Andrea Santangeli
email: andrea.santangeli@helsinki.fi

Paul M Dolman

P Acevedo

Density and habitat preferences of male little bustard across contrasting agro-pastoral landscapes in Sardinia (Italy)

Keywords: Semi-natural steppe, Grazing management, Extensive agriculture, Pastoralism, Agricultural intensification, Tetrax tetrax

The little bustard Tetrax tetrax has undergone severe range contraction within Europe due to abandonment of traditional agro-pastoral activities. Previous studies of habitat selection have mainly focused on extensive cereal and cropland mosaics, while the species' ecology in pastoral landscapes is understood less well, and data are completely lacking from the Sardinian population. We conducted distance sampling surveys of displaying males across three contrasting landscapes in Sardinia and modelled habitat preference at both the landscape and local (within pastures and recent fallows) scale. Abbasanta, with a balance of pasture and cropland, the greatest isolation from roads and shortest vegetation, supported the highest little bustard densities (95%CI 2.7-3.4 males/100 ha). Significantly lower densities were found in two landscapes with lower isolation from roads and taller vegetation within grasslands: Campeda (0.1-0.2 males/100 ha), comprising cropland and pasture in similar proportions to those found at Abbasanta, and Campidano (0.3-0.4 males/100 ha) that was dominated by cereal agriculture. At the landscape level, males preferred pastures and recent fallows over arable lands. At the local scale, within grasslands, probability of occurrence was greater with shorter vegetation, more legume and green herb cover and at points remote from roads. Shorter vegetation in grasslands resulted from high grazing pressure, and habitat suitability for breeding males depends strongly on extensively grazed grasslands. Conservation efforts for this species should focus on maintaining traditional agro-pastoral practices which maintain large areas of extensively grazed pastures and recent fallows located far from roads.

Introduction

Throughout Europe, agricultural landscapes of great antiquity that were created and maintained by traditional farm management practices continue to support important biodiversity wherever they have survived [START_REF] Tucker | Priorities for bird conservation in Europe: the importance of the farmed landscape[END_REF][START_REF] Donald | Agricultural intensification and the collapse of Europe's farmland bird populations[END_REF][START_REF] Petretti | Conservation of steppe birdlife in Italy[END_REF]. In such landscapes, intensification of agricultural practices in mixed-farming areas, coupled with progressive loss of grazed steppe-like grasslands through abandonment of traditional livestock rearing, pose serious threats to biodiversity [START_REF] Tucker | Priorities for bird conservation in Europe: the importance of the farmed landscape[END_REF][START_REF] Chamberlain | Changes in the abundance of farmland birds in relation to the timing of agricultural intensification in England and Wales[END_REF]. Within western Europe, relict-grazed steppe-like grassland landscapes remain in southern France, Spain and parts of Italy (mostly within Sardinia), where pastoralism has a fundamental role in maintaining high conservation value landscapes [START_REF] Tucker | Habitats for birds in Europe: a conservation strategy for the wider environment[END_REF]. The distribution and habitat selection of species of conservation concern may be affected by ecological processes at multiple spatial scales [START_REF] Vergara | Responses of Chilean forest birds to anthropogenic habitat fragmentation across spatial scales[END_REF]. It is therefore important to understand the effect of environmental features and land-use effects within agropastoral regions at both landscape and local scales.

Within Europe, the little bustard Tetrax tetrax is associated with agro-ecosystems comprising low-intensity arable and pastoral areas, particularly mosaics of extensively grazed semi-natural pasture, fallow land and legume crops [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF]. Little bustard is classified as vulnerable in Europe [START_REF] Tucker | Birds in Europe: their conservation status[END_REF] following severe declines throughout most of its range including its loss from many European countries, attributable to agricultural intensification [START_REF] Goriup | Birds in Europe: their Conservation Status[END_REF][START_REF] Hoyo | Handbook of the birds of the world[END_REF]. Spain and Portugal support approximately three quarters of the European population; however, within the last 10 years dramatic local declines of up to 75% have occurred in Spain (García de la [START_REF] García De La Morena | El sisón común en España. I Censo Nacional[END_REF][START_REF] Juana | The dramatic decline of the little bustard Tetrax tetrax in Extremadura (Spain)[END_REF].

Elsewhere in Europe, three remnant populations occur, with two disjunct populations in France and a third nonmigratory population in Sardinia; while the species is now virtually extinct in mainland Italy [START_REF] Petretti | Conservation of steppe birdlife in Italy[END_REF]. The Sardinian population has a long history of geographic isolation, and in contrast to agricultural populations elsewhere (e.g. [START_REF] Jolivet | L'outarde canepetière Tetrax tetrax en France. Statut del l'espèce à la fin du XX siécle[END_REF], has been reported to be generally stable over the last two decades [START_REF] Petretti | Conservation of steppe birdlife in Italy[END_REF]. In both Sardinia and mainland Italy, little bustard mainly occur in extensively grazed pastures [START_REF] Petretti | Conservation of steppe birdlife in Italy[END_REF]. With decreases in traditional livestock grazing, pasture cover in Sardinia was reduced from 64% in 1960 to less than 10% in 2000 and from 7.5% to 2.5% in coastal mainland Italy during the same period [START_REF] Falcucci | Changes in land-use/landcover patterns in Italy and their implications for biodiversity conservation[END_REF]). This was the major cause of loss of little bustard populations in mainland Italy and presents the major threat to the species in Sardinia (De [START_REF] Juana | Little bustard (Tetrax tetrax)[END_REF][START_REF] Petretti | Conservation of steppe birdlife in Italy[END_REF].

In contrast to the large body of work on little bustards in cropland dominated areas, both in northern France [START_REF] Salamolard | Habitat selection by little bustard Tetrax tetrax in a cultivated area of France[END_REF] and in Iberia [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Silva | Summer habitat selection and abundance of the threatened little bustard in Iberian agricultural landscapes[END_REF], there has been much less work on the ecology of populations in pastoral-dominated landscapes (but see [START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF], despite their importance for the species [START_REF] Cramp | The birds of the Western Palearctic hawks to bustards[END_REF]. In Sardinia, although the little bustard is known to occur in both pastoral and agricultural areas, habitat preferences have not been studied. Both landscape-scale landuse and within-field features have been shown to be important for displaying males [START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF][START_REF] Traba | Selection of breeding territory by little bustard (Tetrax tetrax) males in central Spain: the role of arthropod availability[END_REF]), but these nested scales have rarely been considered within a single study (but see [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF] or across contrasting landscapes that differ in composition and disturbance levels.

In this study, we examine habitat preferences of male little bustard in agro-pastoral landscapes at two different spatial scales in order to identify important features to aid conservation management of the species. We consider features at the landscape scale, including the availability of pasture and recently fallowed fields, land-use heterogeneity and human disturbance and at the local scale, including grazing pressure, vegetation structure and composition, and localised human disturbance. We also com-pared density estimates for displaying males among three contrasting landscapes of western Sardinia that differ in land-use composition and structure, recent land-use trends and disturbance levels. From available knowledge of habitat preference of the species, we predicted that males' occurrence would be determined by both large-scale landscape features and small-scale habitat characteristics, with densities higher in the landscape with a balance of pasture and cropland cover and low disturbance levels [START_REF] Lane | Habitat preferences of great bustard Otis tarda flocks in the arable steppes of central Spain: are potentially suitable areas unoccupied?[END_REF][START_REF] Suarez-Seoane | Large-scale habitat selection by agricultural steppe birds in Spain: identifying species-habitat responses using generalized additive models[END_REF], and predicted lower densities in the more homogeneous landscapes exposed to greater levels of disturbance.

Methods

Study landscapes

Three key areas in western Sardinia previously known to support little bustard [START_REF] Schenk | On the distribution, number and conservation of the little bustard in Sardinia (Italy)[END_REF] were selected for study (Fig. 1). Each study area lies within, or includes, one of the four Special Protection Areas (SPAs) designated for the species (under EC Directive 79/409/ CEE). To allow land-use extent to be mapped, a contiguous study area was selected at each site, rather than sampling non-contiguous locations or grid cells across the entirety of each landscape. Boundaries of study areas where delimited to cover as much as possible of the large SPAs at Campeda (CA: ITB023050: 19,578 ha of SPA; 59% of which within the study area) and Abbasanta (AB: ITB023051: 20,103 ha of SPA; 24% within the study area), while the study area delimited at CO comprised all of the designated SPA (Campidano: ITB043054, 1,564 ha) and a further 12,203 ha of the surrounding Important Bird Area IBA (178-Central Campidano, Fig. 1).

CA and AB are upland areas with undulating topography and Meso-Mediterranean sub-humid climate [START_REF] Rivas-Martinez | Clasificacion bioclimatica de la tierra[END_REF] and are dominated by semi-natural steppe, comprising extensive pastures, fallows, arable fields (mainly dry cereals) and hay meadows. Pastures (hereafter indicating grazed land, long-established fallow and natural grasslands), recent fallows and cropped fields are evenly distributed across AB, while at CA croplands are concentrated in the northwestern river valleys, with the central dry stony plateau dominated by pasture, with scattered hay meadows (Fig. 2). In contrast, the lowland area of CO has flat topography and a drier Thermo-Mediterranean climate [START_REF] Rivas-Martinez | Clasificacion bioclimatica de la tierra[END_REF]. It comprises a cereal pseudosteppe landscape (sensu [START_REF] Suarez | Farming in the drylands of Spain: bird of pseudosteppes[END_REF], dominated by cereal fields, interspersed with irrigated crops such as lucerne, and occasional pastures (Fig. 2).

