Christiane Deval

Frédéric Capel

Brigitte Laillet

Cécile Polge

Daniel Bechet

Daniel Taillandier

Didier Attaix

Lydie Combaret

DHA-supplementation prior to fasting prevents muscle atrophy in mice

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DHA-supplementation prior to fasting prevents muscle atrophy in mice

Deval Christiane 1,2 , Capel Frédéric 1,2 , Laillet Brigitte 1,2 , Polge Cécile 1,2 , Béchet Daniel 1,2 , Taillandier Daniel 1,2 , Attaix Didier 1,2 § , Combaret Lydie

Presenting author

Background and aims

Muscle wasting prevails in numerous diseases (e.g. diabetes, cardiovascular and kidney diseases, COPD,…) and increases healthcare costs. A major clinical issue is to devise new strategies preventing muscle wasting. We hypothesized that a longterm docosahexaenoic acid (DHA) supplementation prior to fasting may preserve muscle mass in vivo.

Methods

Six-wk-old C57BL/6 mice were fed a DHA-enriched or a control diet for 8 weeks and then fasted for 48 h. The effect of DHA on i) muscle energy stores (glycogen, triglycerides (TG)), ii) muscle mass, and iii) Akt and AMPK signaling pathways involved in the control of protein and energy metabolism has been addressed. The regulation of the formation and the fate of lipid stores has been also evaluated.

Results

Feeding mice a DHA-enriched diet prior to fasting elevated muscle glycogen contents without any change in TG levels, reduced muscle wasting, blocked the 55 % decrease in Akt phosphorylation, and reduced by 30-40% the activation of AMPK, ubiquitination or autophagy. The DHA-enriched diet fully abolished the fasting induced-mRNA over-expression of the endocannabinoid receptor-1. Finally, DHA prevented or modulated the fasting-dependent increase in muscle mRNA levels for Rab18, PLD1 and perilipins, which determine the formation and fate of lipid droplets, in parallel with muscle sparing.

Conclusion:

These data suggest that long-term DHA supplementation increased energy stores that can be efficiently mobilized, and thus preserved muscle mass in response to fasting through the regulation of Akt-and AMPK-dependent signaling pathways for reducing proteolysis activation. Whether a nutritional strategy aiming at increasing energy status may shorten recovery periods in clinical settings remains to be tested.


Centre

Reestimating the role of leaching for loss of water-soluble vitamins in processed vegetables

Health benefits of fruit and vegetables are mainly linked to their contribution to the diet for fibers, minerals (K, Ca) and vitamins, especially vitamin C (ascorbic acid), vitamin B9 (folate) and provitamin A (carotenoids). Vegetables in particular, are almost systematically consumed after processing, whether industrial (freezing, canning…) or domestic (boiling, steaming, microwaving…). Two mechanisms explain water-soluble vitamins losses from vegetables during heat-treated in presence of water, namely degradation and leaching.

Structure of the folate vitamers Ascorbic acid

Folates were analyzed along an industrial canning chain (5 repetitions), at 5 sampling points.

During industrial processing, total folate content decreased significantly, by 30% in green beans after sterilization, with 20% of the initial amount being transferred into the covering liquid just after canning. This loss by transfer was confirmed by analysis of raw materials, canned green beans and covering liquid 1 month after processing, over a production season: concentrations were equilibrated between vegetable and covering liquid, with limited chemical degradation.

Leaching is highly relevant to water-soluble vitamins losses when heat treatments occur in presence of water or a covering liquid. Leaching takes place particularly at the start of any heating step. This is modulated by vegetable's shape and size and temperature. Quantifying losses in water-soluble vitamins therefore requires more emphasis on the determination of diffusion parameters. Losses in vitamin C were followed in fresh peas and Brussel sprouts, submitted to the same thermal treatment either sous-vide (hermetically sealed pouches) or immersed in water in open nets. This allowed to compare the losses due to thermal degradation and to leaching. Much higher losses were observed when the vegetables were in contact with water, specially for peas, the smaller vegetable Fick's second law and first order degradation kinetics models were used to quantify the relative impact of the two mechanisms.

Impact of common domestic cooking practices was assessed for frozen vegetables. Higher losses were observed for cooking in boiling water for both vitamin C and folates and for all vegetables.

  mg/kg as folic acid)

  Delchier N., Reich M. & Renard C.M.G.C.: Impact of cooking methods on folates, ascorbic acid and lutein in green beans (Phaseolus vulgaris) and spinach (Spinacea oleracea). LWT-Food Science and Technology 49 (2012) 197-201. Delchier, N., Ringling C., Le Grandois J., Aoude-Werner D., Galland R., Georgé S., Rychlik M. & Renard C.M.G.C.: Effects of industrial processing on folate content in green vegetables. Food Chemistry 139 (2013) 815-824 Delchier N., Ringling, C., Maingonnat, J.-F., Rychlik, M. & Renard C.M.G.C. Mechanisms of folate loss during processing: diffusion vs heat degradation. Food Chemistry, in press

  PC: pressure cooker; BW: boiling water; ST: steaming; MW: microwave Catherine M.G.C. Renard, Nicolas Delchier, Hannah Brick, Caroline Garcia, Sylvie Bureau, ---Measured loss by degradation; Measured loss by degradation plus leaching; Estimated impact of leaching Concentration ratio to initial C/C0


de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Integrative and multiscale study of fleshy fruit texture

Marc Lahaye INRA, UR1268 Biopolymères Interactions Assemblages, Nantes, France

Fleshy fruit texture is a key quality trait that determines consumer preferences and food processing. It encompasses different descriptors, which depends on structural elements at the tissue and cell levels, such as cell wall characteristics, tissue architecture and water distribution. In this presentation, our recent work on apple texture will be reviewed.

Expression of cell wall and turgor related genes was analysed along with cell wall polysaccharides composition and structure as well as mechanical properties and sensory texture evaluations. Beside cell wall pectin modifications, changes in hemicellulose structure and expression of turgor pressure related genes were observed during apple development and ripening. The results showed that mealiness, which is a major texture defect for consumer and processing, could be associated with different cell wall chemical characteristics and likely involve distinctive water flux in cells.

Non-destructive assessment of tissue architecture and water distribution in apple tissue by quantitative magnetic resonance imaging was combined with analysis of mechanical, histological and biochemical changes in texture-contrasted varieties during storage. The results revealed that soft to mealy fruit had different air space volumes and water cellular distributions.

The outcome of these studies emphasize that the control of apple texture requires taking into account complex combinations of cell wall, tissue architecture and water distribution determinants. This complexity makes genetic improvement for this key quality trait challenging as well as the development of non-destructive measurements of specific texture descriptors.


est destinée au dépôt

Nanocellulose : applications in food ?

Bernard Cathala

To cite this version: Bernard Cathala. Nanocellulose : applications in food ?. Cellulose Nanomaterials -A Path Towards Commercialization, United States Department of Agriculture -USDA (USA) (USDA). USA. National Nanotechnology Initiative (NNI)., May 2014, Washington, United States. 10 p. hal-02799544 10μm 10μm 10μm 5μm 10μm 10μm 10μm 10μm 10μm 10μm 10μm 10μm 10μm 5μm 5μm 5μm


Virginie Hugouvieux
email: virginie.hugouvieux@supagro.inra.fr

Walter Kob

Structuring of polymer gels via catalytic reactions

Keywords: polymer, gelation, simulations, catalysis, cluster

Certain biopolymers transform into a gel because of a catalytic reaction in which freely moving catalysts, called enzymes, convert repulsive monomers of the polymers into attractive ones. These attractive monomers stick to each other and make that the polymer sol slowly transforms into a gel. Important examples for such systems occur in plant cell walls where some polysaccharides (pectins) undergo gelation due to the presence of an enzyme (pectin methylesterase) [1].

We have developed a simple model (bead-spring polymers+soft particles) that can serve to describe such systems. Using large scale computer simulations we study how the polymer sol (tB  0 in Fig. 1) becomes unstable due to the increasing fraction of attractive monomers and transforms at high temperatures into a dense liquid (i.e. phase separation, tB  7.10 4 ) or into a gel at low temperatures (tB  3.10 3 ).

We find that before the system reaches at long times one of these two phases, it forms at intermediate times a rather regular cluster phase (tB  10 2 ). We rationalize this finding by a competition between two mechanisms: Nucleation of the attractive particles to form clusters and the slow relaxation dynamics of the coarsening of the polymer sol. We show that the temperature dependence of the time scale of these two processes can be understood semi-quantitatively from the bare potential between the constituent particles, i.e., monomers and catalysts, which governs the reactions. Finally we discuss how this cluster phase can be stabilized, thus giving rise to a gel with a non-trivial internal structure.

