Ruben Ladeira

Dr Julien Tap
email: julien.tap@inrae.fr

Dr Muriel Derrien
email: muriel.derrien@danone.com

Christian Milani

Ke-Cui Yang Production

Exploring Bifidobacterium species community and functional variations with human gut microbiome structure and health beyond infancy

Keywords: Human gut microbiome, Bifidobacterium longum, Bifidobacterium adolescentis, partitions, function, MAGs, health

Aim:

The human gut Bifidobacterium community has been studied in detail in infants and following dietary interventions in adults. However, the variability of the distribution of Bifidobacterium species and intra-species functions have been little studied, particularly beyond infancy. Here, we explore the ecology of Bifidobacterium communities in a large public dataset of human gut metagenomes, mostly corresponding to adults. Methods: We selected 9.515 unique gut metagenomes from curatedMetagenomicData. Samples were partitioned by applying Dirichlet's multinomial mixture to Bifidobacterium species. A functional analysis was performed on > 2.000 human-associated Bifidobacterium metagenome-assembled genomes (MAGs) paired with participant gut microbiome and health features.

Results:

We identified several Bifidobacterium-based partitions in the human gut microbiome differing in terms of the presence and abundance of Bifidobacterium species. The partitions enriched in both B. longum and B. adolescentis were associated with gut microbiome diversity and a higher abundance of butyrate producers and were more prevalent in healthy individuals. B. bifidum MAGs harboring a set of genes potentially related to phages were more prevalent in partitions associated with a lower gut microbiome diversity and were genetically more closely related.

Conclusion:

This study expands our knowledge of the ecology and variability of the Bifidobacterium community, particularly in adults, and its specific association with the gut microbiota and health. Its findings may guide the rational selection of Bifidobacterium strains for gut microbiome complementation according to the individual's endogenous Bifidobacterium community. Our results also suggest that gut microbiome stratification for particular genera may be relevant for studies of variations of species and associations with the gut microbiome and health.

INTRODUCTION

Bifidobacterium is a gut microbiome component, the abundance of which varies with age and health status. Many studies have shown that various metabolic, immune, and intestinal disease states coincide with Bifidobacterium depletion from the human gut microbiota [START_REF] Pokusaeva | Carbohydrate metabolism in Bifidobacteria[END_REF] . Bifidobacterium species metabolize a wide range of simple and complex glycans, some dietary and others host-derived [START_REF] Rossi | Folate production by probiotic bacteria[END_REF] , and produce various metabolites, such as organic acids, B vitamins [START_REF] Sakurai | Production of indole-3-lactic acid by bifidobacterium strains isolated fromhuman infants[END_REF] , tryptophan-derived metabolites [START_REF] Duranti | Bifidobacterium adolescentis as a key member of the human gut microbiota in the production of GABA[END_REF] , and neurotransmitters, such as GABA [START_REF] Zeng | A compendium of 32,277 metagenome-assembled genomes and over 80 million genes from the early-life human gut microbiome[END_REF] . As such, Bifidobacterium performs a number of roles in interactions between the gut microbiota and the host. The prevalence of Bifidobacterium species varies considerably during the lifespan of an individual human, particularly between infancy and adulthood. The abundance and prevalence of B. bifidum, B. breve, B. longum subsp. Longum, are typically high in the infant gut microbiota across different populations with variation for B. longum subsp. infantis [START_REF] Derrien | The gut microbiota in the first decade of life[END_REF][START_REF] Derrien | Gut microbiome function and composition in infants from rural Kenya and association with human milk oligosaccharides[END_REF][START_REF] Matsuki | Distribution of bifidobacterial species in human intestinal microflora examined with 16S rRNA-gene-targeted species-specific primers[END_REF] . The prevalence of Bifidobacterium generally exceeds 90% in healthy adults, with just a few species present per subject [START_REF] Derrien | Insights into endogenous Bifidobacterium species in the human gut microbiota during adulthood[END_REF] , mostly B. adolescentis and B. longum subsp longum (B. longum), which are able to metabolize complex dietary carbohydrates [START_REF] Bolte | Long-term dietary patterns are associated with pro-inflammatory and anti-inflammatory features of the gut microbiome[END_REF] . The Bifidobacterium content of the human gut microbiome has been well studied in both healthy and diseased individuals and following dietary interventions, but little is known about the variability of Bifidobacterium species between subjects and its effects on gut microbiome composition, function, and human health. Most cross-sectional or interventional studies performed to date have had a limited sample size, with potentially restricted variation in the gut microbiome, including that for Bifidobacterium communities. Over the last decade, large cross-sectional cohorts (> 1,000 subjects) have been established and studied to disentangle the specific associations between intrinsic and extrinsic factors and the gut microbiome. For instance, the abundance of Bifidobacterium has been shown to depend on genetics, specifically lactase persistence/nonpersistence [START_REF] Asnicar | Microbiome connections with host metabolism and habitual diet from 1,098 deeply phenotyped individuals[END_REF] , and dietary factors, such as carbohydrates [START_REF] Asnicar | Microbiome connections with host metabolism and habitual diet from 1,098 deeply phenotyped individuals[END_REF][START_REF] Nayfach | New insights from uncultivated genomes of the global human gut microbiome[END_REF] . Despite their significant relevance for identifying the major factors underlying gut microbiome variation, some single cohorts may display a lack of gut microbiome variation as a function of health status, lifestyle, age, and taxonomic and functional resolution.

