territory, an increasing number of registrations of greater distance from each other are used to set the territory. To minimize those effects and to reduce observer variation, which is common when TM results were analysed (Best 1975[START_REF] Svensson | Interpersonal variation in species map evaluation in bird census work with the mapping method[END_REF]), an automated territory clustering approach is helpful [START_REF] Gerß | Automatische Revierabgrenzung bei Siedlungsdichteuntersuchungen[END_REF][START_REF] Scheffer | An automated method for estimating the numbers of bird-territories from an observation map[END_REF] especially when combined with a GIS [START_REF] Witham | Use of Geographic Information System to facilitate analysis of spot-mapping data[END_REF]. Such an approach can help to standardize and automate territory interpretation and to find the "correct" number of territories. Therefore, it should incorporate species-specific standards, e.g. minimum number of registrations to count a territory, maximum distance between registrations at which they will be used to set a territory. However, these standards can not diminish drawbacks that arise from ignoring differences between species detection probabilities. As shown in our study, the detection probability differs between species. This suggests that using TM and simply setting a fixed number of registrations for a species until it qualifies for a territory cannot compensate for the huge differences in species detectability. It is not clear to us where this static number of registrations used for TM has its primary and scientific origin, and what its underlying logic, data and tests are. Compared to this static value, DS instead estimates empirically a more realistic correction factor, based on the true survey circumstances of each actual detection event [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF].

The unexpected significantly high negative correlation between detection probabilities and densities estimated by DS suggests that DS may overestimate quiet or cryptic species, especially if patchily distributed, relative to large and conspicuous species. This could be for

Comparison of Distance Sampling and Territory Mapping

Gottschalk -15 the reason that DS assumes a uniform distribution within one habitat for a given study area [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF]. If a high number of birds were counted at small distance, an overestimation of the abundance of these species by DS might be possible, especially if species occur in clusters or when density varies throughout the study area. To reduce the influence of this biased information on the distribution, the pattern of habitat characteristics can be used by modelling abundance covariate effects in DS models to reach reliable density estimates [START_REF] Marques | Improving estimates of bird density using multiple covariate distance sampling[END_REF][START_REF] Royle | Modeling abundance effects in distance sampling[END_REF]). Furthermore, to control for variation in detection probability, sampling points can be visited more frequently or placed at higher densities within areas where quiet or cryptic species might occur and vegetation structure varies (Buckland et al. 2004). The DS software is helpful for designing appropriate survey strategies in such studies. In our study, significantly higher densities of Goldcrest and Firecrest were estimated using DS compared to TM. The detection probability curves for these species showed steeply declining curves and low effective detection radii. According to these curves, a detection probability of 0.5 for Goldcrest and Firecrest can be reached at a distance of 29 m and 34 m, respectively. In practical terms, this means that every second individual would not be detected at these distances. As recommended in [START_REF] Bibby | Bird census techniques[END_REF], TM was conducted by an observer walking lines 50 m apart; resulting in a 21 m and 16 m survey gap respectively. However, to detect more individuals of species having a low detection probability, a closer line-spacing would be needed to reduce the number of missed birds. But even this increased sampling effort does not guarantee the registration of all individuals. [START_REF] Diefenbach | Variability in grassland bird counts related to observer differences and species detection rates[END_REF] found that as many as 60% of the birds more than 50 m from the observer were missed. However, detection probability is known to be dynamic and differ by habitat [START_REF] Mcshea | Variable song rates in three species of passerines and implications for estimating bird populations[END_REF]Rappole 1997, Schiek 1997), with lowest detection probability in broad-leaved forests [START_REF] Pacifici | Effects of vegetation and background noise on the detection process in auditory avian point-count surveys[END_REF]. Our results confirm these findings as they clearly showed significant differences in the probability to detect a species between the four habitats and the lowest EDR in beech forests. This finding has wider implications for bird surveys and Comparison of Distance Sampling and Territory Mapping Gottschalk -16 monitoring to be taken into account, by choosing the correct distance between walking routes according to the EDR of that species that might be found in that habitat and which has the lowest detection probability.

In our study, bird surveys conducted by DS were less time-intensive than TM.

However, using our survey design (which was not specifically designed for rare birds, e.g. lacking many smaller transects or adjusted sampling, and therefore with no assurance of sufficient detections for patchily-distributed species) the amount of effort taken for DS was sufficient to calculate densities for only 15 out of 60 species detected. Additional reliable density estimates would be possible if more birds were detected of those species. However, these additional detections, especially of rarer bird species, would significantly increase survey effort and therefore clearly reduce efficiency of DS. To calculate the abundance of a bird species at least two registrations are required using TM, but at least 20 detections using DS. [START_REF] Barraclough | Distance Sampling: a discussion document produced for the Department of Conservation. Science & Research Internal Report 175[END_REF] stated that the greatest drawback of DS is the number of detections required. In an extreme case, if only a single pair of a less known bird species occurs in one study plot, it has to be recorded several times, e.g. through repeated visits to the survey location, before the precision of the abundance estimated by DS is adequate. If the bird species is not well known, then pooling the distance data across groups of species with a similar relationship between detectability and distance, as recommended by [START_REF] Buckland | Estimating bird abundance: making methods work[END_REF], is not really possible. If confidence values are not needed, TM is more advantageous for roughly estimating density of rarer birds. But DS is known to be less efficient in relatively small study areas especially if densities of rarer birds must be sampled [START_REF] Buckland | Estimating bird abundance: making methods work[END_REF]).

However, if the species is rare, then both methods take more time; with TM, most sites will give "zero", so lots of sites will be needed, and using 5 minute-point counts might be more efficient to detect rare species.

The absence of replication of our study in other landscapes might represent a limitation to the number of species analysed and to general conclusions. However, our data Comparison of Distance Sampling and Territory Mapping Gottschalk -17 showed significant differences between the two methods, suggesting that results are still sensitive to the method employed and demonstrating the need for recommendations on how survey techniques can be further optimized, and "truth" is to be found. If an exact statistical estimation of the species" density is needed, careful use of DS is more convincing as it provides the coefficient of variation as well as the 95% confidence intervals for each calculated density. Such statistical values for bird survey data are fundamental for sciencebased and sustainable management [START_REF] Walters | Adaptive management of renewable resources[END_REF]). Ideally 60 detections, or at least a robust detection curve for each species, should be used to obtain precise DS estimates [START_REF] Buckland | Introduction to Distance Sampling: estimating abundance of biological populations[END_REF]. If this number of detections cannot be reached, and when the survey design cannot be adjusted to obtain a reliable detection curve, approximate or pooled data from other studies or similar species can be used to estimate a detection curve and to be used to estimate densities. Although those density values lack exact confidence values, for small study plots this presents a pragmatic use. On the one side, a bird census using TM in habitats containing species of low perceptibility could be optimized by walking routes spaced less than 50 m apart which reduces the risk of missing elusive species. On the other side, a higher survey effort increases the chance of double-counting for highly abundant and conspicuous birds.

Keeping TM flexible by adapting it to species and site specific requirements can be an important advantage of TM though, and which distinguishes this method from other, more standardized methods. However, this makes the method less comprehensible and therefore less reliable especially for monitoring purposes as it is more driven by the observers" right assessment of local conditions. The number of registrations at which a territory of a species will be counted has to be treated with caution when using TM. From our results we cannot recommend a minimum number of two or three registrations if eight effective visits of the study plot were conducted.

Instead, the number of registrations required to count a territory should be adjusted to the species-specific detection probabilities. Based on the detection probabilities we can Comparison of Distance Sampling and Territory Mapping Gottschalk -18 recommend eight visits and a mean of four registrations to count a territory. However, if detection probabilities of each species is known a species-specific treatment would be more reliable. Generally, the missing confidence interval or any other statistically-based quality assessment largely reduces the serious usability of TM for estimating densities and for science-based management. f %CV = coefficient of variation of the densities estimated by DS.

g 95% CL = 95% confidence limits.

h No. of detections = number of detections used to estimate densities by DS.

  each of four habitats: beech forest (Fagus sylvatica), coniferous forest, open woodland, and Comparison of Distance Sampling and Territory Mapping Gottschalk -8 farmland. Study sites where TM got conducted were limited to 25 ha to avoid census times exceeding the morning peak of bird activity. To reduce possible edge effects, the shapes of the study sites were chosen for compactness. The beech forest study site was located on the north-eastern slope of the Hoherodskopf (710-760 m) and consisted mainly (86%) of 50-year-old beech trees. Small patches of older beech trees were present, along with maple trees (Acer pseudoplatanus and A. platanoides) and common spruce (Picea abies). The coniferous forest was located in the southern slope of the Hoherodskopf (630-675 m); most (95%) of the area was covered by spruce, with small openings of grassland. The open woodland was located on the north-western slope of the Hoherodskopf (660-725 m) and consisted of European mountain ash (Sorbus aucuparia) and white willow (Salix alba), grassland and patches of myrtle blueberry (Vaccinium myrtillus) on open areas. The farmland was located on a northwest slope (480-535 m), and consisted of approximately 55% grassland and 40% barley (Hordeum vulgare) crops. Additionally, single trees, hedges, and grassland field boundaries were present. Counting methods. We visited beech forest, open woodland, and farmland eight times between 29 th March and 17 th July 2006 and the coniferous forest between 4 th April and 25 th June 2008. Survey work was repeated a minimum of one week after the previous visit.

