Martin Hachet

Benoit Bossavit

Aurélie Cohé

Toucheo: Multitouch and Stereo Combined in a Seamless Workspace

Keywords: ACM Classification: H5.1 [Information interfaces and presentation]: Multimedia Information Systems. -Artificial, augmented, and virtual realities. H5.2 [Information interfaces and presentation]: User Interfaces. -Input devices and strategies Design, Human Factors Multitouch, stereoscopic display, 3D user interfaces

We propose a new system that efficiently combines direct multitouch interaction with co-located 3D stereoscopic visualization. In our approach, users benefit from well-known 2D metaphors and widgets displayed on a monoscopic touchscreen, while visualizing occlusion-free 3D objects floating above the surface at an optically correct distance. Technically, a horizontal semi-transparent mirror is used to reflect 3D images produced by a stereoscopic screen, while the user's hand as well as a multitouch screen located below this mirror remain visible. By registering the 3D virtual space and the physical space, we produce a rich and unified workspace where users benefit simultaneously from the advantages of both direct and indirect interaction, and from 2D and 3D visualizations. A pilot usability study shows that this combination of technology provides a good user experience.

INTRODUCTION

Recent years have witnessed a keen interest for new input and output technologies. In particular, a widespread boom has occurred surrounding multitouch interaction on the one hand, and stereoscopic visualization on the other hand, noticeably with the extremely rapid market penetration of these technologies for the general public mass market. Direct multitouch interaction has shown many benefits for interaction with digital content. With good mappings, the intuitiveness of multitouch interfaces favors fast and easy interaction. At the same time, stereoscopic visualization, that has been dedicated to specific audiences and applications for a long time, is now becoming a standard display, and it tends to be well accepted by large audiences. In this work, we lean on these current trends and we merge the benefits of both multitouch interaction and stereoscopic visualization into a new original unified workspace. Our system combines efficient direct multitouch interaction with co-located 3D stereoscopic visualization. Hence, users benefit from an immersive visualization of data while being able to interact with both 2D and 3D elements thanks to well-known, or new, touch-based 2D metaphors and widgets, within a seamless workspace. Figure 1 illustrates this workspace, where a user manipulates a 3D stereoscopic object from 2D gestures on a touchscreen.

In this paper, we first review the related work and technologies, and discuss their advantages and drawbacks. Afterwards, we describe both the hardware and software components of our setup. We then focus on interaction techniques and we present the dedicated transformation widget we have designed. A user study allows us to highlight the strengths and weaknesses of our setup. Finally, we describe an applicative scenario before concluding.

RELATED WORK

Interacting with 3D content on touchscreens is an inspiring challenge that has recently motivated interesting research work. Hancock et al. [START_REF] Hancock | Sticky tools: Full 6DOF force-based interaction for multitouch tables[END_REF] proposed a technique where users manipulate 3D objects with one, two, or three fingers in shallow depth. Reisman et al. [START_REF] Reisman | A screen-space formulation for 2d and 3d direct manipulation[END_REF] extended the well known Rotate-Scale-Translate (RST) multi-touch technique to 3D. Martinet et al. have compared and improved these techniques [START_REF] Martinet | The effect of dof separation in 3d manipulation tasks with multitouch displays[END_REF]. In their approaches, Cohé et al. [START_REF] Cohé | tbox: A 3d transformation widget designed for touch-screens[END_REF] and Kin et al. [START_REF] Kin | Eden: A professional multitouch tool for constructing virtual organic environments[END_REF] adapted standard desktop interfaces to the multitouch paradigm. Hilliges et al. added above-the-surface interac-

Preliminary version

tion paradigms [START_REF] Hilliges | Interactions in the air: adding further depth to interactive tabletops[END_REF]. Even if these techniques enhance interaction with 3D scenes, they generally face the problem of 3D content being visualized on 2D flat screens.

Stereoscopic visualization associated with head tracking favors good perception of 3D content. This technology has been extensively used in VR, and it is now becoming widely available. One of the problems with such a display is the difficulty for the user to interact with, let alone to touch. Dedicated input devices and interaction techniques have been designed (see VR and 3DUI literature, e.g. [START_REF] Bowman | 3D User Interfaces: Theory and Practice[END_REF]), but they are generally not integrated within widely distributed applications. Mirror-based display systems, such as the one presented by Schmandt [START_REF] Schmandt | Spatial input/display correspondence in a stereoscopic computer graphic work station[END_REF], enable co-location between the interaction space and the visually perceived 3D scene. This allows exploiting proprioception and hand-eye coordination [START_REF] Arsenault | The importance of stereo and eye-coupled perspective for eye-hand coordination in fish tank vr. Presence: Teleoper[END_REF]. These systems are generally used with haptic devices, e.g. [START_REF] Komerska | Haptic task constraints fo 3d interaction[END_REF]. Such configurations allow one to directly address 3D points within the registered virtual and physical spaces, as well as to feel virtual shapes. With a semi-transparent mirror, the user can see both the real and the virtual environment at the same time. This has been exploited in AR scenarios, e.g. [4][17]. However, the applicative scenarios for such configurations are generally restricted to specific use, and the level of interaction is quite poor. As far as we know, mirror-based stereoscopic systems have never been exploited in conjunction with an additional touchscreen.

Recently, Benko et al. have proposed to combine touchscreens with stereoscopic visualization [START_REF] Benko | Crossdimensional gestural interaction techniques for hybrid immersive environments[END_REF]. Their system is based on a DiamondTouch table used in conjunction with SeeThrough AR head-worn displays, and 3D tracked gloves. This system manages both 2D images projected on the table, and 3D visualization through the head-worn displays. Our approach shares some similar concepts with their work, and it is complementary. Whereas they have conceived their system to favor collaborative work, we have concentrate on a setup that maximizes the consistency and the seamless integration of 2D and 3D stereoscopic displays. Head-worn displays as used in [START_REF] Benko | Crossdimensional gestural interaction techniques for hybrid immersive environments[END_REF] have substantial advantages, but they also suffer from many weaknesses, including a limited field-ofview and resolution, problems of lag and calibration, and so on. Our proposal oversteps these limitations by leaning on a new original optical configuration. Users wear light glasses, and they can interactively observe high-resolution monoscopic and stereoscopic content with a good visual comfort. We have also introduced new 9 DOF interaction widgets that take benefit of such a rich visualization space.

De la Riviere et al. have demonstrated a stereoscopic multitouch system [START_REF] De La Rivière | 3d multitouch : when tactile tables meet immersive visualization technologies[END_REF]. In their approach, two head-tracked users directly interact on a large stereoscopic multitouch table. This setup revealed many promising uses. On the other hand, it suffers from some limitations. In particular, content occlusion is problematic when visualizing stereoscopic images. Selection is also a critical issue when stereoscopic visualization and touch input are jointly used in a unique setup. Valkov et al. [21][20] have focused on this problem, and they give recommendations for 2D touching of 3D stereoscopic objects. In their work, Coffey et al. [6][7] proposed an immersive setup where users interact on a hori-zontal touchscreen, while visualizing virtual environments displayed from a large vertical display located in front of them. Veit et al. [START_REF] Veit | Experimental analysis of the impact of touch screen interaction techniques for 3d positioning tasks[END_REF] used a similar approach with a twoside workbench. The limitation of such approaches is that users need to change the focus between the interaction plane, and the 3D display. With our setup, users can refocus at two different depths without changing their viewing direction. This allows concentrating at visual content displayed at given depth, while keeping the context provided by the second depth plane.

All those attempts show that the conjunction of multitouch interaction and stereoscopic visualization is very promising. On the other hand, the previous work also shows that merging these technologies may be hard to perform, both in terms of hardware developments and design of efficient interaction techniques. Indeed, their achievements seem to be conflicting: immersive visualization makes the screen disappear and gives the feeling an object is located ahead of or behind the screen, while the touch input enables efficient interaction with any content that is displayed on the screen or the interaction plane. Moreover, the very nature of stereoscopic images conflicts with touch input, as two images are generated to force the eyes convergence and the associated depth perception, which is contradictory with single images displayed at screen depth to enable efficient and precise touch.

SETUP Figure 2: Physical setup

To overcome the limitations of the systems described in the previous section, we propose a new original setup. It relies on a multitouch screen on which users see a visualization and interact with monoscopic content. The stereoscopic visualization is provided by another 3D screen that is hung upsidedown at the top of the system, and that is reflected on a semitransparent mirror located between the user's head and the multitouch screen (see Figure 2). Hence, stereoscopic objects are perceived as if they were displayed above the hands, between the mirror and the touchscreen. The geometry of the system provides a large volume above the touchscreen where stereoscopic objects can be perceived. Head tracking ensures that virtual objects are displayed from a correct point of view, and that the parallax is physically correct for any of the real or virtual object within the workspace. By mapping the virtual stereoscopic volume to the physical space, we can then produce a rich and consistent interactive visualization space.