Fig. 1 :

 1 Fig. 1: Time evolution of the polymer structure (catalysts: green beads; repulsive monomers: blue beads; attractive monomers: red beads)


Alain Roques

Marie-Anne Auger-Rozenberg

Marie-Anne Auger

Recent progress in the early warning and detection of potentially-invasive alien insects through sentinel plantings in other continents and extensive surveys in major European botanical gardens

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Recent progress in the early warning and detection of potentially invasive alien insects through sentinel plantings in other continents and extensive surveys in major European botanical gardens Alain ROQUES & Marie-Anne AUGER-ROZENBERG INRA UR 0633, Zoologie Forestière, Orléans-F-45000, France Identifying appropriate and effective tools for the prior warning and early detection of exotic insect arrivals is urgently needed but extremely challenging because the potential invaders are often unknown, sometimes even to science. Recent sentinel plantings of European trees in China demonstrated one method of creating lists of insects that may need to be prevented from entering Europe but the experiments also revealed many limitations. Extensive surveys of arboreta provide an alternative as illustrated by findings of alien seed insects. Botanical gardens at the Paris Museum of Natural History and at Kew were sampled during 2013-2015 for wild rose hips, Rosa spp. Up to 32 species in Paris and 70 in Kew were surveyed. X-rays of ca. 60000 seeds were extracted and revealed a significant infestation by insect larvae (up to 28% in Paris but only 6% in Kew). In addition to a native seed chalcid, adult emergence revealed infestations by 4 alien species of Megastigmus not previously recorded in Western Europe. Two species confirmed by molecular analyses originated from North America and one came from Eastern Europe but the origin of the fourth is currently unknown. The exchange/trade in seeds is the most likely pathway.


Jérôme Roussel

Pascal Bertolino

GRAPH-BASED DEINTERLACING

Keywords: spatial deinterlacing, Laplacian extrema, graph, edge interpolation

This article presents a new algorithm for spatial deinterlacing that could easily be integrated in a more complete deinterlacing system. The spatial interpolation process often fails to reconstruct edges that are diagonal or close to horizontal, leading to highly visible artifacts. Our solution aims at preserving the linear structure continuity. It connects pieces of horizontal edges coming from neighboring lines to build graphs which are then simplified in branches that represent the linear structure of the scene. These branches give the exact direction in which the interpolation has to be done. For the rest of the image, a traditional directional spatial interpolation gives satisfactory results already. Although the number of pixels interpolated with this method is relatively small, the overall image quality is subjectively well improved.

INTRODUCTION

Interlaced video still is widely used by broadcasters. Interlacing has been chosen to reduce bandwidth in order to conciliate framerate and resolution. However, new flat panels like plasma or L.C.D are progressive ones and thus require the display of the whole image at time t. There is thus a high interest in deinterlacing methods that allow a conversion from interlaced to progressive. They can be classified in two major families, the methods without motion compensation and the ones with motion compensation [START_REF] Sugiyama | A method of de-interlacing with motion compensated interpolation[END_REF][START_REF] Sugiyama | Improvement of motion compensated inter-field interpolation method for de-interlacing[END_REF]. We will focus here on the methods without motion compensation and more precisely the adaptive methods. They consist in going towards the temporal method in areas where there is no movement and towards the spatial method in moving areas [START_REF] Koivunen | Motion detection of an interlaced video signal[END_REF][START_REF] Lin | Motion adaptive interpolation with horizontal motion detection for deinterlacing[END_REF][START_REF] Yu | Motion adaptive deinterlacing with accurate motion detection and anti-aliasing interpolation filter[END_REF]. The spatial interpolation process has its own limitations, mainly on the rendering of edges close to horizontal [START_REF] Almog | Spatial de-interlacing using dynamic time warping[END_REF][START_REF] Ballester | An inpainting based deinterlacing method[END_REF] and the technique proposed in this article addresses this particular point. In the first part, we are going to make a state of the art showing that the limitation mentioned cannot be easily overcome. Then, in a second part we will describe the different steps of our method, based on the construction and simplification of the Laplacian extrema. Finally, we will present our results.

EXISTING METHODS

Many methods and solutions have been proposed to perform spatial interpolation [START_REF] De Haan | Deinterlacing -an overview[END_REF]. Majority of these methods try to make the interpolation along the direction of contours, using the so-called E.L.A method (Edge-based Line Averaging) [START_REF] Doyle | Interlaced to sequential conversion for edtv applications[END_REF]. This latter method detects the best direction Dir for interpolation within a window centered on the missing pixel and then makes the interpolation according to the found direction :

f (i, j) = f (i -1, j -Dir) + f (i + 1, j + Dir) 2 . (1)
f represents the interlaced input image and f the interpolated output image. (i, j) are the spatial coordinates where i represents the line position number and j the column position number. f is defined only for half of the lines, i.e. for i even or odd.

Although it leads to a better interpolation of contours, the method still has several limitations. Actually, the correlation is done at the local level and remains quite sensitive to noise. The edge direction thus happens to be wrong which can lead to very annoying artifacts since it disrupts the structure of thin lines or edges. Many alternatives of this method [START_REF] Chen | Efficient deinterlacing algorithm using edge-based line average interpolation[END_REF] make it possible to correct wrong direction interpolation for a majority of image pixels, for instance by computing the correlation between groups of pixels instead of pixel to pixel (figure 1). However, the results of these methods always remain dependent on the size of the window used, that determines the maximum angle allowed for the reconstruction of the contours. On the other side, the larger the window is, the higher the risk of bad interpolation [START_REF] Yoo | Direction-oriented interpolation and its application to de-interlacing[END_REF]. Different existing features and metrics try to control an adequate window size and introduce weights to reduce the number of false directions [START_REF] Park | New edge dependent deinterlacing algorithm based on horizontal edge pattern[END_REF][START_REF] Byun | Edibased deinterlacing using edge patterns[END_REF]. But methods used to calculate these weights significantly increase the complexity of the solution. Actually, all these alternatives only bring a final minor improvement and do not allow to reconstruct correctly lines and edges close to horizontal. This point is even more annoying in real time where moving horizontal lines do not only look disrupted, but also instable and highly flickering. The method proposed here overcomes the limitation of the current directional spatial interpolation method since it is not based anymore on a searching window. Typically it overrules the wrong interpolation of edges close to horizontal, keeping the results of the traditional spatial interpolation where the results are already satisfactory. It can be added to a classical spatial deinterlacing method that can itself be integrated in a motion adaptive one.

EXTREMA EXTRACTION

With the existing methods, the objects that suffer a wrong reconstruction are the ones for which the contours are ramps or thin edges that are diagonally oriented, and the problem becomes stronger as the contour direction approaches the horizontal. Such contours are disconnected or aliased by the deinterlacing process. This shortcoming is all the more visible that the visual system is very sensitive to continuity. To detect the corresponding contour pixels, a vertical Laplacian filter (i.e since only the vertical direction is involved, this is the second derivative of the intensity in the vertical direction) is applied to the image : It removes the low frequencies and keeps the high frequencies in the vertical direction. The detection of the extrema is carried out locally on the known lines of the image by comparing the Laplacian value of a pixel ∆f (i, j) with its two neighboring values ∆f (i-2, j) and ∆f (i+2, j).

Any resulting pixel has one of the three types : extremum (either minimum or maximum) or not extremum (figure 2, middle). A threshold T allows to reject extrema whose absolute value is too weak to be relevant. Let H be the set of maxima pixels and L the set of minima pixels :

H = ∆f (i, j) > 0 ∆f (i, j) > max(∆f (i -2, j), ∆f (i + 2, j)) + T L = ∆f (i, j) < 0 ∆f (i, j) < min(∆f (i -2, j), ∆f (i + 2, j)) -T
On the same line, the extrema of the same type can form connected components (called segments in the sequel) according to the horizontal 2-connectivity. Since the continuation of the method is not founded on the traversing and the processing of pixels but on the traversing and the processing of segments, the traditional two-dimensional image structure is not suitable any more and is replaced with a higher level structure : namely the segment. Each segment will be a vertex of a graph and will be characterized by its coordinates (line, starting column), length, type (minimum or maximum) and neighborhood.

GRAPH CONSTRUCTION

The chosen data structure is a compromise between the memory size needed and the complexity to traverse H and L. From a data-processing point of view, it is important to note that the data structure used to store the graph is an array of rows (one row per known line of the image). Each row contains the chained list of the segments extracted in the corresponding line. Using chained lists is an easy and flexible way to handle any number of segments in each line, but a 2D array of segments could also be used.

In a first step, the image is browsed line by line and the data structure is filled with the segments and each segment is linked to its neighbors (figure 3). These links are the edges of the graph and are actually materialized by pointers.

A segment on a line i has at most 4 neighbors of the same type : NorthWest and NorthEast on line i -2 and SouthWest and SouthEast on line i + 2. By definition of the Laplacian, neighbors of the same type situated North or South of the segment cannot exist. Neighbors situated on the same line as the segment are not taken into account because no interpolation will be needed between them since the content of line i already exists. A consequence is that the link between two segments has either the direction NorthWest to SouthEast (or (\)) or NorthEast to SouthWest (or (/)).

The distance between two neighboring segments of the same type S 1 and S 2 is the Euclidean distance d(S 1 , S 2 ), cal-culated between the closest extremities of S 1 and S 2 . Two neighboring segments S 1 , S 2 are linked if they are close enough compared to their respective length L 1 , L 2 , i.e if they satisfy the following condition :

d(S 1 , S 2 ) < min(L 1 , L 2 ) * k (2)
In our experiments, k is fixed to 0.7 which is the value of Kell factor.

GRAPH SIMPLIFICATION

Once the segments have been interconnected, depending on the image content and the threshold T , many complex graphs with cycles can be obtained, each one with many segments and links. Each graph must be split into the most likely branches, i.e in such a way that only the longest linear branches are kept since they represent contours in the image rather than noise or texture. For a given segment type, a branch is either a sequel of consecutive segments linked by (\) or a sequel of consecutive segments linked by (/).