  Comparison of Distance Sampling and Territory MappingGottschalk -12 significantly from those estimated by DS (Wilcoxon Matched Pair Test, n=22, z = 2.68, p = 0.013 and n=22, z = 2.19, p = 0.028).Detection probabilities differed among habitats (one-way ANOVA, F 3,19 = 3.6, p = 0.032) and species (Fig.1). Lowest EDR was generally found in beech forest (mean = 79 m, range = 71 -110 m, n = 7) followed by coniferous forest (mean = 87 m, range = 29 -132 m, n = 10), farmland (mean = 100 m, range = 94 -106 m, n = 3) and open woodland (mean of 137 m, range = 124 -150 m, n = 3). Detection probability decreased from more loud and conspicuous species (e.g. Tree Pipit, EDR = 150 m and Common Chaffinch, EDR = 138 m) to the more elusive species in the study plots (e.g. Goldcrest EDR = 34 m and

  b CF = coniferous forest, BF = beech forest, F = farmland, OW = open woodland. c TM = densities (territories / 10 ha) estimated by territory mapping. d reg. = Number of registrations used to count a territory. e DS = densities (birds / 10 ha) estimated by distance sampling.

i

  EDR = Effective detection radius [m] estimated by DS. Comparison of Distance Sampling and Territory Mapping Gottschalk -24

Fig. 1 :Fig. 2 :

 12 Fig. 1: Detection functions of 11 bird species. The curves show significant differences

  Figures

Table 1 .

 1 Densities estimated using two bird survey methods in four different habitats. Bold type displays those differences when DS generated 494

	495	higher densities
										Δ	Δ	Δ	Δ			No of
	Common name a	Scientific name a	Study site b	TM c 2 reg. d	TM 3 reg.	TM 4 reg.	TM 5 reg. DS e	DS.-TM 2 reg.	DS.-TM 3 reg.	DS.-TM 4 reg.	DS.-TM 5 reg. %CV f 95% CL g	detec-tions h EDR i
	Common Wood Pigeon Columba palumbus	CF	5.8	5.6	3.9	2.9	1.7	-4.1	-3.9	-2.2	-1.2	24.4	0.7-2.8	56
	Common Wood Pigeon Columba palumbus	BF	3.6	3.4	2.6	2.2	1.7	-1.9	-1.7	-0.9	-0.5	38.2	0.9-2.5	36
	Eurasian Skylark	Alauda arvensis	F	7.7	7.7	7.3	6.9	7.3	-0.4	-0.4	0	0.4	26.2 3.3-11.3	149
	Tree Pipit	Anthus trivialis	OW	4.6	4.2	3.0	2.1	1.3	-3.3	-2.9	-1.7	-0.8	27.6	0.6-2.0	46
	Eurasian Blackcap	Sylvia atricapilla	BF	6.2	5.2	3.6	2.4	3.9	-2.3	-1.3	0.3	1.5	27.8	1.8-6.1	36
	Eurasian Blackcap	Sylvia atricapilla	CF	6.0	5.4	3.7	2.7	5.8	-0.2	0.4	2.1	3.1	21.1	2.0-9.6	47
	Common Whitethroat	Sylvia communis	F	2.9	2.7	2.0	1.3	1.4	-1.5	-1.3	-0.6	0.2	34.5	0.7-2.1	20
	Common Chiffchaff	Phylloscopus collybita	CF	2.3	2.1	2.1	1.9	1.6	-0.7	-0.6	-0.6	-0.4	33.9	0.8-2.3	32
	Firecrest		Regulus ignicapillus	CF	22.7	13.5	10.4	8.1	27.6	4.9	14.1	17.2	19.5	20.6 9.4-45.8	34
	Goldcrest	Regulus regulus	CF	12.1	8.9	6.0	4.4	18.2	6.1	9.3	12.2	13.8	29.8 8.3-28.1	31
	Common Blackbird	Turdus merula	CF	8.1	7.9	6.0	4.6	2.1	-6.0	-5.8	-3.9	-2.5	29.1	0.9-3.3	53
	Common Blackbird	Turdus merula	BF	6.4	5.6	4.0	3.0	3.9	-2.6	-1.7	-0.1	0.9	22.9	1.5-6.3	64
	European Robin	Erithacus rubecula	CF	8.1	7.9	6.4	4.2	4.0	-4.1	-3.9	-2.4	-0.2	23.3	2.2-9.6	38
	European Robin	Erithacus rubecula	BF	8.1	7.5	5.2	4.2	5.9	-2.2	-1.6	0.7	1.7	22.4	2.2-9.6	43
	Great Tit		Parus major	OW	4.0	2.7	1.9	1.7	1.2	-2.8	-1.5	-0.7	-0.5	34.2	0.6-1.8	35
	Coal Tit		Parus ater	CF	6.4	5.6	3.3	2.7	2.8	-3.6	-2.8	-0.5	0.1	25.8	1.1-4.5	31
	Winter Wren	Troglodytes troglodytes CF	13.5	12.7	12.1	10.2	5.1	-8.4	-7.6	-7.1	-5.1	21.6	1.9-8.3	91
	Winter Wren	Troglodytes troglodytes BF	4.6	4.2	3.0	2.6	3.7	-0.9	-0.5	0.7	1.1	24.7	1.5-5.9

a The taxonomy followed ITIS (www.itis.gov).

Table 2 :

 2 Time required to survey birds in four different habitats of 25 ha area.

	507
		Average survey time per visit (minutes)		508
		beech	coniferous	open	farmland Mean	509
	survey method	forest	forest	woodland			510
	distance sampling	78	71	68	64	70	511
	territory mapping	138	193	100	119	137	512

Acknowledgments

Our study was funded by the German Science Foundation (DFG) within the "Sonderforschungsbereich 299." We are grateful to all colleagues working in this project for continuous discussion and support. Especially, we would like to thank M. Spiegel for conducting the field work. E. Green, S. Oppel, G. Ritchison, and two anonymous reviewers kindly provided helpful comments on the manuscript. The study complies with the current laws of Germany.


Juan Carlos Blanco

Fernando Ballesteros

Alicia García-Serrano

Juan Herrero
email: herreroj@unizar.es

Carlos Nores
email: cnores@uniovi.es

Guillermo Palomero

Behaviour of brown bears killing wild ungulates in the Cantabrian Mountains, Southwestern Europe

Keywords: Ursus arctos, Brown bear, Predation, Wild ungulates, Cantabrian Mountains, Spain

Although brown bears (Ursus arctos) are known to be major predators of ungulates in North America and Northern Europe, there is little documentation regarding bear predation on wild ungulates in Southern Europe. We describe search, detection, killing and prey consumption behaviour by brown bears during seven attacks on <1-month roe deer, red deer and chamois fawns in spring in the Cantabrian Mountains, north-western Spain. As soon as the bears detected a fawn by their smell or their mother's presence, they switched from routine foraging on plants and insects to an intensive search for the fawns, mainly using smell to comb a 0.5-1 ha area for 15-45 min. They killed the fawns either while the latter were resting or after a brief chase. The bears usually took their prey to dense vegetation, consuming it immediately. In four cases, 5-month-old cubs accompanying the female did not participate in the hunt. We also document the apparently non-predatory killing of a 40-kg wild boar by a female bear with cubs surprised by a sudden encounter. They did not eat the boar after the attack.

Introduction

Brown bears (Ursus arctos) can be important predators on ungulates in North America, where they mainly prey on moose (Alces alces), caribou (Rangifer tarandus) and elk (Cervus elaphus). Bears may also kill adult ungulates, but mainly concentrate on calves in the first 4 weeks after birth in spring and early summer [START_REF] Barber-Meyer | Elk calf survival and mortality following wolf restoration to Yellowstone National Park[END_REF][START_REF] Zager | The role of American black bears and brown bears as predators on ungulates in North America[END_REF]. In a review of black (Ursus americanus) and brown bear predation on ungulates in North America, [START_REF] Zager | The role of American black bears and brown bears as predators on ungulates in North America[END_REF] suggested that predation is additive at low ungulate densities and can become compensatory as prey density approaches carrying capacity. In Sweden, bears killed 26% of moose calves, 92% of them <5 month old. Bear predation was probably additive to other natural mortality [START_REF] Swenson | Predation on moose calves by European brown bears[END_REF].

