P Drieux

G Chollon

A Allemand

S Jacques

CHEMICAL VAPOR DEPOSITION AND CHARACTERIZATION OF THICK SILICON CARBIDE TUBES FOR NUCLEAR APPLICATIONS

SiC/SiC composites have an outstanding mechanical behavior under irradiation, but their high porosity excludes their use as nuclear fuel claddings in the future nuclear power plants. A complementary thick and tight SiC sheath could be a solution to ensure the first barrier towards fissile materials.

The aim of this work is to make long, free standing and high strength SiC tubes. A few hundred micrometers thick tubular coatings were produced by chemical vapor deposition at atmospheric pressure, from CH3SiHCl2/Ar/H2 mixtures. Their chemical compositions and microstructures were studied by electron probe microanalysis, Raman spectroscopy and scanning electron microscopy.

The deposition rate, composition and microstructure were investigated as a function of the substrate temperature and the gas flow rates. A Fourier transformed infrared spectroscopy analysis of the gas phase was carried out at the reactor outlet. The Si/C ratio, the SiC degree of crystallization and the surface morphology are strongly related to the maturation of the gas phase and the deposition regime.

INTRODUCTION

The structural components of a nuclear fusion or fission reactor must show a low induced radioactivity after neutron irradiation. Tubular SiC/SiC composites [START_REF] Lorrette | Progress in developing SiC/SiC composite materials for advanced nuclear reactors[END_REF]2] are materials of prime interest to be part of the nuclear fuel claddings of future power plants [START_REF] Jones | Recent advances in the development of SiC/SiC as a fusion structural material[END_REF] thanks to their low neutron activation characteristics [START_REF] Dyomina | Low-activation characteristics of V-alloys and SiC composites[END_REF], but also their high strength and fracture toughness at high temperature [START_REF] Raffray | Design and material issues for high performance SiCf/SiC-based fusion power cores[END_REF].

The optimization of SiC/SiC composites for nuclear purposes requires high thermal properties as well as excellent gas impermeability [START_REF] Riccardi | Issues and advances in SiCf/SiC composites development for fusion reactors[END_REF]. The latter point has led to the idea of covering the composite with a dense and impermeable SiCβ sheath having high mechanical properties and stability at high temperature and under neutron irradiation [START_REF] Feinroth | Multi-Layered Ceramic Tube for Fuel Containment Barrier And Other Applications In Nuclear And Fossil Power Plants[END_REF].

Well known exemplary SiC coating materials, in terms of mechanical properties, are SiC monofilaments [START_REF] Chollon | High temperature properties of SiC and diamond CVDmonofilaments[END_REF]. They are prepared by chemical vapor deposition at atmospheric pressure (APCVD) on a hot filament substrate, using chlorosilanes (e.g. dichloromethylsilane: DCMS) diluted in hydrogen as the SiC precursor [START_REF] Cheng | The microstructure of sigma 1140+SiC fibres[END_REF]. A failure strength up to 6 GPa and a strain of about 1.5 % can indeed be reached in the high performance SCS-Ultra from Specialty Materials Inc. The aim of the present study is to prepare SiC tubes by APCVD, with a composition, microstructure and morphology compatible with high strength and strain at failure.

The present paper reports on our latest results on the elaboration and characterization of thick SiC tubular coatings. Efforts have been mainly concentrated on the adjustment of the stoichiometry and the microstructure of the SiC deposit. The influence of the deposition temperature, the initial composition of the gas mixture and the total flow rate on the physical and chemical properties and growth kinetics of the coatings have been investigated. Fourier transform infrared (FTIR) analyses of the gas phase at the reactor outlet have also been performed to follow the decomposition of the precursor.

EXPERIMENTAL APPROACH

CVD process

The laboratory-scale reactor for the APCVD of thick tubular SiC coatings is presented in Fig. 1. It consists of a horizontal silica glass tube (L = 1000 mm, ∅int = 46 mm) connected to gas inlet/outlet flanges at both ends. Tubular SiC deposits can be obtained by two ways: an outer coating on a solid graphite substrate or an inner coating inside a silica tube.

In reference to the filaments process, a graphite solid cylinder (L = 200 mm, ∅ext = 10 mm) was first used as a substrate. In this case, the CVD reactor operated in a coldwall configuration, since the substrate was directly heated by induction, while the external silica chamber remained relatively cold. The first experiments led to a main difficulty related to the cold-wall configuration and the horizontal position of the cylinder: a thickness gradient was indeed observed between the upper and lower parts of the rod, due to the convection of the gases. Furthermore, the removal of the graphite substrate is difficult and the final morphology of the SiC coating is strongly affected by the roughness and the porosity of the graphite substrate. The inside of a silica tube (L = 500 mm, ∅int = 8 mm) was used as an alternative substrate, the gases being forced to flow through it. The silica tube was brought to high temperature by a surrounding graphite susceptor heated by induction. This unit is placed inside the original 46 mm diameter silica tube. This new reactor was then in a hot-wall configuration, since the whole 8 mm diameter reaction chamber, including the substrate walls, was isothermal in the hot deposition area (4 cm).

Dichloromethylsilane (H3CSiHCl2, DCMS) was used as the SiC precursor [START_REF] Cheng | The microstructure of sigma 1140+SiC fibres[END_REF]. The DCMS vapor saturation was obtained from a H2 carrier gas going through a bubbler maintained at 20°C (PDCMS = 48 kPa). The DCMS concentration was adjusted by further diluting the initial saturated DCMS/H2 mixture in a secondary H2 flow.

As the final aim of this work is the manufacturing of relatively long lengths (several tens of cm), the induction heat station was set on a motorized bench allowing the displacement of the induction coil, and therefore of the hot area, along the substrate. Although the study presented here was conducted without any displacement of the hot area, the travelling speed can range from 0.25 to 5 cm/min.

The surface temperature (T) of the graphite substrate (in the cold-wall configuration) or the graphite shell outer surface (in the hot-wall geometry) was monitored in situ during the CVD process with a high-resolution dichromatic pyrometer (Impac ISQ 5).

Deposition conditions

For all the experiments, the reactor was operated at atmospheric pressure, while the temperature and total gas flow Qtot ranged from 1000 to 1100°C and from 200 to 850 sccm (standard cm 3 .min -1 ), respectively. A list of representative experiments is given in Table 1. Runs at higher temperatures were not carried out in the hot wall geometry due to the risk of silica substrate softening.

Deposition of the pyrolytic carbon interlayer

The first coatings deposited directly on the silica tubes showed a strong chemical bonding that resulted, on cooling, to multiple cracking. A thin layer of pyrolytic carbon (PyC) was therefore deposited at the surface of the SiO2 tube. This interlayer was deposited in the same reactor by injection of pure propene at low pressure (QC3H6 = 100 sccm, T = 1000°C, P = 3 kPa). Its anisotropic structure was found to promote the SiC/SiO2 interface delamination.

Characterization of the coatings

The morphology of the coatings was characterized with a field emission gun scanning electron microscope (SEM) (FEI Quanta 400F).

Electron probe microanalyses (EPMA) (SX 100 from CAMECA, France) were conducted on cross-sections of CVD-coated tube fragments. The Si, C and O concentrations were assessed using the wavelength dispersive spectroscopy (WDS) mode (10 kV, 10 nA). Punctual and line scan measurements were performed along the cross section of the coatings, with a spatial resolution of approximately 1 μm 3 .

Raman microprobe (RM) analyses (Labram HR from Horiba Jobin Yvon) were also conducted. The excitation source was a He-Ne laser (λ=632.8 nm). The power was kept below 1 mW to avoid heating of the sample. The lateral resolution of the laser probe was close to 1 μm and the depth analyzed was less than 1 µm. As for EPMA, punctual and line scan analyses were recorded along the cross section of the deposit.

Analysis of the gas phase

A semi quantitative study of the gas phase was carried out by Fourier transform spectroscopy (FTIR) at the reactor outlet (Fig. 2). The laser beam of the spectrometer (Nicolet 550) was led through a room temperature analysis cell (with two ZnSe windows at both ends) installed at the reactor outlet (absorption configuration). While the reaction chamber was set at atmospheric pressure, the pressure in the gas cell was adjusted at 1 kPa to avoid saturation of absorption peaks on the FTIR spectra. At the outlet of the gas cell, the infrared beam was focused on the HgCdTe detector placed below. The whole infrared beam path was enclosed and constantly purged with a nitrogen flow in order to limit the absorption of atmospheric compounds. A background spectrum was first recorded with a pure H2 flow in the reactor and subtracted to the reacting gases spectrum. The apparatus allowed the acquisition of spectra in the wave-number range 650 -4000 cm -1 , with a resolution of 1 cm -1 .

RESULTS AND DISCUSSION

The PyC interfacial layer deposited at low pressure before SiC (with a thickness of 350 nm) does not introduce major surface flaws in the SiC coating (Fig. 3). The EPMA and SEM measurements at opposite sides confirmed the chemical composition and thickness homogeneity of the SiC coatings deposited in the hot-wall geometry.

A number of thick SiC tubes were obtained with this APCVD process, and were successfully extracted from the silica substrate by taking advantage of the difference of thermal expansion coefficients between the silica substrate and the deposited SiC and the weak SiO2/SiC interface provided by a PyC interlayer.

It is well known [START_REF] Féron | In situ kinetic analysis of SiC filaments CVD[END_REF][START_REF] Chollon | Transient stages during the chemical vapour deposition of silicon carbide from CH3SiCl3/H2: impact on the physicochemical and interfacial properties of the coatings[END_REF] that one of the key parameters in the deposition of SiC is the precursor dilution ratio α:

DCMS H Q Q 2 = α (1)
The Si/C atomic ratio increases significantly with α (Table I). For α increasing from 0 to 12, the silicon atomic concentration in the coating varies from 54 %at. to 67 % at. The oxygen concentration was found to remain lower than 2 %at. in all specimens. Furthermore, a higher dilution of DCMS in H2 results in a sharp decrease of the deposition rate. At 1025 °C, the deposition rate remains below 50 µm/h when α is higher than 4. Such a deposition rate is far too low to consider a continuous deposition process for the production of thick and long tubes within a reasonable time. The Raman Spectra (Fig. 4) show no clear crystallized SiC features (expected at 750-1000 cm -1 ) in the deposits, but a strong band at 500cm -1 , typical of free silicon. The intensity of the Si band varies with α in very good agreement with the Si/C ratio. However, even for α = 0, free silicon is still present in the coating at this temperature and the SiC phase is poorly crystallized, as shown by the very broad and weak characteristic features.

A gradual improvement of the SiC crystalline state can be noticed when the deposition temperature increases. For coatings 4, 9, 10 and 11, corresponding to deposition temperatures increasing from 1025°C to 1100°C, the emergence of two bands in the 700 -1000 cm -1 region, for the highest deposition temperatures confirms the coarsening of the SiC phase (Fig. 5). In parallel, the narrow peak at 520 cm -1 corresponding to crystallized silicon disappears as well as the surrounding bump due to amorphous Si. The Si/C atomic ratios obtained by EPMA are very consistent with the Raman analyses. It is worthy of note that a Si concentration of 49 %at is obtained at 1100 °C, indicating that pure SiC can be deposited in these conditions, without any additional carbon source in the gas phase.

Only few changes of the deposition rate can be noticed with the variation of temperature. The deposition process is apparently not thermally activated, suggesting a mass transfer limited regime [START_REF] Astarita | Regimes of mass transfer with chemical reaction[END_REF]. However, since the temperature increase probably results in parallel in important changes of the gas phase composition in the hot area, the assumption of a precursor supply limitation should be considered with care.

The influence of the total gas flow rate (Qtot) was also examined (see deposits 4, 7 and 8 in Table I, obtained at Qtot = 425 sccm, 850 sccm and 213 sccm, respectively). The gas velocity is an important parameter especially in the case of a mass transfer regime where the deposit morphology can be very sensitive to the gas flow around the substrate. In addition, Qtot may affect the decomposition rate of precursors through the residence time varying along the position in the hot area. At low flow rates (Q8 < Q4 < Q7, see Table 1), the carbon concentration, as measured by EPMA, tends to increase (from 25 %at. to 52 %at.). These results are consistent with the Raman analyses (Fig. 6) showing that the peak assigned to crystalline Si tends to disappear at low Qtot while simultaneously, the SiC characteristic bands gradually appear. The deposition rate increases significantly with Qtot from 33 µm/h at 213 sccm to 237 µm/h at 850 sccm. This behavior is consistent with the previous study on the influence of temperature. The increase of the deposition rate with Qtot is indeed indicative of a regime limited by diffusion. In parallel, the change in the composition of the deposit suggests a major effect of the maturation of gas phase, i.e. of the decomposition rate of DCMS into various intermediates, which may act as effective precursors of Si and C for the coating.