Regarding the taxonomy of Grossman and Wigdor [START_REF] Grossman | Going deeper: a taxonomy of 3d on the tabletop[END_REF], the input space is 2D Planar, and both 2D Table (perceived as 2D) and 2D Heads Up (percieved as 3D Volume) are jointly exploited.

With this setup, users see in the same space the floating 3D objects, their hands, and the data displayed on the touchscreen. Our system inherently provides relevant occlusions clues. Indeed, the stereoscopic objects appear above the user's hands, while the monoscopic content remains below (see Figure 1). Contrary to standard immersive 3D systems that only rely on affecting the eyes convergence to simulate 3D objects depth, our proposal also takes advantage of eye accomodation, and its capability to focus at two different depths. This is illustrated in Figure 3. By changing the lighting conditions inside the system, as well as the brightness of the two screens, we are able to modify the ratio between reflection and transparency and, consequently, we can adjust the visibility of the 2D images, 3D images, and physical objects (e.g. hands). Our current implementation is based on two 32" 1080p LCD screens, which are capable of stereoscopic visualization with shutter-glasses. We chose identical screens for colorimetric consistency purposes, even if stereo is enabled on the top screen only. A PQLab 32" multitouch sensor that is mounted on the bottom screen is used to detect finger gestures. The semi-transparent mirror reflects 70% of the light. Headtracking is currently achieved with an electromagnetic sensor, but it will be replaced by an optical tracking solution to avoid wires. Our software plateform is based on Ogre3D and QT. It runs on a unique PC equipped with a Nvidia Quadro Fx 3800 graphic board. The user is sitting when interacting.

INTERACTION TECHNIQUES

The monoscopic touchscreen provides a direct multitouch interaction surface. Consequently, we benefit from the large variety of user interfaces that were previously designed for such systems. In particular, we take advantage of well known multitouch gestures for zooming and panning operations, for item selection, and even for application control. For example, Figure 3 illustrates the use of a web browser that is displayed on the touchscreen, and Figure 4 shows a standard color editor. Such widgets are embedded inside the immersive workspace, and they can be manipulated with standard touch gestures. Users are therefore able to interact in a fast and easy way with the content beneath the fingers, as they would do with a standard touchscreen.

By default, the 3D objects are displayed and move on the monoscopic screen. They are therefore seen as 2D projection of 3D objects just like on any standard screen. A double tap gesture on a 3D object makes it appear as a stereoscopic object floating above the surface. A simple z-translation animation gives the impression that the object comes out of the touchscreen. The 3D representation of the object on the touchscreen disappears, and it is replaced by a 3D transformation widget located below the floating object. This widget is used to select the floating object, and to manipulate it. A virtual ray, similar to Glueck et al.'s position pegs [START_REF] Glueck | Multiscale 3d reference visualization[END_REF], enforces the link between the manipulated 3D object and its associated transformation widget, as can be seen in Figure 3.

It strengthens the direct relationship between the finger gestures and the object motion, giving an impression of "pseudodirect" interaction, even if the object is not manipulated directly as in, e.g., [START_REF] Reisman | A screen-space formulation for 2d and 3d direct manipulation[END_REF]. The 3D transformation widget is composed of a central disk and additional virtual controllers (see Figure 5). Dual-touch gestures on the central disk of the widget result in RST operations on a plane parallel to the touchscreen (x-y plane). x and y rotations are controlled by way of virtual rods located on the sides of the disk. The fingers crossing the rod would launch an inertial rotation around the rod axis, allowing the control of a single degree of freedom without affecting the others. Similarly, scaling widgets allow the control of the x and y scale factors. For z translations and scaling operations, we use an approach inspired from the balloon metaphor [START_REF] Benko | Balloon selection: A multifinger technique for accurate low-fatigue 3d selections[END_REF]. The user first touches the center of the disk with one finger, and then she or he adjusts the height and the size with a second finger. Hence, users are able to control 9 DOF (plus uniform scaling) of the manipulated 3D object. Figure 5 summarizes these control gestures. Note that all the transformations occur in the camera view frame. The transformation widget can be manipulated with multiple fingers at the same time. Two widgets can also be controlled with two hands for simultaneous manipulation of various objects.

USABILITY TESTING

We conducted an experiment to obtain user feedback about our prototype. Sixteen volunteers (13 males, 3 females, mean age 27) participated in this study. None of them was familiar with stereoscopic visualization nor with large multitouch interaction. The participants were asked to complete a 3D docking task implying 9 Degree-Of-Freedom (3 translations, 3 rotations, 3 scaling), as illustrated in Figure 6. Note that we did not focus on precise manipulation in this pilot study, therefore the matching between the target and the manipulated object was quite tolerant. Each participant had to complete the task five times, with various target sizes, orientations and locations. Before starting, the participants were shown the basic usage of the interface. The whole experiment took approximatively between five and ten minutes by participant. After the experiment, the participants were asked to complete a questionnaire based on a five-point Likert scale. We also collected free-form comments. In this experiment, we were interested in the general feeling of the participants regarding the system. The participants reported they liked the system with an average score of 4.6, which demonstrates a stimulating and satisfying overall experience. The score for the global intuitiveness is 4. Almost none of the participants reported fatigue or sickness (1.1).

The perception of the depth was rated with an average score of 4.5, and the perception of the 3D objects obtained a 4.6. We did not use shadows nor specific shading to enhance depth perception. Even with our current basic experimental environment, participants did not encounter problems with the perception of the 3D environment. The need to move the head for observing the 3D scene has been rated as low (1.8).

The participants tended to keep a fixed head-position. This may indicate that head-tracking is not mandatory once the initial head position has been detected. Note that we need to know the initial head position to correctly map the stereoscopic objects with the monoscopic ones. The low use of head movements may be explained by the inexperience of the participants. Also, the completion of the task did not re-quire extensive head motion. The related mean scores are illustrated in Figure 7. We were also interested in interactions between the visualization of floating objects, the hands, and the widget displayed on the touch screen. The participants answered they were able to visualize well their hand with a score of 4.6. When asking if the hands were too much (resp. too little) visible, the score was 1.6 (resp. 1.5). The hands have not appeared to disturb the visualization of the 3D objects (1.4), nor the 2D objects (2). These results tend to show that the simultaneous visualization of real and virtual elements at different depths do not perturb users (see Figure 8). The simultaneous visualization of their hands and virtual environments may even help users complete the task. This should be studied in more depth in future experiments. Concerning the widget and the manipulation metaphors (Figure 9), the participants reported they were mainly able to perform what they wanted to (4.2). The RST interface has been rated as easy to use with a score of 4.3. The score for the rotation (resp. scaling) widgets was 4 (resp. 3.9). The z widgets obtained a lower score (3.5). We think that one of the reasons is that the finger movements were sometimes stopped by the borders of the touchscreen. To solve this limitation, we afterwards replaced the initial z movements by circular movements. After having caught a dedicated handle, the user can now turn it around the center of the widget, as she or he would do with a crank handle, to modify the z parameter.

Another problem was linked to the selection of the widget due to the parallax between the displayed 2D images and the sensitive surface where the touch inputs were detected.

Because we know the position of the user's head, we could solve this problem by warping the input so it could fit exactly with the perceived display space. Note that the participants have practiced no more than 10 minutes with the widget before assessing it. From our experience, the use of the widget become very efficient after a short period of practice. This needs to be better evaluated in future longer experiments. 3D docking tasks are generally hard to perform. One of the reason is the difficulty to perceive well the 3D structure of the objects and their relative positioning. Another one is linked to the control of 9 DOF. Our proposal address both of these difficulties. The participants had few problems with perceptual issues thanks to the immersive aspect of our system. They manage to complete the task quite easily thanks to the 2D widget and the associated touch gestures.

LIMITATIONS

In the free-form comments section of the experiment, some participants said that the overall environment was too dark. Indeed, the light coming from the stereoscopic screen is attenuated by the reflection on the semi-transparent mirror, as well as by the shutter glasses needed to produce stereo. In the future versions of our prototype, we will seek for screens that maintain a comfortable overall brightness. We also need to study in more depth colorimetric issues to ensure a perfect consistency taking into account the perturbation induced by the reflection. None of the subjects complained about the possible overlapping between the stereoscopic and the monoscopic content, maybe because the test application we proposed was quite simple. We nevertheless think that such conflicts may be problematic. Because we know the position of the viewer and the position of the 3D objects, we can compute the part of the 2D content that is supposed to be visible (resp. invisible). Hence, we can avoid conflicts, and provide very plausible environments.