Apart from Scandinavia and the Russian taiga (literature reviewed by [START_REF] Swenson | Predation on moose calves by European brown bears[END_REF], we know of no study Communicated by C. Gortázar Electronic supplementary material The online version of this article (doi:10.1007/s10344-010-0464-z) contains supplementary material, which is available to authorized users. related to brown bear predation on ungulates in temperate and southern Europe beyond food habits studies based on scat analysis and the finding of a chamois (Rupicapra pyrenaica) killed by a bear in the Apennines [START_REF] Fico | A case of brown bear predation on Abruzzo chamois[END_REF]). In the three major studies on bear feeding habits conducted in the Cantabrian Mountains (Spain), few wild ungulates remains were found in scat. Roe deer only appeared in 1.7 and red deer (Cervus elaphus) in 1.5% of 929 scats analysed by Clevenger et al. (1992a). The authors suppose that most ungulates were taken as carrion because roe deer hardly appeared in scats found in spring, when the females give birth. Roe deer (Capreolus capreolus) was the only species to appear in over ten scats of the 261 analysed by [START_REF] Braña | Hábitos alimenticios y configuración de la dieta del oso pardo en la Cordillera Cantábrica[END_REF]. Finally, out of the 1,500 bear scats analysed by [START_REF] Naves | Brown bear food habits at the border of its range: a long-term study[END_REF], roe deer appeared in 45, chamois in five, red deer in two and wild boar (Sus scrofa) in one. The authors did not ascertain whether the ungulates in the scat were prey or carrion.

In this note, we describe the searching behaviour, detection, killing and consumption of wild ungulate fawns by brown bears in the Cantabrian Mountains, a context with roe and red deer, chamois and wild boar as potential prey. To our knowledge, this is the first time such behaviour has been described in the scientific literature in Europe. Preliminary descriptions by our research team have been published in popular and technical books (e.g. Palomero et al. 2007a).

Study area and methods

In the Cantabrian Mountains (North-western Spain), there are two critically endangered bear subpopulations partially separated by a strip of 50-80 km of low-quality habitat. The western subpopulation (2,800 km 2 ) had 18 and 17 females with cubs of the year in 2007 and 2008, respectively. In the eastern subpopulation (2,100 km 2 ) there were three and two females with cubs of the year in 2007 and 2008 [START_REF] Palomero | Veinte años tras las osas con crías de la cordillera Cantábrica[END_REF]. The bears appear to prefer habitat that is a complex of deciduous forests, shrubs and grasslands at 1,100-1,400 m, i.e. above areas of intense agricultural activity, but below the subalpine zone (see Clevenger et al. 1992b andPalomero et al. 2007b for a detailed habitat description).

We monitored the Cantabrian Mountains bear population from 1988 to June 2010 using the number of females with cubs as an index in order to ascertain population trends (Palomero et al. 2007b) and study their behaviour. The main method for detecting bears (mainly family groups) was to scan with telescopes from vantage points, starting at dawn. Observations were conducted from a distance of 300-1,500 m for the whole time that they were in sight, sometimes during the entire daylight period. No bears were radio collared. All the data we present here were collected by the authors or Fundación Oso Pardo (Brown Bear Foundation) rangers under the authors' supervision. To avoid inaccuracies, accounts from other observers have been excluded.

Results

Between 1988 and June 2010, we observed 141 different bear family groups (females with cubs) over 958 days. The average time per observation day was 119 min (range, <1 min -10 h 20 min; data from 2009,113 observation days). We also observed solitary bears (probably a few hundred) over periods lasting from a few seconds to the entire daylight period. Over the last 23 years, on seven occasions we observed bears predating on fawns in spring (five roe deer, one red deer and one chamois). Furthermore, a female bear with two cubs was seen to kill a wild boar weighing approximately 40 kg. As it did not appear to kill the boar to eat, we did not include the incident as a case of predation.

Bears predating fawns

We observed the first four cases of predation between 10 and 14 June 2000 in Degaña Hunting Reserve (Asturias). Over 5-days monitoring, we saw a female with three cubs of the year hunt and eat small roe deer fawns on 4 days. On 15 June, the female disappeared after a male bear killed the three cubs (Palomero et al. 2007a). In addition, we observed three other solitary bears predating roe deer Attacks and predation on fawns always occurred in spring. In the Cantabrian Mountains, roe deer usually give birth between mid-May and mid-June, whereas red deer and chamois do so in May [START_REF] Blanco | Mamíferos de España, Vol II Cetáceos, Artiodáctilos, Roedores y Lagomorfos[END_REF]. Attacks always occurred a few days or weeks after the fawns' birth (Table 1).

The hunting sequence was very similar in all seven cases. While routinely foraging amongst bracken, ferns and woodland patches, grazing, digging roots and turning over rocks, the adult bears detected the presence of deer and began actively searching for fawns. Prior to predating on the chamois fawn, the bear was foraging in subalpine habitat (treeless, steep mountain slopes of grassland and rocks).

On four occasions, their search seemed to be triggered by the flight of an adult female deer or a herd of chamois. On three occasions, the bears sniffed the air before starting to search, the smell of the fawns or female bears or/and sighting the females likely triggering their behaviour shift from routine foraging on plants and invertebrates (Table 1).

During the search, the bears combed a 0.5-1 ha area, actively zigzagging and often doubling back, sniffing the ground like a bloodhound or raising their head to sniff the air. They also sometimes stood up partially or fully on their hind legs to scan above the line of vegetation. On the five occasions that hunting was timed, the search lasted 45, 45, 40, 15 and 20 min (the last case included a 20-minute pause).

The hunting bears detected the fawn as follows: when the fawn was running about 30 m away (one/seven); when the roe deer female that was resting with the fawn ran out (two/seven); and, apparently, when a fawn was located lying down (four/seven). On one occasion, after being flushed out, the roe deer female attempted a distraction feint, but failed to distract the bear's attention (Table 1).

On one occasion, a female bear pursued a fawn that emerged a few metres away from her, knocking it down approximately 20 m away and then putting it in her mouth. Just prior to capture, the roe deer fawn tried to dodge, but was easily caught. On six other occasions, the bears caught the resting fawn without a chase. On three other occasions, the bear stopped briefly in front of the resting fawn (perhaps to pinpoint it) and then pounced before putting it in its mouth. One parabolic leap resembled that of a red fox (Vulpes vulpes) pouncing on voles (Microtus sp.). In the three remaining cases, the bears took the fawns from the ground in their mouth without any apparent pausing or pouncing. In all but one case, the fawns appeared to die instantly. In one case, the bear was seen shaking the fawn in its mouth.

After catching the fawns, the bears quickly carried them in their mouth several dozen metres away to an area of thick vegetation cover and ate them immediately. Only the bear that captured the chamois fawn in subalpine habitat consumed it in the open. Average prey comsumption time was 35 min (range: 20-53 min; n=5). Rapid removal of the dead fawn may be to prevent possible harassment by wolves (Canis lupus), corvids or raptors (Table 1).

In the four cases that hunting involved a female bear with cubs about 5 months old, the latter tried to follow the female at a short distance, but were sometimes left behind. The cubs never took part in the fawn kill. In three cases, the cubs also ate the dead fawn after it had been moved to cover. In two of those cases they did so in an orderly way, but the other case involved squabbling. During the fourth attack, the cubs were separated from the female during the search and she ate the prey alone.

In five of the seven cases the bears rested after consuming the prey. In the remaining two cases, two solitary bears began a new search for fawns, which ended unsuccessfully 15 and 25 min later.

During the hunts, the bears showed no inclination to attack adult deer, chamois or a second fawn. In both cases that the fawn's mother was near the hunting bear, the latter appeared uninterested in attacking her. In one case, when twin roe deer fawns ran clumsily through the scrub, the female bear killed one and ignored the second. Attack on a wild boar following a sudden encounter Our observation of a female bear killing a 13-month-old wild boar (estimated weight 40 kg) should perhaps not be regarded as a case of predation. On 22 April 2006 at 07.20 in Somiedo Natural Park (Asturias), a female was resting on a rocky slope among ash and walnut trees with her two second-year cubs (estimated weight: 35-40 kg). The juveniles suddenly started running fast towards the trees, and the female followed. The juveniles separated and one encountered a young wild boar. The female positioned herself above the boar on the slope, cutting off its escape and then attacked it with a bite to the back of the head, shaking it and leaving it lifeless on the ground. She immediately left. One juvenile approached the corpse, sniffed it briefly and followed the mother. Monitoring throughout the rest of the day revealed that they did not approach the dead boar again.

The fact that the bears did not eat the boar suggests it was a defensive attack rather than predation. The female probably launched a swift attack to protect the frightened juveniles. This case recalls those described in North America by [START_REF] Herrero | Bear attacks. Their causes and avoidance[END_REF], when female grizzlies with cubs attacked when surprised during sudden encounters with people.

Discussion

Cases of bears hunting neonate deer and chamois in spring or early summer in the Cantabrian Mountains fit with the brown bear predatory behaviour on ungulates described in North America [START_REF] Zager | The role of American black bears and brown bears as predators on ungulates in North America[END_REF] and Scandinavia [START_REF] Swenson | Predation on moose calves by European brown bears[END_REF]. Although there are a large number of observations in North America of bears predating adult ungulates [START_REF] Zager | The role of American black bears and brown bears as predators on ungulates in North America[END_REF], we have never observed a single case. The attack on a 40-kg wild boar was probably a preventive attack to protect the cubs from a perceived threat.