Landscape composition data from 2000 (Table 2) were extracted from regional land-use maps based on CORINE Land Cover classes (scale 1:25,000), considering pasture (all natural grasslands and long-established fallows), cropland (including cereal and other crops, recent fallows and hay meadows) and pooling unsuitable land-cover types (villages, infrastructures, vineyards and woodlands). Landuse data from 2000 were updated during fieldwork in 2008, with the aid of hand-held GPS, to quantify changes in these three broad land-cover categories (pasture, cropland, unsuitable) and to quantify the frequency of specific cropland types (recent fallow, hay meadow, wheat/barley, oat and legume crops) within the three study areas. Recent fallows were identified during fieldwork in 2008 by the presence of therophytic and absence of perennial geophytic plants (mainly of the genus Ferula and Asphodelus).

Survey of male density

Male little bustard display activity is concentrated between April and June in Sardinia [START_REF] Petretti | Notes on the lek behaviour of the little bustard in Italy[END_REF]. Surveys were conducted from 1 May to 8 June 2008, between 0600 to 0900 hours and 1800 to 2100 hours (GMT+1:00), when male display activity is greatest [START_REF] Juana | Distribution and conservation status of the little bustard Tetrax tetrax in the Iberian Peninsula[END_REF], except during rain or strong wind which reduces display activity and detection [START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF].

To allow maximal areal coverage, a combination of point counts and distance transects were used. At CO, a network of secondary roads and tracks covering all the landscape allowed the surveyor to drive between sampling locations; consequently at this site only point transects were performed to maximise survey coverage. At CA and AB, it was necessary to conduct surveys on foot so that a combination of point and line transects were adopted. Transect lines of approximately 4.5 km length, separated by a minimum distance of 500 m, were walked slowly (2 km/h), and point counts of 5 min duration were performed along each transect at intervals of 600 m, following De [START_REF] Juana | Distribution and conservation status of the little bustard Tetrax tetrax in the Iberian Peninsula[END_REF] and [START_REF] Faria | Breeding habitat modelling of the little bustard Tetrax tetrax in the site of community importance of Cabrela (Portugal)[END_REF]. Spacing of transects and point counts was based on male call detection at up to 250 m in landscapes with heterogeneous topography [START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF]. The distance (radial from points and perpendicular from line transects) of each detected male was estimated to the closest 10 m following extensive preliminary training and the male's position was mapped with the aid of a hand-held GPS. Where visibility impaired accurate location of calling individuals, they were approached until visually detected.

Effective strip width (ESW) of distance transects and effective detection radius (EDR) of point counts were calculated in DISTANCE 5.0 [START_REF] Thomas | Research Unit for Wildlife Population Assessment[END_REF]. Halfnormal, hazard-rate and uniform/cosine detection functions were compared and the best model selected according to the Akaike Information Criterion (AICc and AIC for ESW and EDR respectively; [START_REF] Buckland | Introduction to distance sampling: estimating abundance of biological populations[END_REF]. Point count data were right truncated at 10% of observations as this improved model fit. Transect and point count locations were buffered in ArcGIS by using the mean value of the best model for ESW and for EDR; buffers were dissolved and summed and the composite area covered by each survey calculated. This was repeated for upper and lower 95%CI values of EDR and ESW. Density and population estimates (mean and 95%CI) for each study area were calculated on the basis of males detected, areas surveyed and extent of available habitat cover.

Habitat sampling

Habitat was sampled at 96 locations with displaying male little bustard, and at 120 control (absence) points, randomly located across each site while excluding unsuitable land-cover types for the species (woodland, vineyard or infrastructure). Control points were located further than 150 m from the nearest territory, considering mean male territory area of 6.7 ha in France [START_REF] Jiguet | Lek mating systems: a case study in the little bustard Tetrax tetrax[END_REF]. At each point, 15 variables were measured (Table 1), seven of those related to landscape scale and eight to small-scale (within-grassland) habitat characteristics. The landscape variables recorded included: land-use type, land-use heterogeneity and number of bushes and trees within 150 m from the point centre. Land-use heterogeneity, consisting of the counted number of the five different landuse categories, was again recorded within 150 m from the point centre based on known male territory area of 6.7 ha (approximately 150 m radius; [START_REF] Jiguet | Lek mating systems: a case study in the little bustard Tetrax tetrax[END_REF]. Points with high heterogeneity were scarce and were pooled into a single class with value greater than two for computational purposes (Table 1). In addition, in view of the known sensitivity of bustards to human disturbance and infrastructure (Lane et al. [START_REF] Suarez-Seoane | Large-scale habitat selection by agricultural steppe birds in Spain: identifying species-habitat responses using generalized additive models[END_REF], we also considered four proxies of disturbance (distance to roads, tracks, buildings and wires) measured from 1:25,000 military maps derived from 1999 and updated in the field in 2008 (Table 1).

Small-scale habitat variables included percentage cover of each of six vegetation classes estimated within nine quadrats (of 1 m × 1 m), with one located at the centre and others at 10 and 20 m along the four cardinal directions (following [START_REF] Silva | Summer habitat selection and abundance of the threatened little bustard in Iberian agricultural landscapes[END_REF]; Table 1). Vegetation height was also recorded by taking 20 measures using a sward stick (15 cm radius and 70 g weight) randomly placed within 20 m radius of the point centre. We also recorded whether sheep faecal pellets were present in each of eight strip transects (of 2 m × 10 m) along the four cardinal directions from the point centre. An index of grazing pressure was considered ranging from zero (no pellets recorded) to eight (pellets present on each strip transect) following [START_REF] Silva | Summer habitat selection and abundance of the threatened little bustard in Iberian agricultural landscapes[END_REF]. All data were collected by the same observer (AS).

Data analysis

Displaying male habitat preference was analysed by constructing two multivariate logistic regression models with bustard presence or absence as the response. The first model, including all 216 sample points, considered the seven landscape-scale variables (Table 1): land-use, landuse heterogeneity (the number of the five different landuses counted within 150 m from the sample point; categories-low=1, medium=2 and high=≥3), number of bushes and trees, and the four measures of human disturbance (distance to roads, tracks, buildings and wires). The second model considered only the 155 sample (presence and absence) points located within pasture and recent fallows and included the four measures of human disturbance (distance to infrastructures in Table 1) in addition to the seven small-scale variables of habitat structure and quality: vegetation height and the percentage cover of bare or rocky soil, oat, wheat or barley, legumes, herbs and other vegetation material. Wheat and barley were pooled into a single cover category as both were scarce and have similar structure and phenology in our study landscapes.

Land-use and land-use heterogeneity were entered in the first model as categorical predictors, and study area was controlled for in both models as a categorical fixed effect. Inter-correlation among candidate variables was examined using Pearson's correlation. It is appropriate to simultaneously include inter-correlated predictors within a multivariate model, provided the correlation coefficient is less than approximately 0.7-0.8 [START_REF] Freckleton | On the misuse of residuals in ecology: regression of residuals vs. multiple regression[END_REF]. All pairwise correlations among our variables were below 0.6, with the exception of vegetation height which was strongly correlated with grazing pressure (r=-0.71; pooling sample points within pastures and recent fallows). We therefore excluded grazing pressure from the second model based on its lower contribution to the model (judged by -2 loglikelihood ratio, LLR) compared to vegetation height.

Whether the distance to infrastructures (roads, tracks, buildings and wires) differed among the five land-use types and between the three study areas was analysed by means of analysis of variance (ANOVA) using random points only and including an interaction term for land-use type with study area. None of the four distance variables differed significantly among land-use types.

Full initial logistic models including all candidate variables were simplified by backward elimination, removing one variable at a time and assessing the change in Log-Likelihood Ratio (-2LLR, with P≥0.05). Model goodness-of-fit was assessed by classification accuracy, with a probability threshold of 0.5 [START_REF] Hosmer | Applied logistic regression[END_REF] and by calculating the area under the receiver operating characteristics curve (AUC) with associated standard error. Spatial autocorrelation of model residuals were examined by Moran's I test and visual investigation of spline correlogram (plotting correlation coefficients with 95% pairwise bootstrap confidence intervals as a function of distance). As residuals of the landscape-scale model were spatially autocorrelated, we also built a spatial generalised linear mixed model (using the proc glimmix procedure in SAS) following [START_REF] Dormann | Methods to account for spatial autocorrelation in the analysis of species distributional data: a review[END_REF]. Within pastures and recent fallows, vegetation height (square root transformed to fit normality) was related to grazing pressure considering all random and presence points in a general linear model, with grazing pressure as covariate, study area and land-use (classes: pasture and recent fallow) as categorical fixed factors. The interactions between grazing pressure and study area, and study area with land-use were tested in the model and removed if non-significant. The two logistic models were built using SAS version 9.2 (SAS Institute, Cary, NC, USA). PASW Statistics 18 (PASW Statistics 18 Inc. 2010) was used for all other tests.