To perform the simplification process, each link must be aware of the length of its branch so that any link must be removed if needed, just with a local decision. To weight each link with the right length, two browsing passes are necessary in the array of rows. It is much more simple than a true graph traversal that would complicated to handle and time consuming. The first pass browses the array from top to bottom while the second pass does it from bottom to top (note that doing it conversely would provide the same result). In a given row each segment is processed independently, since the only information needed is the kind of links it has and their weight.

For instance, the total length of a (\) branch is obtained by starting from a segment with no NorthWest link, by following the (\) branch, and from 1, adding 1 to the branch length each time a segment is crossed. When the end of the branch is reached, the total length is known and must be propagated back to the whole branch during a second ascending pass. Now the graphs are ready to be simplified : A segment may have 0, 1, 2, 3 or 4 links (figure 4). If it has zero or one link (it is alone in a graph or this is the extremity of a branch), there is nothing to do. If it has 2 links of the same direction, it belongs to one branch and there is nothing to do. If it has 2 links of different directions or if has 3 or 4 links, this means that it belongs to two branches. In these cases, among the 2, 3 or 4 links, the one(s) corresponding to the shorter branch is (are) removed (figure 2, right). A thresholding on the branch length can also be done to process the contours at a given resolution.

INTERPOLATION

The interpolation is carried out using a forward (from West to East) traversing of each remaining branch. Let S 1 and S 2 be two connected segments described by their lengths ). The pixels to be interpolated with our method correspond to the segment S I whose extremities X start I and X end I are linearly interpolated from the extremities of the segments S 1 and S 2 (figure 5) :

Xstart I = Xstart 1 + Xstart 2 -Xstart 1 2 (3)
X end I = X end 1 + X end 2 -X end 1 2 (4)
f (i, j) = 1 2 f Y S 1 , Xstart 1 + (j -Xstart I ) × (L1 -1) (L I -1) + 1 2 f Y S 2 , Xstart 2 + (j -Xstart I ) × (L2 -1) (L I -1) (5)
j ∈ Xstart I , X end I L I is the size of the segment to interpolate X end I -X start I + 1. Y S1 and Y S2 represent the ordinates i -1 and i + 1 (or i + 1 and i -1).

RESULTS

Our algorithm has been tested on different sequences and still images which have been interlaced. On the lighthouse image, the ELA method [START_REF] Park | New edge dependent deinterlacing algorithm based on horizontal edge pattern[END_REF] does not rebuild the horizontal edges and a strong aliasing appears, while our method reconstructs the continuity of these edges (figure 6). In the Barbara image, the ELA method makes a wrong interpolation by being mistaken about direction of the trouser strips, while our method reconstructs them as the original image (figure 7). Note that the method is not curvature dependent and taht it can rebuilt straight or curved contours. However, differences between ELA and our method are not high on PSNR because only few pixels are modified as we can note in table 1. As far as processing time is concerned, the method only performs very simple computations and the amount of chained list data to be analyzed is rather small.

CONCLUSION

Our method is not based on the principle of existing methods. It thus does not suffer from their limitations. Our method aims at correcting the most unpleasant artifacts for the human eye by detecting them directly. It is based on the continuity of object borders in order to reconstruct them. Finally, our method can be added to all the traditional methods to improve their weak point without a high added cost.

Fig. 1 .

 1 Fig. 1. Principle of modified E.L.A.

Fig. 2 .Fig. 3 .

 23 Fig. 2. Laplacian Extrema detection (middle) and the resulting branches (right)

Fig. 4 .Fig. 5 .

 45 Fig. 4. The 6 link configurations (apart from a rotation)

Fig. 6 .Fig. 7 .

 67 Fig. 6. Comparison of the original image (left column), ELA (middle column) and our method (right column)


Consistent conjectures in a dynamic model of non-renewable resource management

Keywords: Dynamic game, dynamic resource management, non probabilistic beliefs, conjectural variations

We consider a model of non-renewable resource extraction where players do not know their opponents' utility functions and form conjectures on the behavior of others. Two forms of beliefs are introduced, one based on the state of the resource, the other on this state and on the others' strategy (their consumption). We focus on consistent equilibria, where beliefs must be consistent with observed past plays. Closed form expressions of the optimal policies are derived. Comparisons are made with the full information benchmark case. With strategy and state based beliefs, the agents may behave more (respectively, less) agressively than in the non-cooperative benchmark depending on the initial consumption level. When initial consumption is low, the optimal consumption path lies below that of the cooperative benchmark. We conclude the analysis by discussing the impact of public policies on the agents' choice of consumption patterns, and the robustness of the results for the case of renewable resources.

Introduction

The interplay between external and strategic effects is a main feature of many economic issues. Non-cooperative behavior in the problem of resource management is shown to lead to overconsumption of the resource compared to a socially efficient (joint) management. Different studies have confirmed this issue in problems of resource management [START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF], Fischer and Mirman (1992,1996), or [START_REF] Houba | Can negotiations prevent fish wars?[END_REF]). Several conclusions follow. Efficient consumption plans do exist, but this typically requires the use of discontinuous strategies [START_REF] Dutta | The tragedy of the commons?[END_REF]), or the threat of punishment [START_REF] Benhabib | The joint exploitation of a productive asset: A game-theoretic approach[END_REF]). In this literature decision takers are assumed to be perfectly rational and to have perfect knowledge of the different characteristics of the setting. In other words, the information is assumed to be complete. In practice, agents might lack information regarding the other people's preferences, and their rationality might be bounded.

We revisit the problem of non-renewable resource management where information is incomplete, and consequently depart from the traditional complete information approach adopted in the above studies. In our procedure decision makers know the evolution rule of the resource but do not have information about their opponents' preferences and operate based on simple beliefs about their opponents' behaviour. An important feature is that players rely on nonprobabilistic beliefs, which we also refer to as conjectures (see for instance [START_REF] Figuières | Theory of Conjectural Variations[END_REF]). Basically, players assume that a variation of their own consumption has a first order linear effect on the consumption of others.

There are different reasons for which we depart from the traditional Bayesian framework. The first one is that the Bayesian framework typically assumes the knowledge of the agents' utility functions and of the distribution of their potential types. In the present setting we want to analyse a situation where agents might not be perfectly rational. Moreover, they do not need knowledge of the utility function of others and of any distribution of types. The second reason is that empirical evidence is often inconsistent with the Bayesian framework. The notion of conjectures is used in the empirical literature as it seems to provide a general framework to model imperfect competition (see [START_REF] Slade | Empirical Games: the oligopoly case[END_REF], Lopez de Haro et al. (2007) among other examples). As such it is often used to analyse oligopoly problems [START_REF] Kalashnikov | Conjectural variations equilibrium in a mixed duopoly[END_REF]) or environmental issues (problems of pollution control as in McKitrik (1999) for example). The theoretical literature on conjectural variation equilibrium has thus focused on several questions in order to provide theoretical foundations of this notion. A part of the literature tries to provide support for this notion in static games by introducing different forms of consistency. However, [START_REF] Lindh | The inconsistency of consistent conjectures. Coming back to Cournot[END_REF] shows that endogeneizing conjectures by using notions of consistency does not rationalize conjectural equilibria in a static setting. The only way to rationalize this type of equilibrium is to develop a dynamic notion.

Another part of the literature focuses on the notion of conjectural variations in evolutionary or repeated settings. Some papers provide support for conjectural variations by considering evolutionary games with myopic agents and showing that consistent conjectures are evolutionary stable. This is shown by [START_REF] Muller | Conjectural variations and evolutionary stability: a rationale for consistency[END_REF] in linear quadratic oligopoly models (and further generalized by [START_REF] Possajennikov | The evolutionary stability of constant consistent conjectures[END_REF]). [START_REF] Dixon | The evolution of consistent conjectures[END_REF] provide an evolutionary process converging to consistent conjectures in an oligopolistic model. Other papers consider a repeated game setting where dynamic notions of conjectural variations are introduced and analysed. Recent contributions are [START_REF] Friedman | Bounded rationality, dynamic oligopoly, and conjectural variations[END_REF], and Jean-Marie and Tidball (2005), where the main focus is on repeated oligopoly games. Finally, learning procedures based on conjectures are developed in Jean-Marie and Tidball (2006) and [START_REF] Quérou | Incomplete information, learning, and natural resource management[END_REF]. In these papers agents assume that a variation of their own strategy has a first order linear effect on the strategy of others, and beliefs are updated based on observations over time.

Still, it has been acknowledged that a fully consistent theory of conjectures should focus on models characterized by a dynamic evolution rule. The present paper focuses on such a situation and makes an interesting contribution to the literature by analysing a fully dynamic problem of natural resource management (fishery management is a good example). More specifically, both [START_REF] Friedman | Bounded rationality, dynamic oligopoly, and conjectural variations[END_REF] and Jean-Marie and Tidball (2005) analyse purely repeated games. By contrast we analyze a discounted discretetime, infinite-horizon extraction game with conjectures, where the state of the resource is subject to a dynamic evolution rule and players use feedback strategies. Our main goal is to understand the influence of such simple beliefs on the agents' behavior. Specifically, we want to know if agents can learn to behave cooperatively in such a setting.

To our knowledge, this is the first contribution in this area that allows for a tractable analysis of the (optimal) consumption plans. Our contribution has two parts. First, we fix the issue of logical consistency by dealing with an explicitly dynamic notion of conjectures. Secondly, we show that agents can learn to behave cooperatively even though they rely on simple beliefs such as conjectures. Initial conditions (levels of consumption at the beginning of the process) are shown to play an important role in the agents' ability to adopt a cooperative behavior.