Some participants also reported that they found the system bulky, and that they had difficulties to reach the rear part of the touchscreen because of the mirror. We will take these comments into consideration and we will work on er-gonomics issues for the design of our future prototype. In particular, we will explore different touchscreen and mirror orientations. We will also work on different form-factors that would avoid the system to be perceived as bulky.

Finally, one of the limitations of our setup is the size of the stereoscopic volume. Compared to large VR screen installations [START_REF] Coffey | Slice wim: a multi-surface, multi-touch interface for overview+detail exploration of volume datasets in virtual reality[END_REF] and HMD-based configurations [START_REF] Benko | Crossdimensional gestural interaction techniques for hybrid immersive environments[END_REF], this volume is reduced with our setup. Note however that this volume is bigger than if the stereoscopic display was coming directly from a touchscreen having the same dimensions.

APPLICATION SCENARIO

To illustrate the benefit of our system, we have developed a demo application for a cultural heritage scenario where archaeologists have to reassemble broken objects by manipulating numerous virtual fragments. To complete this task, archaeologists first need to easily access libraries of objects. Then, they may want to sort, arrange, or annotate them. To do this, direct-touch interaction on a multitouch screen is particularly well suited. Similar tasks in fully immersive VR environments may be harder to complete. On the other hand, archaeologists need to precisely perceive all the shape irregularities or the texture variations of the 3D fragments to understand how they can fit together. This can be favored thanks to the immersive aspect of our system. Both stereoscopic visualization and head-tracking provide powerful visualization. The 3D transformation widget we have designed allows the full manipulation of any 3D object. By using two hands, archaeologists can manipulate two objects at the same time to estimate how well they match together (see Figure 10). This would be difficult to complete with a standard multitouch configuration as depth perception may be a true problem. While manipulating stereoscopic objects, archaeologists can still access various standard controls such as color editors, 2D documents, and so on. We believe that numerous other scenarios where the conjunction of both efficient interaction and good visual immersion would benefit from our setup, e.g. in architecture and medicine scenarios. with both hands for a reassembly task. This figure is rendered in stereo with the views for both eyes visible.

CONCLUSION AND FUTURE WORK

We have proposed a new system that combines both multitouch input and immersive visualization into a single unified workspace. This system relies on the strengths of each tech-nology. It favors at the same time fast and easy interaction, 3D manipulation and camera controls, 2D system control, bimanual interaction, and immersive visualization. The original design of our setup takes advantage of eye accomodation and its capability to focus at different depths.

Our first usability testing was very positive. The participants enjoyed their experience with the system. This study has shown that the blending of stereoscopic display, monoscopic display, and real elements in the same workspace does not cause perceptual conflicts. It has also shown that our setup allows novice users to complete a 9-DOF docking task without many difficulties. User comments have highlighted some limitations of the system. Our future work will consist in improving the design of our prototype by leaning on ergonomics studies. In addition, we want to continue improving and inventing new interaction techniques to push forward the strengths of our system. We also plan to add a depthcamera to mix touch-based and 3D spatial interaction inside the same workspace.

Despite some current areas of improvement, we are convinced that, thanks to its unique blend of complementary displays and simple multitouch interaction schemes, our proposal could bridge the gap between powerful immersive virtual environments and efficient 2D desktop paradigms within a unified workspace.

Figure 1 :

 1 Figure 1: 3D stereoscopic objects and 2D monoscopic content are combined in a unified multitouch workspace.

  (a) Focus on the 3D object (b) Focus on the touchscreen

Figure 3 :

 3 Figure 3: The setup based on two physical displays provides natural focus separation.

Figure 4 :

 4 Figure 4: Both standard 2D widgets and 3D widgets are used in the same workspace.

Figure 5 :

 5 Figure 5: Gestures and corresponding transformations. (a) The widget is displayed on the touchscreen, below the floating object. A virtual ray links both the object and the widget. (b) One or two fingers on the central disk controls RST movements on the x-y plane. (c) Virtual rods are used for x and y rotations. (d) x and y scaling are controlled with dedicated widgets. (d) When touching the center of the disk, a new control widget appears. It allows controlling z translations and scaling.

Figure 6 :

 6 Figure 6: The 3D docking task. The spheres move from red to green when the alignment is correct within a tolerance area.

Figure 7 :

 7 Figure 7: General feeling. Mean scores (1-fully disagree, 5-fully agree) and standard errors (1 SE above and below the mean).

Figure 8 :

 8 Figure 8: Interactions between the visual elements.

Figure 9 :

 9 Figure 9: Manipulation widget.

Figure 10 :

 10 Figure 10: Two virtual fragments being manipulated

Acknowledgements

This work was supported by the ANR project ANR-09-CORD-013 InSTInCT -http://anr-instinct.cap-sciences.net. 3D models: courtesy of VCG-ISTI (Buddha) and Archeovision (virtual fragments).


Benoit Bossavit

Jean-Baptiste De La Rivière

Toni Da Luz

Mathieu Courtois

Cédric Kervegant

Martin Hachet

An Immersive Multitouch Workspace

Introduction

Multitouch interaction has some unique strengths, one of them being that direct touch on 2D screens makes it fast and easy. Unfortunately, because of stereo disparity and content occlusion, direct touch interaction becomes an issue as soon as 3D stereoscopic content is visualized. We propose a new system that combines efficient direct multitouch interaction with co-located 3D stereoscopic visualization. In our approach, users benefit from well-known 2D metaphors and widgets displayed on a monoscopic touchscreen while visualizing 3D objects floating above the surface at an optically correct distance (see Figure 1(a)).

Setup

Our system is composed of a 32" multitouch screen on which users visualize and interact with monoscopic content. The stereoscopic visualization is provided by a 3D screen that is hung upside-down at the top of the system, and that reflects on a semi transparent mirror located between the user's head and the multitouch screen. Hence, stereoscopic objects are perceived as if they were displayed above the hands, between the mirror and the touchscreen. Head tracking ensures that virtual objects are displayed from a correct point of view. By mapping the virtual stereoscopic volume to the physical space, we can then produce a rich and consistent interactive visualization space. With this setup, the user sees in the same space the floating 3D objects, their hands, and the data displayed on the touchscreen. Our system inherently provides relevant occlusions clues. Indeed, the stereoscopic objects appear above the user's hands, while the monoscopic content remains below. Contrary to standard immersive 3D systems that only rely on affecting the eyes convergence to simulate 3D objects depth, our proposal also takes advantage of the eye accommodation and its capability to focus at different depths.

Interaction techniques

The monoscopic touchscreen provides a direct multitouch interaction space. Consequently, we benefit from the large variety of multitouch interfaces that were previously designed for such interfaces. Users are therefore able to interact in a fast and easy way with the content beneath the fingers, as they would do with a standard touchscreen. We have also designed a special widget that takes advantage of our setup to support the manipulation of 3D objects floating above the surface (see Figure 1(b)). It is displayed on the touchscreen, below the 3D object. Dual-touch gestures on the central disk of this widget result in Rotation-Scale-Translate (RST) operations on a plane parallel to the touchscreen, while additionnal rotation as well as scaling widgets allow the control of the remaining degreesof-freedom. Hence, users are able to control 9 DOF (plus uniform scaling) of the manipulated 3D object. Moreover, bi-manual interaction allows the manipulation of two objects at the same time. Since the 3D object always remains above the user's fingers and directly reacts to any motion of the control disk located below, this * e-mail:jb.delariviere@immersion.fr † e-mail:Martin.Hachet@inria.fr provides a near-direct interaction paradigm. Hence, users benefit from the advantages of both direct and indirect interaction.

User Experience

Such a setup combines immersive visualization and multitouch input within a seamless workspace, leading to specific strengths: fast and accurate interaction, easy 3D manipulation, immersive visualization, bimanual input, 2D system control, and so on. It provides a compelling experience to the user. We propose to focus our Siggraph demonstration on a virtual archeology reassembly experience, as such a task is a good example that illustrates the benefit of our setup. Attendees will indeed be able to browse through numerous 3D archaeological fragments and sort them out. To experiment with the solving of such a large puzzle, they will enjoy our occlusion free and head-tracked 3D visualization setup, and they will play with our specific multitouch widget to try different fragment combinations out. They will also enjoy bi-manual input. In addition to this scenario, we will also propose other demo applications to illustrate complementary use cases, in particular in the scope of architecture and medical visualization. The combination of multitouch paradigms and immersive visualization in a unified workspace opens new, unexplored, interfaces for interactive applications in these areas.

Conclusion

We propose to combine both multitouch input and immersive visualization into a single setup that relies on the strengths of each technology, while taking advantage of the specific capabilities that are introduced by a unique design. We believe the resulting proposal combines various research issues (2D/3D visualization, 2D/3D interaction, immersion, direct and indirect multitouch input...), and it may inspire new usages. It will offer a new interactive and immersive experience to attendees and may contribute to push such technologies forward.

