R Caillibotte
email: remi.caillibotte@gm.univ-montp2.fr

Y Leredde

F Vidussi

C Ulses

P Marsaleix

C Estournel

B Mostajir

Impact of oysters as top predators on microbial food web dynamics: a modelling approach with parameter optimisation. Running page head: Modelling oyster impact on MFW

Keywords: Oysters, microbial food web, biogeochemical modelling, parameter optimisation, bacteria, phytoplankton, zooplankton, mesocosm

Impact of oysters as top predators on microbial food web dynamics: a modelling approach with parameter optimisation. Running page head: Modelling oyster impact on MFW

Introduction

Bivalve activities exert a non-neutral influence on coastal ecosystems (e.g., [START_REF] Dupuy | Retention of ciliates and flagellates by the oyster Crassostrea gigas in French Atlantic coastal ponds:protists as a trophic link between bacterioplankton and benthic suspension-feeders[END_REF][START_REF] Kooijman | Dynamic Energy and Mass Budgets in Biological Systems[END_REF][START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF][START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF], and an increase in the number of shellfish farms in specific areas will certainly impact their nearby ecosystems. The roles of bivalves can be summarised by three main functions [START_REF] Richard | Changes in planktonic microbial components in interaction with juvenile oysters during a mortality episode in the Thau lagoon (France)[END_REF]): filtration [START_REF] Dupuy | Feeding rate of the oyster Crassostrea gigas in a natural planktonic community of the Mediterranean Thau Lagoon[END_REF][START_REF] Trottet | Impact of suspended mussels (Mytilus edulis L.) on plankton communities in a Magdalen Islands lagoon (Québec, Canada): A mesocosm approach[END_REF], excretion [START_REF] Mazouni | Influence of suspended oyster cultures on nitrogen regeneration in a coastal lagoon (Thau, France)[END_REF][START_REF] Richard | Influence of suspended scallop cages and mussel lines on pelagic and benthic biogeochemical fluxes in Havreaux-Maisons Lagoon, Îles-de-la-Madeleine (Quebec, Canada)[END_REF][START_REF] Jansen | Seasonal variability in nutrient regeneration by mussel Mytilus edulis rope culture in oligotrophic systems[END_REF], and biodeposition [START_REF] Callier | Sedimentation rates in a suspended mussel farm (Great-Entry Lagoon, Canada): biodeposit production and dispersion[END_REF][START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF][START_REF] Robert | Dose-dependent response of a benthic system to biodeposition from suspended blue mussel (Mytilus edulis) culture[END_REF]. Filtration by oysters selectively removes suspended living or non-living particles from the water column [START_REF] Gosling | How bivalves feed[END_REF][START_REF] Bayne | Feeding[END_REF]. Several studies considered the type and size of particles trapped by oysters, ranging from 3 to 5 μm [START_REF] Barillé | No influence of food quality, but ration-dependent retention efficiencies in the Japanese oyster Crassostrea gigas[END_REF][START_REF] Dupuy | Retention of ciliates and flagellates by the oyster Crassostrea gigas in French Atlantic coastal ponds:protists as a trophic link between bacterioplankton and benthic suspension-feeders[END_REF][START_REF] Kooijman | Dynamic Energy and Mass Budgets in Biological Systems[END_REF] to 500 μm [START_REF] Barillé | No influence of food quality, but ration-dependent retention efficiencies in the Japanese oyster Crassostrea gigas[END_REF][START_REF] Dupuy | Feeding rate of the oyster Crassostrea gigas in a natural planktonic community of the Mediterranean Thau Lagoon[END_REF][START_REF] Gosling | How bivalves feed[END_REF] and references therein), including nano-(3-20 μm) and microplankton (20-200 μm), comprising phyto-, protozoo-and metazooplankton. Oyster biodeposition, resulting from faeces and pseudofaeces production [START_REF] Haven | Aspects of biodeposition by oysters and other invertebrate filter feeders[END_REF], can also influence the ecosystem as it affects the composition of total particulate matter and nutrient recycling. [START_REF] Souchu | Influence of shellfish farming activities on the biogeochemical composition of the water column in Thau lagoon[END_REF], comparing suspended particulate matter and dissolved nutrient distributions between oyster farms and nearby areas not used for cultivation in a Mediterranean lagoon (Thau, France), showed that nitrogen in the biodeposits, which accumulated at the water-sediment interface, was recycled by mineralisation to ammonium, stimulating phytoplankton biomass as a result.

In exploited areas, bivalves can modify microbial plankton community structure (e.g., [START_REF] Froján | Microbial plankton community in the Ría de Vigo (NW Iberian upwelling system): impact of the culture of Mytilus galloprovincialis[END_REF][START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF]. For example, a mesocosm study was conducted in the northwestern Mediterranean lagoon of Thau (France) to assess the structural and functional impacts of oysters on the microbial food web (MFW) [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF]. In this confined environment, the authors analysed the autotroph:heterotroph C biomass ratio (A:H) structural index, indicating that the MFW became more heterotrophic in the presence of oysters as top predators.

Several modelling studies have simulated the evolution of oysters or their influence on biogeochemical cycles and other MFW components. Two main categories can be distinguished with diverse approaches and goals. The first category of models is focused on oyster evolution and behaviours in a specified environment, comprising models using the dynamic energy budget (DEB) theory [START_REF] Kooijman | Energy budgets can explain body size relations[END_REF][START_REF] Kooijman | Dynamic Energy and Mass Budgets in Biological Systems[END_REF][START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF] and other population growth models (e.g., [START_REF] Gangnery | Assessing the production and the impact of cultivated oysters in the Thau lagoon (Mediterranee, France) with a population dynamics model[END_REF][START_REF] Gangnery | Growth model of the Pacific oyster, Crassostrea gigas, cultured in Thau Lagoon (Méditerranée, France)[END_REF][START_REF] Gangnery | Modelling oyster population dynamics in a Mediterranean coastal lagoon (Thau, France): sensitivity of marketable production to environmental conditions[END_REF]. The second category includes models commonly used to investigate bivalve-environment interactions. This category concerns biogeochemical models integrating oysters, where biogeochemical models describe MFW processes generally forced by the physical environment (e.g., temperature and light). Several studies included the nitrogen and/or phosphorus cycles [START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF][START_REF] Zaldívar | Long-term simulation of main biogeochemical events in a coastal lagoon: Sacca Di Goro (Northern Adriatic Coast, Italy)[END_REF]. In parallel with these developments and because of the strong coupling between physics and biology, biogeochemical models are regularly coupled with 3D oceanic circulation models (e.g., [START_REF] Auger | Interannual control of plankton communities by deep winter mixing and prey/predator interactions in the NW Mediterranean: Results from a 30-year 3D modeling study[END_REF][START_REF] Plus | Phytoplankton versus macrophyte contribution to primary production and biogeochemical cycles of a coastal mesotidal system. A modelling approach[END_REF][START_REF] Ulses | Budget of organic carbon in the North-Western Mediterranean open sea over the period 2004-2008 using 3-D coupled physical-biogeochemical modeling[END_REF]). Other developments have been made in coupling ecological models of bivalve algorithms with hydrodynamic models to simulate food availability and thus the carrying capacities of bivalve populations [START_REF] Duarte | Mathematical modelling to assess the carrying capacity for multi-species culture within coastal waters[END_REF][START_REF] Marinov | An integrated modelling approach for the management of clam farming in coastal lagoons[END_REF] or the impact of bivalves on the spatial distribution of phytoplankton [START_REF] Spillman | A spatially resolved model of seasonal variations in phytoplankton and clam (Tapes philippinarum) biomass in Barbamarco Lagoon, Italy[END_REF][START_REF] Cugier | Assessing the role of benthic filter feeders on phytoplankton production in a shellfish farming site: Mont Saint Michel Bay, France[END_REF][START_REF] Ibarra | Coupling 3-D Eulerian bio-physics (ROMS) with individual-based shellfish ecophysiology (SHELL-E): A hybrid model for carrying capacity and environmental impacts of bivalve aquaculture[END_REF].

The present study focused on the interaction between oysters and MFW components and fluxes, using a 0D modelling approach. This work was part of the development of a 3D coupled hydrodynamic-biogeochemical model of an exploited coastal northwestern Mediterranean lagoon (Thau, France). The Thau lagoon is a shallow marine lagoon connected to the sea by three narrow channels. It has been highly exploited for bivalve culture and, in particular, the culture of the nonnative oyster Crassostrea gigas, which has been cultivated in the lagoon since 1972 [START_REF] Hamon | Les malaïgues de l'étang de Thau. Tome 1. Description des malaïgues[END_REF]. The originality of this work lies in the insertion of a new oyster compartment into the highly comprehensive Eco3m-S biogeochemical model, presented in detail in [START_REF] Auger | Functioning of the planktonic ecosystem on the Gulf of Lions shelf (NW Mediterranean) during spring and its impact on the carbon deposition: a field data and 3-D modelling combined approach[END_REF].