We consider two different types of beliefs, one where agents assume that the strategy of others is a function of the state of the resource at the previous period (state based beliefs), the other where the conjecture is a function of the state of the resource and of the other agents's strategy (state and strategy based beliefs). Beliefs are required to be consistent with observed policies (as in Jean-Marie and Tidball (2005)). 1 Consistency requires that the paths corresponding to the players' beliefs must be consistent with observed past plays. Jean-Marie and Tidball (2005) show (in theorem 5.1) in a finite horizon, repeated game setting that, with state based beliefs 2 the consistent equilibrium coincides with the feedback Nash equilibrium (under complete information). This result provides support (based on rational grounds) for this notion of consistency. We will prove (in proposition 4.1) that the same result is valid in the present setting (keeping in mind that we consider a fully dynamic game here). We will further develop the contribution by analysing the case of state and strategy based beliefs.

Let us describe the main parts of the analysis. The procedure and consistent solutions are defined. Convergence is studied, and the resulting policy is compared to the benchmark of complete information both from an economic and environmental point of view. It is proved that, depending on the agents' initial levels of consumption, there are situations where consistent solutions yield better outcomes in terms of environmental sustainability compared to the case of complete information Cournot Nash equilibrium. With strategy and state based beliefs, the agents behave more (respectively, less) agressively than in the non-cooperative benchmark when initial consumption is high (respectively, intermediate). When the initial consumption is low, the optimal consumption path lies below that of the cooperative benchmark. Finally, we discuss the impact that public policies (such as taxes) could have on the management of the resource in such a setting. Indeed such policies are usually present in problems of natural resource management. As such, they are to some extent part of the process, and it seems logical to discuss their impact on the analysis. We highlight that taxes could provide agents with appropriate incentives to decrease their initial level of consumption, so that the process generates a cooperative (or even more sustainable) management of the resource.

The above results show that agents relying on simple beliefs can learn to manage a natural resource cooperatively. However, this is shown to rely on the level of initial consumption that agents choose. This highlights how public policies could help decision making by ensuring that agents adopt the appropriate levels of initial consumption. These conclusions seem to us quite 1 Other studies have defined notions of consistency in dynamic games, as [START_REF] Friedman | Oligopoly and the theory of games[END_REF], [START_REF] Fershtman | Conjectural equilibrium and strategy spaces in differential games[END_REF], or [START_REF] Laitner | Rational duopoly equilibria[END_REF]. These notions raise issues regarding the computability of solutions. We may refer to [START_REF] Friedman | Bounded rationality, dynamic oligopoly, and conjectural variations[END_REF] who propose another possible model too. 2 The state refers to the usual definition when a repeated game is considered.

consistent with the way people make decisions (history dependent policies, coexistence of different types of behavior). Firstly, path dependency has been highlighted by [START_REF] Bischi | Expectation-stock dynamics in multi-agent fisheries[END_REF] in other situations as well (where agents are not perfecly informed about fish stocks). 3 Secondly, the conclusions are interesting as well if one interprets the present process of forming conjectures as a potential heuristic that agents could decide to adopt to guide their decision making in such a setting. Indeed, the present process would enable them to avoid the burden of a more sophisticated analysis of strategic interactions.

The paper is organized as follows. The model and the main results of the benchmark case of full information are introduced in Sections 2 and 3 (respectively). The situations with state based and state and strategy based beliefs are analyzed in Section 4. In Section 5 we discuss the robustness of our results for renewable resources, the case of more than two agents, and the influence of initial consumption levels on the outcome of the process. Section 6 concludes. Technical details of one proof are provided in an appendix in Section 7.

The benchmark case of complete information

The model of non-renewable resource extraction is briefly introduced in the first sub section. 4 The players' utility functions and the specifics of the model are described and the notations explained.

Dynamics

We consider a two-agent problem of natural resource extraction, where the resource is subject to a dynamic evolution rule. The time horizon of the problem is infinite. Let x t be the stock of natural resource at time t. If c i,t denotes agent i = 1, 2 extraction at time t, then the evolution rule is given by

x t+1 = x t -c 1,t -c 2,t , x(0) = x 0 . (1)
We focus in Sections 3 and 4 on the case of a non renewable resource. This is made for analytical tractability (in order to derive closed form expressions of the optimal policies). In Section 5 we will discuss how the results extend to the case of renewable resources by relying on numerical works.

Let us discuss the assumptions of the present model. First, the biological rule and extraction technology are assumed to be linear. This is assumed in models of resource extraction to ensure that the situation remains analytically tractable. The two assumptions imply that a (unique) interior solution to the problem of optimal extraction will exist. 5 In the present contribution this leads to the existence of an (unique) interior optimal consumption path. 6This enables us to make a precise comparison with the complete information model. 7Second, players share the same extraction technology. Again, this ensures that the optimal consumption policies can be characterized analytically. The present contribution constitutes the first application of the present notion of consistent beliefs to such dynamic models. Analytical results are required to highlight its main characteristics (link between initial conditions and optimal consumption policies, comparison with the benchmark case). Applying it to situations with heterogeneous agents is the next step of the project, but this will rely on a numerical approach.

In the next sub section the agents' utility functions are defined, and we elaborate on the implications resulting from this specification.

The agents' payoff functions

Each agent derives utility from present and future consumptions. A given agent's consumption policy influences obviously the consumption of the other by its influence on the evolution of the resource. As such we consider an infinite horizon dynamic game. As in the contribution of [START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF] we will assume that the relationship (for agent i) between consumption at period t and the level of instantaneous utility derived is given by the following specification:

u i (c i,t ) = log(c i,t ),
where c i,t denotes the consumption of agent i at period t. We use such a specification for several reasons. Firstly, logarithmic utility functions are used frequently in problems of resource management (see, among other examples, a recent contribution by [START_REF] Koulovatianos | Optimal growth and uncertainty: Learning[END_REF] on optimal growth and learning). Secondly, this specification will enable us to derive closed-form expressions of the optimal consumption plans and of the state dynamics in the present setting. This is consistent with the main goal of the present paper, which is to highlight the main characteristics of the process introduced in Section 4 (in terms of the economic efficiency of the consumption plan and of the sustainability of the management of the resource) by comparing our findings with those of the benchmark case in terms of the long run management strategies that result from each contribution.

Agents are assumed to choose their optimal consumption policies in order to maximize the discounted sum of their instantaneous utility derived from consumption, that is, to maximize

∞ t=0 β t u i (c i,t ),
taking into account the evolution of the resource (and the influence of the other agents' consumption strategy on this evolution as well). In the above specification of payoffs we assume that 0 < β < 1 denotes the measure of the agents' time preferences. In the case of [START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF] the information is assumed to be complete, while, in the present contribution, agents do not know their opponent's utility function and form beliefs about their behavior. We will later assume two simple forms of such beliefs and we will solve for the above specification of the dynamic game.

In the next sections we will characterize the optimal policies corresponding to the complete information benchmark case,8 then we will define the notions of state based and strategy and state based beliefs (as introduced in Jean-Marie and Tidball ( 2005)) and we will apply these notions to the present model. This will enable us to derive comparisons between our findings and the benchmark case.

The full information benchmark case

We first consider the complete information benchmark case. We characterize the optimal consumption policies (and related state dynamics) under joint and non-cooperative managements (in sections 3.1 and 3.2 respectively).

The cooperative case

In this section the information is assumed to be complete. A given agent's problem is:

max {c 1,t ;c 2,t } ∞ t=0 β t log(c 1,t ) + log(c 2,t ) subject to x t+1 = x t -c 1,t -c 2,t , x 0 given.
We restrict to feedback strategies (we assume that c i,t = a i x t ) for any player i = 1, 2. Plugging this expression into the above equality, we obtain

x t+1 = (1 -a 1 -a 2 )x t , x t = (1 -a 1 -a 2 ) t x 0 .
Solving the problem for a 1 and a 2 , the optimal consumption policies are:

a 1 = a 2 = (1 -β)/2, x c t = β t x 0 , c c i,t = 1 -β 2 β t x 0 ,
where c c i,t denotes agent i's optimal consumption policy (at period t) under joint management.

The non-cooperative case: Nash equilibria

For players i = 1, 2, the problem is specified as follows:

max {c i,t } ∞ t=0 β t log(c i,t ) subject to x t+1 = x t -c 1,t -c 2,t , x 0 given.
We consider feedback strategies again. The problem becomes:

max {a i } ∞ t=0 β t log(a i ) + tlog(1 -a 1 -a 2 ).
Solving for a 1 and a 2 , we obtain the following non-cooperative consumption policies:

a 1 = a 2 = 1 -β 2 -β , x N t = β 2 -β t x 0 , c N i,t = 1 -β 2 -β β 2 -β t x 0 ,
where c N i,t denotes the optimal consumption policy of agent i (at period t) under non-cooperative management of the resource. Notice that

x c t = β t x 0 > β 2 -β t x 0 = x N t , ∀t. (2)
Strategic considerations lead to a more intensive extraction of the resource than under joint management. This is the main conclusion provided by Levhari and Mirman (1980) in the case of a renewable resource.

We would like to stress the following point. We could have used the Bellman equation associated to the above non-cooperative problem in order to solve for the optimal consumption policy at each period of the game. Instead we use the linear structure of the game. The same method will be used in Section 4.1 to derive the consistent solution with state based beliefs when information is incomplete.