Results

Landscape-and local-scale habitat characteristics

Both the current extent of land-cover classes, and their recent history, differed markedly among the three study areas. CO was dominated by cropland, while pasture was extremely scarce, both in the 2008 field survey and in the earlier 2000 land-cover maps (Table 2). Despite the continued dominance of cropland at CO, its extent had slightly reduced through conversion to unsuitable land-uses for the little bustard (mostly vineyards). During field survey in 2008, 26% of the cropland sampled at random points within CO comprised recent fallows. In contrast, in 2008 landscapes at both CA and AB had approximately half the cover of cropland than at CO, and a much greater extent of pasture than at CO, with pasture having similar extent to croplands (Table 2). However, they differed in recent history. Land-cover extent had been stable at CA between 2000 and 2008. In contrast, at AB cropland extent had reduced greatly; from approximately three quarters in 2000 to 43% of the landscape in 2008, with a corresponding expansion of pasture that had been almost absent in 2000 (Table 2). Additional recent fallows were common at AB, comprising 43% of random points within cropland, compared to 9% at CA.

Random points at AB were significantly further from roads (mean±SE 697±63 m) than those at CA and CO (474±119 m and 332±64 m respectively; site effect in ANOVA controlling for land-use and land-use interaction: F 2, 119 =6.105, P=0.003, Tukey tests P<0.05). Random points in AB were also more distant from tracks (239± 26 m) than in CO (109±17 m; F 2, 119 =3.521, P=0.033, Tukey test P<0.05), but distance to tracks was similar between random points in AB and CA (161±29 m in CA; Tukey test P=0.248). Distance of random points to wires and buildings was similar among the three study areas (P= The percentage of each crop type within the cropland land-use is shown from field data collected in 2008 only 0.388 and 0.254, respectively) when controlling for the effect of land-use type.

Considering pastures and recent fallows, vegetation height was negatively related to grazing pressure (GLM controlling for site and pasture versus fallow: r 2 =0.56; grazing: F 1, 154 =124.63, P<0.001) and differed significantly among study areas (fixed effect: F 2, 154 =5.57, P=0.005), with pastures and recent fallows at AB having shorter vegetation (mean=22.2 cm±1.1SE) than those in CO (mean = 33.4 cm ± 4.3SE, P = 0.013) and CA (mean = 28.3 cm±4.0SE, P=0.034). Vegetation was also shorter in pasture (mean 21.1 cm±1.2 SE) than in recent fallows (32.4 cm±2.2, respectively; F 1, 154 =4.497, P=0.036). Nonsignificant interactions between grazing pressure and study area and study area versus land-use type were excluded from the reduced model.

Comparison of male little bustard density among contrasting landscapes

The 31 transects and 675 point counts provided 112 observations (45 and 67 for each method, respectively) of displaying male little bustard across the three sites. For distance transects, the half-normal detection function was best supported (ESW=226.5 m±11.4SE, AICc=528.80) and was used for density estimation; a hazard-rate model performed almost as well (ΔAICc=0.3) and provided a very similar though less precise ESW (234.2 m±35.2SE). For point counts, the uniform function with cosine adjustment was supported (EDR=246.9±12.2SE, AIC=718.4), the next best function (half-normal) had considerably larger AIC (ΔAIC=2.4).

A large population was detected in AB, with densities 23 and 10 times higher than CA and CO, respectively (Table 3); the lack of overlap of 95%CI indicates that density differed significantly among the three sites.

Male little bustard habitat preference

Landscape scale

The multivariate logistic model built pooling data from across the three areas showed moderate internal model fit (AUC=0.79±0.03SE) and spatially autocorrelated residuals (Moran's I test, P<0.05) with positive correlation between points occurring within a distance of approximately 1,000 m. A spatial model that explicitly incorporates spatial correlation had a marginally lower model fit (AUC=0.73±0.03SE). The spatial model retained only the land-use variable as significant (Table 4a), with preference for pasture and new fallows (Fig. 3a). The non-spatial model included effects of both land-use and land-use heterogeneity (Table 4b). This model predicts male little bustards to occur significantly more often in pasture and recent fallows (Fig. 3b; amounting 70 and 17 male detections, respectively) than in either arable (only one male detected), hay meadow or other land-use classes (two and six detections, respectively). Males' occurrence in the non-spatial model was also significantly lower at points with either low or medium land-use heterogeneity, compared to points with high land-use heterogeneity within 150 m (Fig. 3c).

Disturbance variables (distance to buildings, tracks, roads or wires) were not retained in either the spatial or non-spatial model (Table 4a and4b), indicating that they contributed little to explaining incidence of males' presence at this scale.

Local scale, within pastures and recent fallows

The multivariate model considering small-scale habitat elements and wider-scale disturbance variables within pasture and recent fallows (Table 4c) had high classification CI (95%) of surveyed area (with % of suitable habitat covered by survey), density and population estimates are given within brackets. Extent of SPAs and the % of SPA covered by the study area are also shown accuracy (93.5% of all points correctly classified) and very good internal model fit with an AUC of 0.94±0.02SE. Residuals of the model did not show any sign of autocorrelation. Males' probability of occurrence increased with shorter vegetation, higher legume and green herb cover and greater distance to roads (Table 4c).

Discussions

Habitat preference by male little bustard at landscape and local scales

This study confirms and supports the importance of pasture and recent fallows in the landscape for displaying male little bustards [START_REF] Wolff | Landscape context and little bustard abundance in a fragmented steppe: implications for reserve management in mosaic landscapes[END_REF]. It also confirms the importance of small-scale habitat requirements for the species, such as cover of legumes and green herbs [START_REF] Jiguet | Resource defence on exploded leks: do male Little Bustards, Tetrax tetrax, control resources for females?[END_REF][START_REF] Morales | Sexual differences in microhabitat selection of breeding little bustards Tetrax tetrax: ecological segregation based on vegetation structure[END_REF]), short vegetation structure [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF]) and distance from roads, and finally highlights the fundamental role of grazing livestock in maintaining suitable habitat structure for the species in agro-pastoral steppe-like landscapes.

Our results demonstrate the crucial importance of traditional farming systems which include adequate cover of pastures and recent fallows for the little bustard. These land-uses, associated with low-intensity agro-pastoral activities, are rapidly disappearing from most of the Palaearctic but still occur locally in southern Mediterranean regions and are well known as a suitable habitat for the little bustard in Iberia [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Salamolard | Habitat selection by little bustard Tetrax tetrax in a cultivated area of France[END_REF][START_REF] Suarez-Seoane | Large-scale habitat selection by agricultural steppe birds in Spain: identifying species-habitat responses using generalized additive models[END_REF] and southern France [START_REF] Wolff | The benefits of extensive agriculture to birds: the case of the little bustard[END_REF]. A strong preference for high heterogeneity of land-use substrates was also evident in the non-spatial model but not in that incorporating spatial autocorrelation. These contrasting results suggest that the clustering of displaying males in areas of high land-use heterogeneity may be related to ecological requirements of the species, as found in other areas [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Wolff | Landscape context and little bustard abundance in a fragmented steppe: implications for reserve management in mosaic landscapes[END_REF], but may be overlooked if the spatial correlation resulting from the conspecific attraction of breeding males is accounted for.

Contrary to other studies [START_REF] Suarez-Seoane | Large-scale habitat selection by agricultural steppe birds in Spain: identifying species-habitat responses using generalized additive models[END_REF][START_REF] Silva | Managing landscapes for the little bustard Tetrax tetrax: lessons from the study of winter habitat selection[END_REF][START_REF] Osborne | Identifying core areas in a species' range using temporal suitability analysis: an example using little bustards Tetrax tetrax L. in Spain[END_REF][START_REF] Silva | Estimating the influence of overhead transmission power lines and landscape context on the density of little bustard Tetrax tetrax breeding populations[END_REF], we found no effect of proximity to anthropogenic infrastructures on the occurrence of little bustards at the landscape scale. However, at the smaller scale, displaying males located at points further from roads, suggesting that when not constrained by the availability of pastures and new fallows, roads were avoided. At this local scale, bustards also selected areas with higher content of legumes and green herbs and shorter sward height that may relate to food requirements and adequate visibility for displaying males [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Salamolard | Habitat selection by little bustard Tetrax tetrax in a cultivated area of France[END_REF][START_REF] Wolff | Influence of landscape and habitat heterogeneity on the distribution of steppe-land birds in the Crau, southern France[END_REF]. That such clear and strong associations with landuse, vegetation structure and quality were apparent suggests either that the distribution of little bustard is not constrained by lek inertia at these scales or that patterns of local and landscape suitability are stable at timescales relevant to the longevity of individuals and territories. The importance of high grazing intensity in maintaining low vegetation structure suitable for displaying males emphasises the need to preserve sheep rearing activities in our agropastoral landscapes and elsewhere. The extent of pasture areas in Sardinia has rapidly decreased in recent decades [START_REF] Falcucci | Changes in land-use/landcover patterns in Italy and their implications for biodiversity conservation[END_REF], and if this continues, suitable areas for the species may become increasingly scarce and isolated.