A minimal deposition temperature appears necessary to form a near stoichiometric coating with highly crystalline SiC in the reactor. Whereas there are only a few studies on the CVD from DCMS/H2 [START_REF] Féron | In situ kinetic analysis of SiC filaments CVD[END_REF][START_REF] Vincent | Chemically vapour-deposited coatings of silicon carbide on planar alumina substrates[END_REF][START_REF] Brennan | Interfacial studies of chemical-vapor-infiltrated ceramic matrix composites[END_REF], several articles on the dichlorodimethylsilane (DCDMS)/H2 system [START_REF] Brennan | Interfacial studies of chemical-vapor-infiltrated ceramic matrix composites[END_REF][START_REF] Cagliostro | Model for the formation of silicon carbide from the pyrolysis of dichlorodimethylsilane in hydrogen[END_REF][START_REF] Cagliostro | Comparison of the pyrolysis products of dichlorodimethylsilane in the chemical vapor deposition of silicon carbide on silica in hydrogen or argon[END_REF] and an abundant work on the MTS/H2 system [START_REF] Chollon | Transient stages during the chemical vapour deposition of silicon carbide from CH3SiCl3/H2: impact on the physicochemical and interfacial properties of the coatings[END_REF][START_REF] Papasouliotis | Experimental study of atmospheric pressure chemical vapor deposition of silicon carbide from methyltrichlorosilane[END_REF][START_REF] Zhang | CVD of SiC from Methyltrichlorosilane[END_REF][START_REF] Loumagne | Reaction mechanisms of the chemical vapor deposition of SiC-based ceramics from CH3SiCl3/H2 gas precursor[END_REF] can be found in the literature. Féron et al. used DCMS at higher temperatures and atmospheric pressure in a cold wall reactor [START_REF] Féron | In situ kinetic analysis of SiC filaments CVD[END_REF]. They obtained only Si-rich coatings (C/Si at. < 0.7) at much higher deposition rates limited by chemical reactions. This suggests the occurrence of DCMS depletion in the present case, in the hot wall configuration. On the other hand, Cagliostro et al. used DCDMS at similar temperatures and pressure in a hot wall reactor configuration [START_REF] Cagliostro | Model for the formation of silicon carbide from the pyrolysis of dichlorodimethylsilane in hydrogen[END_REF][START_REF] Cagliostro | Comparison of the pyrolysis products of dichlorodimethylsilane in the chemical vapor deposition of silicon carbide on silica in hydrogen or argon[END_REF]. They found a dependence of the deposition rate with the total flow rate for T > 900 °C and a strong silicon excess in the coating, decreasing along the reactor axis (i.e. while the residence time increases). A similar behavior was reported in several studies of the MTS/H2 system in relatively similar T and P conditions. Papasouliotis et al. and Huttinger et al. evidenced, for T = 1000 °C, a decrease of the deposition rate associated with a decrease of the silicon excess along the hot zone [START_REF] Papasouliotis | Experimental study of atmospheric pressure chemical vapor deposition of silicon carbide from methyltrichlorosilane[END_REF][START_REF] Zhang | CVD of SiC from Methyltrichlorosilane[END_REF]. The decreasing of the total flow rate also reduces the deposition rate (by increasing the residence time) [START_REF] Papasouliotis | Experimental study of atmospheric pressure chemical vapor deposition of silicon carbide from methyltrichlorosilane[END_REF].

Since the silica substrate cannot withstand temperatures higher than 1100 °C in the hotwall configuration, only a narrow operating temperature window is allowed. Since the deposition rate is apparently not thermally activated, the gas flow rate (or the residence time) appears as a key parameter for elaboration of long tubes.

The changes in the chemical composition of the deposit indicate an influence of the temperature on the concentration of the effective Si and C precursors brought to the deposition area. This can be explained by a change of the DCMS decomposition rate in the homogeneous (gas phase) state. To support this assumption, complementary FTIR analyses of the gas phase at the reactor outlet were performed.

A room temperature spectrum of the DCMS was recorded as a reference (Fig. 7). This spectrum shows characteristic peaks of chlorosilane products, as in previous works on DCMS [START_REF] Féron | Filament CVD de SiC de composition et microstructure adaptées au procédé Snecma d'enduction par le titane liquide[END_REF]. A table of the vibration modes of the main species from the Si-C-Cl-H system reports the peak frequencies observed experimentally and their assignments from the literature (Table II).

The infrared spectrum of DCMS at 25 °C shows the presence of a weak feature at 3010 cm -1 corresponding to the C-H stretching in the methyl group, a very sharp and intense peak at 2260 cm -1 , due to the stretching of Si-H bond, a low intensity signal corresponding to C-H wagging at 1270 cm -1 , and a conglomerate of intense peaks in the 900 -650 cm -1 region, due to the stretching of Si-C and Si-Cl bonds (Table II). In a second step, an investigation of the temperature effect on the gas phase composition was carried out (Fig. 8).

Whereas the spectra remains unchanged for T ranging from 25 °C to 700 °C, major chemical modifications appear at higher temperatures. As the analysis is carried out ex situ at room temperature, detected molecules are necessarily stable and some of them may result from the recombination of other unstable species (e.g. free radicals). Some part of the DCMS decomposition products can also be consumed during the SiC deposition. However, the apparition of new species at the reactor outlet gives useful information for a better understanding of the APCVD process.

As shown in Fig. 8, the decomposition of dichloromethylsilane in the reactor is accompanied by the formation of HCl, HSiCl3, HSi2Cl2 and CH4. SiCl4, another stable compound, is also expected (642 cm -1 ). However, due to the poor sensitivity of the detector at low frequencies and to other peaks overlapping, this specie could not be followed as accurately as the others. To follow the evolution of the gas phase with temperature, the main peak areas were subsequently plotted versus T (Fig. 9)

The gradual decomposition of DCMS above 700°C (b, d, e, f, g, i) corresponds to the formation of various products in the gas phase. The apparition of methane (j, l), is progressive and tends to a plateau for the highest temperatures. The formation of trichlorosilane (HSiCl3, curves c and k) shows a maximum at 900°C. Its concentration decreases at higher temperatures, indicating consumption, either due to the deposition on the reactor walls, or to the formation of other reactive intermediates. SiH2Cl2 also shows a similar behavior, its concentration decreasing strongly down to nearly zero for temperatures above 900 °C (h). Hence, the formation and consumption of chlorosilanes appears to be strongly related to the silicon incorporation in the coating.

A decomposition route of the DCMS during the APCVD process can be tentatively proposed by the analysis of the above results and the previous conclusions from the literature.

Zhang and Huttinger [START_REF] Zhang | CVD of SiC from Methyltrichlorosilane[END_REF] proposed a mechanism for the deposition of SiC at near atmospheric pressure from MTS/H2 mixture. As for MTS, the Si-C bond is probably quickly dissociated into CH3˙ and HSiCl2˙ in the hot zone [START_REF] Papasouliotis | Experimental study of atmospheric pressure chemical vapor deposition of silicon carbide from methyltrichlorosilane[END_REF][START_REF] Zhang | CVD of SiC from Methyltrichlorosilane[END_REF][START_REF] Loumagne | Reaction mechanisms of the chemical vapor deposition of SiC-based ceramics from CH3SiCl3/H2 gas precursor[END_REF]. This assessment was confirmed by the ab initio calculations of Allendorf et al. [START_REF] Allendorf | Theoretical study of the thermochemistry of molecules in the Si-C-Cl-H system[END_REF]. The Si-H bond of DCMS or HSiCl2˙ is also probably easily broken to form eventually the reactive SiCl2, likely the major silica source for the coating [START_REF] Zhang | CVD of SiC from Methyltrichlorosilane[END_REF]. In parallel, CH3˙ readily recombines into CH4, which has a low surface reactivity. The chlorosilanes and SiCl4 detected by FTIR result from the recombination of the HSiCl2˙ and SiCl2 radicals, in the hot zone or during the cooling. The higher reactivity of SiCl2 compared to CH4 is responsible for the presence of excess silicon, especially at low temperatures and high α ratios. At high temperature (T > 1050 °C), more reactive unsatured hydrocarbons might be formed (FTIR analyses were not carried out at these temperatures), resulting in an increase of the deposition rate and of the carbon ratio in the coating. On the other hand, higher H2 concentrations (higher α) promote the reduction of SiCl2 and therefore the deposition of free silicon, while they stabilize CH4 and prevent the deposition of C.

CONCLUSION

The present paper introduced a method to prepare long and dense monolithic SiC tubes with thick walls by APCVD. Free standing SiC tubes were obtained by deposition inside a SiO2 tube covered with pyrocarbon. An optimization of the process was carried out by analyzing the influence of different parameters such as the precursor dilution in the injected gas mixture, the total flow rate and the deposition temperature. The increase of deposition rate with the gas flow rate, and the low thermal activation of the deposition process indicate that the process is limited by mass transfer. The modification of the gas phase itself with the different parameters leads to significant differences in the final composition of the SiC deposit.

A better understanding of the process was obtained by analyzing the gas phase at the reactor outlet by FTIR spectroscopy. The production of chlorosilanes (SiCl4, HSiCl3 and HSi2Cl2) was highlighted, along with a production of methane (CH4) and HCl. The increase of temperature promotes the reactivity of the carbon precursors, while an important dilution inhibits them and favors the reduction of chlorosilanes, hence resulting in a Si rich deposit. The increase of the deposition rate with Qtot is indicative of a regime limited by diffusion.

The elaboration of longer and thicker SiC tubes is already in progress, and the mechanical testing of the samples, for instance by C-ring tests [28] is planned.

FIGURESFigure 1 .

 1 FIGURES

Figure 3 .

 3 Figure 3. Cross section of the SiO2/PyC/SiC interfacial region

Figure 4 .Figure 5 .Figure 6 .Figure 8 .Figure 9 .

 45689 Figure 4. Influence of α on the Raman spectra


Sarra Nouaoura
email: sarra.nouaoura@enit.utm.tn

Radhouane Fekih-Salem
email: radhouene.fekihsalem@isima.rnu.tn

Nahla Abdellatif
email: nahla.abdellatif@ensi-uma.tn

Tewfik Sari
email: tewfik.sari@inrae.fr

MATHEMATICAL ANALYSIS OF A THREE-TIERED FOOD-WEB IN THE CHEMOSTAT

Keywords: 2010 Mathematics Subject Classification. Primary: 34A34, 34D20; Secondary: 92B05, 92D25 Anaerobic digestion, bifurcation diagram, chemostat, coexistence, stability, three-tiered food-web

A mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food-web is investigated. The model is a sixdimensional system of ordinary differential equations. In our study, the phenol and the hydrogen inflowing concentrations are taken into account as well as the maintenance terms. The case of a large class of growth kinetics is considered, instead of specific kinetics. We show that the system can have up to eight types of steady states and we analytically determine the necessary and sufficient conditions for their existence according to the operating parameters. In the particular case without maintenance, the local stability conditions of all steady states are determined. The bifurcation diagram shows the behavior of the process by varying the concentration of influent chlorophenol as the bifurcating parameter. It shows that the system exhibits a bi-stability where the positive steady state can lose stability undergoing a supercritical Hopf bifurcation with the emergence of a stable limit cycle.

1.

Introduction. The chemostat model is widely used in microbiology and ecology as a mathematical representation of the continuous culture of micro-organisms, that is, the growth of micro-organisms in ecosystems that are continuously fed with nutrients [START_REF] Hoskisson | Continuous culture-making a comeback?[END_REF][START_REF] Monod | La technique de culture continue: Théorie et applications[END_REF][START_REF] Wade | Perspectives in mathematical modelling for microbial ecology[END_REF]. Several textbooks on the mathematical analysis of this model with one and more species are available [START_REF] Harmand | The Chemostat: Mathematical Theory of Microorganism Cultures[END_REF][START_REF] Smith | The Theory of the Chemostat: Dynamics of Microbial Competition[END_REF]. The chemostat model predicts that coexistence of two or more microbial populations competing for a single nonreproducing nutrient is not possible. Only the species with the lowest 'break-even' concentration survives. This result, known as the Competitive Exclusion Principle (CEP), has a long history in the literature of bio-mathematics and the reader may consult [START_REF] Rapaport | A new proof of the competitive exclusion principle in the chemostat[END_REF] and the references therein.

Although the theoretical prediction of the CEP has been corroborated by the experiences of Hansen and Hubell [START_REF] Hansen | Single-nutrient microbial competition: qualitative agreement between experimental and theoretically forecast outcomes[END_REF], the biodiversity found in nature as well as in biological reactors seems to contradict the CEP. This has led to a great deal of mathematical research aimed at extending the chemostat model to better match theory and observations. Among the mechanisms that promote the coexistence of species, we can cite : the crowding effects (see [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF][START_REF] Leenheer | Crowding effects promote coexistence in the chemostat[END_REF] and the references therein), the role of density-dependent growth functions (see [START_REF] Fekih-Salem | A density-dependent model of competition for one resource in the chemostat[END_REF] and the references therein), more complex food webs (see [START_REF] Ballyk | A nutrient-prey-predator model: Stability and bifurcations[END_REF][START_REF] Hsu | Analysis of a model of two parallel food chains[END_REF][START_REF] Wolkowicz | Successful invasion of a food web in a chemostat[END_REF] and the reference therein), the presence of inhibitors that affects the strongest competitor (see [START_REF] Bar | The operating diagram for a model of competition in a chemostat with an external lethal inhibitor[END_REF][START_REF] Dellal | Global analysis of a model of competition in the chemostat with internal inhibitor[END_REF][START_REF] Dellal | The operating diagram of a model of two competitors in a chemostat with an external inhibitor[END_REF][START_REF] Weedermann | Analysis of a model for the effects of an external toxin on anaerobic digestion[END_REF], and the references therein), the commensalistic relationship where a second species (the commensal) needs the first one (the host) to grow while the host species is not affected by the growth of the commensal one (see [START_REF] Benyahia | Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes[END_REF][START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF][START_REF] Dimitrova | Nonlinear stabilizing control of an uncertain bioprocess model[END_REF][START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF] and the references therein), the syntrophic relationship where two microbial species depend on each other for survival (see [START_REF] Daoud | Steady state analysis of a syntrophic model: The effect of a new input substrate concentration[END_REF][START_REF] El-Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF][START_REF] Fekih-Salem | Analyse mathématique d'un modèle de digestion anaérobie à trois étapes[END_REF][START_REF] Sari | The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat[END_REF][START_REF] Sari | A model of a syntrophic relationship between two microbial species in a chemostat including maintenance[END_REF][START_REF] Xu | Maintenance affects the stability of a two-tiered microbial 'food chain' ?[END_REF] and the reference therein).