Bears predating on neonate ungulates in the Cantabrian Mountains showed opportunistic behaviour. The search for ungulate neonate was triggered when the bears perceived a good opportunity during their regular foraging on plants and invertebrates. Predation on ungulates is apparently a learned behaviour [START_REF] Zager | The role of American black bears and brown bears as predators on ungulates in North America[END_REF]. Observations of brown bears in Yellowstone National Park [START_REF] French | Predatory behavior of grizzly bears feeding on elk calves in Yellowstone National Park, 1986-88[END_REF] indicate that the feeding habits of individual bears within a population occur along a gradient from highly predatory to little or no predatory activity. The female bear with cubs that we saw hunting roe deer fawns on four days in May 2000 is (apart from the female with cubs that killed the wild boar) the only one observed during 958 days' monitoring (approximately1,900 observation hours) females with cubs from 1988 to June 2010. She was in mixed habitat of broom stands, fern stands and grassland, which is ideal birthing habitat for roe deer females, in the area of the Cantabrian Mountains with the highest density of roe deer (Teresa Corominas, pers. com.). That female could be a specialist hunter of roe deer neonates.

The traditional basic predatory ethogram comprises three phases: search, pursue and capture [START_REF] Macarthur | On optimal use of a patchy environment[END_REF]. The hunting bears we observed spent most of their energy on the search of fawns hidden in vegetation. The pursue phase was present in only one of seven huntswhen a roe deer fawn left its refuge and ran 20 metres before the bear caught it-and lasted a few seconds. Capture involved no great effort in all cases. The extreme vulnerability of neonate ungulates makes the bear predatory behaviour much simpler than that of the ethogram proposed by [START_REF] Macnulty | A proposed ethogram of large-carnivore predatory behavior, exemplified by the wolf[END_REF] for wolves and other large carnivores.

Our data do not allow assessment of the impact of bear predation on ungulate populations. In the Cantabrian Mountains, there are approximate densities of five and one bear/100 km 2 in the western and eastern subpopulations, respectively. Densities of ungulates are very variable, but 1.6-6.6 roe deer, 2.2-6.3 red deer, 5-14 chamois and 1.2-3.0 wild boar/km 2 have been estimated in different parts of the Cantabrian bear range [START_REF] Blanco | Mamíferos de España, Vol II Cetáceos, Artiodáctilos, Roedores y Lagomorfos[END_REF][START_REF] Nores | Estimación de la población de jabalí (Sus scrofa) por recuento de grupos familiares[END_REF][START_REF] Pérez-Barbería | La evolución de la población del rebeco en la Cordillera Cantábrica[END_REF]. Excluding the data on solitary bears, for which monitoring effort has not been quantified, we have only seen one female hunting ungulates out of 141 different family groups observed in 23 years, and 0.4 hunts/100 monitoring days devoted to females with cubs, many in spring. These figures suggest that the impact of bear predation on the ungulate population is probably very small. However, in order to be sure, ungulate mortality would have to be studied, requiring fawns to be tagged [START_REF] Barber-Meyer | Elk calf survival and mortality following wolf restoration to Yellowstone National Park[END_REF][START_REF] Swenson | Predation on moose calves by European brown bears[END_REF]).

  (15 May 2005, Somiedo Nature Park, Asturias), red deer (4 May 2007, Fuentes Carrionas Nature Park, Palencia) and chamois fawns (27 May 2010, Somiedo N. P.).

Table 1

 1 Summary of eight cases of bears killing ungulates

	Bear	Ungulate	Date	Time+	Fawn	Search	Capture	Prey consumption
		killed			detection	time
	Female with 3	Roe deer fawn	10.6.2000	9.37	Scent. Mother	-	Leapt on resting fawn	40 min in thick
	cubs a				present			vegetation
	Female with 3	Roe deer fawn	11.6.2000	13.33	Mother ran	20 min Fleeing fawn caught in	No data
	cubs a						paws after 20-m chase
	Female with 3	Roe deer fawn	12.6.2000	14.30	Mother ran	40 min Unknown	20 min in thick
	cubs a							vegetation
	Female with 3	Roe deer fawn	14.6.2000	19.28	Scent	53 min Pounced on resting fawn 53 min in thick
	cubs a							vegetation
	Solitary bear	Roe deer fawn	31.5.2005	8.00	Unknown	-	Pounced on resting fawn In thick vegetation
	Solitary bear	Red deer fawn	4.5.2007	9.00	Scent	15 min Caught resting fawn in	30 min in thick
							mouth	vegetation
	Solitary bear	Chamois fawn	27.5.2010	7.50	Herd ran.	45 min Caught resting fawn in	30 min on open
					Scent		mouth	mountain slope
	Female with 2	Wild boar	22.4.2006	7.20	Cubs ran	0 min	Boar grasped in paws	Boar not eaten
	cubs b	(13 months old)					and bitten on back of
							the neck
	+ Time hunt started

a Same family group; cubs 5 months old b Cubs 15 months old

Acknowledgements We thank the rangers of the Fundación Oso Pardo (Brown Bear Foundation) for their enthusiastic efforts. The environmental agencies of Galicia, Asturias, Castilla y León and Cantabria, the Spanish Ministry of the Environment, the European Union Life Projects and the Fundación Biodiversidad provided funds for the long-term bear monitoring. T. Corominas (Biodiversity and Landscape Directorate of Asturias) provided unpublished figures on roe deer densities, Dr. Y. Cortés and two anonymous referees reviewed the manuscript and L. Ashcroft improved the English.


Zakharov-Kuznetsov Equation

Francis Ribaud

Stéphane Vento

Well-posedness results for the 3D

Keywords: KdV-like equations, Cauchy problem AMS Classification: 35Q53, 35B65, 35Q60

We prove the local well-posedness of the three-dimensional Zakharov-Kuznetsov equation ∂ t u + ∆∂ x u + u∂ x u = 0 in the Sobolev spaces H s (R 3 ), s > 1, as well as in the Besov space B 1,1 2 (R 3 ). The proof is based on a sharp maximal function estimate in time-weighted spaces.

Introduction and main results

In this paper we consider the local Cauchy problem for the three-dimensional Zakharov-Kuznetsov (ZK) equation

u t + ∆u x + uu x = 0, u(0) = u 0 , (1.1)
where u = u(t, x, y, z), u 0 = u 0 (x, y, z), t ∈ R and (x, y, z) ∈ R 3 . This equation was introduced by Zakharov and Kuznetsov in [START_REF] Zakharov | On three dimensional solitons[END_REF] to describe the propagation of ionic-acoustic waves in magnetized plasma. The formal derivation of (ZK) from the Euler-Poisson system with magnetic field can be found in [START_REF] Linares | The Cauchy problem for the 3D Zakharov-Kuznetsov equation[END_REF].

Clearly, the Zakharov-Kuznetsov equation can be considered as a multidimensional generalization of the well-known one-dimensional Korteweg-de Vries equation

u t + u xxx + uu x = 0, u(0) = u 0 .
(1.2)

We stress out the attention of the reader that contrary to some other generalizations of the Korteweg-de Vries equation (like the Kadomtsev-Petviasvili equations) the (ZK) equation is not completely integrable and possesses only two invariant quantities by the flow. These two invariant quantities are the L 2 (R 3 ) norm

N (t) =
H(t) = 1 2 R 3
(∇u(t, x, y, z)) 2 u(t, x, y, z) 2 3 dxdydz .

Hence, it is a natural issue to study the Cauchy problem for the (ZK) equation in the Sobolev space H1 (R 3 ) since any local well-posedness result in this space would provide global well-posedness 1 .

Another difference with usual generalization of the KdV equation is that the resonant function associated to the (ZK) equation seems too complex to develop a Bourgain approach, see [START_REF] Linares | Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation[END_REF]. Indeed, this function is defined in the hyperplane ξ1 + ξ2 + ξ3 = 0 by

h( ξ1 , ξ2 , ξ3 ) = ξ 1 | ξ1 | 2 + ξ 2 | ξ2 | 2 + ξ 3 | ξ3 | 2 , ξj = (ξ j , η j , µ j ) ∈ R 3 ,
and its zero set is not so easy to understand. This make again a sharp difference with the Kadomtsev-Petviasvili equations where the use of the resonant function allows to derive local well-posedness (for the KP-II equation, see [START_REF] Bourgain | On the Cauchy problem for the Kadomtsev-Petviashvili equation[END_REF]) or local ill-posedness (for the KP-I equation, see [START_REF] Molinet | Well-posedness and illposedness results for the Kadomtsev-Petviashvili-I equation[END_REF]). This explains why we follow the approach of Kenig, Ponce and Vega introduced in [START_REF] Kenig | Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle[END_REF] for the study of the generalized KdV equation rather than a Bourgain approach.