Differences in little bustard density, land-use composition and structure among the contrasting landscapes The differences in densities observed between the three sites in Sardinia can be largely explained in terms of differences in the availability of suitable land-use types, such as pasture and new fallow, and disturbance patterns.

As predicted, the highest male density occurred at AB which is the least impacted by road disturbance and provides a balance of cropland and pasture cover that has previously been found to represent an optimal combination for little bustard [START_REF] Martinez | Habitat selection by the little bustard Tetrax tetrax in cultivated areas of central Spain[END_REF][START_REF] Wolff | Landscape context and little bustard abundance in a fragmented steppe: implications for reserve management in mosaic landscapes[END_REF].

Again in agreement with predictions, lower density was found at CO that is likely the result of the scarcity of pasture and dominance of cereal fields in addition to the higher road disturbance compared to AB. However, the low density at CA that had a broadly similar balance of cropland and pasture to AB may at least partly relate to the taller, and thus less suitable, vegetation within pastures and recent fallows, as well as higher road disturbance. In the case of CA, other factors than those included in this study may have been responsible for the low numbers recorded. One possible explanation may be due to the social organisation of the species where displaying males attract each other [START_REF] Jiguet | Lek mating systems: a case study in the little bustard Tetrax tetrax[END_REF] so that larger leks may be more attractive than smaller ones; thus, in Campeda the small population may be less attractive to settling males, despite suitable habitat being available in the area.

Implications for management and conservation of little bustards in agro-pastoral landscapes Livestock rearing in pasturelands interspersed by extensively cultivated areas is known to support an important community of birds and other taxa in agro-pastoral landscapes [START_REF] Lemaire | Challenges for grassland science: managing research priorities[END_REF]. In most parts of Europe, large areas once used for mixed farming are now subject to a rapid intensification of both cropping and dairy practices that has led to local and even regional specialisation with the most fertile soils used for cereal cropping and less fertile areas devoted to intensive animal production [START_REF] Lemaire | Challenges for grassland science: managing research priorities[END_REF]. This has led to the disappearance of grasslands from most cereal production regions of North-western Europe and associated biodiversity losses [START_REF] Robinson | Post-war changes in arable farming and biodiversity in Great Britain[END_REF]. For little bustard, the disappearance of pastures and fallow lands across Europe has been identified as the major driver of national extinctions and population declines across its Palaeartic range [START_REF] Hoyo | Handbook of the birds of the world[END_REF][START_REF] Juana | The dramatic decline of the little bustard Tetrax tetrax in Extremadura (Spain)[END_REF]. The clear preference for pasture and recent fallows in Sardinia and the clear importance of grazing livestock in maintaining short vegetation for displaying males indicate that traditional agro-pastoral activities should be preserved in order to prevent further losses of little bustard populations. Such measures will also benefit the wider assemblage of species of conservation concern, such as stone curlew Burhinus oedicnemus, calandra lark Melanocorypha calandra and griffon vulture Gyps fulvus, that depend on traditional low-intensity land-use and that are currently under pressure from agricultural intensification [START_REF] Tucker | Priorities for bird conservation in Europe: the importance of the farmed landscape[END_REF].

As human demands for rural development are consistently increasing, the role of agri-environmental schemes may prove fundamental to enhance the conservation of low density farmland species that require strategic intervention at the landscape scale. However, the recent abolition of obligatory set aside within the European Union may exacerbate declines in grasslands within the agro-pastoral mosaic that is likely to increase the pressure on the already imperilled species associated with traditional farming ecosystems.

Fig. 2

 2 Fig. 2 Location of displaying male observations, transects and point counts across suitable habitat within each of the three study sites. The surveyed areas shown as buffers around transects and point counts

Fig. 3

 3 Fig. 3 Estimated coefficients (±SE) of (a) land-use type from the spatial model, (b) land-use and (c) heterogeneity variables in logistic non-spatial model of little bustard incidence at the landscape scale. The reference class set in the logistic models (other land-use in panel a and b, high heterogeneity class in panel c) are also shown by black thick lines along the vertical (x=0) axis

Table 1

 1 Landscape elements (bold font) and local scale (within semi-natural grassland and recent fallows) habitat variables recorded at the 216 survey points across three study areas within Sardinia

	Variable	Source of data	Description of the variable
	Land-use	Field data
	Pasture		Natural pastures and long fallow land (left set aside for more than 1 year)
	Recent fallow		Land left set aside for 1 year
	Arable		All cultivated land which is not irrigated
	Hay meadow		Unploughed cropland
	Other		Ploughed land, xerophilic steppe (>60% bare or stony ground), irrigated land
	Land-use heterogeneity	Field data	The number of land-use categories counted within 150 m of the point (classes: low=1,
			med=2 and high=3 to 5)
	Distance to roads	Military maps	Distance to the nearest asphalted road
	Distance to tracks	Military maps	Distance to the nearest dirt track
	Distance to buildings	Military maps	Distance to the nearest actively used building
	Distance to wires	Military maps	Distance to the nearest aerial power line
	Bushes/trees	Field data	The number of woody plants (height>2 m) present within 150 m of the point
	Vegetation height	Field data	Measured using a sward stick to the nearest cm
	Grazing pressure	Field data	Range 0-8, calculated as the number of eight transects (each 2×10 m) searched for sheep
			faecal pellet groups.
	Bare/rocky soil cover	Field data	Estimated % of surface with no vegetation cover
	Oat cover	Field data	Estimated % of surface covered by oat
	Wheat/barley cover	Field data	Estimated % of surface covered by wheat or barley
	Legume cover	Field data	Estimated % of surface covered by leguminosae plants (e.g. Trifolium and Medicago spp.)
	Green herbs cover	Field data	Estimated % of surface covered by green non-leguminosae herbs (particularly Cruciferae
			and Compositae)
	Other vegetation cover	Field data

Estimated % of surface covered by dry grass or other vegetation (mainly Poa and Cardus spp.) not included in the previous classes

Table 2

 2 Total areas and percentage extent of pasture, cropland (including legume crops, recent fallows, hay fields and dry cereal fields) and habitat unsuitable for little bustard in vegetation maps from 2000 (extracted from CLC2000 dataset) and 2008 (updated in the field; also shown in Fig.2) in each of three study areas within Sardinia

		AB		CA		CO
	Year	2000	2008	2000	2008	2000	2008
	Total area (ha)	4,912		11,555		15,305
	Land-use (% area):
	Unsuitable	20.2	12.2	24.2	24.6	14.8	26.3
	Pasture	3.6	44.4	43	45.4	1.4	1.1
	Cropland	76.2	43.4	32.8	30	83.8	72.6
	Crop type (% cropland):
	Lucerne		10		9.1		13.0
	Oat		2.5		9.1		8.7
	Barley/wheat		7.5		18.2		30.4
	New fallow		42.5		9.1		26.1
	Hay meadow		30		45.5		17.4
	Other		7.5		9.1		4.4

Table 3

 3 Extent of study areas, suitable habitat available and the area covered by combined distance transect and point surveys are given. Numbers of calling birds detected and estimates of the density and total numbers of males within each of three study areas within Sardinia are also presented

	AB	CA	CO

Table 4

 4 Minimal multivariate logistic regression results. Results show (a) landscape-scale spatial model and (b) landscape-scale non-spatial model of male preference (considering points across all land-use types: n=96 presences, 120 absences), and (c) local scale preference, within pastures and recent fallows (including habitat structure and disturbance variables: n=87 presences, 68 absences)

	Variables	B	SE	-2LLR	P	χ²	df	P	AUC±SE
	(a) All land-use types (spatial model)								0.73±0.03
	Land-use (presence)			6.19	<0.001
	Site			2.42	0.091
	Constant	-2.033	0.639
	(b) All land-use types (non-spatial model)					64.7	8	<0.001	0.79±0.03
	Land-use (presence)			22.492	<0.001
	Land-use heterogeneity			10.725	0.005
	Site			9.978	0.007
	Constant	-0.330	0.729
	(c) Pasture and recent fallow					116.0	6	<0.001	0.94±0.02
	Road distance (m)	0.002	0.001	9.159	0.003
	Vegetation height (cm)	-0.190	0.033	32.665	<0.001
	Legume cover (%)	0.053	0.026	3.947	0.047
	Green herbs cover (%)	0.044	0.015	9.074	0.003
	Site			1.251	0.535
	Constant	1.359	0.810

Estimates (B) and standard error (SE) of coefficients are shown for continuous variables only in model (c); coefficients of category levels for variables in models (a) and (b) are shown in Fig. 3. Significance, assessed by χ 2 test of the change in -2LLR following variable removal from the minimal model, is presented for all retained variables

Acknowledgements We would like to thank Nigel Collar for encouragement throughout this study and the Agriturismo Nurag'elighe for logistic support. We are grateful to Mauro Aresu and Andreina Concas for their valuable assistance in field work, Alberto Cardillo for the assistance with spatial analysis and to all the landowners who kindly granted us access to their land. A. Santangeli was supported by a grant from the Sir Philip Reckitt Educational Trust. Two anonymous referees are also acknowledged for their constructive comments that improved the paper.