An optimisation method assimilating mesocosm observations described in [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] was performed to determine the best control parameters for the model. The Eco3m-S biogeochemical model was used to simulate complex ecosystem dynamics composed of several decoupled cycles of biogenic elements (carbon, nitrogen, phosphorus, and silica). The new oyster compartment was implemented to investigate the impact of the insertion of oysters as top predators of the MFW. The mesocosm study of [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] showed very good experimental results regarding MFW structure and functioning resulting from the introduction of oysters, but these results could not be extrapolated. The advantage of the new model is that it provides high-frequency results and deeper insight into interactions and fluxes within the MFW.

In further studies, the model could also be used to elucidate the potential response of the ecological system to new environmental conditions and forcings. Future studies involving this new model will use 3D coupled hydrodynamic-biogeochemical modelling of the same study area. Specific questions to be addressed by the current study were (1) What are the essential biogeochemical processes modified within the MFW by the insertion of oysters? (2) What are the direct and indirect impacts of the insertion of oysters on the different components of the MFW?

Materials and methods

Experimental data

A mesocosm study was conducted at the Mediterranean Centre for Marine Ecosystem Experimental Research (MEDIMEER) [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] based at the marine station of Sète (southern France). The purpose was to determine the MFW structural and functional responses to oysters as top predators. Four mesocosms filled simultaneously with screened (<1000 μm) natural Thau lagoon surface water were studied over 10 days in October 2005 near the MEDIMEER pontoon. Two of the mesocosms contained only natural water ('Control'), and two contained 10 Crassostrea gigas each ('Oyster'). These mesocosms mimicked a water column without including a sedimentary compartment. The water column was kept homogeneous by using a pump to assure water column mixing.

The present modelling study does not reproduce all observations described in [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] because only a few of them were relevant for the model. Temperature and photosynthetically active radiation (PAR: 400-700 nm) were used as external forcings. Six experimental observations, detailed below, were used to initialise the numerical model on day 0 (beginning of the mesocosm experiment) and for parameter optimisation thereafter.

Temperature (fig. 1) was measured every 10 minutes throughout the mesocosm experiment. PAR (fig. 1) was recorded every 10 minutes, between 14:00 on day 0 and 23:50 on day 7. Then, on day 8, measurements were stopped due to sensor failure. However, to complete our model's forcings, the observed values obtained on day 7 were replicated on days 8 and 9. Nutrient concentration data (nitrate (μM): fig. 2 [START_REF] Sime-Ngando | Significance of planktonic ciliated protozoa in the Lower St. Lawrence Estuary: comparison with bacterial, phytoplankton, and particulate organic carbon[END_REF][START_REF] Vidussi | Effects of experimental warming and increased ultraviolet B radiation on the Mediterranean plankton food web[END_REF]. More details on this study can be found in [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF].

Model description and initial simulation

The Eco3m-S biogeochemical model used in this study is presented in detail in [START_REF] Auger | Functioning of the planktonic ecosystem on the Gulf of Lions shelf (NW Mediterranean) during spring and its impact on the carbon deposition: a field data and 3-D modelling combined approach[END_REF]. It was built to simulate the temporal evolution of a complex ecosystem composed of several decoupled cycles of biogenic elements (carbon, nitrogen, phosphorus, and silica), with a mechanistic approach for biogeochemical processes. A schematic view of the model is presented in fig. 4. The Eco3m-S model includes 36 variables, including photoautotroph and heterotroph biomasses, non-living particulate organic matter (POM) concentration, dissolved organic matter concentration, and nutrient and oxygen concentrations (detailed in Table 1). It also includes 166 fundamental control parameters, in the case without oysters, defining the modelled biogeochemical processes. As an example, the model requires 10 parameters to define the grazing function of the first zooplankton size class (5-50 μm).

The model is composed of several plankton groups characterised by their functional type, resulting from an analysis of experimental and modelling information on the biogeochemical functioning of the NW Mediterranean Sea. Autotrophs included three size classes of phytoplankton, namely, 0.7-2 μm (Phy1), 2-20 μm (Phy2) and 20-200 μm (Phy3, largely dominated by silicifiers such as diatoms), while heterotrophs included four size classes, namely, 0.3-1 μm (mainly bacteria), 5-50 μm (Zoo1, mostly bacterivorous flagellates and small ciliates), 50-200 μm (Zoo2, mainly ciliates and large flagellates), and >200 μm (Zoo3, metazooplankton dominated by copepods). To ease the comparison with the total Chl a concentration measurements, only one group of phytoplankton (Phy) was considered in our study, representing the entire phytoplankton community. The initial parameters of this group were similar to the parameter values of the third phytoplankton size class (Phy3) because diatoms dominated the phytoplankton community during the mesocosm experiment. Optimal parameters obtained by our method thus correspond to the total phytoplankton community. However, the three types of zooplankton were maintained in the model as they predate on different groups of living organisms (bacteria, phytoplankton and even smaller zooplankton) or non-living POM (see fig.

5).

Four types of dissolved inorganic nutrients were considered: nitrate (NO3 -), ammonium (NH4 + ), phosphate (PO4 3-) and silicate (SiO4 2-). The distinct roles of nitrate and ammonium, which are involved in new and regenerated production, were taken into account. Dissolved organic matter (DOM), in the forms of carbon (C), nitrogen (N) and phosphorus (P), was also considered as it was consumed by heterotrophic bacteria. Non-living POM, in the forms of C, N, P, and Si, was divided into two size classes: <50 μm (light) and >50 μm (heavy). Light non-living POM was also in the form of chlorophyll, resulting from phytoplankton death and zooplankton egestion.

The model could also include particulate inorganic matter, which contributes to turbidity and light absorption.

Experimental observations of nutrients, DOC, bacteria and Chl a on day 0 fixed some of the initial conditions required for running the model. Without any information on non-living matter, initial non-living POM concentrations were set to zero at the beginning of the experiment. Since the experimental study did not provide zooplankton biomasses, the initial conditions for Zoo1, Zoo2 and Zoo3 were fixed after calibration. The oxygen concentration was set to be sufficient (200 mmol m -3 ) to avoid limiting biogeochemical processes because the water column was considered well mixed. Temperature and PAR measured during the mesocosm experiment were used in the model to force biogeochemical processes.

Once initial conditions were established and forcings were defined, an initial simulation was performed to simulate the 10 days of the experiment, using parameters suitable for the NW Mediterranean Sea, where the model is usually implemented [START_REF] Auger | Functioning of the planktonic ecosystem on the Gulf of Lions shelf (NW Mediterranean) during spring and its impact on the carbon deposition: a field data and 3-D modelling combined approach[END_REF]. The model can provide daily variations in the 36 variables at a high frequency, typically every 10 minutes, for applied simulation. In fact, this interval of time corresponds to the time step chosen to resolve the differential equations of the model. Biogeochemical fluxes such as primary production, grazing and respiration were extracted as outputs of the model.

The results obtained for this initial simulation were then compared with the available data (nitrate: fig. 2-A fig. 2-F). The model results provided the same orders of magnitude as the experimental data, even though the two sets of magnitudes did not exhibit similar patterns. For instance, the N, P and Si nutrient concentrations were excessively high. However, the N concentrations were similar to the experimental data at the end of the experiment. It should also be noted that bacterial C biomass was reduced to zero after day 4 of the experiment. Because bacteria were involved in the decomposition process and assimilation of DOM, the very low simulated bacterial biomass led to discrepancies between model results and experimental data in terms of nutrients and DOM.

The model also provided additional information such as ammonium concentration (fig. 6-A) and zooplankton biomasses (fig. 6-B) for which no corresponding experimental data were measured.

Optimisation method

To reduce the discrepancies between experimental observations and biogeochemical model results, an optimisation method suggested by [START_REF] Prunet | Assimilation of surface data in onedimensional physical-biogeochemical model of the surface ocean. 1. Method and preliminary results[END_REF] was adopted. The mesocosm experiment provided a total of 54 experimental observations by the measurement of six variables, once a day, for 9 days (see 2.1; day 0 was used as the initial condition). The Eco3m-S model included 166 parameters controlling biogeochemical processes. The aim of this mathematical method was to modify, namely, optimise, the control parameter values in order to minimise a cost function representing the gap between observations and model results.