Some of these models were constructed to have a better understanding of anaerobic digestion (AD), which is an important process used in the treatment of wastewater and waste, including a large number of species that coexist in a very complex relationship. The full anaerobic digestion model (ADM1) developed in [START_REF] Batstone | The IWA Anaerobic Digestion Model No 1 (ADM1)[END_REF] includes 32 state variables and a large number of parameters. Therefore, the qualitative analysis and the control of this model are very difficult because of its complexity and its dependence on many operational variables. The so-called AM2 model developed in [START_REF] Bernard | Dynamical model development and parameter identification for an anaerobic wastewater treatment process[END_REF] represents a two-tiered food-web and provides satisfactory prediction of the AD process by using the parameter identification theory and experimental data. In [START_REF] Fekih-Salem | Analyse mathématique d'un modèle de digestion anaérobie à trois étapes[END_REF], a three-tiered food-web model including enzymatic degradation of a substrate and commensalistic relationship is considered. In [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF], the authors consider a three-tiered food-web with three microbial species (chlorophenol and phenol degraders and hydrogenotrophic methanogen) that encapsulates the essence of the AD process. The corresponding model represents an extension of the model describing the interactions between propionate degraders and hydrogenotrophic methanogens in a two-tiered feeding chain [START_REF] Xu | Maintenance affects the stability of a two-tiered microbial 'food chain' ?[END_REF]. For a recent review of mathematical modeling of anaerobic digestion, the reader is refereed to [START_REF] Wade | Not Just Numbers: Mathematical Modelling and Its Contribution to Anaerobic Digestion Processes[END_REF].

The aim of our paper is to consider the three-tiered microbial 'food-web', developed in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] which is written as follows:

                     Ẋch = (Y ch f 0 (S ch , S H2 ) -D -k dec,ch )X ch Ẋph = (Y ph f 1 (S ph , S H2 ) -D -k dec,ph )X ph ẊH2 = (Y H2 f 2 (S H2 ) -D -k dec,H2 )X H2 Ṡch = D S in ch -S ch -f 0 (S ch , S H2 ) X ch Ṡph = D S in ph -S ph + 224 208 (1 -Y ch ) f 0 (S ch , S H2 ) X ch -f 1 (S ph , S H2 ) X ph ṠH2 = D S in H2 -S H2 -16 208 f 0 (S ch , S H2 ) X ch + 32 224 (1-Y ph )f 1 (S ph , S H2 ) X ph -f 2 (S H2 )X H2 , (1)
where X ch , X ph and X H2 denote, respectively, the chlorophenol, phenol and hydrogen degrader concentrations; f 0 , f 1 and f 2 are the corresponding growth rates; S ch , S ph and S H2 are the chlorophenol, phenol and hydrogen substrates concentrations; S in ch , S in ph and S in H2 are the substrate concentrations in the feed bottle; k dec,ch , k dec,ph and k dec,H2 represent the decay rates; D is the dilution rate of the chemostat; Y ch , Y ph and Y H2 are the yield coefficients.

The choice of the application of chlorophenol-mineralising food-web in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] is due to the availability of experimental data but the study of model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF] applies to any other similar microbial food chain. Indeed, the parameter values related to hydrogen are deduced from that of ADM1 [START_REF] Batstone | The IWA Anaerobic Digestion Model No 1 (ADM1)[END_REF]. For chlorophenol and phenol, they were chosen based on a combination of literature data (see [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] and the references therein). Recently, a rigorous mathematical analysis of this model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF] was done in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] with general growth rates but only the chlorophenol inflowing concentration has been taken into account. Using the linear change of variables given by [START_REF] Daoud | Steady state analysis of a syntrophic model: The effect of a new input substrate concentration[END_REF] and [START_REF] Leenheer | Crowding effects promote coexistence in the chemostat[END_REF], model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF] can be written as follows:

               ẋ0 = (µ 0 (s 0 , s 2 ) -D -a 0 )x 0 ẋ1 = (µ 1 (s 1 , s 2 ) -D -a 1 )x 1 ẋ2 = (µ 2 (s 2 ) -D -a 2 )x 2 ṡ0 = D s in 0 -s 0 -µ 0 (s 0 , s 2 )x 0 ṡ1 = D s in 1 -s 1 + µ 0 (s 0 , s 2 )x 0 -µ 1 (s 1 , s 2 )x 1 ṡ2 = D s in 2 -s 2 -ωµ 0 (s 0 , s 2 )x 0 + µ 1 (s 1 , s 2 )x 1 -µ 2 (s 2 )x 2 (2)
where s i , i = 0, 1, 2 are the three substrates; x i are the three microbial species; µ i are the specific growth rates given by ( 9); a i are the mortality rates; s in i are the concentrations of the three substrates in the feed device. All the yield coefficients in ( 1) are normalized to one except of ω.

In [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], system (2) can have at most three types of steady states when s in 0 > 0 and s in 1 = s in 2 = 0: the washout steady state, a coexistence steady state of three species and a steady steady state where only the hydrogen degrader is extinct. The local stability analysis is achieved when the maintenance is excluded from system [START_REF] Ballyk | A nutrient-prey-predator model: Stability and bifurcations[END_REF] where this six-dimensional model is reduced to a three-dimensional one. A numerical evidence shows that, when maintenance is included, the positive steady state can destabilize through a supercritical Hopf bifurcation with the appearance of a stable periodic orbit [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF]. In [START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF], when maintenance is excluded, the emergence of a supercritical Hopf bifurcation was analytically determined. In [START_REF] El-Hajji | Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat[END_REF], the threetiered model of [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] was studied by neglecting the part of hydrogen produced by the phenol degrader and the mortality rates.

When in addition to s in 0 > 0, the phenol and hydrogen inflowing concentrations are taken into account (s in 1 > 0 and s in 2 > 0), it was proven in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] that system (1) can have up to eight steady states, a fact that was confirmed in [START_REF] El-Hajji | Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat[END_REF] and [START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF] when a larger class of growth functions is considered, but the maintenance terms are neglected. In [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] most of the results on the existence and stability of steady states of model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF] were obtained only numerically.

In this paper, we generalize [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] by allowing a larger class of growth functions and by giving rigorous proofs for the results on the existence of steady states obtained in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] for system [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF]. For this class of growth function, we generalize [START_REF] El-Hajji | Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat[END_REF][START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF] by allowing maintenance terms and we generalize [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] by allowing phenol and hydrogen inflowing concentrations. More precisely, our main objective is to determine the existence of steady states of model (2) in the general case including maintenance and the inflow of the three substrates. Moreover, we analyze the asymptotic behavior of the system without maintenance and we apply our theoretical results to the three-tiered microbial model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF]. Actually, the results on the stability of steady state in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] were obtained when maintenance terms are included. In a forthcoming publication [START_REF] Nouaoura | Mathematical analysis of a threetiered model of anaerobic digestion[END_REF], we study theoretically this case, where system (2) cannot be reduced to a three-dimensional one.

The paper is organized as follows. The next section presents general assumptions for the growth rates and the mathematical analysis of the existence of steady states of model [START_REF] Ballyk | A nutrient-prey-predator model: Stability and bifurcations[END_REF] with respect to the operating parameters. In Section 3, the asymptotic behavior analysis of (2) was done in the particular case without maintenance. Considering specific growth rates, numerical simulations are presented in Section 4 as an application of our theoretical results to model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF]. Finally, conclusions are drawn in Section 5. The proofs of all the propositions and theorems are reported in Appendix A. The parameter values and some auxiliary functions are presented in Tables in Appendix B.

2. Assumptions and steady states. Considering model (2), we make the following general assumptions on the growth functions which are continuously differentiable (C 1 ). (H1) For all s 0 > 0 and s 2 > 0, 0 < µ 0 (s 0 , s 2 ) < +∞, µ 0 (0, s 2 ) = 0, µ 0 (s 0 , 0) = 0.

No growth can occur for species x 0 without substrates s 0 and s 2 . (H2) For all s 1 > 0 and s 2 0, 0

< µ 1 (s 1 , s 2 ) < +∞, µ 1 (0, s 2 ) = 0.
No growth can occur for species x 1 without substrate s 1 . (H3) For all s 2 > 0, 0 < µ 2 (s 2 ) < +∞, µ 2 (0) = 0.

No growth can occur for species x 2 without substrate s 2 . (H4) For all s 0 > 0 and s 2 > 0, ∂µ0 ∂s0 (s 0 , s 2 ) > 0, ∂µ0 ∂s2 (s 0 , s 2 ) > 0. The growth rate of species x 0 increases with substrates s 0 and s 2 . (H5) For all s 1 > 0 and s 2 > 0, ∂µ1 ∂s1 (s 1 , s 2 ) > 0, ∂µ1 ∂s2 (s 1 , s 2 ) < 0. The growth rate of species x 1 increases with the substrate s 1 but is inhibited by the production of s 2 . (H6) For all s 2 > 0, µ 2 (s 2 ) > 0.

The growth rate of species x 2 increases with substrate s 2 . (H7) The function s 2 → µ 0 (+∞, s 2 ) is monotonically increasing and the function s 2 → µ 1 (+∞, s 2 ) is monotonically decreasing. The maximum growth rate of the species x 0 and x 1 increases and decreases, respectively, with the concentration of substrate s 2 .

First, we show that the solutions of model ( 2) are nonnegative and bounded, which is a prerequisite for any reasonable model of the chemostat.

Proposition 1. For any nonnegative initial conditions, all solutions of system (2) remain nonnegative and are bounded for all t 0. Moreover, the set

Ω = (x 0 , x 1 , x 2 , s 0 , s 1 , s 2 ) ∈ R 6 + : ωx 0 + x 1 + x 2 + 2s 0 + 2s 1 + s 2 2s in 0 + 2s in 1 + s in 2
is positively invariant and a global attractor for [START_REF] Ballyk | A nutrient-prey-predator model: Stability and bifurcations[END_REF].

A steady state exists (or is said to be 'meaningful') if and only if all its components are nonnegative. This predicts eight possible steady states, labeled below as in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF]: • SS1 (x 0 = 0, x 1 = 0, x 2 = 0): the washout of all three microbial populations.

• SS2 (x 0 = 0, x 1 = 0, x 2 > 0): only the hydrogen degraders are maintained.

• SS3 (x 0 > 0, x 1 = 0, x 2 = 0): only the chlorophenol degraders are maintained.

• SS4 (x 0 > 0, x 1 > 0, x 2 = 0): only the hydrogen degraders are washed out.

• SS5 (x 0 > 0, x 1 = 0, x 2 > 0): only the phenol degraders are washed out.

• SS6 (x 0 > 0, x 1 > 0, x 2 > 0): all three microbial populations are present.

• SS7 (x 0 = 0, x 1 > 0, x 2 = 0): only the phenol degraders are present.

• SS8 (x 0 = 0, x 1 > 0, x 2 > 0): only the chlorophenol degraders are washed out.

To determine these steady states, we need to define some auxiliary functions that are listed in Table 1. The existence and definition domains of these functions are all relatively straightforward and can be found as in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF]. Following [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], we add a hypothesis on the function Ψ which then assures that there are at most two steady states of the form SS4.

(H8) When ω < 1, the function Ψ has a unique minimum s 2 = s 2 (D) on the interval s 0 2 , s 1 2 , such that ∂Ψ ∂s2 (s 2 , D) < 0 on s 0 2 , s 2 and ∂Ψ ∂s2 (s 2 , D) > 0 on s 2 , s 1 2 . As we will show in Section 4, this hypothesis (H8) is fulfilled with the specific growth rates [START_REF] Dellal | Global analysis of a model of competition in the chemostat with internal inhibitor[END_REF]. Now, we can state our main result.

Table 1. Notations, intervals and auxiliary functions.

