Ayelet Dadon-Pilosof
email: ayelet@ruppin.ac.il.

Keats Conley

Marcelino T Suzuki

Significant SAR11 removal by a hard-bottom community

come

Introduction

Marine microbial communities form the basis of the ocean food web and mediate most of the energy and material fluxes in the ocean [START_REF] Glöckner | Marine Microbial Diversity and its role in Ecosystem Functioning and Environmental Change[END_REF]. Microorganisms constitute a large fraction of the living biomass in the sea [START_REF] Pomeroy | The Microbial Loop[END_REF], and the structure and function of their populations are shaped by a delicate balance between growth and mortality [START_REF] Pernthaler | Predation on prokaryotes in the water column and its ecological implications[END_REF]. Grazing and virus-driven lysis constitute the main sources of mortality [START_REF] Sánchez | Seasonal impact of grazing, viral mortality, resource availability and light on the group-specific growth rates of coastal Mediterranean bacterioplankton[END_REF], countered by the capabilities of microorganisms to avoid grazing [START_REF] Matz | Off the hook -How bacteria survive protozoan grazing[END_REF], survive digestion or resist viral lysis. Grazing or predation on microorganisms by either planktonic or benthic organisms is an important mortality factor in many habitats (e.g., [START_REF] Verity | Feeding In Planktonic Protozoans: Evidence For Non-Random Acquisition of Prey[END_REF][START_REF] Gili | Benthic suspension feeders: their paramount role in littoral marine food webs[END_REF][START_REF] Gorsky | Picoplankton and nanoplankton aggregation by appendicularians-Fecal pellet contents of Megalocercus huxleyi in the equatorial[END_REF][START_REF] Riisgard | Minireview: Ciliary filter feeding and bio-fluid mechanics-present understanding and unsolved problems[END_REF][START_REF] Pernthaler | Predation on prokaryotes in the water column and its ecological implications[END_REF]Patten et al. 2011a). In the pelagic realm, protists commonly dominate the guild of grazers of bacteria [START_REF] Calbet | Phytoplankton growth, microzooplankton grazing, and carbon cycling in marine systems[END_REF][START_REF] Matz | Off the hook -How bacteria survive protozoan grazing[END_REF]. However, in some habitats and seasons, grazing by metazoan and lysis by phages may dominate mortality [START_REF] Hahn | Grazing of protozoa and its effect on populations of aquatic bacteria[END_REF].

Bacteria form some 65-86% of the biomass of microorganisms in the upper ocean [START_REF] Morris | SAR11 clade dominates ocean surface bacterioplankton communities[END_REF]. However, only a small number of bacterial groups dominate that guild [START_REF] Teeling | Substrate-controlled succession of marine bacterioplankton populations induced by a phytoplankton bloom[END_REF]). In the oligotrophic eastern Mediterranean, pico-cyanobacteria (Synechococcus and Prochlorococcus) and a few members of Candidatus Pelagibacter ubique (SAR11) clade dominate microbial communities, accounting for >70% of the total bacterial biomass [START_REF] Partensky | Prochlorococcus, a marine photosynthetic prokaryote of global significance[END_REF][START_REF] Dadon-Pilosof | Surface properties of SAR11 bacteria facilitate grazing avoidance[END_REF]. SAR11 is a clade of heterotrophic bacteria, which constitutes 15-60% of total bacteria in the upper ocean [START_REF] Morris | SAR11 clade dominates ocean surface bacterioplankton communities[END_REF][START_REF] Morris | Basin-scale patterns in the abundance of SAR11 subclades, marine Actinobacteria (OM1), members of the Roseobacter clade and OCS116 in the South Atlantic[END_REF][START_REF] Rappé | Cultivation of the ubiquitous SAR11 marine bacterioplankton clade[END_REF][START_REF] Eiler | Dynamics of the SAR11 bacterioplankton lineage in relation to environmental conditions in the oligotrophic North Pacific subtropical gyre[END_REF][START_REF] Giovannoni | SAR11 Bacteria: The Most Abundant Plankton in the Oceans[END_REF]. It is one of the smallest free-living bacteria in the sea and is thought to be the most abundant group in the world ocean. Within that group, some ecotypes thrive in oligotrophic environments while others in more productive waters [START_REF] Morris | SAR11 clade dominates ocean surface bacterioplankton communities[END_REF][START_REF] Salter | Seasonal dynamics of active SAR11 ecotypes in the oligotrophic Northwest Mediterranean Sea[END_REF]. Members of the SAR11 clade also have the lowest nucleic acid content (LNA) among all non-photosynthetic bacteria [START_REF] Mary | SAR11 dominance among metabolically active low nucleic acid bacterioplankton in surface waters along an Atlantic meridional transect[END_REF]. [START_REF] Dadon-Pilosof | Surface properties of SAR11 bacteria facilitate grazing avoidance[END_REF] reported that some bacteria, especially members of the SAR11 clade, can effectively evade grazing from both pelagic and benthic tunicates and this lack of grazing pressure on SAR11 could partially explain its abundance and ubiquity.

A diverse guild of benthic invertebrate suspension feeders, including sponges, bivalves, cirripedians cnidarians, bryozoans, and tunicates, often dominates subtidal hard substrates [START_REF] Topçu | In situ investigation of Spongia officinalis (Demospongiae) particle feeding: Coupling flow cytometry and stable isotope analysis Nur Eda[END_REF]. Their diet ranges from consumption of dissolved organic matter (DOM) through grazing on microorganisms such as phytoplankton, virioplankton, archaea and bacteria, as well as feeding on zooplankton and detritus [START_REF] Gili | Benthic suspension feeders: their paramount role in littoral marine food webs[END_REF][START_REF] Topçu | In situ investigation of Spongia officinalis (Demospongiae) particle feeding: Coupling flow cytometry and stable isotope analysis Nur Eda[END_REF]. Hard-bottom subtidal communities along the Mediterranean Sea are diverse and are undergoing dramatic changes in the recent decades due to the combined effect of global warming, overfishing, and the introduction of invasive species [START_REF] Rilov | The restructuring of Levant reefs by aliens, ocean warming and overfishing: implications to species interactions and ecosystem functions[END_REF].

Within these communities, retention of small particles is an adaptive advantage since picoplankton often dominate the planktonic community biomass [START_REF] Topçu | In situ investigation of Spongia officinalis (Demospongiae) particle feeding: Coupling flow cytometry and stable isotope analysis Nur Eda[END_REF].

Grazing by benthic suspension feeders on microorganisms is also an important component of the benthic-pelagic coupling in coral reefs [START_REF] Yahel | Phytoplankton distribution and grazing near coral reefs[END_REF][START_REF] Genin | Intense benthic grazing on phytoplankton in coral reefs revealed using the control volume approach[END_REF][START_REF] Genin | Intense benthic grazing of phytoplankton in a coral reef[END_REF][START_REF] Patten | Uptake of picophytoplankton, bacterioplankton and virioplankton by a fringing coral reef community (Ningaloo Reef, Australia)[END_REF]. This grazing pressure on the microbial community is not necessarily uniform as it depends on the spatial heterogeneity of the distribution of different suspension feeders and their respective diets [START_REF] Yahel | Size independent selective filtration of ultraplankton by hexactinellid glass sponges[END_REF][START_REF] Yahel | In situ evidence for pre-capture qualitative selection in the tropical bivalve Lithophaga simplex[END_REF][START_REF] Hanson | Estuarine , Coastal and Shelf Science Selective uptake of prokaryotic picoplankton by a marine sponge ( Callyspongia sp .) within an oligotrophic coastal system[END_REF][START_REF] Dadon-Pilosof | Surface properties of SAR11 bacteria facilitate grazing avoidance[END_REF]. Differential capture of particles from the ambient water based on their size, concentration or morphological features is therefore expected to be reflected in the prey distribution [START_REF] Gili | Benthic suspension feeders: their paramount role in littoral marine food webs[END_REF].

The goal of the current study was to indirectly evaluate the effect of the whole benthic community removal on the distribution of microorganisms across a shallow, subtropical rocky coast. Following the study of [START_REF] Yahel | Phytoplankton distribution and grazing near coral reefs[END_REF] across a coral reef, a study that preceded the demonstration of the role of phytoplankton grazing in the trophic dynamics of coral reefs [START_REF] Genin | Intense benthic grazing of phytoplankton in a coral reef[END_REF][START_REF] Monismith | Flow effects on benthic grazing on phytoplankton by a Caribbean reef[END_REF], we sought to test the hypothesis that differential removal on microbes is also reflected in the cross-shore distribution of bacteria and other picoplankton.