The ecosystem model was considered an application 𝑀: ℝ 𝑚 → ℝ 𝑛 , where 𝑛 is the number of experimental observations (here, 𝑛 = 54) and 𝑚 is the number of parameters (𝑚 = 166 in the model without oysters). With the application 𝑀 being non-linear, the component (𝑝 𝑖 ) 1≤𝑖≤𝑚 of the control parameter vector 𝑃 was optimised iteratively to reduce the gap between the observations denoted (𝑑 𝑗 ) 1≤𝑗≤𝑛

and the corresponding simulation results denoted (𝑐 𝑗 ) 1≤𝑗≤𝑛

. The error associated with observation 𝑑 𝑗 was denoted 𝑟 𝑗 . This optimisation procedure finally consisted of minimising a quadratic cost criterion (equation 1):

𝐹(𝑃) = ∑ ( 𝑑 𝑗 -𝑐𝑗 𝑟 𝑗 ) 2 𝑛 𝑗=1 ( 1 )
The value of the cost function was used to determine if the model results became more similar to the experimental observations with the iterative optimisation method. In the present study, for the first iteration, the control parameter values were extracted from the literature (see [START_REF] Auger | Functioning of the planktonic ecosystem on the Gulf of Lions shelf (NW Mediterranean) during spring and its impact on the carbon deposition: a field data and 3-D modelling combined approach[END_REF]) and corresponded to the initial simulation. 𝑃 0 was then defined as the first guess of the iterative method. 𝑃 𝑘 ∈ ℝ 𝑚 was defined as the control parameter vector for iteration number 𝑘.

The vector of the difference between observed and simulated results (𝑑 𝑗 -𝑐 𝑗 ), denoted 𝑅 𝑘 ∈ ℝ 𝑛 , was introduced. To find the optimal parameters, the aim was to determine 𝛥𝑃 = 𝑃 𝑘+1 -𝑃 𝑘 , which solved equation 2: ) after small explicit perturbations of each parameter 𝛿𝑝 𝑖 and compilation of deviations of the model results 𝛿𝑐 𝑗 . This step required running the model (𝑚 + 1) times at every iteration: a first simulation run without perturbation and then 𝑚 simulation runs with only parameter 𝑚 perturbed. For this reason, the optimisation method could have a high computational cost. However, in the present work, the duration of a simulation run was very short and was not limiting in the process.

𝑀′(𝑃 𝑘 ) × 𝛥𝑃 = 𝑅 𝑘 (
The system must be dimensionless for both parameters and observations. To render the system dimensionless regarding parameters, a parameter weight diagonal matrix 𝑆, containing the a priori variances of the control parameters, was inserted (equation 4).

𝑆 𝑖,𝑖 = 𝜎 𝑖 2 , for 1 ≤ 𝑖 ≤ 𝑚 ( 4 )
In the present study, 𝑆 was practically chosen to reduce 𝑆 -1 2 × 𝑃 0 to the vector identity. A weight diagonal matrix 𝑊 containing the a priori weight of each observation was also introduced to render the system dimensionless regarding observations (equation 5).

𝑊 𝑗,𝑗 = 1 𝑟 𝑗 2 , for 1 ≤ 𝑗 ≤ 𝑛 ( 5 )
𝑊 was assumed to be diagonal here, which meant that the observations were uncorrelated. The advantage of using these two matrices is the ability to choose a more or less important weight for a specific parameter or observation. Then, equation 6 was obtained from equation 2, introducing the new weight matrices:

𝑊 -1 2 × 𝑀′(𝑃 𝑘 ) × 𝑆 1 2 × 𝑆 -1 2 × 𝛥𝑃 = 𝑊 -1 2 × 𝑅 𝑘 ( 6 )
Equation 6 was an under-determined system 𝐴 × 𝑄 = 𝑇, with a number of equations (𝑛 = 54)

stricly smaller than the number of unknowns (𝑚 = 166). The matrix

𝐴 = 𝑊 -1 2 × 𝑀′(𝑃 𝑘 ) × 𝑆 1 2
was thus singular, and various combinations of parameters resulted in the same simulation.

Matrix 𝐴 can be decomposed by singular value decomposition (SVD) as follows:

𝐴 = 𝑈 × 𝐿 × 𝑉 𝑡 ( 7 )
𝐿 is a diagonal matrix of eigenvalues, 𝑈 is a base of eigenvectors in the space of data, and 𝑉 is a base of eigenvectors of parameters. As 𝐴 was singular, some of the singular values were zero and defined the null space of the system and thus the rank. In practice, due to numerical computation, those values were not strictly zero, and the rank was difficult to define. It was typically between 5 and 20. The collinearities between parameters led to a rank deficiency problem, and the pseudoinverse was influenced by small eigenvalues. The system was then truncated at rank 𝑟 using the sphericity test of Gonzalez [START_REF] Vicente | Recherche de la dimension de l'espace latent en ACP[END_REF]. Without this truncation, the new estimated parameters could be far from their starting point, which was not desirable in terms of keeping the model realistic. The solution was finally calculated using equation 8.

𝛥𝑃 = 𝑆 1 2 × 𝑉 𝑟 × 𝐿 𝑟 × 𝑈 𝑟 × 𝑊 -1 2 × 𝑅 𝑘 ( 8 )
𝑉 𝑟 , 𝐿 𝑟 , and 𝑈 𝑟 were the truncated matrices associated with 𝑉, 𝐿 and 𝑈, respectively. The diagonal elements of matrix (𝑉 𝑟 × 𝑉 𝑟 𝑡 ) were estimators of the determination of each control parameter and were called parameter resolutions. The matrix (𝑈 𝑟 × 𝑈 𝑟 𝑡 ) gave similar information for the data, and its diagonal elements were called data resolutions. They showed the contribution of each observation as useful information to the system (e.g., [START_REF] Wunsch | The North Atlantic general circulation west of 50∘W determined by inverse methods[END_REF].

The optimisation procedure involved all 166 parameters. However, in practice, most of them had a weak influence on the results. For example, as noted above (section 2.2), the oxygen concentration was sufficient to avoid limiting biogeochemical processes. Then, the modification of parameters linked with oxygen processes did not affect our results, and the resolutions of these parameters were always null.

System of equations for oysters

To insert the new compartment for oysters into the ecosystem model, a method inspired by [START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF] was applied, assuming two main functions. The first function was the filtration of all living (zooplankton, phytoplankton, and bacteria) and non-living (non-living POM) particles. The second function of the new compartment was biodeposition (e.g., pseudofaeces). These two processes defined the temporal evolution of oyster biomass

([𝑂𝑦𝑠𝑡𝑒𝑟] 𝐶 ), expressed as carbon concentration (mmol C m -3 ). 𝑑[𝑂𝑦𝑠𝑡𝑒𝑟] 𝐶 𝑑𝑡 = ∑𝑓𝑖𝑙𝑡 𝑋 -𝑏𝑖𝑜𝑑𝑂 ( 9 )
𝑓𝑖𝑙𝑡 𝑋 is the filtration by oysters of the living and non-living particulate matter 𝑋 present in the water column (Bac, Zoo1, Zoo2, Zoo3, Phy, and light and heavy non-living POM). Bacteria were also considered in the filtration process as they can be attached to larger non-living particulate matter [START_REF] Frikha | Evolution annuelle et saisonnière de la microbiomasse d'une claire à huîtres : importance des populations bactérioplanctoniques[END_REF], found in De Crignis 2007), but the efficiency of this filtration was assumed to be very low (see Table 2). 𝑏𝑖𝑜𝑑𝑂 corresponded to the biodeposition of non-living POM (light and heavy). The first modelling results showed no impact of oyster excretion on the water column for this specific experimental period. Therefore, this process was not modelled in this study to reduce the number of control parameters needed and simplify the optimisation method. The variables and fluxes presented here were first calculated in units of carbon, as were the processes in other compartments. To include a filtration efficiency 𝑒𝑓𝑓𝑂 𝑋 for each filtrated particle 𝑋, the filtration term was expressed as in equation 10.

𝑓𝑖𝑙𝑡 𝑋 = 𝛼 𝑓𝑖𝑙𝑡 × 𝑒𝑓𝑓𝑂 𝑋 × [𝑋] 𝐶 ( 10 )
The coefficient 𝛼 𝑓𝑖𝑙𝑡 depended on temperature, oyster biomass and the maximum filtration rate of oysters 𝑓𝑖𝑙𝑡 0 .