Definition s i = M i (y, s 2 ) i = 0, 1
Let s 2 0. s i = M i (y, s 2 ) is the unique solution of µ i (s i , s 2 ) = y, for all 0 y < µ i (+∞, s 2 )

s 2 = M 2 (y) s 2 = M 2 (y)
is the unique solution of µ 2 (s 2 ) = y, for all 0 y < µ 2 (+∞)

s 2 = M 3 (s 0 , z) Let s 0 0. s 2 = M 3 (s 0 , z)
is the unique solution of µ 0 (s 0 , s 2 ) = z, for all 0 z < µ 0 (s 0 , +∞)

s i 2 = s i 2 (D) i = 0, 1 s i 2 = s i 2 (D) is the unique solution of µ i (+∞, s 2 ) = D + a i , for all D + a 0 < µ 0 (+∞, +∞), µ 1 (+∞, +∞) < D + a 1 < µ 1 (+∞, 0), resp. I 1 , I 2 I 1 = D 0 : s 0 2 < s 1 2 , I 2 = D ∈ I 1 : s 0 2 < M 2 (D + a 2 ) < s 1 2 Ψ(s 2 , D) Ψ (s 2 , D) = (1 -ω)M 0 (D + a 0 , s 2 ) + M 1 (D + a 1 , s 2 ) + s 2 , for all D ∈ I 1 and s 0 2 < s 2 < s 1 2 φ 1 (D) φ 1 (D) = inf s 0 2 <s2<s 1 2 Ψ(s 2 , D), for all D ∈ I 1 φ 2 (D) φ 2 (D) = Ψ (M 2 (D + a 2 ), D), for all D ∈ I 2 φ 3 (D) φ 3 (D) = ∂Ψ ∂s2 (M 2 (D + a 2 ), D), for all D ∈ I 2 J 0 , J 1 J 0 = max 0, s in 0 -s in 2 /ω , s in 0 , J 1 = 0, s in 1 ψ 0 (s 0 ) ψ 0 (s 0 ) = µ 0 s 0 , s in 2 -ω s in 0 -s 0 , for all s 0 max 0, s in 0 -s in 2 /ω ψ 1 (s 1 ) ψ 1 (s 1 ) = µ 1 s 1 , s in 2 + s in 1 -s 1 , for all s 1 ∈ 0, s in 1 + s in 2 ϕ i (D) i = 0, 1 ϕ i (D) = M i (D + a i , M 2 (D + a 2 )), resp., for all, D ∈ D 0 : s 0 2 < M 2 (D + a 2 ) , D ∈ D 0 : M 2 (D + a 2 ) < s 1 2
Theorem 1. Assume that Hypotheses (H1) to (H6) hold. The steady states SS1, SS2,. . ., SS8, of (2) are given in Table 2. Assume also that Hypothesis (H7) holds.

The necessary and sufficient conditions of existence of the steady states are given in Table 3. When they exist, all steady states (except SS4) are unique.

• If ω 1, when it exists, SS4 is unique.

• If ω < 1, assuming also that (H8) holds, the system has generically two steady states of the form SS4.

Remark 1. If ω < 1, equation Ψ(s 2 , D) = (1 -ω)s in 0 + s in 1 + s in 2 has two solutions s * 1
2 and s * 2 2 if and only if (1 -ω)s in 0 + s in 1 + s in 2 > φ 1 (D), so that ∂Ψ ∂s2 s * 1 2 , D < 0 and ∂Ψ ∂s2 s * 2 2 , D > 0 (see Fig. 1). We denote by SS4 1 and SS4 2 the steady states of type SS4 corresponding to s * 1 2 and s * 2 2 , respectively. These steady states coalesce when (1 -ω)s in 0 + s in 1 + s in 2 = φ 1 (D). In the particular cases, where s in 1 = 0 or s in 2 = 0, some of the steady states described in Theorem 1 do not exist and the existence conditions of the existing steady states can be simplified. More precisely, we have the following result.

Table 2. Steady states of (2). All functions are defined in Table 1.

s 0 , s 1 , s 2 and x 0 , x 1 , x 2 components SS1 s0 = s in 0 , s1 = s in 1 , s2 = s in 2 and x0 = 0, x1 = 0, x2 = 0 SS2 s0 = s in 0 , s1 = s in 1 , s2 = M2(D + a2) and x0 = 0, x1 = 0, x2 = D D+a 2 s in 2 -s2 SS3 s1 = s in 1 + s in 0 -s0 and s2 = s in 2 -ω s in 0 -s0 , where s0 is a solution of ψ 0 (s 0 ) = D + a 0 and x0 = D D+a 0 s in 0 -s0 , x1 = 0, x2 = 0 SS4 s0 = M0(D + a0, s2) and s1 = M1(D + a1, s2), where s2 is a solution of Ψ(s2, D) = (1 -ω)s in 0 + s in 1 + s in 2 and x0 = D D+a 0 s in 0 -s0 , x1 = D D+a 1 s in 0 -s0 + s in 1 -s1 , x2 = 0 SS5 s0 = ϕ0(D), s1 = s in 1 + s in 0 -s0, s2 = M2(D + a2) and x0 = D D+a 0 s in 0 -s0 , x1 = 0, x2 = D D+a 2 s in 2 -s2 -ω s in 0 -s0 SS6 s0 = ϕ0(D), s1 = ϕ1(D), s2 = M2(D + a2) and x0 = D D+a 0 s in 0 -s0 , x1 = D D+a 1 s in 0 -s0 + s in 1 -s1 , x2 = D D+a 2 (1 -ω)(s in 0 -s0) + s in 1 -s1 + s in 2 -s2 SS7 s0 = s in 0 and s2 = s in 2 + s in 1 -s1, where s1 is a solution of ψ1(s1) = D + a1 and x0 = 0, x1 = D D+a 1 s in 1 -s1 , x2 = 0 SS8 s0 = s in 0 , s1 = ϕ1(D), s2 = M2(D + a2) and x0 = 0, x1 = D D+a 1 s in 1 -s1 , x2 = D D+a 2 s in 1 -s1 + s in 2 -s2
Table 3. Existence conditions of steady states of (2). All functions are given in Table 1.

Existence conditions SS1 Always exists SS2 µ2 s in 2 > D + a2 SS3 µ0 s in 0 , s in 2 > D + a0 SS4 (1 -ω)s in 0 + s in 1 + s in 2 φ1(D), s in 0 > M0(D + a0, s2), s in 0 + s in 1 > M0(D + a0, s2) + M1(D + a1, s2) with s2 solution of equation Ψ(s2, D) = (1 -ω)s in 0 + s in 1 + s in 2 SS5 s in 0 > ϕ0(D), s in 2 -ωs in 0 > M2(D + a2) -ωϕ0(D) SS6 (1 -ω)s in 0 + s in 1 + s in 2 > φ2(D), s in 0 > ϕ0(D), s in 0 + s in 1 > ϕ0(D) + ϕ1(D) SS7 µ1 s in 1 , s in 2 > D + a1 SS8 s in 1 > ϕ1(D), s in 1 + s in 2 > ϕ1(D) + M2(D + a2)
Proposition 2. If s in 1 = 0 then, SS7 and SS8 do not exist. If s in 2 = 0, SS2, SS3 and SS5 do not exist. If s in 1 = s in 2 = 0, we have: • The steady states SS2, SS3, SS5, SS7 and SS8 do not exist.

• If ω 1, SS4 and SS6 do not exist. If ω < 1, SS4 and SS6 exist, respectively, if and only if

(1 -ω)s in 0 φ 1 (D) and (1 -ω)s in 0 > φ 2 (D). ( 3
)
Remark 2. Assume that s in 1 = s in 2 = 0. Then, only the steady states SS1, SS4 and SS6 can exist. The existence conditions (3) of SS4 and SS6, respectively, are equivalent to the following conditions given in Lemmas 3 and 4 of [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF]:

s in 0 F 1 (D) := φ1(D) 1-ω and s in 0 > F 2 (D) := φ2(D)
1-ω . Hence, we recover the results of [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] where the study is restricted to the case s in 1 = s in 2 = 0. Notice that, in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], the steady states SS1, SS4 and SS6 were labeled SS1, SS2 and SS3, respectively.

3. Study of the model without maintenance. In this section, we determine the existence and stability conditions of steady states of model ( 2) in the particular case without maintenance. For convenience, we shall use the abbreviation LES for Locally Exponentially Stable. We define the function φ 4 by

φ 4 D, s in 0 , s in 1 , s in 2 = (EIx 0 x 2 + EGφ 3 (D)x 0 x 1 )(Ix 2 + (G + H)x 1 +(E + ωF )x 0 ) + (Ix 2 + (G + H)x 1 + ωF x 0 )GIx 1 x 2 , (4)
where 5) and the functions E, F , G, H and I, are evaluated at the components of SS6 given in Table 2 with a 0 = a 1 = a 2 = 0.

E = ∂µ0 ∂s0 (s 0 , s 2 ), F = ∂µ0 ∂s2 (s 0 , s 2 ), G = ∂µ1 ∂s1 (s 1 , s 2 ), H = -∂µ1 ∂s2 (s 1 , s 2 ), I = µ 2 (s 2 ) (
Theorem 2. Assume that Hypotheses (H1) to (H8) hold. Without maintenance, the steady states and their existence conditions of (2) are given in Tables 2 and3 by putting a 0 = a 1 = a 2 = 0. In addition, the necessary and sufficient conditions of local exponential stability of the steady states are given in Table 4.

Table 4. Maintenance free case: the stability conditions of steady states of (2). All functions are given in Table 1 with a 0 = a 1 = a 2 = 0, while φ 4 is defined by (4).

Stability conditions

SS1 µ0 s in 0 , s in 2 < D, µ1 s in 1 , s in 2 < D, µ2 s in 2 < D SS2 s in 0 < ϕ0(D), s in 1 < ϕ1(D) SS3 µ1 s in 1 + s in 0 -s0, s in 2 -ω s in 0 -s0 < D, s in 2 -ωs in 0 < M2(D) -ωϕ0(D), with s0 solution of equation ψ0(s0) = D SS4 (1 -ω)s in 0 + s in 1 + s in 2 < φ2(D), ∂Ψ ∂s 2 (s2, D) > 0, φ3(D) > 0, with s2 solution of equation Ψ(s2, D) = (1 -ω)s in 0 + s in 1 + s in 2 SS5 s in 0 + s in 1 < ϕ0(D) + ϕ1(D) SS6 φ3(D) 0, or φ3(D) < 0 and φ4 D, s in 0 , s in 1 , s in 2 > 0 SS7 s in 1 + s in 2 < M3 s in 0 , D + M1 D, M3 s in 0 , D , s in 1 + s in 2 < M2(D) + ϕ1(D) SS8 s in 0 < ϕ0(D)
Remark 3. If ω 1, when SS4 exists, its stability condition ∂Ψ ∂s2 (s 2 , D) > 0 is always satisfied (see Lemma 3). If ω < 1, when SS4 1 exists, it is unstable (see Remark 1). When SS4 2 exists, its stability condition ∂Ψ ∂s2 (s 2 , D) > 0 is always satisfied.

4. Application to a chlorophenol-mineralising three-tiered microbial 'food web'. The aim of this section is to illustrate the theoretical results of this paper in the case of the chlorophenol-mineralising three-tiered microbial model (1) considered in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] where the specific growth rates take the form:

f 0 (S ch , S H2 ) = k m,ch S ch K S,ch +S ch S H 2 K S,H 2 ,c +SH 2 , f 1 (S ph , S H2 ) = k m,ph S ph K S,ph +S ph 1 1+SH 2 /K I,H 2 , f 2 (S H2 ) = km,H 2 SH 2 K S,H 2 +SH 2 . ( 6
)
The biological parameter values are provided in Table 10. They were previously used in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF][START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF]. Following [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], model (1) can be rescaled to obtain model (2) using the following change of variables:

x 0 = Y Y0 X ch , x 1 = Y4 Y1 X ph , x 2 = 1 Y2 X H2 , s 0 = Y S ch , s 1 = Y 4 S ph , s 2 = S H2 , (7)
where Y = Y 3 Y 4 . The inflowing concentrations are given by:

s in 0 = Y S in ch , s in 1 = Y 4 S in ph , s in 2 = S in H2 , (8)
the decay rates are a 0 = k dec,ch , a 1 = k dec,ph , a 2 = k dec,H2 (with units d -1 ), the yield coefficients are

Y0 = Y ch , Y1 = Y ph , Y2 = YH 2 , Y3 = 224 208 (1 -Y0), Y4 = 32 224 (1 -Y1),
with ω = 16 208Y . When the yield coefficients are those given in Table 10, we have ω 0.53. The growth functions take the form:

µ 0 (s 0 , s 2 ) = m0s0 K0+s0 s2 L0+s2 , µ 1 (s 1 , s 2 ) = m1s1 K1+s1 1 1+s2/K I , µ 2 (s 2 ) = m2s2 K2+s2 , (9)
where

m 0 = Y 0 k m,ch , K 0 = Y K S,ch , L 0 = K S,H2,c , m 1 = Y 1 k m,ph , K 1 = Y 4 K S,ph , K I = K I,H2 , m 2 = Y 2 k m,H2 , K 2 = K S,H2 .
For the specific growth functions (9), various functions defined in Table 1 are listed in Table 11. From the expression of Ψ in Table 11, a straightforward calculation shows that, for all s 2 ∈ s 0 2 , s 1 2 ,

∂ 2 ψ ∂s 2 2 (s 2 , D) = (1-ω)2K0(D+a0) m0-D-a0 L0+s 0 2 (s2-s 0 2 ) 3 + 2K1(K I +s 1 2 ) (s 1 2 -s2) 3 ,
which is positive since ω < 1 and m 0 > D + a 0 . Thus, the function s 2 → Ψ(s 2 , D) is convex and fulfills (H8) (see Figure 1). Furthermore, model ( 1) is of the form (2) where the growth functions (9) satisfy Hypotheses (H1) to (H8). Consequently, the results of our paper apply to model [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF].

s2 s 0 2 s 1 2 s * 1 2 s * 2 2 φ1(D) Ψ(s2, D) (1 -ω)s in 0 + s in 1 + s in 2 s2
Figure 1. Curve of the function Ψ(., D), where s * 1 2 and s * 2 2 are the solutions of equation Ψ(s

2 , D) = (1 -ω)s in 0 + s in 1 + s in 2 .
Using the steady states in Table 2 and the auxiliary functions given in Table 11 in the case of specific growth functions (9), a straightforward computation shows that all components of the steady states of model ( 2) are the same as those in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] where another rescaling model and change of variables are used. However, the existence conditions in Table 3 of SS3, SS4 and SS7 were not established in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] (see Appendix C.3, C.4 and C.7) where they were checked only numerically, by considering the roots of polynomials of degree 2 or 3 (see formulas (C.3), (C.11) and (C.49)), respectively. All other existence conditions were established in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF]. Hence, our analytical study of model ( 2) permits to give rigorous proofs for [START_REF] Abdellatif | Competition for a single resource and coexistence of several species in the chemostat[END_REF].