Acknowledgments

This work was supported by the ANR project ANR-09-CORD-013 InSTInCT -http://anr-instinct.cap-sciences.net.

Figure 1 :

 1 Figure 1: The setup and a 3D object being manipulated.


Tewfik Sari
email: tewfik.sari@uha.fr

Miled El-Hajji
email: miled.elhajji@enit.rnu.tn

J Ér

Ôme Harmand
email: harmand@supagro.inra.fr

Miled El

Jérôme Harmand

ANALYSIS OF A MATHEMATICAL MODEL OF SYNTROPHIC BACTERIA IN A CHEMOSTAT

Keywords: 2000 Mathematics Subject Classification. 92A15, 92A17, 34C15, 34C35, 34E18 Syntrophic relationship, Mathematical modelling, Coexistence, Asymptotic stability, Anaerobic digestion

A mathematical model involving a syntrophic relationship between two populations of bacteria in a continuous culture is proposed. A detailed qualitative analysis is carried out. The local and global stability analysis of the equilibria are performed. We demonstrate, under general assumptions of monotonicity, relevant from an applied point of view, the asymptotic stability of the positive equilibrium point which corresponds to the coexistence of the two bacteria. A syntrophic relationship in the anaerobic digestion process is proposed as a real candidate for this model.

Introduction

A synthrophic relationship between two organisms refers to a situation where the species exhibit mutualistism but where, at the opposite of what happens in a purely symbiotic relationship, one of the species can grow without the other. Such a situation can be mathematically formalized as follows. Assume that a first species denoted X 1 grows on a substrate S 1 forming an intermediate product S 2 . This intermediate product is required by a second species X 2 to grow. The limiting substrate of the second bacteria being the product of the first bioreaction, the second bacteria cannot grow if the first one is not present.

Such interactions are quite common in nature: it is why a number of models have already been proposed in the literature. Katsuyama et al. [START_REF] Katsuyama | A mathematical model of syntrophic cocultures in the chemostat[END_REF], proposed a model involving two mutualistic species for describing pesticide degradation, while a more general case is considered by Kreikenbohm and Bohl [START_REF] Kreikenbohm | A mathematical model of syntrophic cocultures in the chemostat[END_REF]. Since mutualism involves generally species interacting through intermediate products, other studies consider mutualistic relationships in food webs. For instance, Bratbak and Thingstad [START_REF] Bratbak | Phytoplankton-bacteria interactions: an apparent paradox? Analysis of a model system with both copetition and commensalism[END_REF], or more recently, Aota and Nakajima [START_REF] Aota | Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton[END_REF] considered the mutualism between phytoplankton and bacteria through the carbon excretion by the phytoplankton. A model studied by Freedman et al. [START_REF] Freedman | Mathematical models of predator mutualists[END_REF] was proposed to explain the observed coexistence of such species. However, in the previous studies the models are very specific. In particular, the mathematical analyses of the models are realized for specific growth rates that are explicitely given (in most cases as Monod functions).

To extend the study of mutualism to more general systems, we have recently considered more general assumptions notably with respect to the growth rate functions considered in the models in using qualitative hypotheses, cf. [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF] . Furthermore, it was assumed that the species X 1 may be inhibited by the product S 2 that it produces itself while the species X 2 was simply limited by S 2 . An example of such interactions was given by the anaerobic digestion in which mutualistic relationships allow certain classes of bacteria to coexist. A mutualistic relation has been also considered in [START_REF] Hajji | Association between competition and obligate mutualism in a chemostat[END_REF]. See [START_REF] Hajji | Practical coexistence of two species in the chemostat -A slow-fast characterization[END_REF] for another model of coexistence in the chemostat.

In the actual paper, following [START_REF] Hajji | Analyse d'un relation syntrophique: cas d'un chemostat[END_REF], we revisit the model proposed in [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF] in considering two main changes which significantly further extend the range of practical situations covered by the model. First, we assume that there is some S 2 in the influent. In other terms, the limiting substrate S 2 on which the species X 2 grows is not only produced by the species X 1 but is also available even if the species X 1 is not present. The second modification of the model is that the second species is supposed to be inhibited by an excess of S 1 , the limiting substrate on which the first species grows. To illustrate the usefulness of such extensions of the original model by El Hajji et al. [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF], the biological interpretation of these hypotheses within the context of the anaerobic process is given in the appendix.

The paper is organized as follows. In Section 2, we propose a modified system of four differential equations from the original model in [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. The positive equilibria are determined and their local and global stability properties are established. In the case when the system has a unique positive equilibrium, the global asymptotic stability results are demonstrated through the Dulac's criterion that rules out the possibility of the existence of periodic solutions for the reduced planar system, the Poincaré-Bendixon Theorem and the Butler-McGehee Lemma. Hence, in this case, for every positive initial conditions, the solutions converge to the positive equilibrium point which corresponds to the coexistence of the two bacterial species as observed in real processes. Simulations are presented in Section 4, an example of a syntrophic relationship is given in Section 6 as a candidate for this model.

Mathematical model

Let S 1 , X 1 , S 2 and X 2 denote, respectively, the concentrations of the substrate, the first bacteria, the intermediate product, and the second bacteria present in the reactor at time t. We neglect all species-specific death rates and take into account the dilution rate only. Hence our model is described by the following system of ordinary differential equations :

(1)

                   Ṡ1 = D(S in 1 -S 1 ) -k 3 µ 1 (S 1 , S 2 )X 1 , Ẋ1 = µ 1 (S 1 , S 2 )X 1 -DX 1 , Ṡ2 = D(S in 2 -S 2 ) -k 2 µ 2 (S 1 , S 2 )X 2 + k 1 µ 1 (S 1 , S 2 )X 1 , Ẋ2 = µ 2 (S 1 , S 2 )X 2 -DX 2 .
Where S in 1 > 0 denotes the input concentration of substrate, S in 2 > 0 denotes the input concentration of the intermediate product and D > 0 is the dilution rate.

Assume that the functional response of each species µ 1 , µ 2 : R 2 + → R + satisfies :

A1: µ 1 , µ 2 : R 2 + → R + , of class C 1 , A2: µ 1 (0, S 2 ) = 0, µ 2 (S 1 , 0) = 0, ∀ (S 1 , S 2 ) ∈ R 2 + , A3: ∂µ 1 ∂S 1 (S 1 , S 2 ) > 0, ∂µ 1 ∂S 2 (S 1 , S 2 ) < 0, ∀ (S 1 , S 2 ) ∈ R 2 + , A4: ∂µ 2 ∂S 1 (S 1 , S 2 ) < 0, ∂µ 2 ∂S 2 (S 1 , S 2 ) > 0, ∀ (S 1 , S 2 ) ∈ R 2 + .
Hypothesis A2 expresses that no growth can take place for species X 1 without the substrate S 1 and that the intermediate product S 2 is obligate for the growth of species X 2 . Hypothesis A3 means that the growth of species X 1 increases with the substrate S 1 and it is inhibited by the intermediate product S 2 that it produces. Hypothesis A4 means that the growth of species X 2 increases with intermediate product S 2 produced by species X 1 while it is inhibited by the substrate S 1 . Note that there is a syntrophic relationship between the two species.

We first scale system (1) using the following change of variables and notations :

s 1 = k 1 k 3 S 1 , x 1 = k 1 X 1 , s 2 = S 2 , x 2 = k 2 X 2 , s in 1 = k 1 k 3 S in 1 , s in 2 = S in 2 .
The dimensionless equations thus obtained are :

(2)

                   ṡ1 = D(s in 1 -s 1 ) -f 1 (s 1 , s 2 )x 1 , ẋ1 = f 1 (s 1 , s 2 )x 1 -Dx 1 , ṡ2 = D(s in 2 -s 2 ) -f 2 (s 1 , s 2 )x 2 + f 1 (s 1 , s 2 )x 1 , ẋ2 = f 2 (s 1 , s 2 )x 2 -Dx 2 .
Where the functions f 1 , f 2 : R 2 + → R + are defined by

f 1 (s 1 , s 2 ) = µ 1 ( k 3 k 1 s 1 , s 2 ) and f 2 (s 1 , s 2 ) = µ 2 ( k 3 k 1 s 1 , s 2 ).
Hypotheses A1-A4 satisfied by the functions µ 1 and µ 2 translate in the following assumptions of the functions f 1 and f 2 :

H1: f 1 , f 2 : R 2 + → R + , of class C 1 , H2: f 1 (0, s 2 ) = 0, f 2 (s 1 , 0) = 0, ∀ (s 1 , s 2 ) ∈ R 2 + , H3: ∂f 1 ∂s 1 (s 1 , s 2 ) > 0, ∂f 1 ∂s 2 (s 1 , s 2 ) < 0, ∀ (s 1 , s 2 ) ∈ R 2 + , H4: ∂f 2 ∂s 1 (s 1 , s 2 ) < 0, ∂f 2 ∂s 2 (s 1 , s 2 ) > 0, ∀ (s 1 , s 2 ) ∈ R 2 + .
R 4 + , the closed non-negative cone in R 4 , is positively invariant under the solution map of system (2). More precisely Proposition 1. For every initial condition in R 4 + , the solution of system (2) has positive components and is positively bounded and thus is defined for every positive t. The set

Ω = (s 1 , x 1 , s 2 , x 2 ) ∈ R 4 + : s 1 + x 1 = s in 1 , s 2 + x 2 = x 1 + s in 2
is a positive invariant attractor of all solutions of system [START_REF] Bratbak | Phytoplankton-bacteria interactions: an apparent paradox? Analysis of a model system with both copetition and commensalism[END_REF].