Methods

Study site

Sampling was conducted off Michmoret, Israel (32° 24′N, 34° 52′E) in the Eastern Mediterranean Sea. The oligotrophic Israeli shoreline is characterized by extremely low nutrient levels (Nitrate 0.5-1 µM, Phosphate 0.05-0.1 µM, [START_REF] Krom | Eastern Mediterranean sea: A natural laboratory for studying marine biogeochemical processes[END_REF], high salinity (38.3 to 40.0 PSU) and relatively warm waters (16.5 to 30.8°C) [START_REF] Suari | A long term physical and biogeochemical database of a hyper-eutrophicated Mediterranean micro-estuary[END_REF] that is dominated in terms of numbers and biomass by pico-and nanoplankton sized organisms [START_REF] Herut | The seasonal dynamics of nutrient and chlorophyll a concentrations on the SE Mediterranean shelf-slope[END_REF][START_REF] Raveh | The temporal dynamics of coastal phytoplankton and bacterioplankton in the eastern mediterranean sea[END_REF]. Water stratification appears usually in spring following a deep winter mixing that is enhanced by the high salinity of surface waters.

Eight transects were performed between December 2015 and April 2017 (Table S1).

Each transect encompassed eight stations spanning from 1.5 m depth at the hard-bottom subtidal towards the "open-sea" (~ 0.9 km off-shore, water depth >12 m, Fig. 1). During the sampling periods the average temperature (±SD) was 19.7±0.7 ˚C, salinity was high (39.25±0.2 PSU), and the water column was fully oxygenated (dissolved oxygen concentration 216±6 µM). Due to the need to work very close to the bottom at the rocky shoreline, sampling dates were dictated by sea conditions and were limited to days of calm sea (wave height < 30 cm) and weak winds (<6m s -1 ). To get a larger scale context within the eastern Mediterranean shelf offshore of the high-resolution cross-shore transects, six crossshelf transects were sampled during March 2018, spanning from one km offshore, equivalent to the farthest station in the small scale transects, to 42 km west to the shelf edge at 750 m bottom depth (Fig S1).

The shallowest sampling station in each transect was located ~150 m offshore at 1.6-1.8 m depth where Niskin bottle was placed just above the rocky bottom (c.a. 10 cm) to collect a sample. The 1.5 m sampling depth was retained along the transect as the water deepened down to 12 m at the furthest offshore station, ~0.9 km offshore. The four shallowest stations (hereafter Stations 1-4) were located above hard bottom (rocky area), the 5 th station was located at the boundary between rocks and sand, while stations 6 to 8 were above sandy bottom.

Sampling methods

Prior to sampling, each station was marked with a moored surface buoy and profiles of temperature, salinity, fluorescence, and oxygen were measured using a Seabird SBE19plusV2 CTD equipped with an in vivo chlorophyll fluorimeter (Cyclop7, Turner Designs) and a dissolved oxygen sensor (SBE43, Seabird). The CTD was lowered in a horizontal position at a slow pace (about a meter per minute) so that the water column was adequately profiled even in the shallow-most stations. Due to the need to work close to the bottom at the rocky shoreline, sampling was conducted from either a kayak or a small inflatable skiff.

Seawater was collected simultaneously with CTD profiles, using a 5 L Niskin bottle.

At each station, samples were collected directly from the Niskin bottle for all the required analyses: flow cytometry, DNA extraction and Chl-a measurements. Water samples were kept on ice in the dark for further processing in the laboratory within 2-3 hours. Samples for DNA were collected only during three of the eight sampling times.

Sample analysis

Flow cytometry Flow cytometry was used to quantify the concentrations and the cell characteristics of non-photosynthetic microorganisms (hereafter referred to as non-photosynthetic bacteria), and the following four dominant autotrophic groups: Prochlorococcus (Pro), Synechococcus (Syn), pico-eukaryotic algae (PicoEuk), and nano-eukaryotic algae (NanoEuk). We used an Attune® Acoustic Focusing Flow Cytometer (Applied Biosystems) equipped with a syringebased fluidic system that allows a precise adjustment of the injected sample volume and hence high precision of the measurements of cell concentrations (±5%). The instrument's optical unit contained violet and blue lasers (405 and 488 nm, respectively) and was further adapted for the analysis of marine ultra-plankton samples as described below.

Aliquots of 1.8 mL were collected from each water sample and transferred into 2 mL cryotubes (Corning cat No. 430659). Samples were first incubated for 15 min at room temperature with Glutaraldehyde 50% (electron microscopy grade, Sigma-Aldrich, cat No. 340855) at 0.1% (final concentration). Samples were frozen in liquid nitrogen (at least 60 min) and then stored at -80 ˚C until analysis (within a few days).

Each sample was analyzed twice. First, 600 µl of the sample water was analyzed at a high flow rate (100 µL min -1 ) for the determination of ultra-phytoplankton with a dual threshold (trigger) on the red fluorescence channels of the violet and blue lasers. A second run was used to analyze cells with no autofluorescence, i.e., non-photosynthetic microbes. To visualize these cells, a 300 µL aliquot of the sampled water was incubated with the nucleic acid stain SYBR Green I (20-120 min dark incubation at room temperature, 1:10 4 of the SYBR Green commercial stock). For this run, we used a low flow rate of 25 µL min -1 and the instrument was set to high sensitivity mode. Seventy-five µL of the sample water was analyzed with a dual threshold (trigger) on green fluorescence channels of the violet and blue lasers. The taxonomic identification was based on orange fluorescence (Bl2, 574±13 nm) of phycoerythrin and red fluorescence (Bl3, 690±20 nm and VL3, 685±20 nm) of chlorophylls; side-scatter (SSC), provided a proxy of cell surface complexity and cell volume [START_REF] Marie | Enumeration of Marine Viruses in Culture and Natural Samples by Flow Cytometry Enumeration of Marine Viruses in Culture and Natural Samples by Flow Cytometry[END_REF], and forward-scatter (FSC) was a proxy of cell size [START_REF] Cunningham | Narrow-angle forward light scattering from individual algal cells: Implications for size and shape discrimination in flow cytometry[END_REF][START_REF] Simon | Characterization of oceanic photosynthetic picoeukaryotes by flow cytometery[END_REF].

Where possible, the non-photosynthetic bacteria were further divided based on their green fluorescence (proxy for nucleic acid content) and forward scatter (proxy for size) into three groups: LNA, low nucleic acid non-photosynthetic bacteria; HNA-Ls, high nucleic acid low-scatter non-photosynthetic bacteria; HNA-Hs, high nucleic acid high-scatter nonphotosynthetic bacteria [START_REF] Zubkov | Coexistence of dominant groups in marine bacterioplankton community -A combination of experimental and modelling approaches[END_REF]). Similarly, the eukaryotic algae were separated to pico-and nano-phytoplankton [START_REF] Simon | Characterization of oceanic photosynthetic picoeukaryotes by flow cytometery[END_REF]. The size of Synechococcus is still somewhat controversial, indicating a range of 0.3 to 1.2 µm (e.g., [START_REF] Uysal | Chroococcoid cyanobacteria Synechococcus spp. in the Black Sea: pigments, size, distribution, growth and diurnal variability[END_REF][START_REF] Garcia | Interactions between growth-dependent changes in cell size, nutrient supply and cellular elemental stoichiometry of marine Synechococcus[END_REF]. For pico-eukaryotic algae, we followed [START_REF] Worden | Ecology and diversity of picoeukaryotes[END_REF], who suggested a size range of up to 3.0 µm. Larger cells were termed nano-eukaryotic algae (2.0-20 µm). As a rough proxy of cell size, we used the ratio of the median forward scatter of each cell population to that of the median forward scatter of reference beads (Polysciences™, cat# 23517, Flow Check High-Intensity Green Alignment 1.0 µm) that were used as an internal standard in each sample. See [START_REF] Dadon-Pilosof | Prey taxonomy rather than size determines salp diets[END_REF] for a further discussion of the accuracy of size estimates.

Chlorophyll measurements

Water samples (~300 mL) were collected directly from the Niskin bottles at each station into dark volumetric BOD glass bottles (Wheaton 227667) and maintained on ice in dark cool box. In the lab, samples were prefiltered through a 100 μm mesh (to remove large zooplankton and/or aggregates and suspended pieces of benthic algae) and filtered using low vacuum onto a 25 mm Whatman GF/F filter. Filters were kept frozen at -20°C in 20 mL scintillation vials until further processing. To ensure complete chlorophyll a extraction of coastal phytoplankton, we used a hot dimethyl sulfoxide (DMSO) extraction method [START_REF] Burnison | Modified dimethyl sulfoxide (DMSO) extraction for chlorophyll analysis of phytoplankton[END_REF]. Briefly, 2 mL of DMSO were added into each vial containing the frozen filter. Vials were then incubated for 20 min at 65⁰C, then cooled in a dark box to room temperature (approximately 1 hr). Four mL of buffered Acetone (90% Acetone, 10% saturated MgCO3) were added to the vial and thoroughly mixed. Vials were then left to settle for few minutes and 3 mL sample was drown form the vial to a fluorometer cuvette.