In what follows, we consider the input concentrations S in ph = 0 and S in H2 = 2.67× 10 -5 , corresponding to Fig. 3(a) in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF] when D = 0.01 and k dec,ch = k dec,ph = k dec,H2 = 0. As a consequence of Theorem 2, we obtain the following result which determines the existence and the stability of the steady states of (1) with respect to the input concentration S in ch . Proposition 3. Assume that the biological parameters in (1) are given as in Table 10. Assume that S in ph = 0, S in H2 = 2.67 × 10 -5 , D = 0.01 and k dec,ch = k dec,ph = k dec,H2 = 0. Let σ i , i = 1, . . . , 6 be the bifurcation values defined in Table 5. The existence and stability of steady states of (1), with respect to the input concentration S in ch is given in Table 6. The nature of the bifurcations when S in ch crosses the values σ i , i = 1, . . . , 6 is given in Table 7.

Table 5. Definitions of the critical values of σ i , i = 1, . . . , 6.

Definition

Value

σ 1 = M 0 D, S in H2 /Y 0.001017 σ 2 = (φ 1 (D) -S in H2 )/((1 -ω)Y ) 0.009159 σ 3 = ϕ 0 (D)/Y 0.010846 σ 4 = (S in H2 -M 2 (D) + ωϕ 0 (D))/(ωY ) 0.011191 σ 5 = (φ 2 (D) -S in H2 )/((1 -ω)Y ) 0.016575 σ 6 is the solution of equation φ 4 (S in ch ) = 0 0.029877
SS1 SS2 SS3 SS4 1 SS4 2 SS5 SS6 (0, σ 1 ) U S (σ 1 , σ 2 ) U S U (σ 2 , σ 3 ) U S U U U (σ 3 , σ 4 ) U U U U U S (σ 4 , σ 5 ) U U S U U (σ 5 , σ 6 ) U U S U U U (σ 6 , +∞) U U S U U S
Table 7. Nature of the bifurcations corresponding to the critical values of σ i , i = 1, . . . , 6, defined in Table 5. There exists also a critical value σ * 0.029638 corresponding to the value of S in ch where the stable limit cycle disappears when S in ch is decreasing. Type of the bifurcation σ 1 Transcritical bifurcation of SS1 and SS3 σ 2 Saddle-node bifurcation of SS4 , the steady states SS1 and SS2 cannot be distinguished since they have both a zero X ch -component. Since for S in ch < σ 3 , SS2 is LES and SS1 is unstable, the X ch = 0 axis is plotted in blue, which is the color for SS2 in Table 8 Numerical simulations have shown that there exists a critical value σ * ∈ (σ 5 , σ 6 ), which corresponds to the value of S in ch , where the stable limit cycle that appears for S in ch = σ 6 through a supercritical Hopf bifurcation, disappears when S in ch is decreasing. In [START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF], a numerical study of the bifurcation diagram with respect to the parameter D is given in the case without maintenance and s in 1 = s in 2 = 0. Fig. 6 in [START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF] shows that the disappearance of the stable limit cycle occurs through a saddlenode bifurcation with another unstable limit cycle. We conjecture that in our case ): trajectories of X ch corresponding to those in Fig. 4 showing the sustained oscillations in yellow (a) and blue (b) or the convergence to SS3 in green (c). also the stable limit cycle disappears by a confluence with an unstable limit cycle at S in ch = σ * . When S in ch ∈ (σ * , σ 6 ), the system exhibits a bi-stability with convergence either to SS3 or to a stable limit cycle according to the initial condition. Fig. 4 illustrates the three-dimensional space where the trajectories in yellow and blue converge toward the stable limit cycle in red, while the green trajectory converges toward the steady state SS3. Fig. 5 illustrates the time courses of the yellow and blue trajectories in Fig. 4 proving that the oscillations are maintained with a positive periodic solution. Concerning the green trajectory, the period and the amplitude of oscillations are slightly increased until their destruction over time so that the solution converges to SS3. Finally, when S in ch > σ 6 , the system exhibits also a bi-stability which becomes between SS3 and SS6.

(a) X ch SS4 2 SS4 1 SS6 SS1 SS2 SS3 σ2 σ3¡ ¡ ! e e u σ4 σ5 S in ch (b) X ch SS4 2 SS4 1 SS6 SS1 SS2 SS3 σ * σ6 S in ch

Conclusion.

In this work, we have extended model (1) of a chlorophenolmineralising three-tiered microbial 'food web' presented in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF], by considering the model (2) with general growth functions. Our first aim was the theoretical analysis of (2) by providing the existence conditions of all steady states with respect to the operating parameters. Our study considers the effects of the phenol and hydrogen input concentrations, which were neglected in the analytical analysis given in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], together with the effects of maintenance terms, which were neglected in the theoretical analysis given in [START_REF] El-Hajji | Mathematical modelling and analysis for a three-tiered microbial food web in a chemostat[END_REF][START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF]. System (2) can have up to eight types of steady states: the washout steady state which always exists, a coexistence steady state of all degrader populations and six other steady states corresponding to the extinction of one or two degrader populations. Each type of steady state is unique except that of the exclusion only of the hydrogen degraders (SS4) where there are at most two steady states of this type.

In [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF], results on the existence of some steady states of model ( 1) were obtained only numerically without knowing their exact number. Our analytical results on the existence of the steady states of (2) give rigorous proofs for (1).

Our second achievement was to determine theoretically the asymptotic behavior of system (2) in the particular case without maintenance and to analyze numerically the bifurcation diagram by varying the chlorophenol input concentration when s in 1 = 0. It shows that the system exhibits a bi-stability where the coexistence steady state can destabilize undergoing a supercritical Hopf bifurcation with the occurrence of a stable periodic solution. These interest phenomena have been already depicted in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF], in the particular case s in 0 > 0, s in 1 = s in 2 = 0. The possibility of the Hopf bifurcation of the positive steady state is analytically proved in [START_REF] Sobieszek | Rich dynamics of a three-tiered anaerobic food-web in a chemostat with multiple substrate inflow[END_REF], in the case without maintenance. The destabilization of the positive steady state was not detected by the numerical analysis of the operating diagram in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF].

Our works under investigation will focus on the analysis of the stability of steady states of (2) including maintenance and on the operating diagrams which were obtained numerically in [START_REF] Wade | Emergent behaviour in a chlorophenol-mineralising three-tiered microbial 'food web[END_REF]. In fact, the operating diagram is very useful for the biologist as discussed in [START_REF] Bar | The operating diagram for a model of competition in a chemostat with an external lethal inhibitor[END_REF][START_REF] Dellal | Global analysis of a model of competition in the chemostat with internal inhibitor[END_REF][START_REF] Dellal | The operating diagram of a model of two competitors in a chemostat with an external inhibitor[END_REF] because it allows to predict qualitatively the different asymptotic behaviors of the process according to their control parameters.

Appendix A. Proofs.

Proof of Proposition 1. Since the vector field defined by ( 2) is C 1 , the uniqueness of solution to initial value problems holds. From (2), for i = 0, 1, 2,

x i (τ ) = 0, for any τ 0 ⇒ ẋi (τ ) = 0. If x i (0) = 0, then x i (t) = 0 for all t since the boundary face where x i ≡ 0 is invariant in the vector field C 1 by system (2). If x i (0) > 0, then x i (t) > 0 for all t since x i ≡ 0 cannot be reached in finite time by trajectories such that x i (0) > 0 by the uniqueness of solutions. On the other hand, one has s 0 (τ ) = 0, for any τ 0 ⇒ ṡ0 (τ ) = Ds in 0 s 1 (τ ) = 0, for any τ 0 ⇒ ṡ1 (τ ) = Ds in 1 + µ 0 (s 0 (τ ), s 2 (τ ))x 0 (τ ) s 2 (τ ) = 0, for any τ 0 ⇒ ṡ2 (τ ) = Ds in 2 + µ 1 (s 1 (τ ), 0)x 1 (τ ). Similarly to case x i , if ṡi (τ ) = 0, then s i (t) 0 for all t. In addition, if ṡi (τ ) > 0, then s i (t) 0 for all t. Indeed, for example, consider the case of s 0 where D and s in 0 are positive with s 0 (0) 0. Assume that it exists τ > 0 such that s 0 (τ ) = 0 and s 0 (t) > 0 for all t ∈ (0, τ ). It follows that ṡ0 (τ ) 0, which is the desired contradiction with ṡ0 (τ ) = Ds in 0 > 0. Further, by considering z = ωx 0 + x 1 + x 2 + 2s 0 + 2s 1 + s 2 , we obtain from (2)

ż = D 2s in 0 + 2s in 1 + s in 2 -z -ωa 0 x 0 -a 1 x 1 -a 2 x 2 D 2s in 0 + 2s in 1 + s in 2 -z .
Using Gronwall's lemma, we have

z(t) 2s in 0 + 2s in 1 + s in 2 + z(0) -2s in 0 + 2s in 1 + s in 2 e -Dt
, for all t 0. (10)

Consequently,

z(t) max z(0), 2s in 0 + 2s in 1 + s in 2 , for all t 0. (11)
Thus, the solutions of (2) are positively bounded and are defined for all t 0. From [START_REF] Dimitrova | Nonlinear stabilizing control of an uncertain bioprocess model[END_REF], it can be deduced that the set Ω is positively invariant and from [START_REF] Dellal | The operating diagram of a model of two competitors in a chemostat with an external inhibitor[END_REF], it is a global attractor for (2).

To determine the number of SS3, SS4 and SS7-type steady states and the conditions of their existence, we need the following result.

Lemma 3. The equation ψ i (s 0 ) = y, i = 0, 1, has a solution in the interval J i defined in Table 1

, if ω > 1, φ 1 (D) = -∞ and if ω = 1, φ 1 (D) = Ψ s 0 2 (D), D > 0. • When ω < 1,
there exist two solutions which are equal when s in = φ 1 (D).

Moreover, φ 1 (D) > 0.