Proof. The invariance of R 4 + is guaranteed by the fact that : i.

s 1 = 0 ⇒ ṡ1 = D s in 1 > 0, ii. s 2 = 0 ⇒ ṡ2 = D s in 2 + f 1 (s 1 , 0) x 1 > 0, iii. x i = 0 ⇒ ẋi = 0 for i = 1, 2.
Next we have to prove that the solution is bounded. Let z 1 = s 1 + x 1 , then ż1 = -D(z 1 -s in 1 ) from which one deduces :

(3) s 1 (t) + x 1 (t) = s in 1 + (s 1 (0) + x 1 (0) -s in 1 )e -Dt . Thus s 1 (t) and x 1 (t) are positively bounded. Let z 2 = s 2 + x 2 -x 1 , then ż2 = -D(z 2 -s in
2 ) from which one deduces:

(4) s 2 (t) + x 2 (t) -x 1 (t) = s in 2 + (s 2 (0) + x 2 (0) -x 1 (0) -s in
2 )e -Dt . Thus s 2 (t) and x 2 (t) are positively bounded. Hence, the solution is defined for all positive t. From (3) and (4) we deduce that the set Ω is an invariant set which is an attractor.

Restriction on the plane

The solutions of system (2) are exponentially convergent towards the set Ω and we are interested in the asymptotic behavior of these solutions. It is enough to restrict the study of the asymptotic behaviour of system [START_REF] Bratbak | Phytoplankton-bacteria interactions: an apparent paradox? Analysis of a model system with both copetition and commensalism[END_REF] to Ω. In fact, thanks to Thieme's results [START_REF] Thieme | Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[END_REF], the asymptotic behaviour of the solutions of the restriction of (2) on Ω will be informative for the complete system, see Section 5. In this section we study the following reduced system which is simply the projection on the plane (x 1 , x 2 ), of the restriction of system (2) on Ω.

ẋ1 = [Φ 1 (x 1 , x 2 ) -D] x 1 , ẋ2 = [Φ 2 (x 1 , x 2 ) -D] x 2 . ( 5
)
where

Φ 1 (x 1 , x 2 ) = f 1 s in 1 -x 1 , s in 2 + x 1 -x 2 , Φ 2 (x 1 , x 2 ) = f 2 s in 1 -x 1 , s in 2 + x 1 -x 2 .
Thus, for (5) the state-vector (x 1 , x 2 ) belongs to the following subset of the plane, see Fig. 1 :

S = (x 1 , x 2 ) ∈ R + 2 : 0 < x 1 ≤ s in 1 , 0 < x 2 ≤ x 1 + s in 2 . E T x 1 x 2 s in 2 s in 1 s in 1 + s in 2 Figure 1. The set S
The point F 0 = (0, 0) is an equilibrium of [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. Besides this equilibrium point the system can have the following three types of equilibrium points.

• Boundary equilibria

F 1 = (x 1 , 0), where x 1 = x1 is a solution, if it exists, of equation (6) Φ 1 (x 1 , 0) = D,
• Boundary equilibria F 2 = (0, x2 ), where x 2 = x2 is a solution, if it exists, of equation

(7) Φ 2 (0, x 2 ) = D, • Positive equilibria F * = (x * 1 , x * 2 )
, where

x 1 = x * 1 , x 2 = x * 2 is a solution, if it exists, of the system of equations (8) Φ 1 (x 1 , x 2 ) = D Φ 2 (x 1 , x 2 ) = D.
We use the following notations

D 1 = f 1 (s in 1 , s in 2 ), D 2 = f 2 (s in 1 , s in 2 ). The mapping x 1 → Φ 1 (x 1 , 0) is decreasing, and the mapping x 1 → Φ 2 (x 1 , 0) is increasing. If D 1 > D 2 , there exists a unique real number ξ 1 satisfying Φ 1 (ξ 1 , 0) = Φ 2 (ξ 1 , 0), since Φ 1 (0, 0) = D 1 > D 2 = Φ 2 (0, 0), and Φ 1 (s in 1 , 0) = 0 < Φ 2 (s in 1 , 0).
We denote by D 3 ∈]D 2 , D 1 [ the unique real number (see Figure 2, right) such that:

Φ 1 (ξ 1 , 0) = Φ 2 (ξ 1 , 0) = D 3 . y y = Φ2(x1, 0) y = Φ1(x1, 0) 0 D2 D1 D Φ2(x1, 0) x1 x1 s in 1 y y = Φ2(x1, 0) y = Φ1(x1, 0) 0 D2 D1 D3 D Φ2(x1, 0) x1 x1 s in 1 ξ1 Figure 2.
Existence and uniqueness of x1 . On the left, the case

D 1 < D 2 : Φ 2 (x 1 , 0) > D for all D < D 1 . On the right, the case D 1 > D 2 : Φ 2 (x 1 , 0) > D if and only if D < D 3 .
The mapping x 2 → Φ 1 (0, x 2 ) is increasing, and the mapping

x 2 → Φ 2 (0, x 2 ) is decreasing. Hence, if D 1 < D 2 , there exists a unique real number ξ 2 satisfying Φ 1 (0, ξ 2 ) = Φ 2 (0, ξ 2 ), since Φ 2 (0, 0) = D 2 > D 1 = Φ 1 (0, 0), and Φ 2 (0, s in 2 ) = 0 < Φ 1 (0, s in 2 ). We denote by D 4 ∈]D 1 , D 2 [
the unique real number (see Figure 3, right) such that:

Φ 1 (0, ξ 2 ) = Φ 2 (0, ξ 2 ) = D 4 .
The nature of the trivial equilibrium point F 0 is given in the following lemma.

D 2 < D 1 : Φ 1 (0, x2 ) > D for all D < D 2 . On the right, the case D 2 > D 1 : Φ 1 (0, x2 ) > D if and only if D < D 4 . Lemma 1. If D > max(D 1 , D 2 ) then F 0 is a stable node. If min(D 1 , D 2 ) < D < max(D 1 , D 2 ) then F 0 is a saddle point. If D < min(D 1 , D 2 ) then F 0 is an unstable node.
Proof. The Jacobian matrix J of (5), at point (x 1 , x 2 ), is given by:

J =      - ∂f 1 ∂s 1 x 1 + ∂f 1 ∂s 2 x 1 + f 1 -D - ∂f 1 ∂s 2 x 1 - ∂f 2 ∂s 1 x 2 + ∂f 2 ∂s 2 x 2 - ∂f 2 ∂s 2 x 2 + f 2 -D     
.

where the functions are evaluated at

s in 1 -x 1 , s in 2 + x 1 -x 2 .
The Jacobian matrix at F 0 is given by:

J 0 =   f 1 (s in 1 , s in 2 ) -D 0 0 f 2 (s in 1 , s in 2 ) -D   The eigenvalues are D 1 -D and D 2 -D. Thus, if D > max(D 1 , D 2 ) then F 0 is a stable node. It is an unstable node if D < min(D 1 , D 2 ). It is a saddle point if min(D 1 , D 2 ) < D < max(D 1 , D 2 ).
The conditions of existence of the boundary equilibria F 1 and F 2 , and their nature, are stated in the following lemmas.