In the the two-dimensional case, Faminskii [START_REF] Faminskii | The Cauchy problem for the Zakharov-Kuznetsov equation[END_REF] proved the local and global well-posedness of (ZK) for initial data in H 1 (R 2 ). This result was recently improved by Linares and Pastor who obtained in [START_REF] Linares | Well-posedness for the two-dimensional modified Zakharov-Kuznetsov equation[END_REF] the local well-posedness in H s (R 2 ), s > 3/4. The main tool to derive those results is the following L 2

x L ∞ yT linear estimate [START_REF] Faminskii | The Cauchy problem for the Zakharov-Kuznetsov equation[END_REF],

∀s > 3/4, U (t)ϕ L 2 x L ∞ yT ≤ C ϕ H s (R 2 )
, where U (t)ϕ denotes the free operator associated to the linear part of the (ZK) equation. Note also that in [START_REF] Linares | Well-posedness for the ZK equation in a cylinder and on the background of a KdV soliton[END_REF], Linares, Pastor and Saut recently obtained local well-posedness results in some spaces which contains the onedimensional solitary-waves of (ZK) as well as perturbations.

In the three-dimensional case, as far as we know, the only available result concerning the local well-posedness of (ZK) in the usual Sobolev spaces goes back to Linares and Saut who proved in [START_REF] Linares | The Cauchy problem for the 3D Zakharov-Kuznetsov equation[END_REF] the local well posedness in H s (R 3 ), s > 9/8. In this paper we prove the following result, Theorem 1.1. For any s > 1 and u 0 ∈ H s (R 3 ), there exist T > 0 and a unique solution u of (1.1) in

X s T ∩ C b ([0, T ], H s (R 3 )).
Moreover, the flow-map u 0 → u is Lipschitz on every bounded set of H s (R 3 ).

To derive our result, the main issue is to prove the local in time linear estimate

∀T < 1, ∀s > 1, U (t)ϕ L 2 x L ∞ yzT ≤ C ϕ H s (R 3 ) , (1.3)
where U (t)ϕ denotes the free linear operator associated to the (ZK) equation. Moreover, having a short look on the proof of Theorem 1.1, we note that any improvement of the linear estimate (1.3) will immediately give the corresponding improvement for Theorem 1.1. Unfortunately we prove in Section 3 that the L 2 x L ∞ yzT linear estimate (1.3) fails when s < 1. Hence, this seems to indicate that the case s = 1 could be critical for the wellposedness of the (ZK) equation.

Concerning the "critical case" s = 1, we have unfortunately not been able to prove the local well-posedness in the natural energy space H 1 (R 3 ). Nonetheless, working with initial data in the Besov space B 1,1 2 (R 3 ), we have the following Theorem 1.2. For any u 0 ∈ B 1,1 2 (R 3 ), there exist T > 0 and a unique solution u of (1.1) in

X T ∩ C b ([0, T ], B 1,1 2 (R 3 )).
Moreover, the flow-map u 0 → u is Lipschitz on every bounded set of B 1,1 2 (R 3 ).

This result clearly improves Theorem 1.1 in view of the well-known embeddings

∀s > 1, H s (R 3 ) ֒→ B 1,1 2 (R 3 ) ֒→ H 1 (R 3 ).
To get Theorem 1.2, our main ingredient is to prove the unusual weightedin-time linear estimate for phase localized functions

∀α ≥ 3/8, t α ∆ k U (t)ϕ L 2 x L ∞ yzT 2 k ∆ k ϕ L 2 .
This estimate, combined with the standard Kato smoothing estimate (3.2), allows us to perform a fixed point argument on the Duhamel formulation of (ZK). This paper is organized as follows. In Section 2 we introduce our notations and define the resolution spaces. Section 3 is devoted to estimates related to the linear part of the equation. Finally we prove the key bilinear estimates in Section 4.

Notation

For A, B > 0, A B means that there exists c > 0 such that A ≤ cB. When c is a small constant we use A ≪ B. We write A ∼ B to denote the statement that A B A. For u = u(t, x, y, z) ∈ S ′ (R 4 ), we denote by u (or Fu) its Fourier transform in space. The Fourier variables corresponding to a vector x = (x, y, z) will be denoted by ξ = (ξ, η, µ). We consider the usual Lebesgue spaces L p , 1 ≤ p ≤ ∞ and given a Banach space X and a measurable function u : R → X, we define u L p X = u(t) X L p . For T > 0, we also set L p T = L p ([0, T ]). Let us define the Japanese bracket x = (1 + |x| 2 ) 1/2 so that the standard non-homogeneous Sobolev spaces are endowed with the norm f

H s = ∇ s f L 2 .
We use a Littlewood-Paley analysis. Let p ∈ C ∞ 0 (R d ) be such that p ≥ 0, supp p ⊂ B(0, 2), p ≡ 1 on B(0, 1). We define next p k ( ξ) = p( ξ/2 k ) for k ≥ 0. We set δ( ξ) = p( ξ/2) -p( ξ) and δ k ( ξ) = δ( ξ/2 k ) for any k ∈ Z, and define the operators P k (k ≥ 0) and ∆ k (k ∈ Z) by F(P k u) = p k u and F(∆ k ) = δ k u. When d = 3, we introduce the operators P x k , P y k , P z k , and

∆ x k , ∆ y k , ∆ z k defined by    P x k u(x, y, z) = F -1 (p k (ξ) u(ξ, η, µ)), P y k u(x, y, z) = F -1 (p k (η) u(ξ, η, µ)), P z k u(x, y, z) = F -1 (p k (µ) u(ξ, η, µ)) and    ∆ x k u(x, y, z) = F -1 (δ k (ξ) u(ξ, η, µ)), ∆ y k u(x, y, z) = F -1 (δ k (η) u(ξ, η, µ)), ∆ z k u(x, y, z) = F -1 (δ k (µ) u(ξ, η, µ))
Furthermore we define more general projections P k = j:2 j 2 k P j , ∆ x ≫j = j:2 j ≫2k ∆ x j etc. For future considerations, note that for u ∈ S ′ (R 3 ) and

p ∈ [1, ∞] we have ∆ k u L p ∆ x k P y k P z k u L p + P x k ∆ y k P z k u L p + P x k P y k ∆ z k u L p . (2.1)
With these notations, it is well known that an equivalent norm on H s (R d ) is given by

u H s ∼ P 0 u L 2 +   k≥0 2 2sk ∆ k u 2 L 2   1/2 . For s ∈ R, the Besov space B s,1 2 (R d ) denotes the completion of S(R d ) with respect to the norm u B s,1 2 = P 0 u L 2 + k≥0 2 sk ∆ k u L 2 .

Linear estimates

Consider the linear ZK equation

u t + ∆u x = 0, t ∈ R, x ∈ R 3 . (3.1)
Let ω( ξ) = ξ(ξ 2 + η 2 + µ 2 ) and U (t) = F -1 e itω( ξ) F, be the associated linear operator.

First we prove a standard "Kato smoothing" estimate for the free evolution of (3.1).

Proposition 3.1. For any ϕ ∈ L 2 (R 3 ), it holds that ∇U (t)ϕ L ∞ x L 2 yzt ϕ L 2 . (3.2)
Proof. The proof is modeled on the corresponding result for the KdV equation [START_REF] Kenig | Oscillatory integrals and regularity of dispersive equations[END_REF] (see also [START_REF] Kenig | Well-posedness of the initial value problem for the Korteweg-de Vries equation[END_REF] and [START_REF] Faminskii | The Cauchy problem for the Zakharov-Kuznetsov equation[END_REF] for the two-dimensional case). We perform the change of variables θ = h(ξ) = ω( ξ) and obtain

U (t)ϕ(x) = R 3 e i(x ξ+tω( ξ) ϕ( ξ)d ξ = F -1 θηµ e ixh(θ) (h -1 ) ′ (θ) ϕ(h(θ), η, µ) (t, y, z).
Therefore, applying Plancherel theorem and returning to the ξ-variable yield

U (t)ϕ(x) L 2 yzt = (h -1 ) ′ (θ) ϕ(h(θ), η, µ) L 2 θηµ = |h ′ (ξ)| -1/2 ϕ(ξ, η, µ) L 2 ξηµ ∼ ∇ -1 ϕ L 2 .
We will use in a crucial way the following maximal estimate for the free evolution when acting on phase localized functions. Proposition 3.2. Let 0 < T < 1 and α ≥ 3/8.