Proof. Recall that J 0 = max 0, s in 0 -s in 2 /ω , s in 0 . If s in 2 -ωs in 0 > 0, one has ψ 0 (0) = µ 0 0, s in 2 -ωs in 0 = 0 and if s in 2 -ωs in 0 0, then ψ 0 s in 0 -s in 2 /ω = µ 0 s in 0 -s in 2 /ω, 0 = 0. Thus, ψ 0 max 0, s in 0 -s in 2 /ω = 0. On the other hand, ψ 0 s in 0 = µ 0 s in 0 , s in 2 . Therefore, there exists a solution s 0 ∈ J 0 satisfying ψ 0 (s 0 ) = y if and only if µ 0 s in 0 , s in 2 > y. Since ψ 0 is monotonically increasing, if it exists, this solution is unique. Indeed, we have

dψ0 ds0 (s 0 ) = ∂µ0 ∂s0 s 0 , s in 2 -ω s in 0 -s 0 + ω ∂µ0 ∂s2 s 0 , s in 2 -ω s in 0 -s 0 ,
which is positive thanks to (H4). Now, recall that J 1 = 0, s in 1 . We have

ψ 1 (0) = µ 1 0, s in 1 + s in 2 = 0 and ψ 1 s in 1 = µ 1 s in 1 , s in 2 .
Therefore, there exists a solution

s 1 ∈ J 1 satisfying ψ 1 (s 1 ) = y if and only if µ 1 s in 1 , s in 2 > y.
Since ψ 1 is monotonically increasing, if it exists, this solution is unique. Indeed, we have

dψ1 ds1 (s 1 ) = ∂µ1 ∂s1 s 1 , s in 2 + s in 1 -s 1 -∂µ1 ∂s2 s 1 , s in 2 + s in 1 -s 1 ,
which is positive thanks to (H5). Let us consider now the existence of solution of the equation Ψ(s 2 , D) = s in . From the definitions of s 0 2 and s 1 2 given in Table 1, we have M 0 D + a 0 , s 0 2 = +∞ and M 1 D + a 1 , s 1 2 = +∞. From the definition of Ψ(s 2 , D) given in Table 1, we have for all ω > 0, lim

s2→s 1 2 Ψ(s 2 , D) = +∞, for ω = 1, Ψ(s 0 2 , D) = φ 1 (D),
for all ω > 1, lim

s2→s 0 2 Ψ(s 2 , D) = -∞, for all ω < 1, lim s2→s 0 2 Ψ(s 2 , D) = +∞.
Moreover, we have for all s 2 ∈ s 0 2 , s 1 2 and

D ∈ I 1 , ∂Ψ ∂s2 (s 2 , D) = (1 -ω) ∂M0 ∂s2 (D + a 0 , s 2 ) + ∂M1 ∂s2 (D + a 1 , s 2 ) + 1. ( 12
)
Let s 2 > 0. Under (H4) and (H5), we have

∂M0 ∂s2 (y, s 2 ) = -∂µ0 ∂s2 (s 0 , s 2 ) ∂µ0 ∂s0 (s 0 , s 2 ) -1
< 0, for all y ∈ (0, µ 0 (+∞, s 2 )),

∂M1 ∂s2 (y, s 2 ) = -∂µ1 ∂s2 (s 1 , s 2 ) ∂µ1 ∂s1 (s 1 , s 2 ) -1
> 0, for all y ∈ (0, µ 1 (+∞, s 2 )).

(

) 13
For ω 1, we deduce that the function s 2 → Ψ(s 2 , D) is monotonically increasing by using [START_REF] El-Hajji | A mathematical study of a syntrophic relationship of a model of anaerobic digestion process[END_REF]. If ω = 1, then Ψ s 0 2 , D > 0. When ω < 1, the remaining assertions follow immediately by using assumption (H8).

Proof of Theorem 1. The steady states are obtained by setting the right-hand sides of equations in (2) equal to zero:

[µ 0 (s 0 , s 2 ) -D -a 0 ] x 0 = 0 (14) [µ 1 (s 1 , s 2 ) -D -a 1 ] x 1 = 0 (15) [µ 2 (s 2 ) -D -a 2 ] x 2 = 0 (16) D s in 0 -s 0 -µ 0 (s 0 , s 2 ) x 0 = 0 (17) D s in 1 -s 1 + µ 0 (s 0 , s 2 ) x 0 -µ 1 (s 1 , s 2 ) x 1 = 0 (18) D s in 2 -s 2 + µ 1 (s 1 , s 2 ) x 1 -ωµ 0 (s 0 , s 2 ) x 0 -µ 2 (s 2 ) x 2 = 0. (19)
Using ( 17)+( 14), ( 18)-( 14)+( 15) and ( 19)+ ω( 14)-( 15)+( 16), one obtains the set of equations

   D s in 0 -s 0 -(D + a 0 ) x 0 = 0 D s in 1 -s 1 + (D + a 0 ) x 0 -(D + a 1 ) x 1 = 0 D s in 2 -s 2 -ω (D + a 0 ) x 0 + (D + a 1 ) x 1 -(D + a 2 )x 2 = 0. ( 20
)
We can solve [START_REF] Monod | La technique de culture continue: Théorie et applications[END_REF] and obtain x 0 , x 1 and x 2 with respect to s 0 , s 1 and s 2 :

x 0 = D D+a0 s in 0 -s 0 , (21)
x 1 = D D+a1 s in 0 -s 0 + s in 1 -s 1 , (22)
x 2 = D D+a2 (1 -ω) s in 0 -s 0 + s in 1 -s 1 + s in 2 -s 2 . ( 23
)
We can also solve [START_REF] Monod | La technique de culture continue: Théorie et applications[END_REF] and obtain s 0 , s 1 and s 2 with respect to x 0 , x 1 and x 2 :

s 0 = s in 0 -D+a0 D x 0 , (24)
s 1 = s in 1 + D+a0 D x 0 -D+a1 D x 1 , (25)
s 2 = s in 2 -ω D+a0 D x 0 + D+a1 D x 1 -D+a2 D x 2 . ( 26
)
• For the steady state SS1, x 0 = x 1 = x 2 = 0. Hence, ( 24), ( 25) and ( 26) result in s 0 = s in 0 , s 1 = s in 1 and s 2 = s in 2 . Thus, SS1 always exists. • For SS2, x 0 = x 1 = 0 and x 2 > 0. Hence, [START_REF] Sari | A model of a syntrophic relationship between two microbial species in a chemostat including maintenance[END_REF] and [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] result in s 0 = s in 0 and s 1 = s in 1 . Therefore, [START_REF] Sari | The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat[END_REF] results in

x 2 = D D+a2 s in 2 -s 2 . Since x 2 > 0, (16) results in µ 2 (s 2 ) = D + a 2 . Using definition of M 2 in Table 1, we have s 2 = M 2 (D + a 2 ).
SS2 exists if and only if x 2 > 0, that is to say s in 2 > M 2 (D + a 2 ), which is equivalent to µ 2 s in 2 > D + a 2 , thanks to (H6).

• For SS3, x 1 = x 2 = 0 and x 0 > 0. Hence, [START_REF] Nouaoura | Mathematical analysis of a threetiered model of anaerobic digestion[END_REF] results in

x 0 = D D+a0 s in 0 -s 0 .
Using this expression together with x 1 = x 2 = 0 in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] and [START_REF] Sbarciog | Determination of appropriate operating strategies for anaerobic digestion systems[END_REF] result in

s 1 = s in 1 + s in 0 -s 0 and s 2 = s in 2 -ω(s in 0 -s 0 ). ( 27
)
Since x 0 > 0, ( 14) results in

µ 0 (s 0 , s 2 ) = D + a 0 . ( 28
)
Replacing s 2 by its expression ( 27) with respect to s 0 in ( 28) results in

ψ 0 (s 0 ) = D + a 0 , (29)
where ψ 0 is the function defined in Table 1. SS3 exists if and only if equation ( 29) has a positive solution and the s 1 , s 2 and x 0 -components are positive. This condition is equivalent to say that 0 < s 0 < s in 0 and s 0 > s in 0 -s in 2 /ω. Therefore, (29) must have a solution in the interval J 0 . Using Lemma 3, ( 29) has a solution in the interval J 0 if and only if µ 0 s in 0 , s in 2 > D + a 0 . If it exists, this solution is unique.

• For SS4, x 0 > 0, x 1 > 0 and x 2 = 0. Hence, ( 21) and ( 22) result in

x 0 = D D+a0 s in 0 -s 0 and x 1 = D D+a1 s in 0 -s 0 + s in 1 -s 1 . (30)
Since x 0 > 0 and x 1 > 0, ( 14) and ( 15) result in µ 0 (s 0 , s 2 ) = D + a 0 and µ 1 (s 1 , s 2 ) = D + a 1 . Using definitions of M 0 and M 1 in Table 1, we obtain

s 0 = M 0 (D + a 0 , s 2 ) and s 1 = M 1 (D + a 1 , s 2 ). ( 31
)
Using [START_REF] Wade | Perspectives in mathematical modelling for microbial ecology[END_REF] together with x 2 = 0 in (26), we have

s 2 = s in 2 -ω s in 0 -s 0 + s in 0 -s 0 + s in 1 -s 1 . ( 32
)
Replacing s 0 and s 1 by their expressions (31) with respect to s 2 in (32), it follows that, s 2 is a solution of equation

Ψ (s 2 , D) = (1 -ω)s in 0 + s in 1 + s in 2 , (33)
where Ψ is the function defined in Table 1. According to Lemma 3, SS4 exists if and only if (1-ω)s in 0 +s in 1 +s in 2 φ 1 (D), and the solution s 2 of ( 33) is such that the x 0 and x 1 -components are positive which is equivalent to s in 0 > M 0 (D + a 0 , s 2 ) and

s in 0 + s in 1 > M 0 (D + a 0 , s 2 ) + M 1 (D + a 1 , s 2 ).
The existence of a unique or two steady states of the form SS4 according to ω follows immediately from Lemma 3.

• For SS5, x 0 > 0, x 2 > 0 and x 1 = 0. Using (21) together with x 1 = 0 in [START_REF] Sari | Generalised approach to modelling a three-tiered microbial food-web[END_REF] results in s 1 = s in 1 + s in 0 -s 0 . Using [START_REF] Nouaoura | Mathematical analysis of a threetiered model of anaerobic digestion[END_REF] and this expression in [START_REF] Sari | The mathematical analysis of a syntrophic relationship between two microbial species in a chemostat[END_REF] results in [START_REF] Fekih-Salem | Analyse mathématique d'un modèle de digestion anaérobie à trois étapes[END_REF] and ( 16) result in µ 0 (s 0 , s 2 ) = D + a 0 and µ 2 (s 2 ) = D + a 2 . Using definitions of M 0 , M 2 and ϕ 0 in Table 1, it follows that

x 0 = D D+a0 s in 0 -s 0 and x 2 = D D+a2 s in 2 -s 2 -ω s in 0 -s 0 . Since x 0 > 0 and x 2 > 0,
s 2 = M 2 (D + a 2 ) and s 0 = ϕ 0 (D).
SS5 exists if and only if its components x 0 , x 2 and s 1 are positive. This condition is equivalent to s in 0 > ϕ 0 (D) and s in 2 -ωs in 0 > M 2 (D + a 2 ) -ωϕ 0 (D).

• For SS6, x 0 > 0, x 1 > 0 and x 2 > 0. Then, as a consequence of ( 14), ( 15) and ( 16), we obtain:

µ 0 (s 0 , s 2 ) = D + a 0 , µ 1 (s 1 , s 2 ) = D + a 1 , µ 2 (s 2 ) = D + a 2 .
Using definitions of M 0 , M 1 , M 2 , ϕ 0 and ϕ 1 in Table 1, it follows that s 2 , s 0 and s 1 are given by:

s 2 = M 2 (D + a 2 ), s 0 = ϕ 0 (D), s 1 = ϕ 1 (D), (21)
, ( 22) and ( 23) give the x-components of SS6 in Table 2. SS6 exists if and only if its x-components are positive, that is,

s in 0 > s 0 , s in 1 + s in 0 > s 0 + s 1 and (1 -ω)s in 0 + s in 1 + s in 2 > (1 -ω)s 0 + s 1 + s 2 .
Using the s-components of SS6, these conditions are the same as those in Table 3.

• For SS7, x 0 = x 2 = 0 and x 1 > 0. Hence, [START_REF] Sari | A model of a syntrophic relationship between two microbial species in a chemostat including maintenance[END_REF] results in s 0 = s in 0 . From ( 22), we have

x 1 = D D+a1 s in 1 -s 1 .
Using this expression together with x 0 = x 2 = 0 in (26) results in

s 2 = s in 2 + s in 1 -s 1 . (34)
Since x 1 > 0, then, as a consequence of (15), we obtain:

µ 1 (s 1 , s 2 ) = D + a 1 .
Replacing s 2 by its expression (34) with respect to s 1 results in

ψ 1 (s 1 ) = D + a 1 , (35)
where ψ 1 is defined in Table 1. SS7 exists if and only if equation (35) has a positive solution and the s 2 and x 1 -components of SS7 are positive. This last condition is equivalent to 0 < s 1 < s in 1 , that is, (35) must have a solution in the interval J 1 . Using Lemma 3, there exists a solution s 1 ∈ J 1 , satisfying (35), if and only if µ 1 s in 1 , s in 2 > D + a 1 . If it exists, this solution is unique. • For SS8, x 0 = 0, x 1 > 0 and x 2 > 0. Hence, [START_REF] Sari | A model of a syntrophic relationship between two microbial species in a chemostat including maintenance[END_REF] results in s 0 = s in 0 . Using this expression in [START_REF] Rapaport | A new proof of the competitive exclusion principle in the chemostat[END_REF] and ( 23) results in

x 1 = D D+a1 s in 1 -s 1 , x 2 = D D+a2 s in 1 -s 1 + s in 2 -s 2 .
Since x 1 > 0 and x 2 > 0, as a consequence of ( 15) and ( 16), we have µ 1 (s 1 , s 2 ) = D + a 1 and µ 2 (s 2 ) = D + a 2 . Using definitions of M 1 , M 2 and ϕ 1 (D) in Table 1, it follows that s 2 and s 1 are given by:

s 2 = M 2 (D + a 2 ), s 1 = ϕ 1 (D).
SS8 exists if and only if its components x 1 and x 2 are positive, that is,

s in 1 > s 1 and s in 1 + s in 2 > s 1 + s 2 .
Using the s-components of SS8, these conditions are the same as those in Table 3.