Fluorescence was measured with a calibrated Trilogy fluorometer (Turner Designs) using the non-acidification method [START_REF] Welschmeyer | Improved chlorophyll a analysis: single fluorometric measurement with no acidification[END_REF].

DNA extraction

The relative abundance of prominent microbial taxa (phylotypes) in the seawater was estimated using next-generation sequencing (NGS) of SSU rRNA genes to evaluate any differential removal by suspension feeders benthic community. Ten mL of seawater collected from each station and filtered on a 25 mm, 0.2 μm polycarbonate membrane (GE Healthcare Biosciences, cat. No. 110606) under low vacuum and frozen in 1.5 mL micro-tubes at -20˚C until analysis. DNA from each filter was extracted using the DNeasy 'blood & tissue kit' (QIAGEN,Cat. No. 69504) with the following modifications to the manufacturer's protocol: ATL buffer (180 μL) and 20 μL of proteinase K were added and samples were incubated at 56°C for 1 hr. Then 200 μL of AL Buffer and 200 μL of 95-100% ethanol was added to the sample and the mixture was pipetted into spin columns and placed in a 2 mL collection tube.

Tubes were centrifuged at 6000 RCF for 1 min. The flow-through was discarded and 500 μL of AW1 buffer was added to the column, centrifuged at 6000 RCF for 1 min, and the flowthrough again discarded. This step was repeated for the third time, with 500 μL Buffer AW2 and a spin of 18,000 RCF for 1 min to dry the membrane before elution. For the elution step, the spin column was placed on a new collection tube. Two hundred µL of buffer AE preheated to 56°C was pipetted at three steps (50 µL, 50 µL, and 100 µL) into the column and each step was followed by 6000 RCF centrifugation for 1 min. The sample was then incubated at room temperature for at least a minute and stored at -20°C.

Next-generation sequencing Samples were sequenced by Research and Testing Laboratories (Lubbock TX). The SSU rRNA genes were amplified for sequencing using a forward and reverse fusion primers (515F-Y -926R [START_REF] Parada | Every base matters: assessing small subunit rRNA primers for marine microbiomes with mock communities, time series and global field samples[END_REF]). The forward primer was constructed with (5'-3') the Illumina i5 adapter (AATGATACGGCGACCACCGAGATCTACAC), an 8-10bp barcode, a primer pad, and the 5'-GAGTTTGATCNTGGCTCAG -3' primer. The reverse fusion primer was constructed with (5'-3') the Illumina i7 adapter (CAAGCAGAAGACGGCATACGAGAT), an 8-10bp barcode, a primer pad, and the 5'-GTNTTACNGCGGCKGCTG -3' primer. Primer pads were designed to ensure the primer pad/primer combination had a melting temperature of 63˚C-66˚C, according to methods developed by Patrick Schloss' laboratory (http://www.mothur.org/w/images/0/0c/Wet-lab_MiSeq_SOP.pdf). Amplifications were performed in 25 μL reactions with Qiagen HotStar Taq master mix (Qiagen Inc, Valencia, California), 1μL of each 5 μM primers, and 1 μL of the template. Reactions were performed on ABI Veriti thermocyclers (Applied Biosystems, Carlsbad, California) under the following cycle conditions: 95˚C for 5 min, then 35 cycles of 94˚C for 30 sec, 54˚C for 40 sec, 72˚C for 1 min, followed by one cycle of 72˚C

for 10 min and a final 4˚C hold.

Amplification products were visualized with eGels (Life Technologies, Grand Island, New York). Products were then pooled equimolarly and each pool was size-selected in two rounds using Agencourt AMPure XP (BeckmanCoulter, Indianapolis, Indiana) in a 0.7 ratio for both rounds. Size-selected pools were then quantified using a Qubit 2.0 fluorometer (Life Technologies) and loaded on an Illumina MiSeq (Illumina, Inc. San Diego, California) 2x300 flow cell at 10 pM.

Sequence data analysis SSU rRNA sequences were treated using a qiime2 (v. 2018-8) and biom-format (v.

2.1.6) deployed through a bash scripts (Suppl. File). Briefly demultiplexed forward and reverse reads were imported into qiime2 artifacts and ASV tables were generated using qiime dada2 using options "--p-trim-left-f 19 --p-trim-left-r 20 --p-trunc-len-f 300 --p-trunc-len-r 250" The resulting ASVs were identified using qiime feature-classifier using an in house version of the Silva132 (arb-silva.de) including only the region flanked by the primers, and a taxonomy files extracted from this database (see data availability below for details).

Since the primers we used amplify both 16S rRNA from prokaryotes and chloroplasts as well as 18S rRNA from eukaryotes, the ASV table and sequence files were filtered using qiime taxa to generate three sets of table/sequences, one with 16S rRNA, one with chloroplasts and one with eukaryotic sequences. ASV tables were modified to include taxonomy using sed and gawk and biom-format and exported in ".tsv" format for further analyses. Here we describe only results concerning 2388 prokaryotic ASVs.

Data analysis

Due to temporal changes in microbial communities, and assuming the entire transect represented a single water mass the concentrations along each transect were normalized to the seaward-most station and presented as "% of open sea". The significance level of cross-shore trends was tested using the "Page test" for ordered alternatives [START_REF] Page | Ordered Hypotheses for Multiple Treatments : A Significance Test for Linear Ranks[END_REF]. This nonparametric test is a modified version of the Kruskal-Wallis one-way ANOVA for ranked data.

Nearshore depletion of microbial taxa was tested (each taxon per season) with H1 as an ordered decrease in concentrations from the "open water" toward the shore. Due to missing sampling points, only 3 complete transects toward the hard-bottom were used for the test in each season.

Data collected with the CTD was converted, aligned, and binned (at 0.1 m) with the SBE DataProcessing software (Version 7.2). Since salinity differences along the transects were negligible (<0.01 PSU), density differences were driven solely by temperature. The vertical and horizontal change in temperature and in-vivo chlorophyll fluorescence along the cross-shore transects (hereafter 'anomaly') was calculated within each transect as anomalies (the difference of each data point from the average temperature or fluorescence along the transect). All seasonal anomalies were plotted together using Ocean Data View (Version 5).

Interpolation was made with the Weighted-average gridding function of Ocean Data View using a seeking distance of 0.25 m along the vertical axis and 100 m along the horizontal axis. Due to the low (N=4), data are presented as a seasonal average ± standard error (SE) unless otherwise indicated.

Removal of specific ASVs toward the shore was calculated by multiplying the relative abundance of each ASV and the total bacterial cell counts obtained by flow cytometry and then calculation as a percentage of the open sea station. Normalized removal was calculated by normalizing measured removal to the ASV with the highest removal within the same transect. Implementation of this approach provides a powerful tool to indirectly evaluate the effect of benthic removal of microbial prey at the ASV level. Hereafter, the use of the terms "selectivity" and "preference" are limited to their technical definition [START_REF] Chesson | Measuring preference in selective predation[END_REF][START_REF] Chesson | The estimation and analysis of preference and its Relatioship to foraging models[END_REF], i.e., the removal of a prey type in higher proportion than its proportional presence in the environment, relative to other food types present.

Data availability

Raw sequencing reads are available through the NCBI SRA under accession number PRJNA912166. The analysis pipeline and associated files including scripts, mapping files, taxonomic identification databases and ASV tables are available through a github repository (github.com/suzumar/transect_ms).

Results

During winter, a temperature gradient was found along the transects with colder water at the shallow stations above the hard-bottom (stations 1-5, 19.4±0.03°C, respectively,during winter and, PSU, respectively, during spring).