1. For all ϕ ∈ S(R 3 ) and k ≥ 0, we have

t α ∆ k U (t)ϕ L 2 x L ∞ yzT 2 k ∆ k ϕ L 2 . (3.3) 2.
For any ϕ ∈ S(R 3 ), it holds that

P 0 U (t)ϕ L 2 x L ∞ yzT P 0 ϕ L 2 . (3.4)
Before proving Proposition 3.2, we first show some estimates related to the oscillatory integrals:

I 0 (t, x) = R 3 e i(x ξ+tω( ξ)) p 0 ( ξ)d ξ,
and for k ≥ 1,

I k (t, x) = R 3 e i(x ξ+tω( ξ)) ψ 1 (ξ)ψ 2 (η)ψ 3 (µ)d ξ where (ψ 1 , ψ 2 , ψ 3 ) = (δ k , p k , p k ), (p k , δ k , p k ) or (p k , p k , δ k ). Lemma 3.1. I 0 L 1 x L ∞ yzT 1. Proof. Since |I 0 | 1, it is clear that I 0 L 1 x L ∞ yzT 1 if
we are in the region |x| 1. Thus we may assume that |x| ≫ 1. Define the phase function

ϕ 1 by ϕ 1 (ξ) = tω( ξ) + xξ so that ϕ ′ 1 (ξ) = t(3ξ 2 + η 2 + µ 2 ) + x.
On the support of p 0 , we have |ϕ ′ 1 | |x|. Thus, two integrations by parts yield the estimate

R e φ 1 (ξ) p 0 ( ξ)dξ R p 0ξξ ϕ ′′ 1 + p 0ξ ϕ ′′2 1 ϕ ′3 1 + p 0 ϕ ′′′ 1 ϕ ′3 1 + p 0 ϕ ′′2 1 ϕ ′4 1 |x| -2 . It follows that |I 0 | |x| -2 , which implies that I 0 L 1 x L ∞ yzT 1 as required.
Lemma 3.2. For any α ≥ 3/8 and k ≥ 0, it holds that

t 2α I k L 1 x L ∞ yzT 2 2k .
Proof. We split I k into

I k = R 3 e i(x ξ+tω( ξ)) ψ 1 (ξ)(1 -p 0 (ξ))ψ 2 (η)ψ 3 (µ)d ξ + R 3 e i(x ξ+tω( ξ)) p 0 (ξ)ψ 2 (η)ψ 3 (µ)d ξ := I 1 k + I 2 k . ( 3
(ξ) = tω( ξ) + xξ. (3.6)
Integrating by parts twice with respect to ξ we deduce

R e iϕ 1 ψ 1 (1 -p 0 ) max(|x|, t2 2k ) -2 . It follows that |I 1 k | 2 2k max(|x|, t2 2k ) -2 and next t 1/2 |I 1 k | |x| -3/2 2 k ,
which is acceptable since we integrate in the region |x| ≥ 2 -k . Now we consider the case |x| ∼ t2 2k . Using that

R 2 e itξ(η 2 +µ 2 )+iyη+izµ dηdµ = π t|ξ| e -i y 2 +z 2 4tξ e i π 2 sgn(ξ) ,
we may rewrite I 1 k as

I 1 k = R 2 ψ2 (y -u) ψ3 (z -v) R πi tξ e iϕ 2 (ξ) ψ 1 (ξ)(1 -p 0 (ξ))dξ dudv (3.7) where we set ϕ 2 (ξ) = tξ 3 + xξ -u 2 +v 2 4tξ . Since |ϕ ′′′ 2 | ≥ 6t on the support of 1 -p 0 , Van der Corput's lemma implies that R πi tξ e iϕ 2 (ξ) ψ 1 (ξ)(1 -p 0 (ξ))dξ t -4/3 . It follows that |I 1 k | t -4/3 and t 2α |I 1 k | t 2α-4/3 |x| 2α-4/3 2 -2k(2α-4/3) .
Hence we obtain for α > 1/6 and |x| ≥ 2 -k that

t 2α I 1 k L 1 x L ∞ yzT 2 2k(2α-1/3) 2 -2k(2α-4/3) ∼ 2 2k .
•

I 2 k L 1 x L ∞ yzT 2 2k .
Now we consider the most delicate case |x| ∼ t2 2k and rewrite I 2 k as in (3.7) where ψ 1 (1 -p 0 ) is replaced with p 0 . Let us split p 0 as

p 0 = p -2k + 0 j=-2k δ j . The part p -2k is straightforward since |I 2 k | 1 and we get from |x| ∼ t2 2k 2 2k that I 2 k L 1 x L ∞ yzT 2 2k
. Thus we reduce to estimate

I 2 k,j = R 2 ψ2 (y -u) ψ3 (z -v) R πi tξ e iϕ 2 (ξ) δ j (ξ)dξ dudv for j = -2k, . . . , 0. First consider the case |x| ≪ t -1 2 -2j (u 2 + v 2 ) or |x| ≫ t -1 2 -2j (u 2 + v 2 ). Since ϕ ′ 2 (ξ) = 3tξ 2 + x + u 2 +v 2 4tξ 2 , we have |ϕ ′ 2 | max(|x|, t -1 2 -2j (u 2 + v 2 )
) and an application of the Van der Corput's lemma yields

|I 2 k,j | R 2 | ψ2 (y -u) ψ3 (z -v)|t -1 |x| -3/4 (t -1 2 -2j (u 2 + v 2 )) -1/4 2 -j dudv |x| -3/4 t -3/4 2 -j/2 R | ψ2 (y -u)| |u| 1/2 du.
On the other hand, the change of variables v = 2 k u leads to

R | ψ2 (y -u)| |u| 1/2 du = 2 k R | ψ(2 k y -2 k u)| |u| 1/2 du = 2 k/2 R | ψ(2 k y -v)| |v| 1/2 dv 2 k/2 |v|≤1 dv |v| 1/2 + 2 k/2 |v|≥1 | ψ(2 k y -v)|dv 2 k/2 . Consequently, it is deduced that t 3/4 |I 2 k,j | |x| -3/4 2 -j/2 2 k/2 and t 3/4 I 2 k L 1 x L ∞ yzT 0 j=-2k 2 -j/2 2 k/2 |x| 2 2k dx |x| 3/4 2 2k .
Finally assume that |x| ∼ t -1 2 -2j (u 2 +v 2 ) so that |ϕ ′′ 2 | t2 2k-j . Then we get from Van der Corput's lemma that

|I 2 k,j | (t2 2k-j ) -1/2 t -1 2 -j ∼ t -3/2 2 -k 2 -j/2 .
Hence, we obtain

t 2α |I 2 k | t 2α-3/2 2 -k 0 j=-2k 2 j/2 |x| 2α-3/2 2 -2k(2α-3/2) ,
which is acceptable as soon as α > 1/4. This concludes the proof of Lemma 3.2.

We are now in a position to prove Proposition 3.2.

Proof of Proposition 3.2. Let k ≥ 0 and U k (t) = ∆ k U (t). The proof will follow from a slight modification of the usual AA * argument. Let us define the operator

A k : L 1 T L 2 x → L 2 by A k g = T 0 t α U k (-t)g(t)dt.
We easily check that

A * k h(t) = t α U k (t)h for h ∈ L 2 (R 3
) and moreover,

A * k A k g(t) = T 0 (tt ′ ) α U k (t -t ′ )g(t ′ )dt ′ .
The previous integrand can be estimated by

|(tt ′ ) α U k (t -t ′ )g(t ′ , x)| |t -t ′ | 2α U k (t -t ′ )g(t ′ , x) + |t + t ′ | 2α U k (t -t ′ )g(t ′ , x) := I + II.
Using the Young inequality, the first term is bounded by

I |t -t ′ | 2α I k (t -t ′ ) * x g(t ′ ) (x) (|t -t ′ | 2α I k (t -t ′ ) L ∞ yz ) * x g(t ′ ) L 1 yz (x).
Integrating this into t ′ ∈ [0, T ] and taking the L 2 x L ∞ yzT norm, this leads to

T 0 |t -t ′ | 2α U k (t -t ′ )g(t ′ , x) dt ′ L 2 x L ∞ yzT t 2α I k L ∞ yzT * x g(t) L 1 yzT L 2 x t 2α I k L 1 x L ∞ yzT g L 2 x L 1 yzT . (3.8)
To estimate II, we introduce ǧ(t, x) = g(t, -x) and notice that

U k (t -t ′ )g(t ′ , x) = U k (t + t ′ )ǧ(t ′ , -x).
We infer that

II (|t + t ′ | 2α ) I k (t + t ′ ) L ∞ yz ) * x ǧ(t ′ ) L 1 yz (x)
and we get

T 0 |t + t ′ | 2α U k (t -t ′ )g(t ′ , x)dt ′ L 2 x L ∞ yzT t 2α I k L ∞ yzT * x ǧ L 1 yzT L 2 x t 2α I k L 1 x L ∞ yzT g L 2 x L ∞ yzT . (3.9)
Combining estimates (3.8)-(3.9) and Lemma 3.2 we deduce

A * k A k g L 2 x L ∞ yzT t 2α I k L 1 x L ∞ yzT g L 2 x L yzT 2 2k g L 2 x L ∞ yzT .
The usual algebraic lemma (see Lemma 2.1 in [START_REF] Ginibre | Smoothing properties and retarded estimates for some dispersive evolution equations[END_REF]) applies and yields the first estimate in Proposition 3.2. The second one is obtained by following the same lines and using Lemma 3.1.