Proof of Proposition 2. If s in 1 = 0, then µ 1 s in 1 , s in 2 = 0, so that the condi- tions µ 1 s in 1 , s in 2 > D + a 1 and s in 1 > ϕ 1 (D)
of existence of SS7 and SS8, respectively, cannot be satisfied. Therefore, SS7 and SS8 do not exist. If s in 2 = 0, then µ 2 s in 2 = 0 and µ 0 s in 0 , s in 2 = 0, so that the existence conditions µ 2 s in 2 > D +a 2 , µ 0 s in 0 , s in 2 > D + a 0 of SS2 and SS3 cannot be satisfied, respectively. Moreover, the second existence condition of SS5 implies that

s in 0 < ϕ 0 (D) -M2(D+a2) ω < ϕ 0 (D),
which is in contradiction with the first existence condition of SS5. Therefore, SS2, SS3 and SS5 do not exist. Assume that s in 1 = s in 2 = 0. If ω = 1, the first existence condition of SS4 in Table 3 is written 0 φ 1 (D). This condition cannot be satisfied, since φ 1 (D) = Ψ s 0 2 , D > 0 from Lemma 3. Thus, SS4 does not exist if ω = 1. When ω > 1, we have s 2 is solution of equation

(1 -ω) s in 0 -s 0 = s 1 + s 2 .
Since s 1 > 0 and s 2 > 0, then we have necessarily (1 -ω) s in 0 -s 0 > 0, so that s in 0 -s 0 < 0, which contradicts the positivity of the x 0 -component of SS4 in Table 2. Thus, SS4 does not exist if ω > 1. When s in 1 = s in 2 = 0, the s 2 -component of SS4 becomes the solution of equation

s in 0 = M 0 (D + a 0 , s 2 ) + M1(D+a1,s2)+s2 (1-ω)
.

If 0 < ω < 1, then s in 0 > M 0 (D + a 0 , s 2 ) + M 1 (D + a 1 , s 2 ) > M 0 (D + a 0 , s 2 )
, thus, the second and the third existence conditions of SS4 in Table 3 are satisfied when ω < 1. Therefore, SS4 exists if and only if (1 -ω)s in 0 φ 1 (D). Regarding the steady state SS6 in the particular case s in 1 = s in 2 = 0, the first existence condition in Table 3 becomes

(1 -ω)s in 0 > φ 2 (D), (36)
which is equivalent to

(1 -ω) s in 0 -ϕ 0 (D) > ϕ 1 (D) + M 2 (D + a 2 ). When ω 1, this last inequality cannot hold, since s in 0 > ϕ 0 (D), so that SS6 does not exist. If ω < 1, condition (36) implies that

(1 -ω)s in 0 > (1 -ω)ϕ 0 (D) + (1 -ω)ϕ 1 (D), that is, s in 0 > ϕ 0 (D) + ϕ 1 (D) > ϕ 0 (D)
, which are the second and the third existence conditions of SS6 in Table 3. Thus, (36) is the only existence condition of SS6.

Proof of Theorem 2. In the particular case without maintenance, the steady states and their existence conditions are easily obtained from Tables 2 and3 by putting a i = 0, i = 0, 1, 2. To analyze the local stability, we use the change of variables:

z 0 = x 0 + s 0 , z 1 = x 1 + s 1 -x 0 , z 2 = ωx 0 -x 1 + x 2 + s 2 , (37)
such that model (2) takes the form

               ẋ0 = -Dx 0 + µ 0 (z 0 -x 0 , z 2 -ωx 0 + x 1 -x 2 ) x 0 ẋ1 = -Dx 1 + µ 1 (z 1 + x 0 -x 1 , z 2 -ωx 0 + x 1 -x 2 ) x 1 ẋ2 = -Dx 2 + µ 2 (z 2 -ωx 0 + x 1 -x 2 ) x 2 ż0 = D s in 0 -z 0 ż1 = D s in 1 -z 1 ż2 = D s in 2 -z 2 . (38)
The steady states SS1, SS2,. . ., SS8 of (38) now take the form x 0 , x 1 , x 2 , s in 0 , s in 1 , s in 2 where the x 0 -components of each steady state are given by those in Table 2 with a i = 0. The Jacobian matrix of (38) has an upper block triangular form:

J = J 1 J 2 0 J 3 , where J 2 =   Ex 0 0 F x 0 0 Gx 1 -Hx 1 0 0 Ix 2   , J 1 =   µ 0 -D -(E + ωF )x 0 F x 0 -F x 0 (G + ωH)x 1 µ 1 -D -(G + H)x 1 Hx 1 -ωIx 2 Ix 2 µ 2 -D -Ix 2   , (39)
and J 3 is the 3 × 3 diagonal matrix whose diagonal elements are all -D. Consequently, three eigenvalues of J are given by -D and the three other eigenvalues are given by those of the 3×3 upper-left matrix J 1 . Thus, the stability of the steady state is determined by the sign of the real parts of the eigenvalues of J 1 . Note that, the functions E, F , G, H and I defined by ( 5) are evaluated at the steady state.

We have used the opposite sign of the partial derivative H = -∂µ 1 /∂s 2 , such that all constants involved in the computation become positive.

• For SS1, the characteristic polynomial of

J 1 is P 1 (λ) = (λ -λ 1 )(λ -λ 2 )(λ -λ 3 )
, where

λ 1 = µ 0 s in 0 , s in 2 -D, λ 2 = µ 1 s in 1 , s in 2 -D, λ 3 = µ 2 s in 2 -D.
Therefore, SS1 is LES if and only if λ 1 < 0, λ 2 < 0 and λ 3 < 0, that is, the stability conditions of SS1 in Table 4 hold.

• For SS2, the characteristic polynomial of J 1 is P 2 (λ) = (λ -λ 1 )(λ -λ 2 )(λ -λ 3 ), where

λ 1 = µ 0 s in 0 , M 2 (D) -D, λ 2 = µ 1 s in 1 , M 2 (D) -D, λ 3 = -Ix 2 .
As λ 3 < 0, SS2 is LES if and only if λ 1 < 0 and λ 2 < 0. Since M 0 and M 1 are increasing, these conditions are equivalent to the stability conditions of SS2 in Table 4.

• For SS3, the characteristic polynomial of J 1 is P 3 (λ) = (λ -λ 1 )(λ -λ 2 )(λ -λ 3 ), where

λ 1 = µ 1 s in 1 + s in 0 -s 0 , s in 2 -ω s in 0 -s 0 -D, λ 2 = µ 2 s in 2
-ω s in 0 -s 0 -D, and λ 3 = -(E + ωF )x 0 , where s 0 is the solution in the interval J 0 of equation ψ 0 (s 0 ) = D. As λ 3 < 0, SS3 is LES if and only if λ 1 < 0 and λ 2 < 0. The condition λ 1 < 0 is the first stability condition of SS3 in Table 4. Since M 2 is increasing, the condition λ 2 < 0 is equivalent to

s in 2 -ω s in 0 -s 0 < M 2 (D) ⇐⇒ s 0 < M 2 (D) -s in 2 /ω + s in 0 . (40)
As the function ψ 0 is increasing and ψ 0 (s 0 ) = D, (40) is equivalent to

D < µ 0 M 2 (D) -s in 2 /ω + s in 0 , M 2 (D)
. Since M 0 is increasing, this condition is equivalent to the second stability condition of SS3 in Table 4.

• For SS4, the characteristic polynomial of J 1 is P 4 (λ) = (λ -λ 1 )(λ 2 + c 1 λ + c 2 ), where

λ 1 = µ 2 (s 2 ) -D, c 1 = (E + ωF )x 0 + (G + H)x 1 , c 2 = (E(G + H) + (ω -1)F G)x 0 x 1 ,
and s 2 is defined by [START_REF] Wolkowicz | Successful invasion of a food web in a chemostat[END_REF]. We always have c 1 > 0. Using ( 5) and ( 13), we obtain

∂M0 ∂s2 (D, s 2 ) = -F E and ∂M1 ∂s2 (D, s 2 ) = H G . Using (12), it follows that ∂Ψ ∂s2 (s 2 , D) = F E (ω -1) + H G + 1 = E(G+H)+(ω-1)F G EG . (41)
Since E and G are positive, condition c 2 > 0 is equivalent to ∂Ψ ∂s2 (s 2 , D) > 0. As µ 2 is increasing, SS4 is LES if and only if

s 2 < M 2 (D) and ∂Ψ ∂s2 (s 2 , D) > 0, which is equivalent to (1 -ω)s in 0 + s in 1 + s in 2 < φ 2 (D), φ 3 (D) > 0 and ∂Ψ ∂s2 (s 2 , D) > 0. ( 42
)
Indeed, when ω 1, the mapping s 2 → Ψ(s 2 , D) is increasing for all s 2 ∈ (s 0 2 , s 1 2 ) (see Lemma 3). Hence, the condition

s 2 < M 2 (D) is equivalent to (1 -ω)s in 0 + s in 1 + s in 2 = Ψ(s 2 , D) < Ψ(M 2 (D), D) = φ 2 (D). ( 43
)
In addition, s 2 < M 2 (D) implies that φ 3 (D) > 0. Now, when ω < 1, from Lemma 3 and using Hypothesis (H8), equation [START_REF] Wolkowicz | Successful invasion of a food web in a chemostat[END_REF] has at most two solutions s * 1 2 < s * 2 2 , such that ∂Ψ ∂s2 s * 1 2 , D < 0 and ∂Ψ ∂s2 s * 2 2 , D > 0 (see Fig. 1). Moreover, the mapping s 2 → Ψ(s 2 , D) is increasing for all s 2 ∈ s 2 , s 1 2 . Thus, the condition s * 2 2 < M 2 (D) implies the first and the second conditions of (42). Inversely, if the first condition of (42) or equivalently (43) holds, then

s * 2 2 < M 2 (D) or s 0 2 < M 2 (D) < s * 1 2
. This last condition is in contradiction with the second condition of (42).

• For SS5, the characteristic polynomial of J 1 is P 5 (λ) = (λ -λ 1 )(λ 2 + c 1 λ + c 2 ), where

λ 1 = µ 1 s in 0 + s in 1 -M 0 (D, M 2 (D)), M 2 (D) -D, c 1 = (E + ωF )x 0 + Ix 2 , and c 2 = EIx 0 x 2 . As c 1 > 0 and c 2 > 0, SS5 is LES if and only if λ 1 < 0. Since M 1 is increasing, this stability condition of SS5 is equivalent to that in Table 4. • For SS6, the characteristic polynomial of J 1 is P 6 (λ) = λ 3 + c 1 λ 2 + c 2 λ + c 3 , where c 1 = Ix 2 + (G + H)x 1 + (E + ωF )x 0 , c 3 = EGIx 0 x 1 x 2 c 2 = (E(G + H) + (ω -1)F G)x 0 x 1 + EIx 0 x 2 + GIx 1 x 2 .
From (41), one has E(G + H) + (ω -1)F G = EGφ 3 (D). Since c 1 and c 3 are positive, according to the Routh-Hurwitz criterion, SS6 is LES if and only if

φ 4 D, s in 0 , s in 1 , s in 2 := c 1 c 2 -c 3 > 0, (44)
where the function φ 4 can be written as its expression (4). If φ 3 (D) 0, then condition (44) holds so that SS6 is LES.

• For SS7, the characteristic polynomial of J 1 is P

7 (λ) = (λ -λ 1 )(λ -λ 2 )(λ -λ 3 ),
where

λ 1 = µ 0 s in 0 , s in 2 + s in 1 -s 1 -D, λ 2 = µ 2 s in 2 + s in 1 -s 1 -D, λ 3 = -(G + H)x 1
, where s 1 is the solution in the interval J 1 of equation ψ 1 (s 1 ) = D. Since λ 3 < 0, SS7 is LES if and only if λ 1 < 0 and λ 2 < 0. Since the functions M 2 and M 3 are increasing, the conditions λ 1 < 0 and λ 2 < 0 are equivalent to

s 1 > s in 2 + s in 1 -M 3 s in 0 , D , s 1 > s in 2 + s in 1 -M 2 (D). ( 45
)
Since the function ψ 1 is increasing and ψ 1 (s 1 ) = D, (45) is equivalent to

µ 1 s in 2 + s in 1 -M 3 s in 0 , D , M 3 s in 0 , D < D, µ 1 s in 2 + s in 1 -M 2 (D), M 2 (D) < D.
Since M 1 is increasing, these stability conditions of SS7 are equivalent to those in Table 4.

• For SS8, the characteristic polynomial of J 1 is P

8 (λ) = (λ -λ 1 )(λ 2 + c 1 λ + c 2 ),
where

λ 1 = µ 0 s in 0 , M 2 (D) -D, c 1 = (G + H)x 1 + Ix 2 , c 2 = GIx 1 x 2 .
As c 1 and c 2 are positive, SS8 is LES if and only if λ 1 < 0. Since M 0 is increasing, this stability condition of SS8 is equivalent to that in Table 4.

Proof of Proposition 3. Using the change of variables (8) and from Tables 3 and4, the necessary and sufficient conditions of existence and stability of steady states of (1) are summarized in Table 9 when S in ph = 0 and k dec,ch = k dec,ph = k dec,H2 = 0. Since s in 1 = 0, SS7 and SS8 do not exist, as shown in Proposition 2. Using Table 9, we see that: Table 9. Existence and local stability conditions of steady states of (1), when S in ph = 0 and k dec,ch = k dec,ph = k dec,H2 = 0. All functions are given in Tables 1 and11, while φ 4 and µ i are given by ( 4) and ( 9).