Cyril Imbert

L Silvestre

C 1,α regularity of solutions of degenerate fully non-linear elliptic equations

Keywords: C 1, α estimates, degenerate elliptic equations, fully non-linear elliptic equations, viscosity solutions MSC: 35B45, 35J60, 35J70, 35D40

published or not. The documents may come     C 1,α regularity of solutions of degenerate fully non-linear elliptic equations

Introduction

This paper is concerned with the study of the regularity of solutions of the following non-linear elliptic equation

|∇u| γ F (D 2 u) = f in B 1 (1)
where B 1 is the unit ball of R d and γ > 0, F is uniformly elliptic, F (0) = 0 and f is bounded.

Singular/degenerate fully non-linear elliptic equations. Equation (1) makes part of a class of non-linear elliptic equations studied in a series of papers by Birindelli and Demengel, starting with [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF]. The specificity of these equations is that they are not uniformly elliptic; they are either singular or degenerate (in a way to be made precise).

Birindelli and Demengel proved many important results in the singular case such as comparison principles and Liouville type results [START_REF] Birindelli | Comparison principle and Liouville type results for singular fully nonlinear operators[END_REF], regularity and 1 uniqueness of the first eigenfunction [START_REF]Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF] etc. In the degenerate case, the set of results [START_REF] Birindelli | Eigenvalue, maximum principle and regularity for fully non linear homogeneous operators[END_REF][START_REF]Eigenvalue and Dirichlet problem for fully-nonlinear operators in non-smooth domains[END_REF] is less complete and in particular, there was no C 1,α estimate in the non-radial case (see [START_REF]Uniqueness of the first eigenfunction for fully nonlinear equations: the radial case[END_REF] for the radial case).

Alexandrov-Bakelman-Pucci (ABP) estimate were obtained for such equations independently in [START_REF] Dávila | Alexandroff-Bakelman-Pucci estimate for singular or degenerate fully nonlinear elliptic equations[END_REF] and [START_REF] Imbert | Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations[END_REF]. It was used to derive Harnack inequality in the singular case in [START_REF]Harnack inequality for singular fully nonlinear operators and some existence results[END_REF] and in both cases in [START_REF] Imbert | Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations[END_REF]. From Harnack inequality, it is classical to derive Hölder estimate ( [START_REF]Harnack inequality for singular fully nonlinear operators and some existence results[END_REF] in the singular case, [START_REF] Imbert | Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations[END_REF] in both cases).

Main result. The main result of this paper is the following Theorem 1. Assume that γ ≥ 0, F is uniformly elliptic, F (0) = 0, and f is bounded in B 1 . There exists α > 0 and C > 0 only depending on γ, the ellipticity constants of F and dimension d, such that any viscosity solution u of (1) is C 1,α and

[u] 1+α,B 1/2 ≤ C ||u|| L ∞ + ||f || 1 1+γ L ∞ .
Comments. Getting C 1,α estimates consists in proving that the graph of the function u can be approximated by planes with an error bounded by Cr 1+α in balls of radius r. The proof is based on an iterative argument, in which we show that the graph of u gets flatter (meaning better approximated by planes) in smaller balls. The iterative step, after a rescaling, amounts to show that if p • x + u satisfies (1) in B 1 with osc u ≤ 1, then the oscillation of u, up to a linear function p ′ • x, is smaller in a smaller ball. This is proved by compactness. In order to make such an argument work, the modulus of continuity of u has to be controlled independently of the slopes p and p ′ which can vary from one scale to the other. There is a difficulty since u -p • x does not satisfy any PDE independently of p. The main originality of this paper is to combine the method introduced by Ishii and Lions [START_REF] Ishii | Viscosity solutions of fully nonlinear secondorder elliptic partial differential equations[END_REF] to get Lipschitz estimate in the case of large slopes and the Harnack inequality approach of Krylov-Safonov-Caffarelli [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] adapted in [START_REF] Imbert | Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations[END_REF] to the present framework for small slopes.

An alternative approach to find a modulus of continuity for solutions of the rescaled equation (see [START_REF]Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF] below) for large slopes could be to apply the Harnack inequality from [START_REF] Savin | Small perturbation solutions for elliptic equations[END_REF] to get a uniform Hölder modulus of continuity for |p| large enough instead of the Ishii-Lions method to get a uniform Lipschitz estimate. We chose the latter approach because of its simplicity.

The following example shows that solutions u of (1) cannot be more regular than

C 1,α , even if f is Hölder continuous. Example 1. The function u(x) = |x| 1+α satisfies |Du| γ ∆u = C|x| (1+α)(γ+1)-(γ+2)
where C = (1 + α) 1+γ (d + α -1). In particular, if we choose α = 1/(1 + γ) the right hand side is simply constant. This example shows that even for a constant right hand side and F (D 2 u) = ∆u, we cannot expect in general the solution to be more regular than C 1,α with α < 1.

As far as the authors know, the result of Theorem 1 is new even for the simple equation |∇u| γ ∆u = f (x). For this case we expect the optimal α to be in fact equal to 1/(1 + γ) although we did not work on that issue. For general fully nonlinear equations F (D 2 u) the value of α can get arbitrarily small even in the case γ = 0 (see [START_REF] Nadirashvili | Singular viscosity solutions to fully nonlinear elliptic equations[END_REF] for an example).

The paper is organized as follows. In section 2 we specify the notation to be used in the paper and we review a few well known definitions and results for fully nonlinear elliptic equations. In section 3 we restate Theorem 1 in a simplified form simply by rescaling. In section 3, we also show how the iteration of the improvement of flatness lemma implies the main theorem. The methods of section 3 are more or less standard for proving C 1,α regularity for elliptic equations. In section 4 we find a uniform modulus of continuity for the difference between the solution and a plane appropriately rescaled. Based on this continuity estimates we prove the improvement of oscillation lemma by a compactness argument. In the last section we show a technical lemma that says that viscosity solutions to |∇u| γ F (D 2 u) = 0 are also viscosity solutions to F (D 2 u) = 0. This lemma is used to characterize the limits in the compactness argument for the proof of the improvement of flatness lemma in section 4.

Preliminaries

Notation

For r > 0, B r (x) denotes the open ball of radius r centered at x. B r denotes B r (0). S d denotes the set of symmetric d × d real matrices. I denotes the identity matrix.

For α ∈ (0, 1] and

Q ⊂ R d , we consider [u] α,Q = sup x,y∈Q,x =y u(x) -u(y) |x -y| α , [u] 1+α,Q = sup ρ>0,x∈Q inf p∈R d sup z∈Bρ(x)∩Q |u(z) -p • z|.

Uniform ellipticity

We recall the definition of uniform ellipticity (see [START_REF] Caffarelli | Fully nonlinear elliptic equations[END_REF] for more details). We say that a function F defined on the set of real symmetric matrices and taking real values is uniformly elliptic if there exist two positive constants λ and Λ such that for any two symmetric matrices X and Y , with Y ≥ 0 we have

λ tr Y ≤ F (X) -F (X + Y ) ≤ Λ tr Y.
The constants λ and Λ are called the ellipticity constants. Under this definition F (X) = -tr(X) is uniformly elliptic with ellipticity constants λ = Λ = 1, and

F (D 2 u) = -∆u = f (x) is a uniformly elliptic equation.
The maximum and minimum of all the uniformly elliptic functions F such that F (0) = 0, are called the Pucci operators. We write them P + and P -. Recall that P -has the closed form

P -(X) = -Λ tr X + -λ tr X -,
where tr X + is the sum of all positive eigenvalues of X and tr X -is the sum of all negative eigenvalues of X. With the definition of P + and P -at hand, it is equivalent that F is uniformly elliptic with the inequality

P -(Y ) ≤ F (X + Y ) -F (X) ≤ P + (Y ),
for any two symmetric matrices X and Y .

Two observations

The uniform ellipticity hypothesis on F implies that there exist α 0 ∈ (0, 1) and

C > 0 such that viscosity solutions of F (D 2 u) = 0 in B 1 are C 1,α0 in the interior of B 1 and [u] 1+α0,B 1/2 ≤ C||u|| L ∞ (B1) .
The constants α 0 and C depend on the ellipticity constants and dimension only.

Note that for any constant a > 0, the function a -1 F (aX) has the same ellipticity constants as F . This will be important when rescaling the equation.

Reduction of the problem

In this section, we first show that a simple rescaling reduces the proof of the problem to the case that ||u|| L ∞ ≤ 1/2 and ||f || L ∞ ≤ ε 0 for some small constant ε 0 which will be chosen later. We then further reduce the proof to an improvement of flatness lemma.

Rescaling

We work with the arbitrary normalization ||u|| L ∞ ≤ 1/2 because that implies that osc u ≤ 1 and that will be a good starting point for our iterative proof of C 1,α regularity. Proposition 1. In order to prove Theorem 1, it is enough to prove that

[u] 1+α,B 1/2 ≤ C assuming ||u|| L ∞ (B1) ≤ 1/2 and ||f || L ∞ (B1)
≤ ε 0 for some ε 0 > 0 which only depends on the ellipticity constants, dimension and γ.