𝛼 𝑓𝑖𝑙𝑡 = 𝑓𝑖𝑙𝑡 0 × 𝑓(𝑇) × [𝑂𝑦𝑠𝑡𝑒𝑟] 𝐶 ( 11 )
In addition to other temperature-dependent processes, 𝑓(𝑇) represented the temperature limitation of the biological process. It was expressed with the commonly used 𝑄 10 model (e.g., [START_REF] Sherman | Temperature influence on phytoplankton community growth rates[END_REF].

𝑓(𝑇) = 𝑄 10

(𝑇 𝑟𝑒𝑓 -𝑇)/10 ( 12 )

Biodeposition was calculated as a constant percentage of predation [START_REF] Mazouni | Etude in situ de l'influence des filtreurs (coquillages et épibiontes) sur les flux de matières particulaire et dissoute[END_REF][START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF].

𝑏𝑖𝑜𝑑𝑂 = 𝑏𝑑 % × ∑𝑓𝑖𝑙𝑡 𝑋 ( 13 )
To impact other compartments of the biogeochemical model, the filtration term was included as a sink term for each filtrated particle, and the biodeposition term was included as a source term for light and heavy non-living POM. A total of 14 new parameters (see Table 2) were introduced to the existing Eco3m-S model in order to control the two main functions of oysters.

Results

The results of this study will be presented in two parts. In the first part, the results of the 'Control' optimisation, corresponding to the results of the optimisation procedure applied for the initial Eco3m-S model, before the addition of the new oyster compartment, will be presented. The second part will present the results of the 'Oyster' optimisation, corresponding to the new biogeochemical model including oysters. Each part will be discussed in three steps: a description of simulation results, an analysis of adjusted parameters, and an analysis of data resolutions.

'Control' optimisation

The optimisation method was first performed on the original biogeochemical model, without the oyster compartment, using the experimental observations available for the 'Control' mesocosms (i.e., including the entire MFW but without oysters). The initial simulation presented in section 2.2 was used as the first guess. Similar weights for the observations were imposed (managed by matrix 𝑊, see section 2.3). The optimisation sequence, aiming at minimising the cost function and optimising the parameters of the model, was iterated 47 times (i.e., 47×167 simulation runs)

until the parameters reached a fixed value and plateaued. The value of the cost function ranged from 1.00 initially to 0.08 at the end. The 'Control' optimisation was thus more similar to the experimental observations than the initial simulation was.

Simulation results

Nutrient concentrations for the 'Control' optimisation were more similar to the experimental observations than were the initial simulation results. For nitrate concentration (fig. 2-A), the gap between the 'Control' optimisation results and observations was 0.03 μM on average, whereas the gap was 0.1 μM for the initial simulation. The amplitude of daily variations was also reduced by the optimisation procedure. Between days 3 and 4, the concentration ranged from 0.05 to 0.10 μM, whereas it was 0.33 to 0.76 μM in the initial simulation.

Concerning the phosphate concentration (fig. 2-B), from the beginning until day 3, the decrease in phosphate during the daytime was greater in the 'Control' simulation than in the initial simulation. However, the increase in phosphate concentration during nights was lower for the optimised simulation. The mean concentration over the entire period was 0.18 μM for the optimised simulation, compared to 0.16 μM for the experimental observations.

The silicate concentration (fig. 2-C) for the 'Control' optimisation was lower than that for the initial simulation. The values obtained remained generally larger than the observations, but the trend was well represented. The optimised average concentration throughout the study period was 4.04 μM, compared to 3.65 μM in the initial simulation.

The ammonium concentration (fig. 6-A) increased gradually from day 0 (0.3 μM) until day 5 (3.2 μM) in the 'Control' optimisation. Thereafter, it was stable around a value of 2.6 μM, in contrast to the initial simulation, in which the concentration decreased towards a much lower value (i.e., 0.3 μM at the end of the initial simulation).

The DOC concentration (fig. 2-D) in the optimised simulation decreased less than that in the initial simulation and was nearly constant throughout the experimental period. It ranged from 202

to 235 μM. The maximum experimental values of DOC observed during days 3 and 4 were not captured by the optimised simulation.

Regarding the Chl a concentration (fig. 2-E) for the optimised simulation, the results were more similar to the observations than were those obtained with the initial simulation. The simulated results followed the behaviour of measurements taken during the mesocosm experiment. The decrease in Chl a concentration on day 3 was corrected by the optimisation procedure to fit the experimental points. From day 5 until the end, the amplitude of the daily variations was almost constant for the 'Control' optimisation, with an increase during daytime equal to the decrease during nighttime (approximately 0.6 mg m -3 ). Hence, the simulation did not reproduce the experimental observations on day 7 but was clearly consistent with the last few days.

The most significant improvement was in the bacterial C biomass concentration (fig. 2-F), which showed a good fit with experimental observations for the 'Control' optimisation. In contrast to the initial simulation, the values obtained were not reduced to zero after day 4. Note that the bacterial growth during the first few days was higher in the optimised simulation than in the

Analysis of adjusted parameters

The method used in this study estimated the resolution of each parameter, i.e., the importance of each parameter for making the simulated results more similar to the observations. Parameter resolution was used to sort parameters and identify the key parameters for the optimisation procedure (see Table 3). A small change in these parameters can affect the whole biogeochemical simulation. Therefore, the modification of these key parameters by the optimisation procedure was investigated. Table 3 presents a list of key parameters presenting both a high resolution rate and a significant change.

Seven parameters among the identified key parameters (Table 3) controlled the zooplankton processes. The preference factor of micro-zooplankton for nano-zooplankton and the maximum grazing rate of micro-zooplankton both increased in the process, with changes of +64% and +86%, respectively. This led to higher predation of Zoo1 by Zoo2 in the 'Control' simulation compared to that in the initial simulation. This might explain the shifting in time of the maximum Two of the selected parameters shown in Table 3 correspond to the growth of bacteria. During the optimisation procedure, the net growth efficiency of bacteria greatly increased from 0.3 to 0.99, whereas the maximum uptake rate of bacteria decreased from 4.25× 10 -5 to 1.97× 10 -5 s - 1 . In our model, the growth of bacteria depended on the product of these two parameters. Finally, this modification led to an increase in total bacterial growth in the 'Control' optimisation compared to the initial simulation (see fig. 2-F). These two changes are antagonistic and highlight the fact that the optimisation process modified combinations of parameters and not individual parameters.

The decrease in the maximal nitrification rate from 5.91× 10 -7 to 2.52× 10 -7 s -1 contributed to the diminution of nitrate concentration (fig. 2-A) in the 'Control' optimisation compared to the initial simulation.

The temperature coefficient for decomposition increased from 2.95 to 4.35 ∘ C -1 , and the reference temperature for decomposition greatly decreased. This change impacted the decomposition process, which was temperature-dependent in our model. The decomposition process converts, via the action of bacteria, non-living POM into DOM. The modification of these two parameters, in addition to higher bacterial C biomass (fig. 2-F), led to an increase in DOC concentration (fig.

2-D). Decomposition and other temperature-dependent processes in the model (primary production of phytoplankton, grazing by zooplankton, nitrification, etc.) were modified by the optimisation procedure by changing the reference temperature and/or the temperature coefficient.

These two parameters acted as highly non-linear parameters.

The internal quotas for Phy (N:C, P:C, and Si:C) were also changed during the optimisation process. The range of the (Si:C) ratio increased as the maximal internal ratio was changed from 0.19 to 0.38 mol Si mol C -1 . The phytoplankton community could thus consume more silicate to build their skeleton or frustules (in the case of diatoms). From the initial simulation to the 'Control' optimisation, the silicate concentration (fig. 2-C) decreased, becoming more similar to the observations. The consumption of nitrate by the phytoplankton community varied less (see fig. 2-A) because the interval length of (N:C) quota was reduced. Similarly, the interval length of (P:C) quota increased, allowing more phosphate uptake by the phytoplankton community.

Several parameters presented a very large resolution, with a maximum of 0.78, but none of them had a resolution equal to 1. Thus, a change in a single parameter value was not enough to adjust our model. The optimisation procedure modified a linear combination of parameters.

Analysis of data resolution

The optimisation method allowed the calculation of data resolutions, which represent the contribution of each observation as effective information to optimise the model. Fig. 7-A shows data resolutions for the 'Control' optimisation, for the last iteration of the optimisation procedure.

The data resolution varied between sets of observations, which depended on sampling time.

Therefore, the data resolutions also varied during the experimental period.

Experimental observations of the Chl a concentration were effective information for the optimisation procedure as they had the highest data resolution among the available observations.