Lemma 2. An equilibrium

F 1 = (x 1 , 0) exists if and only if D < D 1 . If it exists then it the unique equilibrium on the positive x 1 semi-axis. If D 1 < D 2 then F 1 is a saddle point for all D < D 1 . If D 2 < D 1 , then F 1 is a saddle point for all 0 < D < D 3 and a stable node for all D 3 < D < D 1 .
Proof. An equilibrium

F 1 = (x 1 , 0) exists if and only if x 1 = x1 ∈]0, s in 1 [ is a solution of (6). Let ψ 1 (x 1 ) = Φ 1 (x 1 , 0). Then ψ ′ 1 (x 1 ) = - ∂f 1 ∂s 1 (s in 1 -x 1 , s in 2 + x 1 ) + ∂f 1 ∂s 2 (s in 1 -x 1 , s in 2 + x 1 ).
By assumption H3, ψ ′ 1 (x 1 ) < 0. Since ψ 1 (0) = D 1 , and ψ 1 (s in 1 ) = 0, equation ( 6) admits a solution in the interval ]0, s in 1 [ if and only if D < D 1 . If this condition is satisfied then (6) admits a unique solution since the function ψ 1 (.) is decreasing, see Figure 2. The Jacobian matrix at F 1 is given by:

J 1 =    - ∂f 1 ∂s 1 x1 + ∂f 1 ∂s 2 x1 - ∂f 1 ∂s 2 x1 0 f 2 -D   
where the functions are evaluated at (s in 1 -x1 , s in 2 + x1 ). The eigenvalues are

f 2 (s in 1 -x1 , s in 2 + x1 ) -D = Φ 2 (x 1 , 0) -D, and - ∂f 1 ∂s 1 x1 + ∂f 1 ∂s 2 x1 < 0. Thus F 1 is a saddle point if Φ 2 (x 1 , 0) > D. If D 1 < D 2 , this condition is satisfied for all D < D 1 . If D 2 < D 1 , it is statisfied for all 0 < D < D 3 , see Figure 2. F 1 is a stable node if D 3 < D < D 1 and D 2 < D 1 . Lemma 3. An equilibrium F 2 = (0, x2 ) exists if and only if D < D 2 .
If it exists then it the unique equilibrium on the positive Proof. An equilibrium F 2 = (0, x2 ) exists if and only if

x 2 semi-axis. If D 2 < D 1 then F 2 is a saddle point for all D < D 2 . If D 1 < D 2 , then
x 2 = x2 ∈]0, s in 2 [ is a solution of (7). Let ψ 2 (x 2 ) = Φ 2 (0, x 2 ). Then ψ ′ 2 (x 2 ) = - ∂f 1 ∂s 2 (s in 1 , s in 2 -x2 ).
By assumption H4, ψ ′ 2 (x 2 ) < 0. Since ψ 2 (0) = D 2 , and ψ 2 (s in 2 ) = 0, equation ( 7) admits a solution in the interval ]0, s in 2 [ if and only if D < D 2 . If this condition is satisfied then (7) admits a unique solution since the function ψ 2 (.) is decreasing, see Figure 3. The Jacobian matrix at F 2 is given by:

J 2 =    f 1 -D 0 - ∂f 2 ∂s 1 x2 + ∂f 2 ∂s 2 x2 - ∂f 2 ∂s 2 x2   
where the functions are evaluated at (s in 1 , s in 2 -x2 ). The eigenvalues are

f 1 (s in 1 , s in 2 -x2 ) -D = Φ 1 (0, x2 ) -D, and - ∂f 2 ∂s 2 x2 < 0. Thus F 2 is a saddle point if Φ 1 (0, x2 ) > D. If D 2 < D 1 , this condition is satisfied for all D < D 2 . If D 1 < D 2 , it is statisfied for all 0 < D < D 4 , see Figure 3. F 2 is a stable node if D 4 < D < D 2 and D 1 < D 2 .
Let us discuss now the conditions of existence of positive equilibria F * , and their number. An equilibrium

F * = (x * 1 , x * 2 ) exists if and only if x 1 = x * 1 , x 2 = x * 2 is a solution of (8) lying in S. One has ∂Φ 1 ∂x 2 = - ∂f 1 ∂s 2 (s in 1 -x 1 , s in 2 + x 1 -x 2 )
. By assumption H3, this partial derivative is positive. Hence, equation Φ 1 (x 1 , x 2 ) = D defines a function x 2 = F 1 (x 1 ) such that F 1 (x 1 ) = 0 when D < D 1 . Recall that x 1 = x1 is the solution of (6) which, according to Lemma 2 exists and is unique, if and only if D < D 1 . One has

F ′ 1 (x 1 ) = - ∂Φ1 ∂x1 (x 1 , F 1 (x 1 )) ∂Φ1 ∂x2 (x 1 , F 1 (x 1 )) = -∂f1 ∂s1 + ∂f1 ∂s2 ∂f1 ∂s2 = 1 - ∂f1 ∂s1 ∂f1 ∂s2 > 1.
Hence the function F 1 is increasing. Since Φ 1 (s in 1 , 0) = 0, the graph Γ 1 of F 1 has no intersection with the right boundary of the domain S, defined by x 1 = s in 1 . This graph separates S in two regions denoted as the left and right sides of Γ 1 , see Figure 4. One has also

∂Φ 2 ∂x 2 = - ∂f 2 ∂s 2 (s in 1 -x 1 , s in 2 + x 1 -x 2 ).
By assumption H3, this partial derivative is positive. Hence, equation Φ

2 (x 1 , x 2 ) = D defines a function x 2 = F 2 (x 1 ) such that F 2 (0) = x2 when D < D 2 .
Recall that x 2 = x2 is the solution of (7) which, according to Lemma 3 exists and is unique, if and only if D < D 2 . One has

F ′ 2 (x 1 ) = - ∂Φ2 ∂x1 (x 1 , F 2 (x 1 )) ∂Φ2 ∂x2 (x 1 , F 2 (x 1 )) = -∂f2 ∂s1 + ∂f2 ∂s2 ∂f2 ∂s2 = 1 - ∂f2 ∂s1 ∂f2 ∂s2 > 1.
Hence the function F 2 is increasing. Since Φ 2 (x 1 , s in 2 + x 1 ) = 0, the graph Γ 2 of F 2 has no intersection with the top boundary of the domain S, defined by x 2 = s in 2 + x 1 . Thus the point at the very right of Γ 2 lies necessarily on the right boundary of S, defined by x 1 = s in 1 . Hence it lies on the right side of Γ 1 , see Figure 4.

The graphs Γ 1 and Γ 2 can intersect or not, see Figures 4, 5 and 6. If they intersect at some point

F * = (x * 1 , x * 2 ) then F * is a positive equilibbrium. If the point A at the very left of Γ 2 Left Right Γ1 • • • Γ2 Γ1 F * 1 F * 2 F * 3 A A Γ2 Γ1 • • F * 1 F * 2 Figure 4.
On the left, the left and right sides of Γ 1 . On the center, the point A at the very left of Γ 2 lies on left side of Γ 1 : there are generically an odd number of intersections (3 in this example). On the right, the point A at the very left of Γ 2 lies on right side of Γ 1 : there are generically an even number of intersections (2 in this example).

lies on left side of Γ 1 then Γ 1 and Γ 2 intersect in at least one point F * = (x * 1 , x * 2 ). They can have multiple intersections. Generically they have an odd number of intersections (see Figure 4, center). If the point A at the very left of Γ 2 lies on right side of Γ 1 then Γ 1 and Γ 2 can intersect or not. Generically they have an even number of intersections (see Figure 4, right). The nature of a positive equilibrium F * is stated in the following lemmas.

Lemma 4. If an equilibrium

F * = (x * 1 , x * 2 ) exists then it is a stable node if F ′ 1 (x * 1 ) > F ′ 2 (x * 1 )
. It is a saddle point if the opposite inequality is satisfied.

Proof. The Jacobian matrix at F * is given by:

J * =      - ∂f 1 ∂s 1 x * 1 + ∂f 1 ∂s 2 x * 1 - ∂f 1 ∂s 2 x * 1 - ∂f 2 ∂s 1 x * 2 + ∂f 2 ∂s 2 x * 2 - ∂f 2 ∂s 2 x * 2     
where the derivatives are evaluated at (s in

1 -x * 1 , s in 2 + x * 1 -x * 2 ). Notice that tr(J * ) = - ∂f 1 ∂s 1 x * 1 + ∂f 1 ∂s 2 x * 1 - ∂f 2 ∂s 2 x * 2 < 0 and det(J * ) = x * 1 x * 2 ∂f 1 ∂s 1 ∂f 2 ∂s 2 - ∂f 1 ∂s 2 ∂f 2 ∂s 1 = x * 1 x * 2 ∂f 1 ∂s 2 ∂f 2 ∂s 2 [F ′ 2 (x * 1 ) -F ′ 1 (x * 1 )] .
By Assumptions H3 and H4, the product of the partial derivatives is negative. Therefore, the determinant is positive if

F ′ 1 (x * 1 ) > F ′ 2 (x * 1
) and negative if the opposite inequality is satisfied. Hence the equilibrium

F * = (x * 1 , x * 2 ) is a stable node if F ′ 1 (x * 1 ) > F ′ 2 (x * 1 )
. It is a saddle point if the opposite inequality is satisfied.

The number of equilibria of (5) and their nature are summarized in the next theorem.