Proof. Given any function u under the assumptions of Theorem 1, we can take

κ = 2||u|| L ∞ + (||f || L ∞ /ε 0 ) 1/(1+γ) -1
and consider the scaled function ũ(x) = κu(x) solving the equation

|∇ũ| γ κF (κ -1 D 2 ũ) = κ 1+γ f (x).
We previously made the observation that the function κF (κ -1 X) has the same ellipticity constants as F (X). But now

||ũ|| L ∞ ≤ 1/2 and || f || L ∞ ≤ ε 0 . There- fore, if [ũ] 1+α,B 1/2 ≤ C,
by scaling back to u, we get

[u] 1+α,B 1/2 ≤ Cκ -1 ≤ C(||u|| L ∞ (B1) + ||f || 1/(1+γ) L ∞
) which concludes the proof.

It is enough to prove that the solution u of ( 1) is C 1,α at 0 that is to say that there exists C > 0 and α (only depending on the ellipticity constants, dimension and γ) such that for all r ∈ (0, 1), there exists p ∈ R d such that osc

Br (u -p • x) ≤ Cr 1+α .
(

) 2
If we start with a function u such that osc B1 u ≤ 1, we already have the inequality for r = 1 with C = 1. In order to get such a result for all r ∈ (0, 1), it is enough to find ρ, α ∈ (0, 1) such that for all k ∈ N there exists p k ∈ R d such that osc

B ρ k (u -p k • x) ≤ ρ k(1+α) .
The inequality (2) follows with C = ρ -(1+α) . This is the reason why we consider r k = ρ k and we aim at proving by induction on k ∈ N the following Lemma 1. There exists ρ, α ∈ (0, 1) and ε 0 ∈ [0, 1] only depending on γ, ellipticity constants and dimension such that, as soon as a viscosity solution u of (1) with ||f || L ∞ ≤ ε 0 satisfies osc B1 u ≤ 1, then for all k ∈ N, there exists

p k ∈ R d such that osc Br k (u -p k • x) ≤ r 1+α k . (3)
The choice of ρ depends on the C 1,α0 estimates for F (D 2 u) = 0. Precisely, since we assume that any viscosity solution u of F (D 2 u) = 0 in B 1 is C 1,α0 , it is in particular C 1,α0 at 0, that is to say there exists C 0 > 0 such that for all r ∈ (0, 1), there exists p ∈ R d such that osc

Br (u -p • x) ≤ C 0 r 1+α0 .
We then pick ρ ∈ (0, 2 -γ-1 ) such that

C 0 ρ α0 ≤ 1 4 . (4)
Given a solution u of F = 0 in B 1 , we also pick

p ρ = p ρ (u) such that osc Bρ (u -p ρ • x) ≤ 1 4 ρ. (5)

Reduction to the improvement of flatness lemma

In order to prove Lemma 1, we prove an improvement of flatness lemma; it is the core of the paper. It basically says that if p • x + u solves (1) in B 1 and the oscillation of u in B 1 is less than 1, say, then the function u can be approximated by a linear function in a smaller ball with an error that is less than the radius of the ball. We make this statement rigourous and quantitative now.

Lemma 2 (Improvement of flatness lemma). There exists ε 0 ∈ [0, 1] and ρ ∈ (0, 1) only depending on γ, ellipticity constants and dimension such that, for any p ∈ R d and any viscosity solution u of

|p + ∇u| γ F (D 2 u) = f in B 1 (6) such that osc B1 u ≤ 1 and f L ∞ (B1) ≤ ε 0 , there exists p ′ ∈ R d such that osc Bρ (u -p ′ • x) ≤ 1 2 ρ.
It is important to remark that the choice of ρ and ε 0 works for all vectors p in the previous Lemma. No constant depends on p.

We now explain how to derive Lemma 1 from Lemma 2.

Proof of Lemma 1. For k = 0, we simply choose p 0 = 0 and ( 3) is guaranteed by the assumption osc u ≤ 1.

We choose α > 0 small such that ρ α > 1/2. We assume now that k ≥ 0 and that we constructed already p k ∈ R d such that (3) holds true. We then consider for x ∈ B 1 ,

u k (x) = r -1-α k [u(r k x) -p k • (r k x)].
The vector p k is such that osc B1 u k ≤ 1. Moreover, u k satisfies

|r -α k p k + Du k | γ r 1-α F (r α-1 D 2 u k ) = f k (x) with f k (x) = r 1-α(1+γ) k f (r k x). In particular, f k L ∞ (B1) ≤ ε 0 as long as α < 1/(1 + γ).
Notice that the function r 1-α F (r α-1 X) has the same ellipticity constants as F (X), therefore the C 1,α0 estimates are conserved by this scaling. Now we apply Lemma 2 and get q k+1 such that osc

Bρ (u k -q k+1 • x) ≤ 1 2 ρ
Because of our choice of α, we then obtain p k+1 such that osc

Br k+1 (u -p k+1 • x) ≤ r 1+α k 1 2 ρ ≤ r 1+α k+1 .
The proof is now complete.

Equi-continuity of rescaled solutions

The proof of Lemma 2 relies on the following lemma in which the modulus of continuity of solutions of ( 6) is controlled.

Lemma 3 (Modulus of continuity independent of p). For all r > 0, there exist β ∈ (0, 1) and C > 0 only depending on ellipticity constants, dimension, γ and r and such that for all viscosity solution u of (6) with osc B1 u ≤ 1 and

||f || L ∞ (B1) ≤ ε 0 < 1 satisfies [u] β,Br ≤ C. ( 7
)
In particular, the modulus of continuity of u is controlled independently of p.

Proof of Lemma 3

This lemma is a consequence of the two following ones.

Lemma 4 (Lipschitz estimate for large p's). Assume u solves (6) with osc B1 u ≤ 1 and ||f || L ∞ (B1) ≤ ε 0 < 1. If |p| ≥ 1/a 0 , with a 0 = a 0 (λ, Λ, d, γ, r), then any viscosity solution u of ( 6) is Lipschitz continuous in B r and

[u] 1,Br ≤ C (8)
where C = C(λ, Λ, γ, d, r).

Lemma 5 (Hölder estimate for small p's). Assume u solves [START_REF]Regularity and uniqueness of the first eigenfunction for singular fully nonlinear operators[END_REF] We now turn to the proof of these two lemmas.

with osc B1 u ≤ 1 and ||f || L ∞ (B1) ≤ ε 0 < 1. If |p| ≤ 1/a 0 ,
Proof of Lemma 4. We rewrite (6) as

|e + aDu| γ F (D 2 u) = f
where e = p/|p| and a = 1/|p| ∈ [0, a 0 ] and

f = |p| -γ f.
Remark that f L ∞ (B1) ≤ a γ 0 ε 0 . We use viscosity solution techniques first introduced in [START_REF] Ishii | Viscosity solutions of fully nonlinear secondorder elliptic partial differential equations[END_REF]. For all x 0 ∈ B r/2 , we look for L 1 > 0 and L 2 > 0 such that

M = sup x,y∈Br u(x) -u(y) -L 1 ω(|x -y|) -L 2 |x -x 0 | 2 -L 2 |y -x 0 | 2 ≤ 0 where ω(s) = s -ω 0 s 3 2 if s ≤ s 0 := (2/3ω 0 ) 2 and ω(s) = ω(s 0 ) if s ≥ s 0 .
We choose ω 0 such that s 0 ≥ 1. We notice that if we proved such an inequality, the Lipschitz constant is bounded from above by any L > L 1 .

We argue by contradiction by assuming that M > 0. If (x, y) ∈ Br × Br denotes a point where the maximum is reached (recall that u is continuous and its oscillation is bounded), we conclude that

L 1 ω(|x -y|) + L 2 |x -x 0 | 2 + L 2 |y -x 0 | 2 ≤ osc B1 u ≤ 1. We choose L 2 = (4/r) 2 , so that |x -x 0 | ≤ r 4 and |y -x 0 | ≤ r 4 .
With this choice, we force the points x and y where the supremum is achieved to be in B r . Remark also that the supremum cannot be reached at (x, y) with x = y, otherwise M ≤ 0. Hence, we can write two viscosity inequalities.