The situation of incomplete information with conjectures

In this section we consider that agents know the evolution rule of the resource but have limited information on the other agent's preferences. They form conjectures on their opponent's behavior. If one interprets an agent's conjecture as his type, then the type describes this agent's belief about the nature of the strategic interaction with his opponent. By contrast with the Bayesian framework, no knowledge about the other agent's form of utility function or about the distribution of types is needed.

We first analyse the case where players use state-based beliefs, that is, where beliefs are a function of the remaining stock of the resource at any given period (in section 4.1). We confirm the result of Jean-Marie and Tidball (2005, theorem 5.1) who show (in a repeated game setting) that the consistent equilibrium with state based conjectures coincides with the complete information feedback Nash equilibrium. Then we analyse the case of strategy and state based beliefs (in section 4.2) and we compare the optimal consumption policies (corresponding to the consistent solution) with those of the full information benchmark case.

The case of state based beliefs

Let us consider in a first step the situation where any player i conjectures that player j's consumption policy at period t is a function of x t only. Specifically, we consider that agent 1's beliefs regarding the behavior of agent 2 have the following form: c 1 2,t = a 2 x t . In other words, agent 1 conjectures that the optimal consumption of agent 2 at any given period t is a function of the stock of the resource at the beginning of the same period. It is implicitly assumed that each agent can observe the stock of the resource at the end of each period. Now we look for consistent solutions, that is, solutions such that the conjectured policy corresponds to the actual optimal policy. This means that conjecture a 2 must be such that the conjectured value of consumption of agent 2 (as implied by the above expression) corresponds to the solution to the optimal problem of agent 2 (for any given period).

It can be checked that a straightforward application of this definition of consistency leads to the following conclusion. 9Proposition 4.1. Let us denote by c f c i,t the (feedback) consistent optimal consumption policy of agent i (i = 1, 2) at period t. When agents use state based beliefs, the optimal consumption policy is characterized, for any period t and any agent i, by: c f c i,t = c N i,t , where c N i,t denotes the non-cooperative optimal consumption policy in the benchmark (full information) case.

The proof is omitted as it is immediately checked that one has to solve the same system of first order conditions in both situations. Because of the assumptions there is a unique optimal policy that satisfies these conditions, which yields the conclusion.

The implication of this result is that the non-cooperative management of the resource is equivalent under both complete and incomplete information. Even though this conclusion might be perceived as disappointing at first sight, there is a notable difference between the two cases. Indeed, requiring consistency enables to get rid of the assumption of perfect knowledge about the agents' utility functions. Thus, the case of state based beliefs provides some support for the notion of consistent conjectures. In this simple case conjectures are rationalized in that the notion of consistency leads to the well know concept of feedback Nash equilibrium.

We will now introduce a second type of beliefs, which are defined as a function of the state of the resource and of the agents' consumption strategy. We will then compare the optimal consumption policies with those of the benchmark case.

The case of state and strategy based beliefs

We first consider a type of beliefs where a conjecture is a function of both the state of the resource and of the consumption pattern. We will see that this definition allows for a rich pattern of potential behaviors. Moreover it will be stressed that the optimal consumption policy (and thus the management of the resource along the equilibrium path) depends on the initial consumption levels. This influence will be characterised as well.

We now proceed with the definition of strategy and state based beliefs. Let us consider the case of player 1. This player conjectures that player 2's consumption decision at period t is given by:

c 1 2,t = a 2 x t + b 2 c 1,t-1 ,
where a 2 and b 2 model the player's beliefs. In other words, agent 1 assumes that the consumption policy of agent 2 at period t is a function of the state of the resource at period t and of agent 1's consumption strategy at period t -1. 10This form deserves further comment. Firstly, it is implicitly assumed that agents observe the level of the resource and the consumption of others at every period. Secondly, it is worth explaining why we choose this form. This is the simplest form of beliefs that can be thought of when beliefs are defined as a function of the characteristics of the situation at each period, namely, the state of the resource and the strategy of the other agents. Since agents are assumed to learn the behavior of others, it seems natural to assume that they define their beliefs as a function of the only characteristic of the other agents that they can observe. We consider the simplest form of relationship to define conjectures by assuming that the consumption of one agent has a first order linear effect on the behavior of the others. A natural implication is that agents are (to some extent) boundedly rational. The other way to think about the definition of the conjectures is to consider the process of forming conjectures as a potential heuristic that agents could decide to adopt to guide individual decision making. With this perspective in mind, the process could be useful if it proves to be reasonably efficient. We will come back to this question during the analysis of the process. 11Let us come to the notion of feedback consistency. If there is a persisting difference between the consumption of player 2 as conjectured by player 1 and the actual optimal policy of player 2, then player 1 would have incentives to stop relying on this scheme. This is why we require that solutions be consistent. Consistent solutions are such that the conjectured policy corresponds to the actual optimal policy. This notion of consistency enables us to introduce an equivalent formulation of the problem. 12Specifically, from Jean-Marie and Tidball (2005) it is known that finding the feedback consistent solution to the present problem is equivalent to finding the solution to:

max {c i t } ∞ t=0 β t logc i,t ,
subject to the following constraints:

x t+1 = x t -a j x t -b j y t -c i,t and
y t+1 = c i,t
where initial stock x 0 and catch c i,-1 are common knowledge at the beginning of the problem. 13 Variable y t is introduced as a second state variable, since this new formulation will be useful to solve the problem. As an example, for i = 1, the dynamics of the equivalent problem can be rewritten as follows:

x t+1 -x t = -a 2 x t -b 2 y t -c 1,t (3)
and

y t+1 -y t = c 1,t -y t . (4)
In the present case the state vector of the dynamics is defined by equations ( 3) and [START_REF] Dutta | The tragedy of the commons?[END_REF]. Specifically, at any period t the state of the process is given by (x t , c 1,t-1 , c 2,t-1 ). Requiring feedback consistency amounts to requiring that the functional relationship that links (x t , c 1,t-1 , c 2,t-1 ) to the belief of each agent regarding the opponent's optimal catch at period t corresponds to the actual optimal policy of this country.

From this definition of consistency the remaining steps in calculations are straightforward. When we derive the optimality conditions associated to the above problem, we will obtain a set of parameters (a 2 , b 2 ) (respectively, a set of pairs (a 1 , b 1 ) for agent 2) that are potential solutions. Then we will look for the specific values that correspond to the feedback consistent solutions.

The above specification of the problem will enable us to derive the closed form expression of the optimal consumption path. Then we will compare the optimal policies with those of the benchmark case provided in Section 3. Specifically, we will prove that the symmetric consistent optimal consumption policy (and resulting stock level) is characterized by

c f c t = [β(1 -a)] t c 0 , x f c t = c f c t -bc f c t-1 a ,
for any period t (c f c t denotes the symmetric (feedback) consistent optimal consumption policy at period t with strategy and state based beliefs). We can now state the main result of this section. We provide an explicit link between the level of initial consumption and the closed form expressions of optimal consumption policies and corresponding state dynamics. This enables us to compare the different policies to the benchmark cases of sections 3.1 and 3.2. 14 The following remark will prove to be useful in this analysis.

Remark 4.1. First notice that we deal with values of c 0 such that x 0 > 2c 0 , otherwise the stock will be exhausted before the beginning of the process. Second, one could notice that, since the discount factor β lies between 0 and 1, we have the following inequalities:

1 -β 2 x 0 < 1 -β 2 -β x 0 < 1 2 x 0 . (5)
These inequalities will be useful in the next result.

The main conclusions are described as follows.

Proposition 4.2. The following results hold:

• If c 0 = 1-β 2
x 0 , then a = 0 and b = β. The feedback consistent solution is given by :

c f c t = βc f c t-1 , x f c t = β t x 0 . In this case x f c t = x c t .
In others words, if the level of initial consumption corresponds to that of the full information cooperative case, then the feedback consistent solution coincides with the benchmark solution under joint management.

• If c 0 = 1-β 2 x 0 then :

c t = [1 -2 c 0 x 0 ] t c 0 , x t = [1 -2 c 0 x 0 ] t x 0 .
Moreover, we can state the following comparisons:

-if c 0 < 1-β 2 x 0 then a < 0 and b > 0 and x f c t > x c t > x N t ; -if 1-β 2 x 0 < c 0 < 1-β 2-β x 0 then a > 0, b > 0 and x N t < x f c t < x c t ; -if 1-β 2-β x 0 < c 0 < 1 2 x 0 then a > 0 and b < 0 and x f c t < x N t < x c t .
Proof. See Appendix.

Let us briefly elaborate on the above result. The optimal consumption policies (and thus, the stock of the resource along the equilibrium path) depend on the initial consumption level. This is an important feature of the present model. The observation of this initial condition can help one to anticipate which of the patterns of consumption will prevail. From another perspective, this implies that one might influence the consumption path that will be chosen by focusing on the initial level of consumption.

Remark 4.2.

There is an interesting interpretation of Proposition 4.2. 15The form of the beliefs c i j,t = a j x t + b j c i,t-1 can be thought about as follows. As player i consumes more today, he believes that the other agent will react and increase his own consumption tomorrow. Thus, agent i will be punished next period by the other player's faster extraction. This creates a situation close to that of tit-for-tat strategies (even though information is incomplete in the present setting), which might enable the players to sustain the full information cooperative path or more environmentally sustainable paths (depending on the initial consumption levels).