Theorem 1.

(1) If D < min(D 1 , D 2 ) then ( 5) admits the trivial equilibrium F 0 which is an unstable node, the boundary equilibria F 1 and F 2 which are saddle points, and at least one positive equilibrium F * . If F * is the unique positive equilibrium then it is a stable node. Generically, the system has an odd number of positive equilibria which are alternatively stable nodes and saddle points, the one at the very left of these positive equilibria is a stable node. 5) admits the trivial equilibrium F 0 , which is a saddle point, and the boundary equilibrium F 2 , which is a stable node. Generically, the system can have an even number of positive equilibria which are alternatively saddle points and stable nodes, the one at the very left of these positive equilibria is a saddle point. (c) If D 2 < D 1 and D 2 < D < D 3 then ( 5) admits the trivial and boundary equilibria F 0 and F 1 which are saddle points and at least one positive equilibrium F * . If F * is the unique positive equilibrium then it is a stable node. Generically, the system has an odd number of positive equilibria which are alternatively stable nodes and saddle points, the one at the very left of these positive equilibria is a stable node. 5) admits the trivial equilibrium F 0 , which is a saddle point, and the boundary equilibrium F 1 , which is a stable node. Generically, the system can have an even number of positive equilibria which are alternatively saddle points and stable nodes, the one at the very left of these positive equilibria is a saddle point. 5) admits the trivial equilibrium F 0 which is a stable node. Generically, the system can have an even number of positive equilibria which are alternatively saddle points and stable nodes, the one at the very left of these positive equilibria is a saddle point.

(d) If D 2 < D 1 and D 3 < D < D 1 then (
(3) If D > max(D 1 , D 2 ) then (

Growth functions of Monod type

In this section we consider growth functions f 1 and f 2 of the following form

(9) f 1 (s 1 , s 2 ) = m 1 s 1 (K 1 + s 1 )(L 1 + s 2 ) , f 2 (s 1 , s 2 ) = m 2 s 1 (K 2 + s 1 )(L 2 + s 2 )
.

Such functions are simply the product of a Monod function in s 1 by a decreasing functions of s 2 . Such functions are currently used in biotechnology when the growth of a functional species is limited by a substrate while inhibited by another one. Such situations are common in water treatment technology like in the denitrification (limited by the nitrate and inhibited by the dissolved oxygen) or in the anoxic or anaerobic hydrolysis (limited by the slowly biodegradable substrates while inhibited by an excess of oxygen) processes which are modeled this way (cf. [START_REF]The IWA Task Group on Mathematical Modelling for Design and Operation of Biological Wastewater Treatment, Activated Sludge Models ASM1[END_REF]).

One can readily check that (9) satisfy Assumptions H1 to H4. By straighforward calculations one has

F 1 (x 1 ) = -Dx 2 1 + m 1 + D K 1 -L 1 + s in 1 -s in 2 -m 1 s in 1 + D K 1 + s in 1 L 1 + s in 2 D(K 1 + s in 1 -x 1 ) F 2 (x 1 ) = Dx 2 1 + m 2 + D L 2 -K 2 + s in 2 -s in 1 + m 2 s in 2 -D K 2 + s in 1 L 2 + s in 2 m 2 -D(K 2 + s in 1 -x 1 ) Hence equation F 1 (x 1 ) = F 2 (x 1 )
giving the abscissa of positive equilibria is an algebraic equation of degree 2. Thus, it cannot have more than two solutions. Hence, the situation depicted on the center of Figure 4, of three positive equilibria, is excluded. However, the situation depicted in the right of Figure 4, with two positive equilibria can occur. For instance, consider the following

• • • • F 1 F 2 F * F 0 D < 3/5 • • • F 1 F * F 0 3/5 < D < 8/9 • • • • F 1 F * 1 F * 2 F 0 8/9 < D < 1 • • F 1 F 0 1 < D < 6/5 • F 0 6/5 < D Figure 5.
Relative positions of the isocline ẋ1 = 0 (in red) and ẋ2 = 0 (in green).

values of the parameters ( 10)

m 1 = 8, m 2 = 4, K 1 = L 2 = 1, L 1 = K 2 = 2, s in 1 = s in 2 = 3 Then D 1 = 6/5, D 3 = 8/9, D 2 = 3/5.
There is another bifurcation value, D = 1 which correspond to the case when the graphs Γ 1 and Γ 2 are tangent, see Figure 6. For this example five cases can occur, see Figure 5: Proposition 2. Consider system [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF] where f 1 and f 2 are given by [START_REF] Katsuyama | A mathematical model of syntrophic cocultures in the chemostat[END_REF] with parameters [START_REF] Kreikenbohm | A mathematical model of syntrophic cocultures in the chemostat[END_REF]. Then

(1) when D < 3/5, the system has four equilibria, F 0 which is an unstable node, F 1 and F 2 , which are saddle points and F * , which is a stable node. This is case (1) of Theorem 1, with a unique positive equilibrium. (2) when 3/5 < D < 8/9, the system has three equilibria, F 0 and F 1 , which are saddle points and F * , which is a stable node. This is case (2.c) of Theorem 1, with a unique positive equilibrium.

(3) when 8/9 < D < 1, the system has four equilibria, F 0 and F * 1 , which are saddle points and F 1 and F * 2 , which are stable nodes. This is case (2.d) of Theorem 1, with two positive equilibria. (4) when 1 < D < 6/5, the system has two equilibria, F 0 , which is a saddle point and F 1 which is a stable node. This is case (2.d) of Theorem 1, with no positive equilibrium. (5) when D > 6/5, the system has one equilibrium, F 0 , which is a stable node. This is case

(3) of Theorem 1, with no positive equilibrium.

In the case when 8/9 < D < 1 a bistability phenomenon occurs. According to the initial condition, both species can coexist at equilibrium F * 2 , or species x 2 goes to extinction at equilibrium F 1 . This phenomenon is illustarted numerically with D = 0.95 in Figure 7.

For the following values of the parameters ( 11)

m 1 = 8, m 2 = 7, K 1 = K 2 = L 2 = 1, L 1 = 3/2, s in 1 = s in 2 =
* 2 , • • • F 1 F * F 0 D = 3/5 • • • F 1 F * F 0 D = 8/9 • • • F 1 F * F 0 D = 1 • F 0 D = 6/5
F 1 F * 1 F * 2 F 1 F * 1 F * 2 F 0 Figure 7.
Numerical solutions in the bistability case D = 0.95 and parameters values [START_REF] Kreikenbohm | A mathematical model of syntrophic cocultures in the chemostat[END_REF]. On the left, the separatrix (in green) of the saddle point F * 1 separate the domain S in two region which are the basins of attraction of the boundary equilibrium point F 1 and the positive equilibrium point F * 2 . On the center, the phase portrait. On the right, the isoclines.

or both species go to extinction at equilibrium F 0 . This phenomenon is illustarted numerically in Figure 8.

Global analysis

Let us establish first that (5) admits no periodic orbit nor polycycle inside S Theorem 2. There are no periodic orbits nor polycycles inside S.

Proof. . Consider a trajectory of (5) belonging to S. Let us transform the system (5) through the change of variables ξ 1 = ln(x 1 ), ξ 2 = ln(x 2 ). Then one obtains the following system : [START_REF] Smith | The theory of the chemostat, Dynamics of microbial competition[END_REF]. On the left, the separatrix (in green) of the saddle point F * 1 separate the domain S in two region which are the basins of attraction of the boundary equilibrium point F 0 and the positive equilibrium point F * 2 . On the center, the phase portrait. On the right, the isoclines.

(12)    ξ1 = h 1 (ξ 1 , ξ 2 ) := f 1 (s in 1 -e ξ1 , s in 2 + e ξ1 -e ξ2 ) -D, ξ2 = h 1 (ξ 1 , ξ 2 ) := f 2 (s in 1 -e ξ1 ,
From Dulac criterion [START_REF] Smith | The theory of the chemostat, Dynamics of microbial competition[END_REF], we deduce that the system [START_REF] Thieme | Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[END_REF] has no periodic trajectory. Hence [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF] has no periodic orbit in S.