Before doing so, we compute the gradient of the test-function for u with respect to x and y at (x, y)

q x = q + 2L 2 x
and q y = q -2L 2 y

where q = L 1 ω ′ (|δ|) δ, δ = x -y and δ = δ/|δ|. To get appropriate viscosity inequalities, we shall use Jensen-Ishii's Lemma in order to construct a limiting sub-jet (q x , X) of u at x and a limiting super-jet (q y , Y ) of u at y such that the following 2n × 2n matrix inequality holds for all ι > 0 small enough (depending on the norm of Z):

X 0 0 -Y ≤ Z -Z -Z Z + (2L 2 + ι)I where Z = L 1 D 2 (ω(|•|))(x-y).
We refer the reader to [START_REF] Barles | Hölder continuity of solutions of second-order non-linear elliptic integro-differential equations[END_REF][START_REF] Barles | Lipschitz regularity of solutions for mixed integro-differential equations[END_REF] for details. Applying the previous matrix inequality as a quadratic form inequality to vectors of the form (v, v) we obtain

(X -Y )v, v ≤ (4L 2 + ι)|v| 2 . (9)
Therefore X -Y ≤ (4L 2 + ι)I, or equivalently, all eigenvalues of X -Y are less than 4L 2 + ι. On the other hand, applying now the particular vector ( δ, -δ), we obtain

(X -Y ) δ, δ ≤ (4L 2 +ι-6ω 0 L 1 |x-y| -1/2 )| δ| 2 ≤ (4L 2 +ι-3 √ 2ω 0 L 1 )| δ| 2 . (10)
Thus, at least one eigenvalue of X -Y is less than (4L 2 + ι -3ω 0 √ 2L 1 ) (which will be a negative number). We next consider the minimal Pucci operator P -. We recall that -P -(A) equals λ times the sum of all negative eigenvalues of A plus Λ times the sum of all positive eigenvalues. Therefore, from ( 9) and ( 10), we obtain

P -(X -Y ) ≥ -λ(4L 2 + ι -3 √ 2ω 0 L 1 ) -Λ(d -1)(4L 2 + ι) ≥ -(λ + (d -1)Λ)(4L 2 + ι) + 3 √ 2ω 0 λL 1 .
We now write the two viscosity inequalities and we combine them in order to get a contradiction.

|e + aq x | γ F (X) ≤ f (x) |e + aq y | γ F (Y ) ≥ f (y).
We will choose a 0 small enough depending on L 1 and L 2 so that |aq x | ≤ 1/2 and |aq y | ≤ 1/2. The constant L 1 will be chosen later and its value does not depend on this choice of a 0 . In particular, we have We now use that F is uniformly elliptic to write

F (X) ≥ F (Y ) + P -(X -Y ).
Combining the previous displayed inequalities and recalling ||f ||

L ∞ ≤ ε 0 yields 3 √ 2ω 0 λL 1 ≤ (λ + Λ(d -1))(4L 2 + ι) + 2 γ+1 ε 0 .
Choosing L 1 large enough depending on λ, Λ, d, γ, and the previous choice of L 1 (which depends on r only), we obtain a contradiction. Note that this choice of L 1 does not depend on the previous choice of a 0 , so we should first choose L 1 large and then a 0 small. The proof of the lemma is now complete.

Proof of Lemma 5. The equation can be written as G(Du, D 2 u) = f with G(q, X) = |p + q| γ F (X).

In particular, if |q| ≥ 2a -1 0 then |p + q| γ ≥ a -γ 0 . In particular,

G(q, X) = 0 |q| ≥ 2a -1 0 ⇒ P + (D 2 u) + a γ 0 |f | A γ ≥ 0 P -(D 2 u) -a γ 0 |f | A γ ≤ 0
where P ± denote extremal Pucci's operators associated with the ellipticity constants of F . We know from [START_REF] Imbert | Alexandroff-Bakelman-Pucci estimate and Harnack inequality for degenerate/singular fully non-linear elliptic equations[END_REF] that there exists β 1 ∈ (0, 1) and C 1 only depending on r, dimension and ellipticity constants of F such that

[u] β1,Br ≤ C 1 osc B1 u + max(2a -1 0 , f L n (B1) ) ≤ C 1 (1 + max(2a -1 0 , ε 0 )).
The proof of the lemma is now complete.

Proof of the improvement of flatness Lemma

With Lemma 3 in hand, we can now turn to the proof of Lemma 2.

Proof of Lemma 2. We argue by contradiction and we assume that there exist sequences

ε n → 0, p n ∈ R d , f n such that f n L ∞ (B1) ≤ ε n , and u n satisfying (6) with (p, f ) = (p n , f n ) such that for all p ′ ∈ R d , osc Bρ (u n -p ′ • x) > 1 2 ρ.
Remark that f n → 0 as n → ∞.

Thanks to Lemma 3, we can extract a subsequence of (u n ) n converging locally uniformly in B 1 to a continuous function u ∞ . Remark that we have in particular for all p

′ ∈ R d , osc Bρ (u ∞ -p ′ • x) > 1 2 ρ. (11)
We are going to prove that u ∞ satisfies F (D 2 u ∞ ) = 0 in B 1 . This will imply that there exists a vector p ρ such that (5) holds true. This is the desired contradiction with [START_REF]Harnack inequality for singular fully nonlinear operators and some existence results[END_REF].

To prove that F (D 2 u ∞ ) = 0 in B 1 , we now distinguish two cases.

If we can extract a converging subsequence of p n , then we also do it for u n and we get at the limit

|p ∞ + ∇u ∞ | γ F (D 2 u ∞ ) = 0 in B 1 .
In particular, we have F (D 2 u ∞ ) = 0 in B 1 (see Lemma 6 in the next subsection).

If now we cannot extract a converging subsequence of p n , then |p n | → ∞ and in this case, we extract a converging subsequence from e n = p n /|p n | and dividing the equation by |p n | we get at the limit

|e ∞ + 0∇u ∞ | γ F (D 2 u ∞ ) = 0 in B 1
for e ∞ = 0 so that we also have in this case F (D 2 u ∞ ) = 0 in B 1 . The proof of the lemma is now complete.

5 Viscosity solutions of |∇u| γ F(D 2 u) = 0

In the previous subsection, we used the following lemma.

Lemma 6. Assume that u is a viscosity solution of

|p + ∇u| γ F (D 2 u) = 0 in B 1 .
Then u is a viscosity solution of F (D 2 u) = 0 in B 1 .

Proof. We reduce the problem to p = 0 as follows. The function

v = u + p • x satisfies |∇v| γ F (D 2 v) = 0 in B 1 .
If we proved the result for p = 0, we conclude that

F (D 2 u) = F (D 2 v) = 0 in B 1 .
We now assume that p = 0. We only prove the super-solution property since the sub-solution property is very similar.

Consider a test-function φ touching u strictly from below at x ∈ B 1 . We assume for simplicity that x = 0. Hence, we have, φ(0) = u(0) = 0 and φ < u in B r \ {0} for some r > 0. We can assume without loss of generality that φ is quadratic: φ(x) = 1 2 Ax • x + b • x. If b = 0, then we get the desired inequality: F (A) ≥ 0.

If b = 0, we argue by contradiction by assuming that F (A) < 0. Since F is uniformly elliptic, this implies that A has a least one positive eigenvalue. Let S be the direct sum of eigensubspace corresponding to non-negative eigenvalues. Let P S denote the orthogonal projection on S. We then consider the following test function ψ(x) = φ(x) + ε|P S x|.

Since φ < u in B r , then u -ψ reaches its minimum at x 0 in Br in the interior of the ball for ε small enough. We claim first that P S x 0 = 0. Indeed, if this is not true, we use the fact that Hence Ax 0 + εe 0 = 0 and we get the following contradiction

F (A) ≥ F (A + εB) ≥ 0.
The proof is now complete.

  then u is β-Hölder continuous in B r and [u] β,Br ≤ C where β = β(λ, Λ, d, r, a 0 ) and C = C(λ, Λ, d, r, a 0 ).

1 2 ≤

 12 min(|e + aq x |, |e + aq y |).

|P s x| = min |e|=1 e

 |e|=1 • P S x and we deduce that for all e ∈ R d such that |e| = 1, the test-function φ(x) + εe • P S x touches u at x 0 and we thus have for all such e's|Ax 0 + εP S e| γ F (A) ≥ 0.Hence, there exists such an e such that Dφ(x 0 ) + εP S e = 0 and we get the contradiction F (A) ≥ 0.Since P S x 0 = 0, ψ is smooth in a neighbourhood of x 0 and we get the following viscosity inequality|Ax 0 + εe 0 | γ F (A + εB) ≥ 0where e 0 = P S x 0 /|P S x 0 | and B ≥ 0 since x → |P S x| is convex. Remark next that (Ax 0 + εe 0 ) • P S x 0 = P S Ax 0 • x 0 + ε|P S x 0 | ≥ ε|P S x 0 | > 0.

Acknowledgements. The authors are grateful to I. Birindelli and F. Demengel for pointing out this open problem and for the fruitful discussions they had together.

Luis Silvestre was partially supported by the Sloan fellowship and NSF grants DMS-1065979 and DMS-1001629.


Ingmar Steiner
email: ingmar.steiner@inria.fr

Korin Richmond

Ian Marshall
email: ian.marshall@ed.ac.uk

Calum D Gray
email: calum.gray@ed.ac.uk

The Magnetic Resonance Imaging subset of the mngu0 articulatory corpus

Keywords: numbers 43, 70, Aj, 43, 70, Jt [AL] Date

HAL is

Introduction

Technology applications that use speech data, for example, text-to-speech (TTS) synthesis, automatic speech recognition (ASR), focus almost exclusively on acoustics. However, the acoustic signal produced by a human speaker crucially depends on the shape of the speaker's vocal tract and the movements of articulators such as the tongue or lips. Techniques such as motion capture or medical imaging can provide valuable articulatory data to supplement the acoustic speech signal. Applications such as TTS synthesis or ASR can exploit such data to improve their modeling of human speech production.