In the cross-shore transects, the concentration of all microbial populations decreased toward shore in a highly significant gradient (Page test, p<0.01, Fig. 3a-d). In most cases, and for most of the microbial populations, cell concentrations were similar along the ~700 m sandy section of the transects, between the open sea and the outer boundary of the rocky area (Stations 6 to 8). Sometimes even an increase was observed along that section (Table S1,

Discussion

This study was designed to evaluate whether the effect of differential removal of microorganisms by benthic suspension feeders can be detected at the community level. Most of the microbial populations were depleted above the hard-bottom area compared to the open sea. Differential removal was observed at low extent (between and within species). A similar trend was also observed for chlorophyll a concentration, suggesting the formation of a depleted boundary layer over the hard-bottom. The hard-bottom community at the study site included a diverse assemblage of suspension feeders, including sponges, bivalves, ascidians, polychaetes, hydrozoans, and bryozoans [START_REF] Rilov | Alien turf: Overfishing, overgrazing and invader domination in south-eastern Levant reef ecosystems[END_REF]. Suspensions feeders on the hardbottom occur in different densities, and hence the competition for available prey is between species, within species and even between different taxa. Niche speciation is expected in such diverse and dense community where different taxa utilize different filtration mechanisms and presumably, different organic carbon sources in their diet. If the depletion of microbial cells is the outcome of grazing by benthic suspension feeders, near-bottom depletion should generate a shore-wise depletion above the hard-bottom section of the transects compared to the sandy section of the transect (e.g., [START_REF] Genin | Intense benthic grazing of phytoplankton in a coral reef[END_REF][START_REF] Jones | Hydrodynamic control of phytoplankton loss to the benthos in an estuarine environment[END_REF]. In case other processes determine the cross-shore trend (e.g., runoff, eutrophication, and coastal pollution), the depleted zone is expected to extend throughout the water column and no difference is expected along the transects. Differential grazing of phytoplankton by dense populations of benthic suspension feeders was also reported in San-Francisco bay where it was attributed to different sinking rates of the microalgae [START_REF] Lucas | Bivalve Grazing Can Shape Phytoplankton Communities[END_REF]. Such a phenomenon is very unlikely in the east Mediterranean where the planktonic community is dominated by very small cells (<10 μm) with negligible sinking rates [START_REF] Siokou-Frangou | Plankton in the open Mediterranean Sea: a review[END_REF]. This body of prior work supports the assumption that the depletion reported here is due to benthic suspension grazing although it was not measured directly.

Physical and biological processes are the major factors controlling changes in particle concentrations throughout the ocean, and advective processes could be responsible for the pico and nanoplanktom depletion we observed. [START_REF] Monismith | Thermally Driven Exchanges between a Coral Reef and the Adjoining Ocean[END_REF] showed that shallow regions nearshore experience larger temperature changes than deeper regions offshore. When the water warms during the daytime (e.g., in the spring), the shallow near-shore water body tends to warm faster than the nearby open sea. The warmer water expands and flows offshore, causing deeper and cooler water to flow onshore at depth to replace it. This 'thermal flow' (Fig 7) leads to an upwelling of deeper water and material to the nearshore region (Fig 7). The opposite process occurs during winter or cold nights, when a faster cooling of the shallow, near-shore waterbody initiates near-shore downwelling of cold surface water that reverses the direction of the "thermal flow" cycle and induces onshore flow of surface water (Fig 7). In the Eastern Mediterranean Sea, the water column is usually stably stratified, and the numbers and biomass of surface water plankton is lower than subsurface layers [START_REF] Suari | A long term physical and biogeochemical database of a hyper-eutrophicated Mediterranean micro-estuary[END_REF]. Onshore transport of surface water from the offshore during the cold phase of the thermal flow cycle and the nearshore downwelling of "plankton poor" seawater is expected to yield a benthic zone of depleted of plankton close to shore [START_REF] Labiosa | The interplay between upwelling and deep convective mixing in determining the seasonal phytoplankton dynamics in the Gulf of Aqaba : Evidence from SeaWiFS and[END_REF]. However, this physical mechanism would not be expected to change the relative abundance of different microbial groups along the transects, nor should it differentially affect rocky and sandy sections of the transects. Moreover, during the warm phase of the thermal flow cycle (in spring for example), local upwelling is expected to bring plankton-rich water onshore, countering removal, and the benthic zone should be as rich if not richer nearshore. Our results showed that nearshore depletion occurred during both the cold and hot 'phase' of the thermal flow, suggesting that a biological process such as filtration is probably the mechanism responsible for the depletion above the hard-bottom section in the transects rather than advection of planktonic poor water. A similar conclusion was reached by [START_REF] Patten | Uptake of picophytoplankton, bacterioplankton and virioplankton by a fringing coral reef community (Ningaloo Reef, Australia)[END_REF], who showed depleted levels (~40% on average) of microbial cells over a reef with negligible removal over a sandy bottom.

Our previous work showed partial grazing resistance of members SAR11 clade to grazing by pelagic and benthic tunicates [START_REF] Dadon-Pilosof | Surface properties of SAR11 bacteria facilitate grazing avoidance[END_REF], and in general, lower retention efficiencies on LNA bacteria than other prey types. Selective grazing and specifically low retention on LNA compare to both HNA bacteria and Synechococcus was also observed in sponges [START_REF] Hanson | Estuarine , Coastal and Shelf Science Selective uptake of prokaryotic picoplankton by a marine sponge ( Callyspongia sp .) within an oligotrophic coastal system[END_REF][START_REF] Jiménez | Nutrient fluxes in marine sponges: methodology, geographical variability and the role of associated microorganisms[END_REF]) and preferential retention of Synechococcus and eukaryotic algae over other prey types was observed in bivalves (Yahel et al. 2009). The results (although using indirect measurements) of this study suggested inversely to what we expected that members of SAR11 were grazed at higher efficiencies than other available prey (Fig 5,6) somewhat reflecting their overall high abundance. This work strengthens the niche speciation assumption, hinting that some taxa within the hardbottom community specialize on grazing SAR11. Besides the obvious members of the suspension feeding community, we must consider the heterotrophic nanoflagellates attached to the rocks [START_REF] Yahel | Size independent selective filtration of ultraplankton by hexactinellid glass sponges[END_REF]). This cryptic community has a large grazing effect that may be part of the explanation for prey depletion above the hard-bottom. It is known that heterotrophic nanoflagellates are important bacterivores in pelagic waters [START_REF] Tophøj | Feeding and growth of the marine heterotrophic nanoflagellates, Procryptobia sorokini and Paraphysomonas imperforata on a bacterium, Pseudoalteromonas sp. with an inducible defence against grazing[END_REF].

The apparent lack of grazing evasion by members of SAR11 could be possibly explained by a different mode of interaction to other filter feeders compared to tunicates. Suspension feeders have different filtration organs, using cilia, mucus, or both to capture and process suspended particles. While some are active suspension feeders and specialize in filtration of small particles, others are passive feeders and specialize in filtration of large particles [START_REF] Gili | Benthic suspension feeders: their paramount role in littoral marine food webs[END_REF]. Different grazing strategies of benthic suspension feeders would potentially increase the opportunities for exploitation of available prey by communities of suspension feeders [START_REF] Gili | Benthic suspension feeders: their paramount role in littoral marine food webs[END_REF]. Specialization of different suspension feeders on different prey would explain the homogeneous decrease of different available prey. While removal by benthic filter feeders is a likely biological explanation for the depletion of pico-and nanoplankton we observed, other processes such adsorption to mucus might also have contributed [START_REF] Decho | Microbial exopolymer secretions in ocean environments: their role (s) in food webs and marine processes[END_REF] Measurements of removal rates of microbial plankton by benthic organisms at the level of the whole community remain challenging despite of decades of studies of this theme [START_REF] Sargent | Organic productivity of an Atoll[END_REF][START_REF] Odum | Trophic Structure and Productivity of a Windward Coral Reef Community on Eniwetok Atoll[END_REF][START_REF] Johannes | The Metabolism of Some Coral Reef Communities: A Team Study of Nutrient and Energy Flux at Eniwetok[END_REF]. More recent studies, assessed spatial gradients of DOC, bacterioplankton and virioplankton concentrations in reef ecosystem resulting in depleted microbial community over the reef compare to negligible removal over sandy bottom nearby [START_REF] Patten | Uptake of picophytoplankton, bacterioplankton and virioplankton by a fringing coral reef community (Ningaloo Reef, Australia)[END_REF][START_REF] Nelson | Depleted dissolved organic carbon and distinct bacterial communities in the water column of a rapid-flushing coral reef ecosystem[END_REF]). An alternative approach utilizes a control-volume approach either by physically enclosing the community in a bell jar for in situ measurements (e.g., [START_REF] Hopkinson | Community metabolism and nutrient cycling at ,Gray's Reef a hard bottom habitat in the Georgia Bight[END_REF] or by enclosing and imaginary "box" over the bottom using vertical arrays of samplers and ADCPs to quantify the fluxes of plankton into and out of the control volume [START_REF] Genin | Intense benthic grazing on phytoplankton in coral reefs revealed using the control volume approach[END_REF][START_REF] Genin | Intense benthic grazing of phytoplankton in a coral reef[END_REF]. A third approach integrates individual rate measurements of dominant benthic suspension feeders with their abundance and size distribution to assess the community flux and its effect on the planktonic community in the overlying water (e.g., [START_REF] Genin | Intense benthic grazing of phytoplankton in a coral reef[END_REF]Lucas et al. 2016 and references therein).