The use of public databases containing large amounts of human gut microbiota shotgun metagenomic data spanning different ages, countries, health statuses, and lifestyles has greatly increased in recent years, providing new insight into the association of the gut microbiota with the host, environmental factors, and the reconstruction of metagenome-assembled genomes (MAGs) [START_REF] Parks | Recovery of nearly 8,000 metagenome-assembled genomes substantially expands the tree of life[END_REF][START_REF] Almeida | A new genomic blueprint of the human gut microbiota[END_REF][START_REF] Pasolli | Extensive unexplored human microbiome diversity revealed by over 150,000 genomes from metagenomes spanning age, geography, and lifestyle[END_REF][START_REF] Leviatan | An expanded reference map of the human gut microbiome reveals hundreds of previously unknown species[END_REF][START_REF] Zhernakova | Population-based metagenomics analysis reveals markers for gut microbiome composition and diversity[END_REF][START_REF] Filippis | Newly explored faecalibacterium diversity is connected to age, lifestyle, geography, and disease[END_REF] . An extensive study by Pasolli et al. yielded over 150.000 MAGs [START_REF] Leviatan | An expanded reference map of the human gut microbiome reveals hundreds of previously unknown species[END_REF] . Large-scale analyses of targeted bacterial species have improved our understanding of their diversity, ecology, and association with health and lifestyle. For instance, the study of Faecalibacterium prausnitzii [START_REF] Karcher | Genomic diversity and ecology of human-associated Akkermansia species in the gut microbiome revealed by extensive metagenomic assembly[END_REF] , Akkermansia muciniphila [START_REF] Tett | The prevotella copri complex comprises four distinct clades underrepresented in westernized populations[END_REF] , and Prevotella copri [START_REF] Hidalgo-Cantabrana | Bifidobacteria and their health-promoting effects[END_REF] has revealed new diversity and specific functional features associated with the host and environmental factors. Despite the increasing availability of metagenomic data, we still know very little about the metabolic contributions of strains within an ecological niche. Strains of Bifidobacterium have long been considered of major interest for use as probiotics [START_REF] Sarkar | Bifidobacteria-Insight into clinical outcomes and mechanisms of its probiotic action[END_REF][START_REF] Holmes | Dirichlet multinomial mixtures: generative models for microbial metagenomics[END_REF] , but the variability and ecology of resident Bifidobacterium species and their association with the host and the gut microbiome have been little explored in large-scale studies. One recent study characterized the strain dynamics, pangenome, and genomic diversity of the main Bifidobacterium species from the human gut in early life with MAGs [START_REF] Derrien | The gut microbiota in the first decade of life[END_REF] . However, functional pangenomic analyses of Bifidobacterium in the adult human gut are lacking.

We performed an exploratory analysis of the ecology of Bifidobacterium based on data from a public database containing human gut microbiome data, mostly for adult subjects. We first confirmed the previously reported associations with health, age, and other factors. We then identified Bifidobacterium partitions of the gut differing in terms of the abundance of Bifidobacterium, species composition, gut microbiome features, and health status. Finally, using MAG-based pangenomic analysis, we showed that the prevalence of some functional features of some Bifidobacterium species differed between health-associated Bifidobacterium partitions. This study paves the way for more precise approaches to guide the selection of Bifidobacterium strains for gut microbiome complementation in adulthood and, ultimately, human health.

METHODS

Pooled metagenomic studies dataset

We extracted taxonomic data from the curatedMetagenomicData (cMD) R package (Pasolli et al.) (version 3.0, release 2021), which consists of manually curated metadata together with all the taxonomic read counts aggregated per species with MetaPhlAn3, for 86 studies (17,959 samples). Gut metagenomes with more than five million reads were retained, and one duplicate study (referred to as "LeChatelierE_2013") was excluded. The read counts for the samples were sum-collapsed by genus. The resulting feature table was rarefied to a depth of 1.000.000 counts per sample for alpha diversity analysis. Filtering for origin (stools), with the selection of one fecal sample per subject (highest number of reads), resulted in 9,515 unique samples (61 studies). This dataset was used for a global description of the abundance of Bifidobacterium and the prevalence of Bifidobacterium species across different metadata curated in cMD: age, lifestyle, antibiotic use status, and health status.

The age categories were as follows: newborn (< 1 year of age), child (age ≥ 1 year and < 12 years), school-age individuals (age ≥ 12 and < 19 years); adult (age ≥ 19 years), senior (> 65 years).

Lifestyle was classified as westernized or non-westernized, and antibiotic use was classified as yes (the month preceding stool sample collection) or no.