Discussion

Proposition 4.2 leads to several predictions. The first part of the result states that the feedback consistent solution coincides with the cooperative solution under complete information provided that the initial level of consumption is the same in both cases. This is intuitively appealing. If agents cooperate initially (even though they might not know it due to incomplete information) then asking for consistency ensures that cooperation will be sustained in the long run.

If the initial consumption is too high, then the present procedure leads to a more aggressive pattern than even in the non-cooperative case under full information. In such a case the effects of strategic behaviors are reinforced by incomplete information. However, provided that agents' consumption level lies initially below a threshold value the previous conclusion is reversed. Specifically, for moderate values, the stock of the resource lies in between non-cooperative and joint management patterns under full information.

When initial consumption is sufficiently low, we obtain the surprising conclusion that the present procedure leads to an under-exploitation of the resource compared to the full information cooperative benchmark (if one focuses on economic efficiency only, that is, on getting the most utility out of consumption). The intuition is as follows. Unlike the situation of complete information, the choice of the initial consumption level influences the shape of the optimal consumption path. The consistency requirement enables one to select a pattern of individual behavior that comes close to pure cooperation. Then, the choice of a sufficiently low initial consumption level enables the agents to select a path that is parallel to the full information cooperative benchmark, but that lies below it (due to the choice of a lower initial consumption level). This result has an interesting implication. There are situations where the management of the resource could be subject to other criteria than economic efficiency. For instance, it could be the case that the focus be on sustainable management. In such a case, the process of forming conjectures would help agents to learn to coordinate on a management path which would be even more appropriate (with regard to sustainability) than the cooperative benchmark. This suggests that in the third case incomplete information makes the emergence of alternative (and more cautious) patterns of behavior plausible, and our procedure takes advantage of this possibility.

There is a final insight to be noticed. In many problems of resource management public policies (taxes, quotas) are either implemented or potentially available. Thus, these policies can be considered as part of the setting. Parts of Proposition 4.2 have implications for the use of such instruments. Indeed, it is shown that a more efficient management of the resource (compared to the non cooperative benchmark) is feasible provided that initial consumption is kept below the level that corresponds to a non cooperative management under complete information. The sustainability of the resource can be further improved if initial consumption is lower than in the cooperative benchmark. Such low levels of initial consumption can be difficult to implement, and it could be the case that some (public) intervention be required to achieve this task. It is easy to understand that an appropriate extraction tax would then contribute to a better management outcome, as it would lead to a decrease in the level of initial consumption, which would then ensure that the agents would learn to adopt a more cooperative pattern of behavior.

Extensions

In the analysis we focus on a quite simple case in order to derive closed form expressions of the optimal consumption policies and of the resulting evolution of the stock of the resource. Specifically, we consider a non-renewable resource, and we study the case where there are only two agents. In this section, we first discuss these assumptions. Secondly, we analyse how agents may possibly choose their initial consumption levels.

The case of renewable resources

In Proposition 4.2 we characterize the optimal consumption path under consistent conjectures. In order to obtain this characterization, we assume that the resource is non renewable. It might be interesting to discuss whether one might expect (from a qualitative point of view) a similar type of result in the case of renewable resources. Thus, we consider the following biological rule:

x t+1 = [x t -c 1,t -c 2,t ] α , x(0) = x 0 ,
where α ∈ (0, 1) denotes the rate of regeneration of the resource. All other specifications and notations remain the same. We focus here on one of the most interesting cases described in Proposition 4.2. Specifically, provided that agents choose an initial consumption level corresponding to cooperative management, the process results in the cooperative consumption path. As a reminder [START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF]), the full information cooperative solution is characterised (for any period t) by:

c c t = 1 2 (1 -αβ)x c t , x c t+1 = (αβx c t ) α .
Now, regarding our process, the expressions of the state variables and of the Hamiltonian function associated to agent i's problem becomes:

x t+1 -x t = [(1 -a j )x t -c i,t -b j y i t ] α -x t , y i t+1 -y i t = c i,t -y i t , and
H = β t logc i,t + Π i t [x t+1 -x t ] + λ i t [y i t+1 -y i t ]
where Π i t and λ i t are the associated adjoint variables. This is a well-defined problem. Optimal consumption policies exist and satisfy, for any period t:

∂H ∂c i,t = 0 = β t c i,t -Π i t α[(1 -a j )x t -c i,t -b j y i t ] α-1 + λ i t , Π i t-1 = Π i t α[(1 -a j )x t -c i,t -b j y i t ] α-1 (1 -a j ), and
λ i t = -Π i t b j α[(1 -a j )x t -c i,t -b j y i t ] α-1 .
The second condition can be rewritten as:

Π i t-1 = -λ i t-1 (1 -a j ) b j .
The non-linear evolution of the resource does not enable us to provide a closed form expression of the optimal consumption policies. Nonetheless, we know that these policies are characterized by the above conditions and the consistency requirement (for i, j = 1, 2 and any period t):

c j,t = a j x t + b j c i,t-1 .
As a result, we adopt the following steps. First, we ask whether, for any rate of renewal 0 < α < 1 and an initial consumption (corresponding to the cooperative solution), the consistent process generates a consumption path that approximates the full information cooperative solution.

In order to answer this question, we rely on numerical simulations. Since we cannot provide closed form expressions, we generate numerically a trajectory that will approximate the consistent optimal policies (with respect to an approximation parameter ε chosen sufficiently small). Then, we check if the generated policy constitutes a good approximation of the cooperative solution.

Let us focus on approximating the consistent optimal policies. The numerically generated policy relies on a certain number of parameters, namely a, b and Π 0 . The first thing to notice is that if the policy is a good approximation, then there will be consistency with respect to the steady state of the resource. This implies that a and b satisfies a first condition so that consistency holds. Now, in order to run the simulations, we have to initialize the process. Anticipating our claim of cooperation and hopping that it will be a good approximation of consistency, we consider the following way to do so. We require that the initial consumption level coincides with the initial full information cooperative level. We use α = 0.5, β = 0.8 and x 0 = 0.5 (the initial cooperative consumption is c c 0 = 0.15). Figure 1 highlights that a good approximation to the consistent trajectory is obtained. Indeed, the variation c t -(ax t + bc t-1 ) becomes small after ten time periods. We are now in a position to assess whether our generated ε-consistent trajectory constitutes a good approximation to the full information cooperative path. Figure 2 (below) illustrate that the first part of our results are robust to the case of renewable resources. Specifically, when initial consumption corresponds to cooperative management under complete information, the consistent process constitutes a good approximation of the entire cooperative path. In a second step, we ask whether the second part of our results in Proposition 4.2 are also robust. In order to illustrate this feature, we generate policies where the initial consumption is allowed to vary. The last graph highlights the robustness of the properties obtained in Proposition 4.2. Specifically, when the initial consumption level is higher (respectively, lower) than at the cooperative solution, agents consume more (respectively, less) than on the cooperative path. In the above graph we set c 0 = 0.18 > c c 0 = 0.15, and we obtain that the stock of the resource resulting from the process remains lower than that of the full information cooperative path. Indeed, we notice that the corresponding line lies below that of the cooperative solution.

To summarize, simulations seem to suggest that the results obtained in the present paper exhibit some robustness when non-linearity (of the evolution of the resource) is taken into account.

The case of more than two agents

The main results (namely Propositions 4.1 and 4.2) have been proved in the case of two agents. We now discuss whether these results are robust when one considers multilateral interactions.

In order to highlight the robustness, we define the form of the conjectures that will be used when there are more than two agents. We will focus on the case of three players so that the exposition is kept as simple as possible.

In the case of state-based beliefs, the form of the conjectures can account for more than one opponent as long as one forms conjectures on the other agents as an aggregate. Let us define agents' conjectures as follows:

• Agent 1 conjectures that c 2,t + c 3,t = c 1 -1,t = a -1 x t ; • Agent 2's conjecture is c 1,t + c 3,t = c 2 -2,t = a -2 x t ;
• The third agent conjectures that

c 1,t + c 2,t = c 3 -3,t = a -3 x t .
With this definition, it is straightforward to check that, focusing on the corresponding first order conditions, the consistency requirement will result in the following equalities:

1 c f c 1,t = 1 c f 2,t c = 1 c f c 3,t .
The first order conditions will then be identical to that of non-cooperative management, resulting in the following consumption policy for any agent j:

c j,t = 1 -δ 3 -2δ x t .
This trajectory coincides with the non-cooperative consumption path. This is the same result as in Proposition 4.1.

Regarding the case of state and consumption-based beliefs, the form of conjectures remains unchanged, the only amendment being that agents form conjectures on the extraction of the other two agents considered as an aggregate. In this case, focusing on the case of agent 1 (as an example), feedback consistency would require that we obtain at every period t:

a -1 x t + b -1 c 1,t-1 = c 2,t + c 3,t .
In the symmetric case where c 1,0 = c 2,0 = c 3,0 = c 0 and focusing on the symmetric solution, the above condition would be rewritten as:

ax t + bc t = c 2,t + c 3,t = 2c t .
With this amendment in mind, the proof of Proposition 4.2 remains entirely valid. One derives the following characterisation of parameters a and b in the case of three agents:

a = x 0 β -x 0 + 3c 0 x 0 β , b = (x 0 -3c 0 )[2c 0 β -3c 0 -x 0 β + x 0 ] βc 0 x 0 ,
and Proposition 4.2 generalises to the present setting. For instance, in the second case, the stock dynamics is characterised as x t = (1-3 c 0 x 0 ) t x 0 , and the comparison with respect to non cooperative management and the cooperative solution remains the same.