Figure 1 :

 1 Figure 1: The location of the transect line in the Eastern Mediterranean (a) transects location along the Israeli shore (b) transect showing the bottom (yellow), sampling stations (white numbered lines), dotted white line (1.5 m depth) representing sampling depth.

  Fig 2a) in comparison to the "open sea" (stations 6-8, 20.1±0.03°C, Fig 2a). During spring, no such temperature gradient was observed along the transects but the beginning of stratification was noticeable, with a half-degree Celsius warmer surface layer (0-3 m, 20.1°C±0.07, Fig 2b) than the deeper water (3-12m, 19.6°C±0.02, Fig 2b). The warmer, nearshore surface layer (10-25 m depth, Fig S1) showed typical coastal enrichment, with higher chlorophyll concentrations (Fig S1) and cell counts (Fig S2) than the open sea. Salinity was similar along the transect (1.5m depth) and with depth (0-12m) during winter and spring

  S2). Most of the depletion occurred above the rocky habitat (Stations 1 to 5). The depletion of the microbial populations over the rocky section range between 25-50% of their percentage of the open sea (Fig 3a-d). In vivo chlorophyll fluorescence (Fig S3) and the concentration of extracted chlorophyll a also showed a strong depletion above the hard-bottom area compared to the open sea (Fig 3e).

Figure 2 :

 2 Figure 2: Temperature anomaly along the cross-shore transects. (a) Winters of 2015-2016, n=4. (b) Springs of 2016-2017, n=4. Sampling stations are shown as vertical black lines. Grey shading approximates the bottom depths based on the bottom depth of each cast. In most cases, the CTD was lower all the way to the bottom (~0.2 mab).

Figure 3 :

 3 Figure 3: Cross-shore transects showing the percent change in the concentrations of several microbial cell populations (counted by flow cytometry) from their "open sea" concentration ~900 m offshore. (a) Bact, non-photosynthetic bacteria, (b) Euk, eukaryotic algae, (c) Syn, Synechococcus sp. (d) PLP, Prochlorococcuslike particles, (e) Extracted chlorophyll a concentration along 3 transects. Blue, n=4, transects performed during winter. Orange, n=4, transects performed during spring. The significance of the Page test for a gradient of decreasing concentrations toward the shore is denoted as **<0.01, ***<0.001

Figure 6 :

 6 Figure 6: Normalized removal of the 20 most abundant ASVs from the open sea toward the shore. (a-c) transects above rocky hard bottom (January 2016, March 2016 and, April 2016). Pink indicates members of SAR11 clade, green indicates autotrophs, and blue indicates other non-photosynthetic bacteria. Dashed vertical line represents the expected retention assuming equal retention probability for all cells. Size of circles represents relative abundance in the open water (Station 8) during sampling., with the two white circles plotted for scaling.

  although members of SAR11 clade are known as free-living and not particle-associated or biofilm forming bacteria (Giovannoni 2017, Haro-Moreno et al. 2020) Evaluation of evasiveness using "in versus out" experiments such those in Dadon-Pilosof et al. (2017) with other taxa, or the measurement of adsorption of different bacteria including cultured Candidatus Pelagibacter ubique to transparent exopolymer particles (Long and Azam 1996) with water from different points in the transect remain as possible future experiments to further clarify our results.

Figure 7 :

 7 Figure 7: Schematic illustration of a near-shore thermal flow cycle. (a) during winter, when cooling of nearshore water drives offshore flow at depth and onshore flow at the surface. (b) during spring, when the shallow water near shore warms and water flow offshore at the surface and onshore below.

Further

  investigation is required to develop and estimate, based on indirect evaluation, cells removal above hard-bottom suspension feeders. Methodology fine tuning is required for estimating the cells removal above the hard-bottom compared to open sea and to discriminate grazing, other biological processes and physical mechanisms. Previous studies showed that SAR11 evade predation of benthic and pelagic ascidians mucus, but this study showed (albeit indirectly) that SAR11 was removed efficiently as other prey cells by hard bottom grazers.Potential grazers might be sponges, bivalves, nanoflagellates attached to the surface of the rocks which filter their prey by using several different mechanisms. Understanding the mechanisms and the variance between the grazers' preferences and/or the microbial cells abilities to evade predation has ramifications on processes affecting the marine food web such as top-down control and, benthic pelagic coupling. Grazing resistance mechanisms are still understudied and should be investigated further to gain knowledge on its effects in the 482 marine ecosystem. 483

Acknowledgements

Research funding was provided by Ruppin Academic center, support provided to A.D.P by the Mediterranean Sea Research Center of Israel. I thank my supervisors Prof. Amatzia Genin and Prof. Gitai Yahel for their advice along the way. Special gratitude to R.

Rosenblatt for availability and technical assistance throughout the field experiments.

Author Contributions:

A.D.P. designed the study and participated in field experiments, data analysis, and manuscript preparation. K.R.C, participated in data analysis, and contributed substantially to drafting the manuscript. M. T.S designed and performed bioinformatics analyses and contributed substantially to drafting the manuscript. All authors discussed the results and commented on the manuscript during its preparation and approved the submitted version of the manuscript.


Mathieu Bajodek

Alexandre Seuret

Frédéric Gouaisbaut

Order of Legendre-LMI conditions to assess stability of time-delay systems

Keywords: Time-delay systems, Stability conditions, LMI, Converse Lyapunov theorem, Polynomial approximation

This paper investigates the stability analysis of time-delay systems through Lyapunov arguments. Using the existence of a complete Lyapunov-Krasovskii functional and relying on the polynomial approximation theory, our main goal is to approximate the complete Lyapunov functional and to take profit of the supergeometric convergence rate of the truncated error part. Necessary and sufficient conditions in the linear matrix inequality (LMI) framework for sufficiently large approximated orders are consequently proposed. Moreover, an estimation of the necessary order is provided analytically with respect to system parameters.

INTRODUCTION

An extension of the Lyapunov necessary and sufficient theorem has been introduced in [START_REF] Datko | Extending a theorem of A. M. Liapunov to Hilbert space[END_REF] to analyze the stability of linear infinite-dimensional systems. This contribution ensures the existence of a complete quadratic Lyapunov functional without giving more details on its design. Focusing now on time-delay systems, such a complete Lyapunov functional has been constructed in [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF]. Nevertheless, the Lyapunov inequalities conditions stay theoretical and cannot be implemented. In parallel, infinite-dimensional systems are often modeled by approximated methods to be analyzed numerically. It is the case of Padé approximations in the frequency domain (see [START_REF] Golub | Matrix Computations[END_REF]) or pseudo-spectral methods in the time domain (see [START_REF] Gottlieb | Numerical analysis of spectral methods: theory and applications[END_REF]). Within this approximation context, most of the results release in the convergence of the solution towards the expected one. However, few results are able to provide a sufficiently high approximation index enough to establish the stability properties using the convergence of the error part.

In the wide variety of pseudo-spectral methods, two categories stand out. From one side, collocation methods amount to interpolate the state. The idea is to create a discretization map such that the error is exactly zero at the interpolated points. In between these points, the approximation functions can be selected to be constant (see [START_REF] Gu | Stability of Time-Delay Systems[END_REF]), linear (see [START_REF] Gu | Complete quadratic Lyapunov-Krasovskii functional: limitations, computational efficiency, and convergence[END_REF]; [START_REF] Medvedeva | Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems[END_REF]), polynomial (see spline methods) or exponential (see [START_REF] Egorov | Necessary stability conditions for linear delay systems[END_REF]). Applying an interpolation leads to necessary conditions of stability as in [START_REF] Egorov | Necessary stability conditions for linear delay systems[END_REF]; [START_REF] Medvedeva | Stability of neutral type delay systems: A joint Lyapunov-Krasovskii and Razumikhin approach[END_REF] or to sufficient conditions of stability based on discretized Lyapunov functional as in [START_REF] Gu | Stability of Time-Delay Systems[END_REF]. Recently, these results have been proven to converge asymptotically using some convergence properties and deliver inner and outer estimations of the stability regions (see respectively [START_REF] Gu | Complete quadratic Lyapunov-Krasovskii functional: limitations, computational efficiency, and convergence[END_REF] and [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF]). From the other side, tau methods amount to approximate the state. The idea is to take support of a complete Hilbert basis in order to proceed an approximation through Galerkin-like methods using, for instance, Fourier trigonometric basis or Legendre polynomials. Simple Lyapunov functionals based on the first coefficient (i.e. the mean value of the state) of such basis have led to linear matrix inequality (LMI) conditions exposed in [START_REF] Fridman | Tutorial on Lyapunov-based methods for time-delay systems[END_REF], which are, however, quite conservative. To reduce this conservatism, hierarchic Lyapunov functionals taking support on the n first Legendre coefficients have been proposed in Seuret andGouaisbaut (2013, 2015). It is only recently that the gap between these approximated functionals and the original complete functional has been closed. Indeed, the authors of [START_REF] Bajodek | On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation[END_REF] haven proven that sufficient Legendre-LMI conditions converge and become necessary for sufficiently large orders. This paper explains the methodology, which allows to obtain the necessary and sufficient stability conditions developed in [START_REF] Bajodek | On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation[END_REF], and improves the estimation of the necessary order n * . After recalling the complete Lyapunov functional of time-delay systems, an intermediate result quantifies the quality of the approximation of the complete Lyapunov functional so that the Lyapunov converse theorem holds. Based on the selection of a Legendre approximation, which ensures exponentially fast uniform convergence, the necessary and sufficient condition of stability for time-delay systems resumes to an LMI of size n * expressed with respect to system parameters. Finally, the LMI test as well as the order n * are computed on an academic example.