The data resolution decreased from 1.00 on day 1 to 0.08 on day 4 and then reached 0.49 and 0.46 on days 5 and 6, respectively. At the end of the period, it remained below 0.25. The nitrate data also had a high resolution at the very beginning and during the last few days of the experiment. The phosphate data had increasing resolution throughout the experimental period, from 0.06 initially to 0.31 at the end. The resolution of the silicate data followed the same trend as that of the phosphate data, except on day 1, when the silicate data were more useful for the optimisation process. The DOC data showed the same increase as the phosphate data. However, on days 2, 3 and 4, the resolution of the DOC data was very low. The resolution of the bacterial data decreased and varied between 0.71 and 0.09 during the experimental period.

On day 1, the Chl a and nitrate observations presented the highest resolutions for the optimisation process. From day 2 to day 6, the Chl a and bacterial C biomass observations were the most influential. During those 5 days, the resolution of other observations remained low in comparison, except that of silicate on day 3. At the end of the period, the resolutions were almost homogeneous among observations, even if the nitrate data had a relatively high resolution.

'Oyster' optimisation

To estimate the best parameter values for the model including the new oyster compartment, an optimisation procedure was performed with 180 parameters (166 initially and 14 added for oysters). Realistic values for the new parameters were chosen according to the literature (see Table 2). The initial filtration rate 𝑓𝑖𝑙𝑡 0 was set to 2.31 × 10 -7 s -1 , as suggested by [START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF], and derived from [START_REF] Mazouni | Nutrient and oxygen exchanges at the water-sediment interface in a shellfish farming lagoon (Thau, France)[END_REF]. The biodeposition rate 𝑏𝑑 % was initially set at 0.3 [START_REF] Mazouni | Etude in situ de l'influence des filtreurs (coquillages et épibiontes) sur les flux de matières particulaire et dissoute[END_REF]. The filtration efficiency coefficients 𝑒𝑓𝑓𝑂 𝑋 were arbitrarily chosen (see Table 2) but were free to change with the optimisation process, as were other parameters. As previously mentioned, only one group of phytoplankton was considered, representing the entire phytoplankton community. However, the model included three filtration efficiencies for phytoplankton that can be used with three size classes of phytoplankton in future developments, but two of them were useless for our study and were thus fixed at zero. Experimental observations from mesocosms including oysters as top predators [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] were assimilated. The first guess of the method included the 166 parameter values resulting from the 'Control' optimisation plus the 14 new parameter values, defined as the 'Oyster' initial simulation. The optimisation process was iterated 75 times (i.e., 75 × 181 simulation runs). The cost function ranged from 1.00 for the 'Oyster' initial simulation to 0.51 for the 'Oyster' optimisation.

Simulation results

The nitrate concentration after optimisation was very similar to that in the observations (fig. 3-A).

The simulated mean value over the entire period was 0.108 μM, compared to 0.104 μM for the observations. Compared to the 'Control' optimisation (fig. 2-A), in the 'Oyster' optimisation, the amplitude of the daily variations was low (fig. 3-A), and over the entire period, the concentration ranged from a minimum of 0.07 μM to a maximum of 0.17 μM.

Similarly, the phosphate concentration (fig. 3-B) of the 'Oyster' optimisation was more similar to that of the observations than was the concentration in the 'Oyster' initial simulation. This similarity was particularly pronounced for the second half of the experimental period, when the concentration varied between 0.02 and 0.29 μM. The daily variations occurring until day 3 were relatively important compared with those on the following days.

The silicate concentration (fig. 3-C) in the optimised simulation was more similar to that of the observations than the concentration in the initial simulation. The trend was also better than that in the initial simulation. The silicate concentration decreased until day 3 and remained almost constant thereafter, with a slight increase to 6.77 μM at the end of the experiment relative to the concentration of 6.35 μM observed in the mesocosms.

With the addition of oysters, the ammonium concentration (fig. 8-A) exhibited an ever-increasing trend in the 'Oyster' optimisation, varying from 1.20 μM in the 'Oyster' initial simulation to 2.73 μM in the optimised simulation on day 2. At the end, the simulated ammonium concentration was 10.91 μM.

The optimised simulated DOC concentration (fig. 3-D) showed a slight decrease and then increased more than that in the 'Oyster' initial simulation to be more similar to that observed in the mesocosms. However, the simulated concentrations were much lower than the observed concentrations. The average DOC concentration over the entire period was 214.07 μM for the optimised simulation and 261.73 μM for the observations.

From day 0 to day 2, the Chl a concentration (fig. 3-E) for the 'Oyster' optimisation exhibited the same increase observed in the mesocosms, with an amplitude of daily variations of 1.2 mg m -3 .

Then, it decreased sharply on day 3 and periodically fluctuated thereafter. The maximum value in the optimised simulation, reached on day 1, was 3.88 mg m -3 . Between day 5 and day 9, the average concentration was 0.98 mg m -3 , compared to 0.20 mg m -3 for the observations.

The bacterial C biomass concentration (fig. 3-F) in the optimised simulation was more similar to that for the observations than that in the initial simulation. It increased from day 0 until the end of day 1, where it reached a maximum of 21.87 mmol C m -3 . Then, a quick decrease was simulated until day 3. However, the simulated bacterial C biomass concentration remained lower than the observed concentration from day 4 until the end of the experiment.

Concerning the zooplankton C biomass concentration (fig. 8-B), after day 3, Zoo2 largely dominated the zooplankton community. Its concentration reached a maximum of 25.00 mmol C m -3 on day 3. The concentration in the optimised simulation followed a similar scheme as that in the 'Control' simulation without oysters, with succession of domination by the different zooplankton classes. However, the spikes appeared earlier in the simulation. The Zoo3 C biomass concentration was increased by the optimisation process. At the end of the experimental period, the C biomass concentrations of Zoo1, Zoo2, and Zoo3 were 1.88 mmol C m -3 , 15.41 mmol C m -3 , and 2.34 mmol C m -3 , respectively.

Fig. 9 shows the temporal evolution of oyster C biomass during the ten days of the experiment, revealing a regular increase from the beginning until the end of the experiment of 9.2 to 16.5 mmol C m -3 . The only information that we had at the beginning of the experiment was the introduction of 10 oysters into each mesocosm. This initial condition was highly speculative, and this led to careful consideration of the optimised value obtained for the filtration rate parameter after optimisation. The adjustment of this initial condition for oyster C biomass should result in an adjustment of oyster parameters.

Analysis of adjusted parameters

The optimisation procedure led to the calibration of the 180 parameters, including the new parameters for oysters, as detailed in Table 2. Table 4 shows that the key parameters changed during the optimisation according to their resolution. The new 'Oyster' parameters did not have the highest resolution values (see Table 2). However, as expected, the filtration rate 𝑓𝑖𝑙𝑡 0 , the filtration efficiency of oysters for Phy and the biodeposition rate 𝑏𝑑 % changed very slightly during the process.

Two of these key parameters concerned bacterial processes (Table 4). The maximum uptake rate for bacteria greatly increased from 1.97 to 2.87× 10 -5 s -1 , contributing to the increase in bacterial C biomass concentration (fig. 3-F). Furthermore, DOM transformation into inorganic nutrients greatly increased. The bacterial (P:C) ratio also changed during the optimisation process, affecting the consumption of phosphate by bacteria.

Six key parameters concerned the processes of the phytoplankton community (Table 4). The modification of two of them during optimisation affected the growth of the phytoplankton community. First, the maximal internal (Chl:N) ratio for Phy increased, which resulted in a faster increase in the Chl a concentration for the 'Oyster' optimisation than for the 'Oyster' initial simulation. In contrast, the increase in the half-saturation constant for nitrate uptake from 2.24 to 4.66 μM tended to decrease the growth of the phytoplankton community at a constant nitrate concentration (fig. 3-A). These two changes had opposite effects, but the results showed higher growth for Phy (fig. 3-E). This discrepancy could be related to the fact that the maximal internal (Chl:N) ratio had a higher resolution value (0.84 against 0.38), i.e., higher importance in the optimisation procedure. The maximal (Si:C) ratio also decreased, from 0.38 to 0.28 mol Si mol C -1 . This modification caused the diminution of the silicate concentration after day 3 (fig. 3-C)

for the optimised simulation compared to the 'Oyster' initial simulation, even if the Chl a concentration remained low (fig. 3-E).