On the influence of initial consumption

Proposition 4.2 highlights what we consider as an interesting feature of the process: the optimal consumption path depends on the initial condition. To us, this is not a weakness of the process as there are many real world economic instances that exhibit path dependency. Moreover, it highlights a potential role for public intervention, which is to influence initial consumption so that the process could then lead the agents to the desired optimal path. It might still be interesting to study if one can provide support to the choice of certain initial consumption levels. First, using the results obtained in Section 4, it is easily checked that the agents' payoffs can be expressed as functions of the initial consumption levels. Specifically, let us denote c i,0 agent i's initial consumption level. The agents' consumption policies can then be derived as:

c i,t = β t ( x 0 -c 1,0 -c 2,0 βx 0 ) t c i,0 .
Plugging this expression into the agents' payoffs, we obtain:

f i (x 0 , c 1,0 , c 2,0 ) = ∞ t=0 β t ln(β t ( x 0 -c 1,0 -c 2,0 βx 0 ) t c i,0 ).
Rewriting, we get:

f i (x 0 , c 1,0 , c 2,0 ) = ln(c i,0 ) 1 -β + [lnβ + ln( x 0 -c 1,0 -c 2,0 βx 0 )] β (1 -β) 2 .
Thus, anticipating the use of the process, we can rewrite the agents' payoffs as an instantaneous function of the initial consumption levels. How can one provide support to specific levels of initial consumption? One might be tempted to consider a one-shot game (before the agents play according to this process) where agents would play either a la Nash or cooperatively, and then solve the corresponding problem. It is easily checked that initial consumption would correspond either to the Nash equilibrium of the game, or to the initial full information cooperative consumption level. However, one should be cautious when providing support to initial consumption levels.

Indeed, agents are assumed to form conjectures in the present paper because they do not know their opponent's utility function. As such, one has to consider that they form conjectures on the other agent's choice of initial consumption too. Moreover, since the problem is now related to instantaneous payoff functions, one has to rely on a different form of conjectures than that used in the present paper. We are going to use a process of forming conjectures introduced in Jean-Marie and Tidball (2006) that is suited to the new specification of the payoff functions.

We refer to Jean-Marie and Tidball (2006, Section 2) for a complete and formal definition of the process that we use. Agents now define their conjecture regarding the opponent's choice of initial consumption with respect to an exogenously given pair of reference consumption levels (c 1,0 , c 2,0 ). This reference point might, for instance, correspond to a status quo . 16 The form of the first agent's conjecture is thus:

c 1 2,0 = c 1,0 + r 2 (c 1,0 -c 1,0 ),
where r 2 denotes the agent's conjecture. Agent 2 adopts a similar form of conjecture regarding the other agent's behavior. It amounts to assuming that an agent conjectures that his initial consumption choice has a first order effect on the initial consumption of his opponent. Plugging this expression into the agents' payoffs, we obtain:

f i (x 0 , c i,0 , c i j,0 ) := f i (c i,0 , c i j,0 ) = ln(c i,0 ) 1 -β +[ln(β)+ln( x 0 -c i,0 -c i,0 -r j (c i,0 -c i,0 ) βx 0 )] β (1 -β) 2 .
The process developed in Jean-Marie and Tidball ( 2006) is based on learning by trial and error. The agents' conjectures r 1 ad r 2 are updated until they become stationary as each agent i tries to make the best choice regarding c i,0 (anticipating that it will be used as the initial consumption level when agents will play according to the process defined in this paper). We propose to check if one can provide support to the choice of certain values of initial consumption levels as the stationary outcomes of the learning process developed in Jean-Marie and Tidball (2006).

In order to obtain the stationary outcomes of this process, one has to derive the first order conditions (for given conjectures r 1 and r 2 ) related to the utility functions f i (c , i,0 c i j,0 ), which enables one to express each agent's optimal choice as a function of the agents' conjectures. Then each agent updates his own conjecture by trial and error until it converges eventually. The stationary values of the conjectures are characterised as fixed points of the updating process. Using the result provided in Jean-Marie and Tidball (2006, Section 3) there are two pairs of stationary values of the agents' conjectures in the present problem:

1. The first pair satisfies r 1 = r 2 = 1, and this pair corresponds to the following choices:

c 1,0 = c 2,0 = 1 2 (1 -β)x 0 .
Thus, the initial consumption level of the full information cooperative solution path is supported as a stationary outcome of the process.

2. The expression of the second solution depends in a complex way on the parameters c i,0 . If one looks for the symmetric solution where c 1,0 = c 2,0 = c 0 , it follows that it is characterised as, for any agent i:

c i,0 = c 0 ,
that is, the consumption levels chosen correspond to those provided by the reference point. Now, if one looks for predictions that are robust to the specification of the process of forming initial conjectures (to any specification of the reference point (c 1,0 , c 2,0 )), then the first pair of initial consumption levels is the only feasible solution. Thus, using the process developed in Jean-Marie and Tidball (2006) as a way to assess which one of the possible outcomes in Proposition 4.2 is a preferred equilibrium, one can conclude that the full information initial consumption level is supported as a preferred choice.

Concluding remarks

We study a problem of resource management under incomplete information when decision takers interact strategically and where each agent's consumption decision has an influence on the evolution of the size of the resource. A learning procedure is developed where agents form conjectures which are updated according to available observations. We consider two types of conjecture, one based only on the state of the resource, the other based on the state and the consumption strategy of the other agents. The solution studied is such that beliefs must be consistent with observed behaviors.

Closed form expressions of the optimal policies are obtained and compared to the benchmark case (as provided in Section 3). It is proved that the optimal policies correspond to the full information, non-cooperative solution when conjectures are based on the state of the resource only. In the second case (with state and strategy based beliefs), the consistent solution is shown to yield better outcomes regarding the resource management in the long run compared to joint management if initial consumption is sufficiently low, or to lead to a more aggressive pattern than the non-cooperative benchmark if initial consumption is too high. For intermediate values of initial consumption, the optimal path lies in between the non-cooperative and cooperative benchmark cases. Finally, several extensions (more than two agents, renewable resources, selection of specific initial consumption levels) are discussed.

From a practical point of view, the present procedure provides an explicit link between the initial conditions and the resulting long run policies. In other words, the dependence of convergence on the initial conditions explains which of the equilibria will prevail given the initial conditions. As incomplete information is a usual characteristic of resource extraction, the knowledge of an explicit link between initial consumption and resulting long run dynamics is a potentially useful property. If one thinks about this process of forming conjectures as a potential heuristic that agents could decide to adopt to guide individual decision making, then this link is useful as it provides a way to induce agents to adopt different patterns of management of the resource.

Even though the present contribution highlights some interesting properties of the learning procedure, we abstract from several issues. A first extension would be to analyse cases where agents could be heterogeneous. For instance, they could use different extraction technologies. The investigation of other forms of conjectures might be interesting in the case of renewable resources. These points are left for future research.

Appendix

In this appendix we now provide the proof of proposition 4.2. We proceed in three steps. First, we derive the optimality conditions associated to the problem. This enables us to characterize the optimal consumption path. Secondly, we characterize the solutions that satisfy our requirement of consistency. Finally, the conclusions of proposition 4.2 follow immediately from this characterization and the results obtained in Section 3.

Let us first derive the optimality conditions associated to the problem. With this new specification (associated to the new state dynamics [START_REF] Dixon | The evolution of consistent conjectures[END_REF][START_REF] Dutta | The tragedy of the commons?[END_REF]), the principle of the maximum can be used. 17 The associated Hamiltonian is (using (3) and ( 4)):

H = β t logc
From conditions (6) we obtain:

c 1,t = β t π t -λ t ; π t = π 0 (1 -a 2 ) t ; λ t = -b 2 π t+1 .
Using these expressions we can rewrite π t -λ t as

π t -λ t = π 0 (1 -a 2 ) t + b 2 π 0 (1 -a 2 ) t+1 = π 0 (1 -a 2 ) t 1 + b 2 -a 2 1 -a 2 .
Thus, we obtain a final expression of the optimal consumption plan:

c 1,t = [β(1 -a 2 )] t 1 -a 2 π 0 (1 + b 2 -a 2 ) . (7)
Using this expression for t = 0, we can deduce an expression of the initial parameter π 0 as:

π 0 = 1 -a 2 c 1,0 (1 + b 2 -a 2 ) . (8)
Now we can rewrite the expression of the state dynamics. Let us denote

A = b 2 + β(1 -a 2 ) β(1 + b 2 -a 2 )π 0 ; (9)
now we can obtain that

b 2 c 1,t-1 + c 1,t = b 2 (β(1 -a 2 )) t-1 1 -a 2 π 0 (1 + b 2 -a 2 ) + (β(1 -a 2 )) t 1 -a 2 π 0 (1 + b 2 -a 2 ) = (β(1 -a 2 )) t A.
Thus, the state dynamics can be rewritten as:

x t+1 = (1 -a 2 )x t -b 2 c 1,t-1 -c 1,t = (1 -a 2 )x t -[β(1 -a 2 )] t A;
or x t = (1-a 2 ) t x 0 -A(1-a 2 ) t-1 t-1 i=0

β i = (1-a 2 ) t x 0 -A(1-a 2 ) t-1 1 -β t 1 -β . (10)
Up to now there are several potential solutions to the problem (that is, several possible pairs of parameters (a 2 , b 2 )). Now we solve for the feedback consistent solutions. We consider symmetric solutions because we deal with a symmetric problem. Basically this amounts to assuming c 1,0 = c 2,0 = c 0 , that is, the initial level of consumption is the same for both agents. Since the problem is symmetric, we focus on a symmetric solution too. This implies c 1,t = c 2,t = c t for any period t, hence a 1 = a 2 = a, and b 1 = b 2 = b. Now we come back to feedback consistency. As said previously, the state of the process is given (for any period t) by (x t , c t-1 ). Feedback consistency requires that an agent's consumption strategy as conjectured by the other agent corresponds to his actual optimal policy. Formally, we must have:

ax t + bc t-1 = c t . (11)
Thus, to obtain the feedback consistent solutions, one can rewrite condition (11) by using the expressions of c t and x t obtained in ( 7) and [START_REF] Friedman | Bounded rationality, dynamic oligopoly, and conjectural variations[END_REF], and then obtain the corresponding values of coefficients a and b.