Theorem 3. Assume that system [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF] has at most one positive quilibrium F * , then for every initial condition in S, the trajectories of system ( 5) converge asymptotically to :

• F * if D < min(D 1 , D 2 ). • F * if D 1 < D 2 and D 1 < D < D 4 • F 2 if D 1 < D 2 and D 4 < D < D 2 . • F * if D 2 < D 1 and D 2 < D < D 3 . • F 1 if D 2 < D 1 and D 3 < D < D 1 . • F 0 if max(D 1 , D 4 ) < D.
Proof. We restrict the proof to the situation where D < min(D 1 , D 2 ). The other cases can be done similarly. Let x 1 (0) > 0, x 2 (0) > 0 and ω the ω-limit set of (x 1 (0), x 2 (0)). ω is an invariant compact set and ω ⊂ S. Assume that ω contains a point M on the x 1 x 2 axis :

• M can't be F 0 because F 0 is an unstable node and can't be a part of the ω-limit set of (x 1 (0), x 2 (0)),

• If M ∈]x 1 , s in 1 ] × {0} (respectively M ∈ {0}×]x 2 , s in 2 ]). As ω is invariant then γ(M ) ⊂ ω which is impossible because ω is bounded and γ(M ) =]x 1 , +∞[×{0} (respectively γ(M ) = {0}×]x 2 , +∞[), • If M ∈]0, x1 [×{0} (respectively M ∈ {0} , ×]0, x2 [). ω contains γ(M ) =]0, x1 [×{0} (respectively γ(M ) = {0}×]0, x2 [). As ω is a compact, then it contains the adherence of γ(M ), [0, x1 ] × {0} (respectively {0} × [0, x2 ]). In particular, ω contains F 0 which is impossible, • If M = F 1 (respectively M = F 2 ). ω is not reduced to F 1 (respectively to F 2 ). By
Butler-McGehee theorem, ω contains a point P of (0, +∞) × {0} other that F 1 (respectively of {0} × (0, +∞) other that F 2 ) which is impossible. Finally, the ω-limit set don't contain any point on the x 1 x 2 axis. System (5) has no periodic orbit inside S. Using the Poincar-Bendixon Theorem [START_REF] Smith | The theory of the chemostat, Dynamics of microbial competition[END_REF], F * is a globally asymptotically stable equilibrium point for system [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF].

Theorem 4. Assume that system (5) has at most one positive quilibrium F * , then for every initial condition in R 4 + , the trajectories of system (2) converge asymptotically to:

• E * if D < min(D 1 , D 4 ). • E * if D 1 < D 2 and D 1 < D < D 4 . • E 2 if D 1 < D 2 and D 4 < D < D 2 . • E * if D 2 < D 1 and D 2 < D < D 3 . • E 1 if D 2 < D 1 and D 3 < D < D 1 . • E 0 if max(D 1 , D 2 ) < D.
Proof. Let (s 1 (t), x 1 (t), s 2 (t), x 2 (t)) be a solution of (2). From ( 3) and ( 4) we deduce that

s 1 (t) = s in 1 -x 1 (t) + K 1 e -Dt and s 2 (t) = s in 2 + x 1 (t) -x 2 (t) + K 2 e -Dt , where K 1 = s 1 (0) + x 1 (0) -s in 1 and K 2 = s 2 (0) + x 2 (0) -x 1 (0) -s in 2 . Hence (x 1 (t), x 2 (t)
) is a solution of the nonautonomous system of two differential equations :

   ẋ1 = f 1 s in 1 -x 1 + K 1 e -Dt , s in 2 + x 1 -x 2 + K 2 e -Dt -D x 1 , ẋ2 = f 2 s in 1 -x 1 + K 1 e -Dt , s in 2 + x 1 -x 2 + K 2 e -Dt -D x 2 . (13)
This is an asymptotically autonomous differential system which converge to the autonomous system [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. The set Ω is attractor of all trajectories in R 4 + and the phase portrait of system reduced to Ω (5) contains only locally stable nodes, unstable nodes, saddle points and no trajectory joining two saddle points. Thus we can apply Thiemes's results [START_REF] Thieme | Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations[END_REF] and conclude that the asymptotic behaviour of the solution of the complete system ( 13) is the same that the asymptotic behaviour described for the reduced system (5) and the main result is then deduced.

6. The anaerobic digestion process : An example of a synthrophic relationship "Methane fermentation" or "anaerobic digestion" is a process that converts organic matter into a gaseous mixture mainly composed of methane and carbon dioxide (CH 4 and CO 2 ) through the action of a complex bacterial ecosystem (cf. Fig. 9). It is often used for the treatment of concentrated wastewaters or to stabilize the excess sludge produced in wastewater treatment plants into more stable products. There is also considerable interest in plant-biomass-fed digesters, since the produced methane can be valorized as a source of energy. It is usually considered that a number of metabolic groups of bacteria are involved sequencially.

One specific characteristic of the anaerobic process is that within such groups, there exists populations exhibiting obligatory mutualistic relationships. Such a syntrophic relationship is necessary for the biological reactions to be thermodynamically possible. In the first steps of the reactions (called "acidogenesis"), some hydrogen is produced. In El Hajji et al. [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF], this production of hydrogen at this reaction step was neglected (compare Fig. 9 with Fig. 1 of [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]). This hypothesis constitue the first novelty with respect to [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. It is to be noticed that an excess of hydrogen in the medium inhibits the growth of another bacterial group called "acetogenic bacteria". Their association with H 2 consuming bacteria is thus necessary for the second step of the reaction to be fulfilled. Such a syntrophic relationship has been pointed out in a number of experimental works (cf. for instance the seminal work by [START_REF] Bryant | Methanobacillus omelianskii, a symbiotic association of two species of bacteria[END_REF]). Let us consider the subsystem of the anaerobic system where the VFA (for Volatile Fatty Acids) are transformed into H 2 , CH 4 and CO2. We can formalize the corresponding biological reactions as a first bacterial consortium X 1 (the acetogens) transforming S 1 (the VFA) into S 2 (the hydrogen) and acetate (cf. Fig. 9). Then, a second species X 2 (the hydrogenotrophic-methanogenic bacteria) grows on S 2 . In practice, acetogens are inhibited by an excess of hydrogen and methanogens by an excess of VFA. Thus, it is further assumed that X 1 is inhibitied by S 2 and X 2 by S 1 . The last inhibition relationship constiute the second novelty with respect to [START_REF] Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. This situation is precisely the one considered within the model [START_REF] Aota | Mutualistic relationships between phytoplankton and bacteria caused by carbon excretion from phytoplankton[END_REF]. We have proposed a mathematical model involving a syntrophic relationship of two bacteria. It results from this analysis that, under general and natural assumptions of monotonicity on the functional responses, the stable asymptotic coexistence of the two bacteria is possible.

x2 s in 2 ξ2Figure 3 .

 23 Figure 3. Existence and uniqueness of x2 . On the left, the case D 2 < D 1 : Φ 1 (0, x2 ) > D for all D < D 2 . On the right, the case D 2 > D 1 : Φ 1 (0, x2 ) > D if and only if D < D 4 .

  F 2 is a saddle point for all 0 < D < D 4 and a stable node for all D 4 < D < D 2 .

( 2 )

 2 If min(D 1 , D 2 ) < D < max(D 1 , D 2 ), four subcases must be distinguished (a) If D 1 < D 2 and D 1 < D < D 4 then (5) admits the trivial and boundary equilibria F 0 and F 2 , which are saddle points and at least one positive equilibrium F * . If F * is the unique positive equilibrium then it is a stable node. Generically, the system has an odd number of positive equilibria which are alternatively stable nodes and saddle points, the one at the very left of these positive equilibria is a stable node. (b) If D 1 < D 2 and D 4 < D < D 2 then (

  3 the bifurcational values are D 1 = 4/3 and D 2 = 21/16. If D > max(D 1 , D 2 ), for instance for D = 3/2, one obtains a bistability phenomenon corresponding to case (3) of Theorem 1, with two positive equilibria. According to the initial condition, both species can coexist at equilibrium F

Figure 6 .

 6 Figure 6. The non hyperbolic cases. When D = 6/5, F 0 and F 2 coalesce. When D = 8/9, F * 1 and F 1 coalesce (saddle node bifurcation). When D = 1, F * 1 and F * 2 coalesce (saddle node bifurcation). When D = 6/5, F 0 and F 2 coalesce.

1 F * 2 F * 1 F * 2 F 0 Figure 8 .

 121208 Figure 8. Numerical solutions in the bistability case D = 1.5 and parameters values[START_REF] Smith | The theory of the chemostat, Dynamics of microbial competition[END_REF]. On the left, the separatrix (in green) of the saddle point F * 1 separate the domain S in two region which are the basins of attraction of the boundary equilibrium point F 0 and the positive equilibrium point F * 2 . On the center, the phase portrait. On the right, the isoclines.

2 CH 4 Figure 9 .

 249 Figure 9. Anaerobic fermentation process

  s in 2 + e ξ1 -e ξ2 ) -D.

	We have	∂h 1 ∂ξ 1	+	∂h 2 ∂ξ 2	= -e ξ1 ∂f 1 ∂s 1	+ e ξ1 ∂f 1 ∂s 2	-e ξ2 ∂f 2 ∂s 2	< 0.

Acknowledgements

The authors acknowledge Inra and Inria for financial support.