In the same manner as in the 'Control' optimisation, seven parameters linked with zooplankton processes are shown in Table 4. The parameters presented both a high resolution and a significant change during the optimisation procedure and concerned different processes. For example, the maximal grazing rate of Zoo2 and the preference factor of Zoo2 for bacteria concerned predation by zooplankton and thus their prey, whereas the fraction of messy feeding for zooplankton concerned their messy feeding and thus light and heavy non-living POM concentrations.

Analysis of data resolution

In the 'Oyster' optimisation, the data resolution results were very different from those in the 'Control' optimisation. According to fig. 7-B, on average, the phosphate and silicate data resolutions were the highest, while the resolutions of the nitrate, Chl a and bacteria data were almost equal. During the experimental period, the resolution for each observation showed numerous variations.

The Chl a data resolution was high on days 1 and 3, at 0.53 and 0.84, respectively, but remained low thereafter. The Chl a data resolution was the highest on day 4. The nitrate data resolution was high on days 1, 3 and 7. It reached a maximum on day 7, at 0.99. The phosphate data resolution was very high on days 2 and 3, reaching 0.98 on day 3. It was almost constant at approximately 0.25 during the last few days of the experiment. The silicate data resolution was also high on days 1, 7 and 9, when it reached a very high value of approximately 0.93. The DOC data resolution remained very low during the experimental period, except on day 9, when it was 0.79. The DOC data made the smallest contribution to the optimisation process. The bacterial data resolution was higher on day 2, at 0.69, than on the other days of the experiment but made a good contribution to the optimisation during the entire period of the experiment.

Discussion

To assess the impact of oysters as top predators on MFW dynamics, a modelling approach with parameter optimisation was applied. The Eco3m-S biogeochemical model, coupled with the optimisation method presented in section 2.3, efficiently reproduced observations made during a mesocosm experiment. The optimisation method estimated a linear combination of parameters giving the best compatibility with the six different observations (nitrate, phosphate, silicate, DOC, Chl a, and bacterial C biomass). Finding a solution for the assimilation process is not a simple curve-fitting procedure because of the complexity and non-linearity of the interactions between the compartments of the model. Once the new biogeochemical model including oysters was validated, the impact of oysters on the MFW was investigated by studying the structural changes in the MFW and the interactions within the MFW.

The potential of the parameter optimisation

The parameter resolution resulting from the optimisation method reflected the importance of each biogeochemical process in the model. Using this new benefit, the biogeochemical processes could be sorted by importance. For example, in our modelling experiment, bacterial processes played a very important role in MFW dynamics. Indeed, in both the 'Control' and 'Oyster' optimisations, the maximum uptake rate for bacteria exhibited the highest resolution of the entire set of control parameters (see Table 3 andTable 4), in line with the major role played by bacteria in the marine carbon cycle (e.g., [START_REF] Cho | Major role of bacteria in biogeochemical fluxes in the ocean's interior[END_REF][START_REF] Azam | Bacteria-organic matter coupling and its significance for oceanic carbon cycling[END_REF].

In contrast, the low resolution of the control parameters of the new oyster compartment, with a maximum of 0.08 (Table 2), showed that changing associated processes was not beneficial for the optimisation process. However, the fact that parameter values did not change during the optimisation might indicate that the choice of the initial values was robust and that these values could thus be used for further studies.

To meet the mathematical requirements of the optimisation method, some of the most influential parameters, i.e., those with the highest resolutions, were modified by optimisation and reached unrealistic values. These unrealistic values allowed the simulated results to be more similar to the experimental observations by minimising the cost function more efficiently. This was the case for the net growth efficiency for bacteria and three other parameters controlling temperature functions: the reference temperature for zooplankton, the reference temperature for decomposition, and the temperature coefficient for decomposition. The optimisation method preferred to change one parameter controlling the temperature functions, affecting several temperature-dependent processes at the same time, over changing multiple parameters controlling only one process each. Note that the parameters controlling temperature functions generally had a high resolution value. Furthermore, unrealistic values could be explained with a deeper analysis of other parameter values. According to the scientific literature, the net growth efficiency for bacteria should be between 0.05 and 0.7 (Del [START_REF] Giorgio | Bacterial growth efficiency in natural aquatic systems[END_REF]. However, considering that bacterial growth depends on the product of this parameter value with the value of the maximal uptake rate of bacteria, the decrease in the maximal uptake rate of bacteria during the optimisation process partially balanced the increase in the net growth efficiency for bacteria. For example, for the 'Control' optimisation, the product varied from 1.28× 10 -5 to 1.95× 10 -5 s -1 .

Indeed, the optimisation process modified a linear combination of parameters and not individual parameters.

Oysters induced structural changes in the MFW

The introduction of oysters as top predators of the MFW led to a more heterotrophic ecosystem.

The structural autotroph:heterotroph C biomass ratio index (A:H, fig. 10) was calculated and compared to results from [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF]. This calculation was simple with the model, which estimated the C biomass of each biological component of the model, in comparison with experimental calculations, which were associated with strong uncertainties because of the C biomass calculation method. The gap between the two simulations was less important than the gaps associated with the experimental study, but the results were in the same direction. Overall, the experimental MFW was well represented by the model, at least structurally, with only six types of experimental observations assimilated. Therefore, the model corroborated the tendency for a transition to a more heterotrophic MFW with the addition of oysters as top predators, which could be useful for testing scenarios with the model. For example, the model could be used to simulate a scenario with a higher quantity of oysters to assess the structural impact of more intensive exploitation.

The main function of oysters in our model was filtering all mesocosm water containing living and non-living organisms. the increase in the biomass of zooplankton and reduction in the biomass of phytoplankton, which are prey of zooplankton, could be in line with top-down control. The simulated predation of phytoplankton by zooplankton was slightly higher in the 'Oyster' optimisation than in the 'Control' optimisation (not shown), but this result does not fully explain the large increase in zooplankton biomass. However, the reduction in phytoplankton biomass is explained by the topdown control of oysters but also by the increased predation of zooplankton. [START_REF] Losey | Positive predator-predator interactions: enhanced predation rates and synergistic suppression of aphid populations[END_REF] showed that the combined influence of two predators could be greater than the sum of their individual impacts. Here, the combined influence of oyster and zooplankton biomasses on phytoplankton biomass was greater than the sum of the individual impacts.

The introduction of a top predator to an ecosystem could lead to unexpected results; in particular, the introduction of an alien top predator could have a devastating impact on native species (e.g., [START_REF] Bytheway | Deadly intentions: naïve introduced foxes show rapid attraction to odour cues of an unfamiliar native prey[END_REF]). In our case, we cannot exclude this possibility for the Thau lagoon ecosystem, where non-native oysters (Crassostrea gigas, a Japanese oyster) have been cultivated

since 1972 [START_REF] Hamon | Les malaïgues de l'étang de Thau. Tome 1. Description des malaïgues[END_REF], because oysters exert strong structural control of the natural lagoon MFW community.

Indirect impact of oysters on bacterial biomass

Simulated bacterial biomass strongly increased in the presence of oysters as top predators of the MFW, in accordance with the study of [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF]. Those authors proposed different hypotheses to explain the increase in bacteria, and some of the hypotheses can be discussed based on the new modelling results acquired here. First, bacteria might benefit from oyster excreta [START_REF] Mazouni | Influence of oyster culture on water column characteristics in a coastal lagoon (Thau, France)[END_REF]), but as the excretion of oysters was not modelled in this study and simulated bacterial biomass was similar to that in the experimental observations, this benefit might not explain the increase. Second, [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] highlighted the assumption that the reduction in virus-like particles due to oyster filtration could favour an increase in bacterial biomass. However, viruses were not modelled, particularly due to the complexity of their interactions with other components of the MFW, and this assumption could not be investigated.