Rewriting condition [START_REF] Houba | Can negotiations prevent fish wars?[END_REF], we have:

c t = [β(1 -a)] t 1 -a π 0 (1 + b -a) = a[(1 -a) t x 0 -A(1 -a) t-1 1 -β t 1 -β ] + b[β(1 -a)] t-1 1 -a π 0 (1 + b -a)
.

Using the expression of A given by ( 9) and rewriting, we obtain: (13) It is easily checked that conditions [START_REF] Jean-Marie | Consistent conjectures, equilibria and dynamic games[END_REF] and ( 13) yield the following solutions:

c t = [β(1 -a)]
a = x 0 β -x 0 + 2c 0 x 0 β , b = (x 0 -2c 0 )(c 0 β -2c 0 -x 0 β + x 0 ) βc 0 x 0 = x 0 -2c 0 x 0 - x 0 -2c 0 c 0 a. (14)
We now prove proposition 4.2. The proof is straigthforward because by construction

c f c t = [β(1 -a)] t c 0 , x f c t = c f c t -bc f c t-1
a .

The optimality of corresponding policies results from the signs of a and b, which in turn follow from [START_REF] Kalashnikov | Conjectural variations equilibrium in a mixed duopoly[END_REF] and [START_REF] Fershtman | Conjectural equilibrium and strategy spaces in differential games[END_REF]. The comparisons follow immediately from the forms of optimal policies derived in sections 3.1 and 3.2. This concludes the proof.

Figure 1

 1 Figure 1 around here.

Figure 2

 2 Figure 2 around here.

Figure 3

 3 Figure 3 around here.

  t 1 -a π 0 (1 + b -a) = (1 -a) t [ax 0 -a (1 -a)(1 -β) b + β(1 -a) βπ 0 (1 + b -a) ] +[β(1 -a)] t { a[b + β(1 -a)] βπ 0 (1 -a)(1 -β)(1 + b -a) + b(1 -a) βπ 0 (1 -a)(1 + b -a) }.Now feedback consistency is equivalent to looking for values of coefficients a and b such that:ax 0 -a (1 -a)(1 -β) b + β(1 -a) βπ 0 (1 + b -a) + β(1 -a)] βπ 0 (1 -a)(1 -β)(1 + b -a) + b(1 -a) βπ 0 (1 -a)(1 + b -a) = 1 -a π 0 (1 + b -a).

  1,t + π t [-a 2 x t -b 2 y t -c 1,t ] + λ t [c 1,t -y t ],where λ t and π t denote the adjoint variables related to the two constraints of the present optimization problem. For any period t the first order conditions are:

	∂H ∂c 1,t	=	β t c 1,t	-π

t + λ t = 0, π t -π t-1 = -∂H ∂x t = a 2 π t , λ t -λ t-1 = -∂H ∂y t = b 2 π t + λ t .

In practice, both the resource stock and the behaviour of others might be uncertain. Thus, both papers are useful complements since they tackle two complementary aspects of uncertainty.

The problem has been analysed by[START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF] in the case of renewable resources. We will refer to this work in the present paper since this type of resource will be considered in Section

As said previously, we will discuss the case where the biological rule is non linear and concave in Section 5.

When agents have perfect foresight, the existence of a Markov perfect equilibrium is ensured.

Let us keep in mind that[START_REF] Levhari | The Great Fish War: an example using a dynamic Cournot-Nash solution[END_REF] do not analyse the case of a non renewable resource. We use the present specification as it is simple and will enable us to derive closed form expressions of the optimal policies.

Because this characterization is straightforward, we will omit the full proofs.

We use the specific structure of the game as it enables us to simply identify the optimal coefficients characterizing the solution.

One should keep in mind that x t is the state of the resource at the beginning of period t. As such, it is observable by the agent before he chooses his optimal consumption policy at this period, together with consumptions at period t -1. Thus, his belief at period t is based on the data that are observable at the beginning of this period.

Let us notice that the notion of strategy and state based beliefs shares a feature in common with the notion introduced by[START_REF] Friedman | Bounded rationality, dynamic oligopoly, and conjectural variations[END_REF]. Specifically, they use the other agents' strategies at period t -1 in order to define the agents' conjectures at period t. We will see that the present paper is consistent with their analysis, as we will check that there is a link between the initial conditions (the consumption levels) and the optimal consumption path.

We elaborate on the characterization of consistent solutions by using the equivalent formulation of the problem.

Knowledge of c i,-1 is equivalent to knowing c i,0 in the present setting because of the consistency requirement.

A general conclusion will be that the resource will be exhausted eventually (since optimal consumption is positive and the resource is non renewable). The important point is to compare the findings under incomplete and full information.

We are grateful to a referee for providing this interpretation.

We refer to Jean-Marie and Tidball (2006) for a discussion of this assumption.

It should be noted that the use of conjectures enables us to move from a dynamic game to a problem of optimal control. This is why we can use the principle of the maximum here.


Cécile Labadie

Christian Ginies

Marie Helene

M H Guinebretière

Catherine M G C Renard

Celine Cerutti

Frederic F Carlin

Hydrosols of Citrus aurantium and Rosa sp. support the growth of a diverse spoilage microflora impacting volatile compounds

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Communication orale

Congrès international « Iseo 2015 » 14-16 Septembre 2015, Lublin, Pologne. E-mail: ce.labadie@hotmail.com; frederic.carlin@avignon.inra.fr Citrus aurantium L. ssp. amara L., Rosa centifolia L. and Rosa damascena Miller hydrosols (also called hydrolats) are hydrodistillation products mainly used as food flavoring agents or ingredient in cosmetics. We analyzed 22 hydrosol samples collected from different manufacturers around the Mediterranean basin and in Eastern Europe, at storage time between 0 (sampling at processing day) and 13 months. These hydrosols contained essential oil (EO), at median concentrations ranging between 116 mg/L in the Rosa centifolia hydrosols mainly composed of 2-phenylethanol and citronellol, and 677 mg/L in the Citrus aurantium hydrosols mainly composed of linalool and alpha-terpineol. Although essential oils have been described for antimicrobial effects, with reported MIC of essential oil compounds ranging between 200 to 1000 mg/L against, there were not in sufficient concentrations to prevent microbial proliferation of hydrosol bacteria, the only micro-organisms detected in the tested hydrosols. Maximal bacterial counts in hydrosols sampled a few days to a few months storage after processing (using a Malassez counting chamber and decimal dilutions on solid medium in Petri dishes) were in the range 106 -107 CFU/mL and did not comply with professional standards which recommends bacterial counts lower than 102 CFU/mL. Only hydrosols stored in a sterile packaging showed the lowest count during a three months period (<5 CFU/mL). This suggests that contaminations likely occur during operations handling hydrosols in the open air, i.e. in non-sterile conditions. The 58 hydrosol bacterial isolates were divided into 4 major branches: a Pseudomonas sp. branch, a Burkholderia cepacia complex branch, and two undefined species branches belonging to Acetobacteraceae and Rhodospirillaceae. These bacteria require low nutrients for growth and are environmental contaminants. A few of them could metabolize alpha-terpineol and geraniol, with a concomitant production of 6-methyl-5-hepten-2-one, and could also metabolize 2phenylethyl acetate with a concomitant production of 2-phenylethanol. Challenge tests with pathogenic or reference micro-organisms (Bacillus cereus; Escherichia coli; Staphylococcus aureus; Pseudomonas aeruginosa; Listeria monocytogenes; Salmonella enterica enterica Typhimurium, Candida albicans and Aspergillus brasiliensis) showed that only P. aeruginosa, and A. brasiliensis were able to multiply at 30°C in at least one of the tested hydrosol. None of these microorganisms has been found among hydrosol isolates. In conclusion packaging in aseptic conditions could contribute to the microbiological stability of hydrosols during long-term storage. If not, a chemical stabilization with preservatives should be considered.

Hydrosols of

Keywords: Hydrosol, orange blossom water, rose water, essential oil, challenge-tests, pathogenic bacteria, spoilage bacteria.

Citrus aurantium and Rosa sp. support the growth of a diverse spoilage microflora impacting volatile compounds

	Cécile Labadie1,2*, Christian Ginies2, Marie-Hélène Guinebretiere2, Catherine Renard2, Céline
	Cerutti1, Frédéric Carlin2*
	1 Albert Vieille SAS, 06227 Vallauris, France
	2 INRA, Avignon Université, UMR408 Sécurité et Qualité des Produits d'Origine Végétale, F-84000 Avignon,
	France

*Corresponding authors.