However, recording articulatory data is unfortunately not as straightforward as recording an acoustic signal. Specialist facilities and expertise are typically required, which make articulatory recordings more expensive. Indeed, the recording process itself can often be tricky, with practical complications that may also increase the burden on the subject. Consequently, the few articulatory corpora which have been made freely available (e.g. [START_REF] Munhall | X-ray film database for speech research[END_REF][START_REF] Westbury | X-Ray Microbeam Speech Production Database User's Handbook Version 1[END_REF][START_REF] Wrench | A multi-channel/multi-speaker articulatory database for continuous speech recognition research[END_REF][START_REF] Narayanan | A multimodal real-time MRI articulatory corpus for speech research[END_REF] have been well received and extensively used by the research community. Each of these contains articulatory and acoustic data for a range of speakers, but the data for any individual speaker may be insufficient for certain applications.

The mngu0 articulatory corpus addresses this shortfall and provides a large amount of articulatory data from a single speaker of British English. The corpus consists of multiple subsets of data acquired in different modalities, including a large number of sentences recorded using video and electromagnetic articulography (EMA), dental casts, and magnetic resonance imaging (MRI) scans of the speaker's vocal tract. Taken together, they provide both high-speed (200 Hz) articulatory movement data during running speech, and the speaker's vocal tract geometry in three dimensions, and offer a valuable resource to the speech community. While the EMA portion of the mngu0 corpus has previously been described and made publicly available [START_REF] Richmond | Announcing the electromagnetic articulography (day 1) subset of the mngu0 articulatory corpus[END_REF], the purpose of this paper is to announce the availability of the corresponding MRI data for this speaker and describe the details of its acquisition.

MRI data

The purpose of MRI scanning the mngu0 speaker was two-fold. First, we wanted to capture the 3D geometry of the speaker's vocal tract, as well as representative configurations of the speaker's articulators for producing a range of speech sounds. A series of volumetric MRI scans was performed for this purpose. Second, we wanted to investigate and capture the effects of coarticulation, for which a set of dynamic MRI scans was performed. Overall, therefore, by adapting the procedure described in [START_REF] Birkholz | Vocal tract model adaptation using magnetic resonance imaging[END_REF], a prompt list was designed that consisted of sustained vowels and consonants, as well as dynamic VCV transitions, elicited by repetitive production of consonant-vowel (CV) syllables.

The scanner used for this study was a GE Medical Systems Signa HDx 1.5T. The speaker was placed in the scanner in supine position and fitted with a head-and-neck radio frequency (RF) coil. All of the scans were completed within one 120 min session, with a short break between the sustained and dynamic scans.

The mngu0 subject is a trained, professional speaker. As well as general advantages in terms of the level of performance obtained, this was beneficial for the MRI scans in particular, as he was able to sustain production of vowels for 20 s on average, and of repetitive utterances for around 10 s to 15 s. The speaker was therefore capable of producing each prompt over the entire duration of a scan.

As Figure 1 shows, the region of interest (ROI) is clearly visible in the resulting image data, extending from the lips to the rear wall of the pharynx in anterior-posterior direction, from the larynx to the nasal cavity in inferior-superior direction, and (in the volumetric scans) laterally between the mandibular joints. This figure also exhibits an aliasing artifact; a segment of the back of the speaker's head and neck appears at the anterior edge of the images, overlapping with the speaker's nose. This occurs when the imaged anatomy extends outside the field of view (FOV). But since we wanted to gain good coverage of the vocal tract, and the aliasing does not impact image quality in the ROI, we chose this configuration and the aliasing can safely be ignored.

Due to the acoustic conditions within the scanning chamber and the lack of a noise-canceling fiber-optic (FO) microphone, no simultaneous acoustic recordings were possible during the MRI session. For this reason, a separate acoustic recording session was conducted as well (see Section 3).

Volumetric scans

Static, 3D scans of 13 sustained vowels and 15 sustained consonants were acquired with a fast gradient echo sequence having 26 sagittal slices of 4 mm thickness, repetition time (TR) 51 ms, echo time (TE) 3 ms, flip angle 30°, and field of view (FOV) 280 mm reconstructed as 256×256 pixels.

The prompts are listed in Table 1a. For each scan, the speaker produced the corresponding prompt, maintaining audible production of the target phone during the entire acquisition.

Figure 1 and Mm. 1 illustrate one raw volumetric scan [ɑ], while Figure 2 and Mm. 2 show a 3D vocal tract mesh extracted from that scan, along with a surface rendering of the face and head (the aliasing artifact has been removed).

Dynamic mid-sagittal scans

Dynamic scans of 16 consonants in three vocalic contexts [ɑ, i, u] were acquired with a fast gradient echo sequence similar to the above, but with a single midline sagittal slice of thickness 10 mm, and TR 4 ms and TE 2 ms, enabling 40 consecutive time frames to be acquired in 10 s.

The prompts are listed in Table 1b. The speaker produced each prompt as repetitive CV syllables, synchronizing production of the target consonants to the scans by timing their stable phase with the noise emitted by the MRI scanner.

An example of the dynamic data for nasals [m, n, ŋ] is displayed in Figure 3. Each panel shows an overlay of 30 MRI frames, providing an "averaged" image that offers an enhanced view of the articulators.

Dental reconstruction

Since the teeth are invisible in MRI,1 a final volumetric scan was acquired using blueberry juice (which has favorable NMR properties, due to its high manganese content) to distinguish oral cavity from teeth. This produced a negative 3D scan of the teeth for subsequent dental reconstruction.

For this dental scan, the speaker lay prone in the scanner, and filled his mouth with the juice by sucking it from a bottle through a flexible tube. While this did produce images clearly showing the teeth, there was no time left in the session for a long acquisition, and therefore the spatial resolution of the dental scan is no higher than that of the other volumetric scans.

Acoustic reference recordings

To compensate for not being able to simultaneously record the acoustic signal, and to give the speaker opportunity to familiarize himself with the prompt list and the general procedure in an informal, non-clinical environment, high-quality acoustic reference recordings were made on the day before the MRI session.

The acoustic recording session took place in a sound-proofed room at the Informatics Forum, University of Edinburgh. The prompts were recorded using a DPA Type 4035 microphone mounted on a headset. The microphone signal was captured directly to hard disk using an EDIROL UA-25 audio interface connected to a laptop computer. The recordings were made with a 96 kHz sampling rate at 24 bit quantization. .

[i]

.

[ɑ]

.

[u]

.

[m]

.

[n]

.

[ŋ]

. . . . . . . . .

Figure 1 :

 1 Figure 1: Cutaway volume rendering of raw volumetric data for [ɑ].

Figure 2 :

 2 Figure 2: 3D vocal tract extracted from [ɑ] scan, shown with surface rendering.

Figure 3 :

 3 Figure 3: Overlay of 30 mid-sagittal frames of [m, n, ŋ] (rows) dynamically produced in vocalic context [i, ɑ, u] (columns). The critical articulators for each consonant (lips, tongue tip, and tongue dorsum, respectively) achieve occlusion, while the tongue body assumes a different target shape in each vocalic context. The velum is lowered in all conditions, allowing the speaker to sustain production of the nasal.

EL106 J. Acoust. Soc. Am. 131 (2), February 2012 © 2012 Acoustical Society of America

J. Acoust. Soc. Am. 131 (2), February 2012 Steiner et al.: Magnetic resonance imaging set mngu0 corpus EL107

EL108 J. Acoust. Soc. Am. 131 (2), February 2012 Steiner et al.: Magnetic resonance imaging set mngu0 corpus

their nuclear magnetic resonance (NMR) properties make them nearly indistinguishable from the surrounding air J. Acoust. Soc. Am. 131 (2), February

Steiner et al.: Magnetic resonance imaging set mngu0 corpus EL109

EL110 J. Acoust. Soc. Am. 131 (2), February 2012 Steiner et al.: Magnetic resonance imaging set mngu0 corpus

J. Acoust. Soc. Am. 131 (2), February 2012 Steiner et al.: Magnetic resonance imaging set mngu0 corpus EL111

Acknowledgments

Imaging was carried out at the Brain Research Imaging Centre, Edinburgh (http://www. bric.ed.ac.uk), Division of Clinical Neurosciences, University of Edinburgh, Western General Hospital, Edinburgh, a core area of the Wellcome Trust Clinical Research Facility and part of the SINAPSE collaboration (http://www.sinapse.ac.uk).

This work was supported by Marie Curie Early Stage Training Site "EdSST" (MEST-CT-2005-020568) and EPSRC grant EP/E027741/1 ("ProbTTS").

Published Online 13 January 2012

The speaker read out the prompt list twice, once standing upright, and again in supine position. This was done to allow comparison in the acoustic domain between these two postures. The supine recordings were made to ensure that the audio matched the articulatory configuration during the MRI scans, where gravity and posture can influence articulation (e.g. [START_REF] Kitamura | Difference in vocal tract shape between upright and supine postures: Observations by an open-type MRI scanner[END_REF].

The acoustic speech data has been manually segmented into prompts.

Distribution

The raw data from the MRI session has been anonymized to protect the privacy of the speaker, but is otherwise unmodified from the DICOM files produced at the MRI facility.

The volumetric scans and dynamic scans will be made freely available for research purposes, as separate downloads at a dedicated website, http://mngu0.org/. The acoustic data and the corresponding segmentation data will likewise be made available.