Our results support the third hypothesis posed by [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF], concerning the reduction in competition between bacteria and phytoplankton for the uptake of nutrients. To quantify competition between bacteria and phytoplankton, nutrient uptake fluxes were analysed. Fig. 12 shows the comparison between 'Control' and 'Oyster' optimisations for ammonium (fig. 12-A)

and phosphate uptake (fig. 12-B) by phytoplankton. For bacteria, the same comparison is shown for ammonium uptake (fig. 12-C) and phosphate excretion (fig. 12-D). The model version of [START_REF] Auger | Functioning of the planktonic ecosystem on the Gulf of Lions shelf (NW Mediterranean) during spring and its impact on the carbon deposition: a field data and 3-D modelling combined approach[END_REF] includes potential control of bacterial growth by phosphorus availability, in addition to limitation by carbon and nitrogen availability. This model formulation is derived from [START_REF] Thingstad | P limitation of heterotrophic bacteria and phytoplankton in the northwest Mediterranean[END_REF]. Bacteria first absorb DOM, but they can also assimilate ammonium and/or phosphate if DON and/or DOP are lacking. Bacteria can also act as decomposers and excrete nutrients, depending on the comparison of DOC:DON and/or DOC:DOP with their internal ratios [START_REF] Kirchman | Carbon versus iron limitation of bacterial growth in the California upwelling regime[END_REF]. All the processes (nutrient uptake, excretion and respiration) make the control of their stoichiometry possible. Our results showed that, except for phosphate uptake, phytoplankton nutrient uptake was lower with the addition of oysters (fig. 12-A and 12-B). Furthermore, in contrast to the pattern observed in the 'Control' optimisation, where no ammonium bacterial uptake was observed (fig. 12-C), bacterial ammonium uptake increased in the 'Oyster' optimisation after day 7. It can be concluded that competition between bacteria and phytoplankton was reduced in our model, especially that for ammonium. In addition, the excretion of phosphate by bacteria was higher with oysters and conveniently balanced the higher phosphate needs of the phytoplankton community. [START_REF] Thingstad | Ability of a "minimum" microbial food web model to reproduce response patterns observed in mesocosms manipulated with N and P, glucose, and Si[END_REF] showed that the dynamics of a water column receiving nitrate, phosphate and DOC continuously shift to lower nutrient competition and higher nutrient regeneration. The introduction of oysters slightly increased the DOC, nitrate and phosphate concentrations in the water column, which could correspond to the case studied by [START_REF] Thingstad | Ability of a "minimum" microbial food web model to reproduce response patterns observed in mesocosms manipulated with N and P, glucose, and Si[END_REF].

Our results showed that the reduction in this competition benefited both bacterial and phytoplankton growth. However, the phytoplankton community was subject to other strong constraints, such as filtration by oysters, as explained above. Then, the larger quantity of nutrients available for bacteria actually resulted in more bacterial biomass in the 'Oyster' simulation.

Impact of oysters on the zooplankton community

The increase in bacterial biomass in the 'Oyster' simulation favoured an increase in bacterivorous zooplankton biomass (mainly Zoo1 but also Zoo2 in a smaller proportion). However, the Zoo1 C biomass slightly decreased with the addition of oysters (fig. 6-B versus 8-B). The maximum Zoo1

C biomass of the 'Oyster' optimisation was 5.46 mmol C m -3 and was reached at the beginning of day 2. On the same date, in the 'Control' simulation, the Zoo1 C biomass was 4.41 mmol C m -3 .

This result supports the idea that the increase in bacterial biomass favoured bacterivorous zooplankton because the growth of Zoo1 was higher in the presence of oysters.

The high Zoo1 C biomass obtained in the presence of oysters led to a direct increase in the C biomass of Zoo2. Thus, Zoo1 served as a link between bacteria and Zoo2. Moreover, our results

showed an increase in Zoo2 in the 'Oyster' simulation that was faster than the increase in Zoo1

(fig. 8-B). Indeed, the predation pressure of Zoo2 on Zoo1 was higher than the predation pressure of Zoo1 on bacteria (not shown).

Therefore, in the presence of oysters, the phytoplankton community was greatly impacted by direct oyster filtration. Furthermore, the reduction in competition between bacteria and phytoplankton for the uptake of nutrients benefited bacteria and increased bacterial biomass. This increase in bacterial biomass favoured bacterivorous zooplankton (Zoo1), which were predated by Zoo2. Finally, this phenomenon resulted in a higher Zoo2 biomass in the 'Oyster' simulation than in the 'Control' simulation. The modelling approach provided a real advantage in investigating the interactions between different components of the MFW. The use of a mechanistic formulation of biogeochemical processes (see section 2.2) renders the extraction of stocks and fluxes simple.

This discussion focused on filtration by oysters (fig. 11), predation by zooplankton and competition between phytoplankton and bacteria for nutrients (fig. 12). These biogeochemical fluxes, easily extracted from the simulation results, provided a great deal of new information for our understanding of the impact of oysters as top predators of the MFW.

Conclusion

In this study, numerical ecosystem simulations with the Eco3m-S model were performed to provide deeper insight into the interactions and fluxes within the MFW. A parameter optimisation method assimilating mesocosm experimental observations described in [START_REF] Mostajir | Microbial food web structural and functional responses to oyster and fish as top predators[END_REF] was used, and the model efficiently reproduced observations with and without oysters. The model provided high-frequency results over the period of the experiment, and fluxes within the MFW were extracted and analysed. For example, the comparison of nutrient uptakes by bacteria and the phytoplankton community brought to light a decrease in competition for nutrients between these two components of the MFW in the presence of oysters. Due to the filtration of all mesocosm water including organisms and non-living particles, the direct and immediate impact of oysters was mainly a reduction in phytoplankton carbon biomass. In addition, our results showed an increase in zooplankton and bacterial biomasses. The mesocosms thus became more heterotrophic with oysters. The reduction in competition for nutrients between bacteria and phytoplankton, as mentioned above, resulted in a higher bacterial biomass in the 'Oyster' simulation. The increase in zooplankton biomass was explained by a strong increase in simulated Zoo2 biomass (50-200 μm, mainly ciliates and large flagellates) due to the grazing of Zoo1 bacterivorous zooplankton (5-50 μm, mostly bacterivorous flagellates and small ciliates), which appeared as an intermediary trophic link. In our study, oysters pushed the system to a more heterotrophic state with higher micro-zooplankton and bacterial C biomasses and lower phytoplankton C biomass.

Modelling, observation and experimentation are complementary for obtaining a good understanding of ecological processes. Observation and experimentation provide a real but partial view of the processes involved, and the ideal scenario in which everything could be observed or monitored persists. Modelling approaches provide estimates more or less close to reality, depending on the quality of known assumptions. However, such approaches often yield information that is more complete and has a greater temporal resolution. Models are fed with theories and observations. The modelling approach introduces a new point of view, which is useful for understanding biogeochemical mechanisms, paying attention to assumptions made beforehand. In our case, we succeeded in reproducing not only the steady state of the observed mesocosm experiment but also the dynamics of the ecosystem over the 10 days of the experimental period. Additional information was provided by the model, such as biogeochemical fluxes within the MFW. This idea could encourage the scientific community to pursue more interactions between modelling, observations and experimental communities.

Analysing the data assimilated during the optimisation process could also be a simple way to define the most interesting data required for the modelling approach. It could help manage experimental efforts in order to conduct a more realistic modelling study. For example, in our study, data available for Chl a or bacteria were very important for the 'Control' optimisation.

Furthermore, our results were very similar to mesocosm observations and suggested that ammonium observations, which were not available, were not essential in this specific case. Data resolutions also contain temporal information that reveals, for example, if it is better to measure each variable once a day or measure one variable at a high frequency and others at a low frequency. The answer strongly depends on the system's dynamics and the complexity of the processes involved. Regarding our results, a good effort to obtain numerous Chl a observations is needed, compared to DOC observations, which could be measured only once a week for example.

Tables Table 1: List of state variables in the Eco3m-S model. Table 2: List of the new parameters for oysters introduced to the Eco3m-S model. ( 1) [START_REF] Chapelle | Modelling nitrogen, primary production and oxygen in a Mediterranean lagoon. Impact of oysters farming and inputs from the watershed[END_REF], ( 2) Le [START_REF] Gall | Ingestion of a bacterivorous ciliate by the oyster Crassostrea gigas:protozoa as a trophic link between picoplankton and benthic suspensionfeeders[END_REF], ( 3) [START_REF] Defossez | Selective feeding in shellfish: size-dependent rejection of large particles within pseudofaeces from Mytilus edulis, Ruditapes philippinarum and Tapes decussatus[END_REF], (4) De [START_REF] Crignis | Sources de nourriture et bio-énergétique de l'huître creuse Crassostrea gigas : les nanoflagellés et les ciliés jouent-ils un rôle dans le budget d'énergie dynamique d'huîtres[END_REF] (5) [START_REF] Dupuy | Feeding rate of the oyster Crassostrea gigas in a natural planktonic community of the Mediterranean Thau Lagoon[END_REF], ( 6) [START_REF] Grant | A spatially explicit ecosystem model of seston depletion in dense mussel culture[END_REF], ( 7) [START_REF] Mazouni | Etude in situ de l'influence des filtreurs (coquillages et épibiontes) sur les flux de matières particulaire et dissoute[END_REF], ( 8) [START_REF] Gall | Influence de la température sur la physiologie de l'huître Crassostrea gigas[END_REF].