Existence conditions Stability conditions

SS1 Always exists µ0 Y S in ch , S in H 2 < D, µ2 S in H 2 < D SS2 µ2 S in H 2 > D Y S in ch < ϕ0(D) SS3 µ0 Y S in ch , S in H 2 > D µ1 Y S in ch -s0, S in H 2 + ω Y S in ch -s0 < D S in H 2 -ωY S in ch < M2(D) -ωϕ0(D) with s0 solution of ψ0(s0) = D SS4 (1 -ω)Y S in ch + S in H 2 φ1(D), Y S in ch > M0(D, s2) + M1(D, s2) with s2 solution of Ψ(s2, D) = (1 -ω)Y S in ch + S in H 2 (1 -ω)Y S in ch + S in H 2 < φ2(D), ∂Ψ ∂s 2 (s2, D) > 0, φ3(D) > 0 SS5 Y S in ch > ϕ0(D), S in H 2 -ωY S in ch > M2(D) -ωϕ0(D) Y S in ch < ϕ0(D) + ϕ1(D) SS6 (1 -ω)Y S in ch + S in H 2 > φ2(D), Y S in ch > ϕ0(D) + ϕ1(D) φ3(D) 0 or φ3(D) < 0 and φ4(D, S in ch , S in H 2 ) > 0
• SS1 always exists and is unstable, since the second stability condition in

S in ch > (M 0 (D, S in H2 ))/Y =: σ 1 .
For S in ch = σ 1 , there is a transcritical bifurcation of SS3 and SS1, which have the same components at σ 1 (see Table 2). Consider the function y = F (S in ch ) defined by:

F (S in ch ) = µ 1 (Y S in ch -s 0 , S in H2 + ω(Y S in ch -s 0 )), (47)
where s 0 depends also on S in ch . The first stability condition of SS3 in Table 9 is written F S in ch < D. Fig. 6 shows that this condition holds for all S in ch > σ 1 , since the maximum of the function F is smaller than 0.0013 and D = 0.01. From the second stability condition, SS3 is LES if and only if showing that F S in ch < 0.0013, for all S in ch > σ 1 .

S in ch > S in
• Recall that ω 0.53 < 1 for the set of parameters given in Table 10. Therefore, equation Ψ(s 2 , D) = (1 -ω)Y S in ch + S in H2 admits two solutions s * 1 2 and s * 2 2 which correspond to two steady states SS4 1 and SS4 2 , respectively. When it exists, SS4 1 is unstable, as stated in Remark 3. From Table 9, the first existence condition of these steady states holds if and only if

S in ch ≥ φ1(D)-S in H 2
(1-ω)Y =: σ 2 .

Fig. 7 shows that the second existence condition of SS4 1 and SS4 2 in Table 9 holds, for all S in ch ∈ [σ 2 , 0.05], since the straight line of equation y = Y S in ch is above the curves of the functions y = M 0 D, s * i 2 + M 1 D, s * i 2 , for i = 1, 2, respectively. SS4 2 is unstable, since the third stability condition does not hold as φ 3 (D) -6513 < 0. Therefore, SS4 1 and SS4 2 exist and are unstable for all S in ch ≥ σ 2 . They disappear for S in ch < σ 2 . For S in ch = σ 2 there is a saddle-node bifurcation. For S in ch = σ 5 there is a transcritical bifurcation of SS4 1 and SS6. • From Table 9, SS5 exists if and only if For S in ch = σ 3 , there is a transcritical bifurcation of SS5 and SS2. For S in ch = σ 4 , there is a transcritical bifurcation of SS5 and SS3. When it exists, SS5 is LES since S in ch < σ 4 0.011191 < ϕ0(D)+ϕ1(D) Y 0.013717. Then, SS6 exists if and only if S in ch > σ 5 . For the stability of SS6, we have φ 3 (D) < 0 and we plot the functions φ 4 with respect to S in ch . Fig. 8 shows that the equation φ 4 (S in ch ) = 0 has a unique solution σ 6 0.029877 such that φ 4 (S in ch ) < 0 for all σ 5 < S in ch < σ 6 and φ 4 (S in ch ) > 0 for all S in ch > σ 6 . To give a numerical evidence of the Hopf bifurcation occurring through the positive steady state SS6 as S in ch varies, we determine the eigenvalues of the matrix J 1 defined by (39) and evaluated at this steady state. Fig. 9(a) shows that one eigenvalue λ 1 S in ch remains negative for all S in ch ∈ (σ 5 , 0.05]. Fig. 9(b) shows that the two other eigenvalues are real and distinct for all S in ch ∈ (σ 5 , σ ) and we denote them by λ 2 S in ch and λ 3 S in ch , then they become a complex-conjugate pair for all S in ch ∈ (σ , 0.05) and we denote them by λ 2,3 S in ch = α S in ch ± iβ S in ch , which becomes purely imaginary for the particular value S in ch = σ 6 such that α(σ 6 ) = 0, with β(σ 6 ) = 0. Moreover, one has dα dS in ch (σ 6 ) < 0. Therefore, SS6 changes its stability through a supercritical Hopf bifurcation with the emergence of a stable limit cycle that we illustrate in Fig. 4.

σ 3 := ϕ0(D) Y < S in ch < S in
Appendix B. Tables. In this section, we provide the biological parameter values in Table 10 and we present in Table 11 various auxiliary functions defined in Table 1 in the case of specific growth functions [START_REF] Dellal | Global analysis of a model of competition in the chemostat with internal inhibitor[END_REF]. s1 ∈ 0, s in 1 + s in 2

1 and SS4 2 σ 3

 3 Figs. 2 and 3 depict the bifurcation diagram of system (1) where X ch is represented as a function of the bifurcation parameter S in ch . Figs. 2(b) and 3 show magnifications of the bifurcation diagram showing the transcritical bifurcations occurring at σ 1 , σ 3 , σ 4 and σ 5 , the saddle-node bifurcation occurring at σ 2 , the Hopf

Figure 3 .

 3 Figure 3. (a) Magnification of saddle-node bifurcation at S in ch = σ 2 and the transcritical bifurcation at S in ch = σ 5 when S in ch ∈ [0.006, 0.02]. (b) Magnification of the appearance and disappearance of stable limit cycles when S in ch ∈ [0.0294, 0.0302].

Figure 4 .

 4 Figure 4. Case S in ch = 0.029639 ∈ (σ * , σ 6 ): bi-stability of the limit cycle (in red) and SS3.

Figure 5 .

 5 Figure 5. Case S in ch = 0.029639 ∈ (σ * , σ 6 ): trajectories of X ch corresponding to those in Fig.4showing the sustained oscillations in yellow (a) and blue (b) or the convergence to SS3 in green (c).

Figure 6 .

 6 Figure 6. Curve of the function y = F S in ch

1 D , s * 1 2 y = M 0 D , s * 2 2 + M 1 D , s * 2 2 S in ch σ2 Figure 7 .

 120212σ27 Figure 7. The green line of equation y = Y S in ch is above the red and blue curves of the functions M 0 D, s * i 2 +M 1 D, s * i 2 , i = 1, 2.

H 2 -

 2 M2(D)+ωϕ0(D) ωY =: σ 4 .

Figure 8 .

 8 Figure 8. (a) Curve of the function φ 4 S in ch for S in ch > σ 5 and the solution σ 6 of equation φ 4 S in ch = 0. (b) A magnification for S in ch ∈ (σ 5 , 0.034).

Figure 9 .

 9 Figure 9. Three eigenvalues of the matrix J 1 evaluated at SS6 as a function of S in ch . Real part of the pair of eigenvalues λ 2,3 for S in ch ∈ (σ , 0.05] where σ = 0.018.

2 L 0 +s 2 M1K 0 +s 0 s 0 2 (D + a0 < m0 s 1 2 ( 1 DD ∈ I1 : s 0 2 < s2 < s 1 2

 2222121 Auxiliary functionDefinition domain M0(y, s2) = yK 0 (L 0 +s 2 )m 0 s 2 -y(L 0 +s 2 ) 0 y < m 0 s (y, s2) = yK 1 (K I +s 2 ) m 1 K I -y(K I +s 2 ) 0 y < m 1 K I K I +s 2 M2(y) = yK 2 m 2 -y 0 y < m2 M3(s0, z) = zL 0 (K 0 +s 0 ) m 0 s 0 -z(K 0 +s 0 ) 0 z < m 0 s 0 D) = L 0 (D+a 0 ) m 0 -D-a 0 D) = K I (m 1 -D-a 1 ) D+a + a1 < m1 Ψ(s2, D)= (1 -ω) (D+a 0 )K 0 (L 0 +s 2 ) m 0 s 2 -(D+a 0 )(L 0 +s 2 ) + (D+a 1 )K 1 (K I +s 2 ) m 1 K I -(D+a 1 )(K I +s 2 ) + s2 ψ0(s0) = m 0 s 0( s in 2 -ω(s in 0 -s 0 )) (K 0 +s 0 )(L 0 +s in 2 -ω(s in 0 -s 0)) s0 ∈ max 0, s in 0 -s in 2 /ω , +∞ ψ1(s1) = m 1 s 1 K I (K 1 +s 1 )(K I +s in 2 +s in 1 -s 1)

Table 6 .

 6 Existence and stability of steady states, with respect to S in ch . The letter S (resp. U) means that the corresponding steady state is LES (resp. unstable). No letter means that the steady state does not exist.

	Interval of S in ch

  .Figure 2. (a) Projection of the ω-limit set in variable X ch as a function of S in ch ∈ [0, 0.05] (b) A magnification of the transcritical bifurcations occurring at σ 1 , σ 3 and σ 4 when S in ch ∈ [0, 0.015].

	0.001	X ch			(a)						7e-06	X ch				(b)
	0.0009
												6e-06
	0.0008
	0.0007											5e-06
	0.0006
												4e-06
	0.0005
												3e-06
	0.0004
	0.0003			SS4 2	SS6						2e-06		SS3			SS5
	0.0002 0.0001			SS4 1								1e-06
	0	SS1 SS2 SS3								0e00		SS1 SS2
		0 σ1	0.02 e u σ5 e u e 0.015 σ2 σ3 0.01 0.005 ¡ ¡ ! ¡ ¡ ! σ4 e	0.025 S in ch	0.03 ¡ ¡ ! σ * e e u σ6 0.035	0.04	0.045	0.05	0		σ1	0.002	0.004	0.006 S in ch	0.008	0.01 σ3σ4 0.012	0.014
												6e-04
	3.5e-04
												5.5e-04
	3e-04											5e-04
												4.5e-04
	2.5e-04
												4e-04
												3.5e-04
	2e-04
												3e-04
	1.5e-04											2.5e-04
												2e-04
	1e-04
												1.5e-04
												1e-04
	5e-05
												5e-05
	0e00											0e00
	0.006	0.008	0.01	0.012	0.014	0.016	0.018		0.02	0.0294	0.0295	0.0296	0.0297	0.0298	0.0299	0.03	0.0301	0.0302

Table 8 .

 8 Colors used in Figs.2 and 3. The solid (resp. dashed) lines are used for LES (resp. unstable) steady states.

	SS1 SS2 SS3	SS4 1	SS4 2	SS5	SS6
	Red Blue Purple Dark Green Magenta Green Cyan

  if and only if µ i s in i , s in 2 > y. The mapping ψ i is monotonically increasing and thus, if it exists, this solution is unique. The equation Ψ(s 2 , D) = s in has a solution if and only if s in φ 1 (D).

• When ω 1, the mapping s 2 → Ψ(s 2 , D) is monotonically increasing, and thus, if it exists, this solution is unique. Moreover

Table 9

 9

	does not hold, as
	µ 2 S in H2	1.0845 > D = 0.01.	(46)
	• SS2 exists, since the existence condition in Table 9 holds from (46). It is LES if
	and only if
	S in ch < ϕ

0 (D)/Y =: σ 3 . • SS3 exists if and only if µ 0 Y S in ch , S in H2 > D, which is equivalent to

  • From Table 9, SS6 exists if and only if

	S in ch >	φ2(D)-S in H 2

Table 10 .

 10 Nominal parameter values. Units are expressed in Chemical Oxygen Demand (COD).

	Parameter	Wade et al. [31]	Unit
	k m,ch	29
	k m,ph	26	kgCODS/kgCODX/d
	km,H 2	35
	K S,ch KS,H 2 ,c K S,ph KI,H 2 KS,H 2	0.053 10 -6 0.302 2.5×10 -5 3.5×10 -6	kgCOD/m 3
	Y ch	0.019
	Y ph	0.04	kgCODX/kgCODS
	YH 2	0.06

Table 11 .

 11 Auxiliary functions in the case of growth functions given by[START_REF] Dellal | Global analysis of a model of competition in the chemostat with internal inhibitor[END_REF].

This work was supported by the Euro-Mediterranean research network TREASURE (http: //www.inra.fr/treasure).