In order to get the desired bounds for data in H s (R 3 ), s > 1, we will use the following estimate. Proposition 3.3. For 0 < T < 1, s > 1 and any ϕ ∈ S(R 3 ), it holds that

U (t)ϕ L 2 x L ∞ yzT ϕ H s . (3.10)
This proposition is a direct consequence of Lemma 3.1 together with the following result. Lemma 3.3. For any ε > 0 and k ≥ 0, it holds that

I k L 1 x L ∞ yzT 2 (2+ε)k . Proof. Setting I i,j,k (t, x) = R 3 e i(x ξ+tω( ξ)) δ i (ξ)δ j (η)δ k (µ)d ξ,
we see that from the estimate

∆ n u L 1 x L ∞ yzT i,j,k:2 i ,2 j ,2 k 2 n ∆ x i ∆ y j ∆ z k u L 1 x L ∞ yzT ,
it suffices to show that

I i,j,k L 1 x L ∞ yzT (1 + M )2 2M (3.11)
for all i, j, k ∈ Z and with M = max(i, j, k) ≥ 0. From the straightforward bound |I i,j,k | 2 i+j+k , we see that we may assume |x| ≥ 2 -m where m = min(i, j, k). In the region |x| ≪ t2 2M or |x| ≫ t2 2M , the phase function ϕ 1 defined in (3.6) satisfies |ϕ ′ 1 | max(|x|, t2 2M ) and integrations by parts lead to

R e iϕ 1 δ i 2 -i |x| -2 .
Therefore we get |I i,j,k | 2 j+k-i |x| -2 and then

I i,j,k L 1 x L ∞ yzT 2 j+k-i+m 2 2M
. Finally consider the case |x| ∼ t2 2M . Rewriting I i,k,j as

I i,j,k = R 2 δj (y -u) δk (z -v) R πi tξ e iϕ 2 (ξ) δ i (ξ)dξ dudv, we immediately obtain |I i,j,k | t -1 2 2M |x| -1 and thus I i,j,k L 1 x L ∞ yzT M 2 2M , as desired.
Up to the end point s = 1, we show next that estimate (3.10) is sharp.

Proposition 3.4. Suppose that for any ϕ ∈ H s (R 3 ) we have

t α U (t)ϕ L 2 x L ∞ yzT ϕ H s , (3.12)
for some α ≥ 0. Then it must be the case that s ≥ 1.

Proof. Let us define the smooth functions ϕ k through their Fourier transforms by

φ( ξ) = δ -2k (ξ)δ k (η)δ k (µ)
for k ≥ 0. Then it is easy to see that

ϕ k H s ∼ 2 ks . (3.13)
On the other hand, for ε > 0 small enough, we set t = ε and y = z = ε2 -k so that |yη

+ zµ + tω( ξ)| ε. Choosing |x| ≪ 2 2k
, we obtain the lower bound

|U (t)ϕ k (x)| = R 3 e i(x ξ+tω( ξ)) ϕ k ( ξ)d ξ = R 3 [e i(yη+zµ+tω( ξ)) -1]e ixξ δ -2k (ξ)δ k (η)δ k (µ)d ξ + R 3 e ixξ δ -2k (ξ)δ k (η)δ k (µ)d ξ 1. It follows that t α U (t)ϕ k L 2 x L ∞ yzT 2 k
where the implicit constant does not depend on k. Therefore, (3.12) and (3.13) imply 2 k 2 ks .

From this, we get for large k that s ≥ 1.

In the sequel of this section we prove retarded linear estimates which will be used later to perform the fixed point argument. Proposition 3.5. Let f ∈ S(R 4 ). Then we have

∇ t 0 U (t -t ′ )f (t ′ )dt ′ L ∞ T L 2 x f L 1 x L 2 yzT . (3.14) Proof. The dual estimate of (3.2) reads ∞ -∞ U (-t ′ )∇f (t ′ )dt ′ L 2 f L 1 x L 2 yzT . (3.15)
Noticing that U (t) is a unitary group on L 2 (R 3 ) we obtain for any fixed t,

∞ -∞ U (t -t ′ )∇f (t ′ )dt ′ L 2 x f L 1 x L 2 yzT . (3.16)
To conclude we substitute in (3.16) f (t ′ ) by χ [0,t] (t ′ )f (t ′ ) and then take the supremum in time in the left-hand side of the resulting inequality.

Proposition 3.6. Let f ∈ S(R 4 ). Then we have

∇ 2 t 0 U (t -t ′ )f (t ′ )dt ′ L ∞ x L 2 yzT f L 1 x L 2 yzT . (3.17)
Proof. We first observe that for any g : R → R,

t 0 g(t ′ )dt ′ = 1 2 R g(t ′ ) sgn(t -t ′ )dt ′ + 1 2 R g(t ′ ) sgn(t ′ )dt ′ ,
where sgn(•) denotes the sign function. Consequently we have

∇ 2 R(t, x) = ∇ 2 t 0 U (t -t ′ )f (t ′ )dt ′ = 1 2 ∇ 2 R U (t -t ′ )f (t ′ ) sgn(t -t ′ )dt ′ + 1 2 ∇ 2 R U (t -t ′ )f (t ′ ) sgn(t ′ )dt ′ = 1 2 ∇ 2 R 1 (t, x) + 1 2 ∇ 2 R 2 (t, x). (3.18)
Taking the inverse space-time Fourier transform, it is clear that

R 1 = F -1 τ ξ sgn(τ -ω( ξ)) f (τ, ξ) .
Hence we get by Plancherel theorem

∇ 2 R 1 L 2 yzt = F -1 ξ | ξ| 2 sgn(τ -ω( ξ)) f (τ, ξ) L 2 ηµτ = K(τ, x, η, µ) * x F yzt (f (x))(η, µ, τ ) L 2 ηµτ (3.19)
where K is the inverse Fourier transform (in ξ) of the tempered distribution defined as the principal value of | ξ| 2 /(τ -ω( ξ)). It follows that

K(τ, x, η, µ) = R e ixξ | ξ| 2 τ -ω( ξ) dξ = R e ix(η 2 +µ 2 ) 1/2 ξ (η 2 + µ 2 ) 3/2 (ξ 2 + 1) τ -(η 2 + µ 2 ) 3/2 ξ(ξ 2 + 1) dξ = R e iyξ ξ 2 + 1 c -ξ(ξ 2 + 1) dξ with y = (η 2 + µ 2 ) 1/2
x and c = τ /(η 2 + µ 2 ) 3/2 . Next, a partial fraction expansion leads to

ξ 2 + 1 c -ξ(ξ 2 + 1) = - α 2 + 1 (3α 2 + 1)(ξ -α) - α 3α 2 + 1 2αξ + α 2 -1 ξ 2 + αξ + α 2 + 1
where α is the unique real root of c -X(X 2 + 1). Therefore, we get

|K(τ, x, η, µ)| α 2 + 1 3α 2 + 1 R e iyξ ξ -α dξ + 2α 2 3α 2 + 1 R e iyξ ξ ξ 2 + αξ + α 2 + 1 dξ + |α(α 2 -1)| 3α 2 + 1 R e iyξ ξ 2 + αξ + α 2 + 1 dξ = K 1 + K 2 + K 3 .
The first term K 1 is bounded by an integral witch is the Fourier transform of a function that behaves near the singular points like the kernel of the Hilbert transform 1/ξ (or its translates) whose Fourier transform is sgn(x). It follows that K 1 is bounded uniformly in α and y. In the same way,

K 2 ∈ L ∞ since K 2 R e iyξ ξ ξ + α 2 2 + 3α 2 4 + 1 dξ 1 3α 2 + 4 R e iyξ ξ ξ √ 3α 2 +4 2 + 1 dξ R e izξ ξ ξ 2 + 1 dξ
where z = y √ 3α 2 + 4. Concerning K 3 , it is easily estimated by

K 3 |α(α 2 -1)| 3α 3 + 1 R dξ ξ 2 + 3α 2 4 + 1 |α(α 2 -1)| √ 3α 2 + 4 (3α 2 + 1)(3α 2 + 4)
1.

We deduce that K ∈ L ∞ (R 4 ). Hence, (3.19) combining with the Young inequality yields

∇ 2 R 1 L ∞ x L 2 yzt K L ∞ F yzt (f (x)) L 1 x L 2 ηµτ f L 1 x L 2 yzt . (3.20)
To treat the contribution of R 2 , we use the smoothing bound (3.2) and its dual estimate (3.15) to get

∇ 2 R 2 L ∞ x L 2 yzt ∇ R U (-t ′ )f (t ′ ) sgn(t ′ )dt ′ L 2 f L 1 x L 2 yzt . ( 3
U (t -t ′ )P 0 f (t ′ )dt ′ L 2 x L ∞ yzT P 0 f L 1 x L 2 yzT , (3.22)
t α t 0 U (t -t ′ )∆ k f (t ′ )dt ′ L 2 x L ∞ yzT ∆ k f L 1 x L 2 yzT . (3.23)
Proof. Combining estimates (3.3) and (3.15) we obtain the non-retarded version of (3.23):

t α T 0 U (t -t ′ )∆ k f (t ′ )dt ′ L 2 x L ∞ yzT ∆ k f L 1 x L 2 yzT . (3.24)
We consider the function H k (t, x) = t α t 0 U (t -t ′ )∆ k f (t ′ )dt ′ that we may always assume to be continuous on [0, T ] × R 3 . From (3.24) it follows that for all measurable function t : R 3 → [0, T ],

t(x) α T 0 U (t(x) -t ′ )∆ k f (t ′ , x)dt ′ L 2 x L ∞ yz ∆ k f L 1 x L 2 yzT .
Replacing now f (t ′ , x) by f (t ′ , x)χ [0,t(x)] (t ′ ) we see that

H k (t(x), x) L 2 x L ∞ yz ∆ k f L 1 x L 2 yzT . (3.25) Since the map t → H k (t, x) is continuous on the compact set [0, T ], there exists α ∈ [0, T ] such that H k (α, x) = sup t∈[0,T ] H k (t, x), therefore the map x → t 0 (x) = inf α ∈ [0, T ] : H k (α, x) = sup t∈[0,T ] H k (t, x)
is well-defined and measurable on R 3 . Hence, choosing t 0 in (3.25) we infer that

H k (t 0 (x), x) L 2 x L ∞ yz ∆ k f L 1 x L 2 yzT , which yields sup t∈[0,T ] H k (t, x) L 2 x L ∞ yz ∆ k f L 1 x L 2 yzT
and ends the proof of (3.23). The proof of (3.22) is similar and therefore will be omitted.