Notations: Throughout the paper, N, R m×p , S m and S m + denote the sets of natural numbers, real matrices of size m × p, symmetric matrices of size m and positive symmet-ric matrices of size m, respectively. For any square real matrix M , M ⊤ denotes the transpose of M , H(M ) stands for M + M ⊤ and tril(M ) stands for the triangular lower part of M . Furthermore, for any M ∈ S m , its minimal and maximal eigenvalues are denoted σ(M ) and σ(M ). The 2-norm of matrix M in R m×p is |M | = σ(M ⊤ M ). The vector u = vec(M ) in R mp×1 collocates the columns of matrix M and 'vec -1 ' denotes the inverse operator. Notation ⊗ refers to the matrix Kronecker product. The set of square-integrable functions from (-h, 0) to R m×p is noted L 2 (-h, 0; R m×p ). Let also H 1 (-h, 0; R m×p ) be the set of functions F , such that F and F ′ are in L 2 (-h, 0; R m×p ). For any φ in R nx × L 2 (-h, 0; R nz ), the norm is given by

∥φ∥ = |φ(0)| 2 + 0 -h |φ(θ)| 2 dθ. Lastly, denote x t such that x t (θ) = x(t + θ), for any θ in [-h, 0].

CONVERSE LYAPUNOV THEOREM FOR TIME-DELAY SYSTEMS

Presentation of the system

Consider a linear time-delay system given by ẋ(t) = Ax(t)

+ A d x(t -h), ∀t ≥ 0, x(θ) = φ(θ), ∀θ ∈ [-h, 0], (1)
where the delay h > 0 and matrices A, A d in R nx×nx are constant and known. The initial condition φ belongs to

D =        φ(θ) = φ 0 if θ = 0, φ 1 (θ) if θ ∈ [-h, 0], such that φ 0 φ 1 ∈ R nx × H 1 (-h, 0; R nx ), φ 1 (0) = φ 0       
.

Definition 1. (GES). The trivial solution to (1) is said globally exponentially stable (GES), if there exist κ ≥ 1 and µ > 0 such that x t , solution to (1) generated by any initial condition φ in D, verifies ∥x t ∥ ≤ κ e -µt ∥φ∥ , ∀t ≥ 0.

(2) Remark 1. It is worth noticing that the regularity properties of the solutions to time-delay systems ensure that such a definition in the sense of ∥•∥-norm also implies that there exist κ ≥ 1 and μ > 0 such that |x(t)| ≤ κ e -μt sup [-h,0] |φ(θ)|, ∀t ≥ 0.

(3)

Existence of a complete Lyapunov functional

Since [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems[END_REF]; [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF], it is well known that a necessary and sufficient stability condition of system (1) is the existence of a complete Lyapunov functional, which is recalled below.

For any W 1 , W 2 and W 3 in S nx + and for any x t in D, define the following Lyapunov functional

V(x t ) = x ⊤ t (0)U (0)x t (0)+2x ⊤ t (0) 0 -h U (θ+h)A d x t (θ)dθ + 0 -h 0 -h x ⊤ t (θ 1 )A ⊤ d U (θ 2 -θ 1 )A d x t (θ 2 )dθ 1 dθ 2 + 0 -h x ⊤ t (θ) θ + h h W 2 + W 3 x t (θ)dθ, ( 4
) where U is a matrix function from [-h, h] to R nx×nx , called the Lyapunov matrix. Interestingly, referring to [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF], Section 2.10), a method is provided to build analytically this matrix. Its expression is given by

U (θ) = vec -1 [ I n 2 x 0 ] e θM N -1 -vec(W ) 0 if θ ≤ 0, U ⊤ (-θ) if θ > 0,
(5) where matrices

W = W 1 + W 2 + W 3 , M = -A ⊤ ⊗In x -A ⊤ d ⊗In x In x ⊗A ⊤ d In x ⊗A ⊤ , N = A ⊤ ⊗In x +In x ⊗A ⊤ A ⊤ d ⊗In x I n 2 x 0 + In x ⊗A ⊤ d 0 0 -I n 2 x e -hM . (6)
Such a complete Lyapunov functional V has been built so that its time-derivative along the trajectories of the system yields

V(x t ) = - 1 h 0 -h xt(0) hxt(θ) xt(-h) ⊤ W1 0 0 * 1 h W2 0 * * W3 xt(0) hxt(θ) xt(-h)
dθ. ( 7)

A necessary and sufficient condition of stability for system (1) based on this functional is formulated below.

Necessary and sufficient condition of stability

The following theorem is obtained by application of the Lyapunov theorem. The proposed formulation is issued from (Gu et al., 2003, Theorem 5.19). Theorem 1. System (1) is GES if and only if there exist positive scalars α 1 , α 2 , α 3 such that V defined by ( 4) satisfies, for any x t in D, inequalities

α 1 ∥x t ∥ 2 + 0 -h x ⊤ t (θ) θ + h h W 2 x t (θ)dθ ≤ V(x t ), V(x t ) ≤ α 2 ∥x t ∥ 2 , V(x t ) ≤ -α 3 ∥x t ∥ 2 , ( 8a
) (8b) (8c)
where V denotes the time-derivative of V along (1). Remark 2. It is worth noticing that in the sense of the stability definition given by (3), the Lyapunov-Krasovskii version of this theorem also holds replacing (8) by

α 1 |x t (0)| 2 ≤ V(x t ) ≤ α 2 sup [-h,0] |x t (θ)| 2 , V(x t ) ≤ -α 3 |x t (0)| 2 . (9a) (9b)
Refer to [START_REF] Chaillet | A relaxed lyapunov-krasovskii condition for global exponential stability of lipschitz time-delay systems[END_REF] for further details.

Proof : To begin with, the sufficiency refers to the application of the Lyapunov theorem. The proof of necessity starts by noting that (8c) already holds by construction of the Lyapunov matrix (7). Indeed, expression (7) leads to α 3 = max(σ(W 1 ), hσ(W 2 )). Equation ( 8b) is also easy to prove since V is quadratic with respect to x t . Let focus now on (8a) and introduce a functional W as follows

W(x t ) = V(x t ) -α 1 ∥x t ∥ 2 - 0 -h x ⊤ t (θ) θ + h h W 2 x t (θ)dθ.
According to (7), differentiating W along the trajectories of (1) leads to

Ẇ(x t ) = -xt(0) xt(-h) ⊤ W1 0 0 W3 xt(0) xt(-h) -α 1 xt(0) xt(-h) ⊤ H(A)+In x A d A ⊤ d -In x xt(0) xt(-h) .
Then, there exists a sufficiently small α 1 > 0 such that W is negative. Integrating the above inequality from t to ∞, we obtain lim 1) is assumed to be GES, inequality W(x t ) ≥ 0 holds, which concludes the proof. □

T →∞ W(x T ) -W(x t ) ≤ 0, ∀t ≥ 0. As x T -→ T →∞ 0 because system (
A modified version of Theorem 1 will be exposed in the next section. Indeed, in practice, the closed-form of the Lyapunov matrix U cannot be calculated precisely enough and the necessary and sufficient LMI conditions (8) need to be simplified to be tested. Then, it will rely on an approximation of the complete Lyapunov functional.