State variables

  -A and 3-A, phosphate (μM): fig. 2-B and 3-B, silicate (μM): fig. 2-C and 3-C) as well as dissolved organic carbon concentration (DOC) data (μM: fig. 2-D and 3-D) were determined with samples taken once a day at 10:00 from day 1 until day 9. Chlorophyll a (Chl a) concentration in mg m -3 (fig. 2-E and 3-E) was measured using high-performance liquid chromatography (HPLC). Bacterial abundances measured by flow cytometry were converted to C biomass (in mmol of carbon m -3 , fig. 2-F and 3-F) using a factor of 20 fg C bacterium -1 (Sime-

  , phosphate: fig. 2-B, silicate: fig. 2-C, DOC: fig. 2-D, Chl a: fig. 2-E, bacteria:

  initial simulation. The concentration reached a maximum of 15.73 mmol C m -3 on day 2 and a minimum of 4.76 mmol C m -3 between days 4 and 5, compared to 15.52 and 4.61 mmol C m -3 for the observations, respectively.Zoo1 and Zoo2 C biomass concentrations (fig.6-B) in the 'Control' optimisation followed a similar increase for the first two days of the experiment, contrary to those in the initial simulation. Then, the Zoo2 C biomass concentration exceeded the Zoo1 concentration, which depicted a link of predation. Between days 2 and 3, the two concentrations reached a maximum (5.26 mmol C m -3 for Zoo1 and 7.97 mmol C m -3 for Zoo2). From day 3 until the end of the experiment, Zoo1 and Zoo2 showed similar patterns, and the gap between the two concentrations remained almost constant. At the end, the Zoo1 and Zoo2 concentrations were 2.72 and 5.32 mmol C m -3 , respectively. The C biomass concentration of Zoo3 was almost the same in the 'Control' optimisation and the initial simulation.

  Zoo2 C biomass concentration (fig.6-B), happening on day 7 in the initial simulation and on day 3 in the 'Control' simulation. Zoo2 predates on Zoo1, and the change in the dynamics of Zoo2 also impacted the dynamics of Zoo1, with a diminution of the maximum Zoo1 C biomass concentration. The maximal grazing rate of nano-zooplankton decrease also participated in this change.

  Fig. 11 shows the filtration of each MFW component, resulting from equation 10. According to the parameters used to define the filtration rate, the most filtrated compartment corresponded to the phytoplankton community. Zooplankton was the second most filtrated, and light non-living POM and bacteria were the third most filtrated. Our model included the direct reduction of phytoplankton biomass by oysters by adding a new filtration term, 𝑓𝑖𝑙𝑡 𝑃ℎ𝑦 . This result was in line with those of Cugier et al. (2010) and Mostajir et al. (2015). Nevertheless, this process also indirectly impacted other components of the MFW, such as zooplankton, which may find fewer prey in the phytoplankton community. The simulated phytoplankton biomass was approximately 30% lower than the phytoplankton biomass in the 'Control' optimisation (fig. 2-E versus 3-E). The bacterial biomass was also affected, being 50% greater than that in the 'Control' optimisation (fig. 2-F versus 3-F). Oysters strongly increased zooplankton biomass by approximately 110% (fig. 6-B versus 8-B). Therefore,

  with X = carbon, nitrogen, phosphorus, or silica mmol X m -3 nl-POXy Heavy (y = h) and light (y = l) non-living particulate organic X (organic detritus), with X = carbon, nitrogen, phosphorus, or silica mmol X m -3 ChlDet Light chlorophyll detritus coming from phytoplankton death or egestion of zooplankton mg Chl m -3

FiguresFigure 1 :

 1 Figures

Figure 2 :

 2 Figure 2: Comparison among the mesocosm experimental study (black circles), initial simulation (blue dashed line), and 'Control' optimisation (red line) for the (A) nitrate concentration (μM), (B) phosphate concentration (μM), (C) silicate concentration (μM), (D) dissolved organic carbon

Figure 6 :

 6 Figure 6: Comparison between the initial simulation (dashed lines) and 'Control' optimisation (solid lines) for the (A) ammonium concentration (μM) and (B) zooplankton C biomass concentration (mmol C m -3 ). The three compartments of zooplankton are represented: Zoo1 (blue), Zoo2 (red), and Zoo3 (green).

Figure 7 :

 7 Figure 7: Data resolution for each type of observation, detailed for each day of the experiment, for the (A) 'Control' optimisation and (B) 'Oyster' optimisation.

Figure 8 :

 8 Figure 8: Comparison between the 'Oyster' initial simulation (dashed lines) and 'Oyster' optimisation (solid lines) for the (A) ammonium concentration (μM) and (B) zooplankton C biomass concentration (mmol C m -3 ). The three compartments of zooplankton are represented: Zoo1 (blue), Zoo2 (red), and Zoo3 (green).

Figure 9 :

 9 Figure 9: Oyster C biomass concentration (mmol C m -3 ) for the 'Oyster' initial simulation (blue dashed line) and 'Oyster' optimisation (red solid line).

Figure 11 :

 11 Figure 11: Filtration by oysters of other components of the MFW: phytoplankton (red), zooplankton (blue), bacteria (green), light non-living POM (purple), and heavy non-living POM (orange).

  2 )𝑀′(𝑃 𝑘 ) is the tangent application of 𝑀 evaluated at point 𝑃 𝑘 . 𝑀′(𝑃 𝑘 ) = (

					𝜕𝑐 𝑗 𝜕𝑝 𝑖	1≤𝑗≤𝑛 1≤𝑖≤𝑚 )	, with a
	dimension of 𝑛 × 𝑚, is also called the Jacobian matrix of the system. This yields the system of
	unknowns 𝛥𝑝 𝑖 , with 1 ≤ 𝑖 ≤ 𝑚:
			∑ 𝑚 𝑖=1	𝜕𝑐 𝑗 𝜕𝑝 𝑖	× 𝛥𝑝 𝑖 = 𝑑 𝑗 -𝑐 𝑗 , for 1 ≤ 𝑗 ≤ 𝑛	( 3 )
	Each component (	𝜕𝑐 𝑗 𝜕𝑝 𝑖	𝛿𝑐 𝑗 𝛿𝑝 𝑖 ) was approximated by (

Table 3 :

 3 The list of key parameters changed during the 'Control' optimisation process assimilating'Control' mesocosm observations.

	Name	Initial value Final Value	Units	Resolution
	Maximum uptake rate of bacteria	4.25×10 -5	1.97×10 -5	s -1	0.78
	Maximal Chl:N ratio of Phy	2.30	2.03	mg Chl mmol N -1	0.77
	Reference temperature for zooplankton processes	18.00	12.50	∘ C	0.64
	Maximal internal Si:C ratio of Phy	0.19	0.38	mol Si mol C -1	0.58
	Maximal internal N:C quota of Phy	0.20	0.19	mol N mol C -1	0.54
	Preference factor of Zoo2 for Zoo1	0.25	0.41	-	0.49
	Half-saturation constant for nitrate uptake for Phy	1.00	2.24	mmol N m -3	0.49
	Maximal grazing rate of Zoo1 at 15.5 ∘ C	4.50×10 -5	3.68×10 -5	s -1	0.40
	Net growth efficiency of zooplankton	0.80	0.75	-	0.38
	Maximal grazing rate of Zoo2 at 15.5 ∘ C	3.00×10 -5	5.60×10 -5	s -1	0.37
	Net growth efficiency of bacteria	0.30	0.99	-	0.36
	Maximal internal P:C quota of Phy	1.90×10 -2	2.85×10 -2	mol P mol C -1	0.34
	Preference factor of Zoo2 for Phy	0.15	0.06	-	0.29
	Reference temperature for decomposition processes	20.00	9.37	∘ C	0.28
	Fraction of messy feeding for zooplankton	0.23	0.31	-	0.27
	Maximal nitrification rate at 0 ∘ C	5.91×10 -7	2.52×10 -7	s -1	0.17
	Temperature coefficient for decomposition	2.95	4.35	∘ C -1	0.16

Table 4 :

 4 The list of key parameters changed during the 'Oyster' optimisation process assimilating 'Oyster' mesocosm observations.

	Name	Initial value	Final Value	Units	Resolution
	Maximum uptake rate of bacteria	1.97×10 -5	2.87×10 -5	s -1	0.84
	Maximal internal Si:C ratio of Phy	0.38	0.28	mol Si mol C -1	0.63

Acknowledgements

This work was supported by the CNRS-INSU project "CHIFRE" and a PhD grant from the Montpellier University. We also thank the "American Journal Expert" company for reviewing the English.