4 Proofs of Theorems (1.2) and (1.1)

In this section we solve (1.1) in the spaces B 1,1 2 (R 3 ) and H s (R 3 ), s > 1. We consider the associated integral equation

u(t) = U (t)u 0 - 1 2 t 0 U (t -t ′ )∂ x (u 2 )(t ′ )dt ′ . (4.1) 4.1 Well-posedness in B 1,1 2 (R 3 ) For u 0 ∈ B 1,1 2 (R 3
), we look for a solution in the space

X T = {u ∈ C b ([0, T ], B 1,1 2 ) : u X T < ∞}
for some T > 0 and where

u X T = N (u) + T (u) + M (u) with N (u) = P 0 u L ∞ T L 2 x + k≥0 2 k ∆ k u L ∞ T L 2 x , T (u) = P 0 u L ∞ x L 2 yzT + k≥0 2 2k ∆ k u L ∞ x L 2 yzT , M (u) = P 0 u L 2 x L ∞ yzT + k≥0 t α ∆ k u L 2 x L ∞ yzT ,
and 3/8 ≤ α < 1/2. First from Propositions 3.1, 3.2 together with the obvious bound

U (t)u 0 L ∞ T L 2 x u 0 L 2 , (4.2)
we get the following linear estimate:

U (t)u 0 X T u 0 B 1,1 2 . ( 4.3)
Now we need to estimate

t 0 U (t -t ′ )∂ x (u 2 )(t ′ )dt ′ X T in terms of the X T -norm of u. Using standard paraproduct rearrangements, we can rewrite ∆ k (u 2 ) as ∆ k (u 2 ) = ∆ k lim j→∞ P j (u) 2 = ∆ k   P 0 (u) 2 + j≥0 (P j+1 (u) 2 -P j (u) 2 )   = ∆ k   P 0 (u) 2 + j k ∆ j+1 u(P j+1 u + P j u)   .
On the other hand, by similar considerations, we see that P 0 (u 2 ) can be rewritten as

P 0 (u 2 ) = P 0 [P 0 (u) 2 ] + P 0   j≥0 ∆ j+1 u(P j+1 u + P j u)   .
Hence, without loss of generality, we can restrict us to consider only terms of the form:

A = P 0 [P 0 (u) 2 ], B = ∆ k   j k ∆ j uP j u   , C = P 0   j≥0 ∆ j uP j u  
for k ≥ 0, since the estimates for the other terms would be similar. By virtue of (3.14)-(3.17)-(3.22) and (3.23), we infer that

t 0 U (t -t ′ )∂ x (u 2 )(t ′ )dt ′ X T P 0 (u) 2 L 1 x L 2 yzT + k≥0 2 k   j k ∆ j uP j u L 1 x L 2 yzT   + j≥0 ∆ j uP j u L 1 x L 2 yzT . (4.4)
The first term in the right hand side is bounded by

P 0 (u) 2 L 1 x L 2 yzT P 0 u L 2 P 0 u L 2 x L ∞ yzT T 1/2 P 0 u L ∞ T L 2 x M (u) T 1/2 u 2 X T . (4.5)
To evaluate the contribution of B, note that

∆ j uP j u L 1 x L 2 yzT = (t -α ∆ j u)(t α P j u) L 1 x L 2 yzT t -α ∆ j u L 2 t α P j u L 2 x L ∞ yzT T µ ∆ j u L ∞ T L 2 x t α P j u L 2 x L ∞ yzT (4.6)
where µ = 1/2 -α > 0. Therefore, since

t α P j u L 2 x L ∞ yzT P 0 u L 2 x L ∞ yzT + j k=0 t α ∆ k u L 2 x L ∞ yzT M (u), (4.7)
we deduce from the discrete Young inequality that

k≥0 2 k   j k ∆ j uP j u L 1 x L 2 yzT   T µ M (u) k≥0   j k 2 k-j (2 j ∆ j u L 2 x L ∞ yzT )   T µ u X T k≥0 2 k ∆ k u L 2 x L ∞ yzT T µ u 2 X T . (4.8)
Using again estimates (4.6)-(4.7), we easily get that the last term in the r.h.s. of (4.4) can be estimated by j≥0

∆ j uP j u L 1 x L 2 yzT T µ M (u) j≥0 ∆ j u L ∞ T L 2 x T µ u 2 X T . (4.9)
Hence, gathering (4.3)-(4.4)-(4.5)-(4.8) and (4.9), we infer that

Gu X T u 0 B 1,1 2 + T µ u 2 X T ,
and in the same way,

Gu -Gv X T ( u X T + v X T ) u -v X T .
Hence for T > 0 small enough, G is a strict contraction in some ball of X T . Theorem 1.2 follows then from standard arguments.

4.2 Well-posedness in H s (R 3 ), s > 1

Let u 0 ∈ H s (R 3 ) with s > 1. For T > 0, we introduce the space X s T = {u ∈ C b ([0, T ], H s ) : u X s T < ∞} where u X s T = N (u) + T (u) + M (u) with

N (u) = P 0 u L ∞ T L 2 x +   k≥0 4 sk ∆ k u 2 L ∞ T L 2 x   1/2 , T (u) = P 0 u L ∞ x L 2 yzT +   k≥0 4 (s+1)k ∆ k u 2 L ∞ x L 2 yzT   1/2 , M (u) = P 0 u L 2 x L ∞ yzT +   k≥0 4 (s-1-ε)k ∆ k u 2 L 2 x L ∞ yzT   1/2
, for some ε > 0 small enough. From Proposition 3.1 together with (3.10) and (4. The estimates for these terms follow the same lines than the B 1,1 2 case, except that we use the Young inequality for ℓ 1 ⋆ ℓ 2 , as well as the bound

P j u L 2 x L ∞ yzT P 0 u L 2 x L ∞ yzT + j k=0 ∆ k u L 2 x L ∞ yzT M (u) +   k≥0 4 (1-s+ε)k   1/2 M (u) M (u),

R 3 u 2

 32 (t, x, y, z) dxdydz , and the Hamiltonian

  .21) Estimates (3.20)-(3.21) together with (3.18) yield the desired bound. Proposition 3.7. Let T ≤ 1, k ≥ 0 and α ≥ 3/8. Then for any f ∈ S(R 4 ), t 0

0 U∆ j uP j u L 1 x L 2 yzT∆

 02 2), U (t)u 0 X s T u 0 H s .Following the arguments given in Subsection 4.1, it is not too hard to see thatt (t -t ′ )∂ x (u 2 )(t ′ )dt ′ j uP j u L 1 x L 2 yzT .

  .5) • Estimate for I 1 k . Since we have |ξ| 1, a rough estimate for I 1 k yields |I 1 k | 2 3k , which gives the desired bound in the region where |x| ≤ 2 -k . Therefore we may assume |x| ≥ 2 -k . If we have either |x| ≪ t2 2k or |x| ≫ t2 2k , then using that |ω ′ | ∼ 2 2k we infer |ϕ ′ 1 | max(|x|, t2 2k ) where ϕ 1 is the phase function ϕ 1

  Estimate for I 2 k . We treat now the low frequencies term I 2 k . The case |x| 1 is easily handled since we have the rough estimate |I 2 k | 2 2k . Thus we only need to consider the region where |x| ≫ 1. In the domain |x| ≪ t2 2k or |x| ≫ t2 2k , we have |ϕ ′ 1 | max(|x|, t2 2k ) where ϕ 1 is defined in (3.6), and thus

R e iϕ 1 p 0 |x| -2 . It follows that |I 2 k | 2 2k |x| -2 and

Note that global existence occurs provided the local existence time only depends on the norm of the initial data in a suitable way. This is usually the case when solving the equation by a standard fixed point procedure.

as soon as 0 < ε < s -1. This leads to

This proves the existence and uniqueness of a local solution u in X s T with T = T ( u 0 H s ) small enough.