MODIFIED CONVERSE LYAPUNOV THEOREM

Legendre polynomials approximation

The Lyapunov matrix U is composed of a complex combination of exponential functions. In order to derive tractable numerical tests, the idea is to introduce approximations of U , which satisfy convergence properties to ease this purpose. Define the Legendre polynomials approximations at any order n in N * as

U (θ+h) = U 1,n ℓ n θ + h h U1,n(θ+h) + Ũ1,n (θ), ∀θ ∈ [-h, 0], U (θ) = U 2,n ℓ n θ + h 2h U2,n(θ) + Ũ2,n (θ), ∀θ ∈ [-h, h],
(10) where matrix ℓ n in R nnx×nx collocates the n first Legendre polynomials scaled on [0, 1]

ℓ n (θ) = [l 0 (θ) . . . l n-1 (θ)] ⊤ ⊗ I nx , ∀θ ∈ [0, 1]. ( 11
)
In this decomposition, the constant matrices U 1,n and U 2,n have been selected as the orthogonal projection of U (θ + h) and U (θ) on the interval [-h, 0] and [-h, h] respectively, and their expressions are given by

U 1,n = 0 -h U (θ + h)ℓ ⊤ n θ + h h dθ I n , U 2,n = 1 2 h -h U (θ)ℓ ⊤ n θ + h 2h dθ I n , (12)
where the normalization matrix of Legendre polynomials is equal to

I n = I In x n
where, for all S ∈ S nx , we define

I S n = 1 h 1 0 ℓ n (θ)ℓ ⊤ n (θ)dθ -1 ⊗ S= diag 1 h , . . . , 2n -1 h ⊗ S. (13)
For any W 1 , W 2 and W 3 in S nx + and x t in D, we introduce a new functional

V n (x t ) = x ⊤ t (0)U (0)x t (0)+2x ⊤ t (0) 0 -h U 1,n (θ+h)A d x t (θ)dθ + 0 -h 0 -h x ⊤ t (θ 1 )A ⊤ d U 2,n (θ 2 -θ 1 )A d x t (θ 2 )dθ 1 dθ 2 + 0 -h x ⊤ t (θ) θ + h h W 2 + W 3 x t (θ)dθ. ( 14
)
Such a functional will be considered as an approximation of the original V. Denoting the error Ṽn = V -V n , we obtain

Ṽn (x t ) = 2x ⊤ t (0) 0 -h Ũ1,n (θ)A d x t (θ)dθ + 0 -h 0 -h x ⊤ t (θ 1 )A ⊤ d Ũ2,n (θ 2 -θ 1 )A d x t (θ 2 )dθ 1 dθ 2 , (15)
where Ũ1,n and Ũ2,n refer to the errors expressed in (10).

Nevertheless, the price to pay is that the functional V n generated by this approximation at order n does not necessarily verify the condition (8) of the complete one. In this sequel, we provide a new theorem stating a necessary and sufficient condition of stability even though the complete Lyapunov functional is not built by U but by its approximation U 1,n and U 2,n . This theorem naturally relies on Ũ1,n and Ũ2,n , which are seen as an approximation error.

A new necessary and sufficient condition of stability

The main result of the paper is presented below. Theorem 2. Let W 1 , W 2 and W 3 in S nx + and assume that the following inequality holds

Ψ n (θ) := W1+H( Ũ1,n(0)Ad) Ψ 1 n (θ) -Ũ1,n(-h)Ad * 1 h W2 Ψ 2 n (θ) * * W3 ≻ 0, (16)
for a given n in N * and for all θ in [-h, 0] with 1) is GES if and only if there exist positive scalars α 1 , α 2 , α 3 such that V n defined by ( 14) satisfies, for any x t in D, inequalities

Ψ 1 n (θ) = A ⊤ Ũ1,n (θ)A d -Ũ ′ 1,n (θ)A d + A ⊤ d Ũ2,n (θ)A d , Ψ 2 n (θ) = A ⊤ d Ũ ⊤ 1,n (θ)A d -A ⊤ d Ũ ⊤ 2,n (θ + h)A d . System (
α 1 ∥x t ∥ 2 ≤ V n (x t ) ≤ α 2 ∥x t ∥ 2 , Vn (x t ) ≤ -α 3 ∥x t ∥ 2 . (17a) (17b)
Proof : Trivially, the sufficiency of the statement is verified by application of the Lyapunov theorem. For the necessity of the proposed statement, let us first note that the right-hand part of (17a) is directly obtained because the Lyapunov functional is quadratic with respect to x t . Let now focus on the two remained inequalities. Concerning (17b), the time-derivative of V n = V -Ṽn along (1) given by ( 7) and ( 15) leads to

Vn (x t ) = - 1 h 0 -h xt(0) xt(θ) xt(-h) ⊤ W1 0 0 * 1 h W2 0 * * W3 xt(0) xt(θ) xt(-h) dθ -2(Ax t (0) + A d x t (-h)) ⊤ 0 -h Ũ1,n (θ)A d x t (θ)dθ +2x ⊤ t (0) Ũ1,n (θ)A d x t (θ) 0 -h + 0 -h Ũ ′ 1,n (θ)A d x t (θ)dθ -2 0 -h x ⊤ t (θ 1 )A ⊤ d Ũ2,n (θ 2 -θ 1 )A d x t (θ 2 ) θ1=0 θ1=-h dθ 2 + 0 -h 0 -h x ⊤ t (θ 1 )A ⊤ d (∂ θ1 +∂ θ2 ) Ũ2,n (θ 2 -θ 1 )A d x t (θ 2 )dθ 1 dθ 2 .
The last term vanishes and the previous expression boils down to

Vn (x t ) = - 1 h 0 -h xt(0) hxt(θ) xt(-h) ⊤ Ψ n (θ) xt(0) hxt(θ) xt(-h) dθ, ( 18
)
where Ψ n is given in ( 16). Therefore if matrix Ψ n is positive definite there exists a sufficiently small α 3 > 0 such that Ψ n (θ) ≻ α 3 I 3nz which yields to (17b).

In order to ensure the left-hand part of (17a), let us introduce the following functional

W n (x t ) = V n (x t ) -α 1 ∥x t ∥ 2 . ( 19
)
According to the previous calculations ( 18), differentiating W n along the trajectories of (1) leads to

Ẇn (x t ) = - 1 h 0 -h xt(0) hxt(θ) xt(-h) ⊤ Ψ n (θ) xt(0) hxt(θ) xt(-h) dθ -α 1 xt(0) xt(-h) ⊤ H(A)+In x A d A ⊤ d -In z xt(0) xt(-h) . (20)
Therefore, if Ψ n is positive definite, then there exists a sufficiently small α 1 > 0 such that Ẇn (x t ) is negative definite. Integrating then Ẇn (x t ) from t to infinity yields lim

T →∞ W n (x T ) -W n (x t ) = -W n (x t ) ≤ 0,
where we have used the assumption that system (1) is GES meaning that W n (x T ) -→ T →∞ 0. Hence, (19) ensures that there exists a sufficiently small α 1 > 0 such that the right-hand part of inequality (17a) holds. □

In the following, inequality (8a) in Theorem 1 and inequality (17b) in Theorem 2 are translated into a numerical condition.

Necessary and sufficient LMI condition of stability

For any (P n , R, S) ∈ S (n+1)nx × S nx + × S nx + , introduce the Lyapunov candidate functional considered in Seuret andGouaisbaut (2014, 2015) V

n (x t ) = xt(0) 0 -h ℓn( θ+h h )xt(θ)dθ ⊤ P n xt(0) 0 -h ℓn( θ+h h )xt(θ)dθ + 0 -h x ⊤ t (θ) θ + h h R + S x t (θ)dθ. ( 21
)
Using an integration by parts, the time-derivative of V n along the trajectories of system (1) is given by

Vn (x t ) = 2 xt(0) 0 -h ℓn( θ+h h )xt(θ)dθ ⊤ P n A d -ℓn(0) x t (-h) + 2 xt(0) 0 -h ℓn( θ+h h )xt(θ)dθ ⊤ P n A 0 ℓn(1) Ln xt(0) 0 -h ℓn( θ+h h )xt(θ)dθ + xt(0) xt(-h) ⊤ R+S 0 0 -S xt(0) xt(-h) - 1 h 0 -h x ⊤ t (θ)Rx t (θ)dθ, ( 22
)
where the derivation matrix of Legendre polynomials is

L n = -tril(ℓ n (1)ℓ ⊤ n (1) -ℓ n (0)ℓ ⊤ n (0))I n , (23)
In this context, the following theorem can be established. Theorem 3. For a given n in N * , assume that inequality ( 16) holds (i.e. Ψ n (θ) ≻ 0, for all θ in [-h, 0]). System (1) is GES if and only if there exist matrices (P n , S, R) ∈ S (n+1)nx × S nx + × S nx + the two following linear matrix inequalities hold

Ξ + n = P n + I S n ≻ 0, Ξ - n = H(P n A n ) + C ⊤ n (R + S)C n - 1 h I R n P n B n * -S ≺ 0, (24)
where matrices are defined by

I S n = 0 0 0 I S n , A n = A 0 ℓn(1) Ln , B n = A d -ℓn(0) , C n = In x 0 ⊤ .
Proof : Firstly, the proof of the sufficiency is provided in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF] using for a given order n in N * the Lyapunov candidate functional given by ( 21) and applying the Lyapunov theorem. Secondly, to prove the necessity, we take support on Theorem 2. For all W 1 , W 2 and W 3 in S nx + , let matrices

P n = U (0) Q n * T n , R = W 2 , S = W 3 , (25)
Q n = 0 -h U (θ + h)A d ℓ ⊤ n (θ)I n dθ, T n = 0 -h 0 -h I n ℓ n (θ 1 )A ⊤ d U (θ 2 -θ 1 )A d ℓ ⊤ n (θ 2 )I n dθ 1 dθ 2 .
Note that (P n , R, S) belongs to S (n+1)nx × S nx + × S nx + using the symmetric property of U . Moreover, the Lyapunov candidate functional V n given by ( 21) is the Legendre approximation ( 14) of the Lyapunov converse functional V. Assume that inequality (16) holds at order n and that system (1) is GES. Applying (8a) in Theorem 1 and (17b) in Theorem 2, there exist positive scalars α 1 , α 3 such for the following inequalities hold

α 1 ∥x t ∥ 2 + 0 -h x ⊤ t (θ) θ + h h R x t (θ)dθ ≤ V(x t ), Vn (x t ) ≤ -α 3 ∥x t ∥ 2 . (26a)
(26b) Then, these inequalities are applied for x t in D n ⊂ D given by

D n =            φ(θ) =        φ 0 if θ = 0, ℓ ⊤ n θ + h h ζ n if θ ∈ (-h, 0), φ -h if θ = -h, such that [φ 0 ζ n φ -h ] ⊤ ∈ R (n+2)nx .            .
Thus, inequalities (26a),(26b) lead to Ξ + n ≻ 0 and Ξ - n ≺ 0, respectively, which concludes the proof.

□

Theorem 3 provides a necessary and sufficient LMI condition for the stability of system (1). Nonetheless, it does not guarantee the existence of an order n such that the LMI must be satisfied. The estimation of this order is reserved to the next section.

ESTIMATION OF THE NECESSARY ORDER

Convergence properties

In the light of [START_REF] Dunkl | Orthogonal polynomials of several variables[END_REF], we provide convergence properties satisfied by Legendre polynomial approximation in order to estimate an order from which the necessary and sufficient stability condition holds.

Firstly, the Lyapunov matrix U associated to W defined in ( 5) is continuous on [-h, h] and satisfies sup

[-h,0) U (d) ≤ |M | d ρ |W | , ∀d ∈ N, (27)
where U (d) is the d th derivative of U , the parameter

ρ = √ n x e h|M | N -1 , (28)
and the matrices M , N and W are given by ( 6). Thanks to this smoothness, the uniform convergence of Ũ1,n towards zero is supergeometric as proposed in Property 1 below. Property 1. For all n ≥ 6, the Legendre approximation error Ũ1,n satisfies the inequalities sup

[-h,0] Ũ1,n ≤ µ n-⌈µ⌉ ρ 1 (n -3 2 ) . . . (⌈µ⌉ + 3 2 ) |W | , (29a)
h sup

[-h,0] Ũ ′ 1,n ≤ µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) |W | , (29b)
where

µ = h |M | 2 , ρ 1 = 2π 3 ⌈µ⌉ ρ, (30)
and matrices M , N and W are given by ( 6).

Proof : The proof can be found in [START_REF] Wang | On the convergence rates of Legendre Approximation[END_REF], Theorem 2.5) using Legendre polynomials properties and d = n -1 -⌈µ⌉ successive integrations by parts. □ Property 2. For any n ≥ 4, the Legendre approximation error Ũ2,n satisfied the inequality

sup θ∈[-h,h] Ũ2,n ≤ ρ 2 √ n -3 |W | , (31)
where

ρ 2 = √ 2πh(1 + √ n x πh e h|M | M 2 N -1 ), (32)
and matrices M , N and W are given by (6).

Proof : The proof is provided in [START_REF] Bajodek | Stability analysis of linear ODE-PDE interconnected systems[END_REF]. □

The convergence properties of Legendre polynomial approximation ensures the existence of an order n * from which Theorem 3 holds, as explained below.

Estimation of the orders

The following theorem estimates the orders for which Theorem 3 can be applied. Theorem 4. Theorem 3 holds for orders n ≥ 6 for which the following inequality holds

1 - µ n-⌈µ⌉ ρ ′ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) - ρ ′ 2 √ n -3 ≥ 0. ( 33
)
where scalars ρ

′ 1 , ρ ′ 2 are equal to    ρ ′ 1 = (3 + |A| + 1 h + |A d |)(2 + h) |A d | ρ 1 , ρ ′ 2 = 2(2 + h) |A d | 2 ρ 2 , (34)
Proof : Choosing matrices W 1 = I nx , W 2 = hI nz and W 3 = I nz , it is worth noticing that Ψ 1 n and Ψ 2 n given by ( 16) are upper bounded by

Ψ 1 n ≤ (|A| + 1 h )µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) + |A d | ρ 2 √ n -3 (2 + h) |A d | , Ψ 2 n ≤ µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) + ρ 2 √ n -3 (2 + h) |A d | 2 .
From Schur complement, inequality ( 16) is satisfied under the conditions

                     1 - (3 + |A| + 1 h )µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) + |A d | ρ 2 √ n -3 (2+h)|A d | ≥ 0, 1- (|A|+ 1 h +|A d |)µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) + 2|A d | ρ 2 √ n -3 (2+h)|A d | ≥ 0, 1 - (1 + |A d |)µ n-⌈µ⌉ ρ 1 (n -7 2 ) . . . (⌈µ⌉ + 3 2 ) + |A d | ρ 2 √ n -3 (2+h)|A d | ≥ 0, (35)
Consequently, the estimated order n * satisfy (33), which concludes the proof. □

Thanks to the approximation of U by Legendre polynomials and leaning on the supergeometric convergence rate of error Ũ1,n , we greatly reduce the estimate of the order n * proposed in [START_REF] Bajodek | On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation[END_REF]. These estimated orders are shown in the last example section.

NUMERICAL EXAMPLE

Consider the system (1) with

A = 0 1 0 0 , A d = 0 0 -k 1 -k 2 , h = 1, (36)
for proportional gains k 1 > 0 and k 2 ≥ 0.

Figure 1 reports the necessary orders n * given by ( 33) that corresponds to system (36) for several gains (k 1 , k 2 ) in [0, 1] × [0, 1.5]. These orders n * still very large to test the necessary and sufficient LMI conditions (24).

As a complement, Figure 2 reports the sufficient order n such that our sufficient condition of stability holds. Increasing the order n from 1 to n * , we are looking for the first order ň from which the LMI condition ( 24) is true.

One remarks that there are lower than the necessary order, which leaves a room for forthcoming improvements.

CONCLUSIONS

In this work, three steps leading to necessary and sufficient LMI conditions of stability are highlighted. First, the complete Lyapunov functional is presented. Second, from Theorem 2, we are now able to quantify the degree of conservatism introduced by an approximation process. More generally, an extension to linear ordinary-partial interconnected differential equations would be promising to regard.

Future works could also be dedicated to the reduction of the necessary order n * from which the necessary and sufficient condition holds. The goal would be to get closer to the sufficient order ň. To do so, we could consider other pair of matrices W 1 , W 2 and W 3 and treat directly the necessary condition Ψ n ≻ 0 or investigate other pseudospectral methods.

Fig. 1 .

 1 Fig. 1. Estimation of the necessary order n * for example (36).

Fig. 2 .

 2 Fig. 2. Sufficient order n to satisfy LMI (24) for example (36).

  Finally, the necessary and sufficient Legendre-LMI test provided by Theorem 3 is established. The size of the LMI test can be calculated thanks to Theorem 4. Following the same methodology, this work paves the way to develop new LMI conditions for larger classes of systems. Future directions of research concern time-delay systems with multiple delays (see Alexandrova and Zhabko (2021)), distributed delays (see Egorov et al. (2017)) or non linear delay systems (see Di Ferdinando et al. (2021)).