Health-related metadata were aggregated into six categories as follows: control: subject known to be healthy; adenoma: patients with all types and subtypes of adenoma; colorectal: patients with colorectal cancers including metastases; metabolic: patients with metabolic conditions including atherosclerotic cardiovascular disease, hypercholesterolemia, hypertension, type 2 diabetes, and impaired glucose tolerance; bowel: patients with inflammatory bowel disease (IBD); arthritis: patients with rheumatoid arthritis or Behçet's disease (BD).

Bifidobacterium-based clustering of the gut microbiome

Samples were partitioned by applying Dirichlet's Multinomial Mixture (DMM) modeling to the microbiota data [START_REF] Huerta-Cepas | EggNOG 5.0: a hierarchical, functionally and phylogenetically annotated orthology resource based on 5090 organisms and 2502 viruses[END_REF] for 32 detected Bifidobacterium species with counts across cMD. We filtered the 9.515 datasets as follows to obtain a final dataset relating to 5.329 subjects for DMM: (1) We retained individuals who had not had antibiotic treatment as declared in the cMD (antibiotic use = no) (N = 216) or without information (N = 3.571) to prevent bias in the diversity calculation; (2) We excluded subjects with a total count < 500, to overcome Bifidobacterium underdetection issues (N = 19); (3) We excluded subjects with no Bifidobacterium species total reads count as a DMM standard (N = 380). DMM models were calculated for different numbers (k) of clusters, k ∈ [1,30], and evaluated with the Bayesian information criterion (BIC) and five different seeds. These methods are based on minimizing a penalized criterion, taking into account model fit and complexity. We chose three random seeds and calculated the minimum k for different model fits, and then selected the most frequently observed. We determined the contribution of each Bifidobacterium species to each DMM cluster from the calculated models.

Functional analysis of Bifidobacterium MAGs

We retrieved 3,973 metagenomic-assembled genomes (MAGs) assigned to 15 Bifidobacterium species from http://opendata.lifebit.ai/table/?project=SGB. The MAGs were previously decontaminated and taxonomically assigned by Mash [START_REF] Leviatan | An expanded reference map of the human gut microbiome reveals hundreds of previously unknown species[END_REF] . The study identifier, sample identifier, assigned species, and completeness were collected for each MAG. Prodigal was used for gene calling for each MAG, and more than six million genes were called. The computation time required for annotation was decreased by clustering the MAG gene against a non-redundant gut Bifidobacterium gene catalog, using CD-HIT at 95% nucleotide identity, with a minimum sequence overlap of 90%. Non-redundant Bifidobacterium genes were annotated with EggNOG 5.0 [START_REF] Yin | dbCAN: a web resource for automated carbohydrate-active enzyme annotation[END_REF] and dbCAN [START_REF] Odamaki | Genomic diversity and distribution of Bifidobacterium longum subsp. longum across the human lifespan[END_REF] version 3 to obtain orthologous genes (OGs) and CAZy families, respectively. Quality was ensured by selecting the MAGs with completeness > 80%. Bifidobacterium species with at least 30 associated MAGs were selected, giving a total of six species. 820 MAGs were assigned to B. longum, 700 to B. adolescentis, 339 to B. bifidum, 178 to B. pseudocatenulatum, 54 to B. catenulatum, and 34 to B. dentium. Pairwise distances were calculated for all MAGS within each species, with Mash v2.347 and the default sketch size. Hierarchical clustering was then performed for each species with the "ward.D2" method and the "pheatmap" R package (1.0.12).

Statistical analysis

The associations between Bifidobacterium partitions and quantitative variables (notably α-diversity and Bifidobacterium abundances) were analyzed with Kruskal-Wallis tests and a post-hoc test (Mann-Whitney test, adjusted for FDR). Pearson's chi-squared test was used to determine whether 1) Bifidobacterium partitions were associated with categorical variables (age, lifestyle, health status) and 2) whether the prevalence of OGs in MAGs for each species was associated with health-associated partitions, adjusted for FDR (within species). DESeq2 (v1.28.1) was used to identify bacterial species for which abundance differed between Bifidobacterium partitions with the "poscounts" normalization option to accommodate the sparsity of microbiota data. The global effects of the Bifidobacterium were estimated in likelihood ratio tests and Wald tests for pairwise comparisons of clusters. A FDR correction for multiple testing was applied to each test to account for the number of species tested. Log 2 fold-changes in expression are expressed as the estimate ± standard error. When specified, FDR corrections were applied with the Benjamini-Hochberg procedure.

RESULTS

Analysis of pooled metagenomic studies recapitulates major findings of human gut Bifidobacterium ecology

We used the "curatedMetagenomicData" (cMD, version 3) database (Pasolli et al.) to study the ecology of the Bifidobacterium community in the human gut microbiome. The cMD provides standardized, curated human microbiome data with several pieces of metadata per participant. We selected only gut metagenomes (one per subject) and obtained 9.515 unique samples [Table 1]. This dataset contains predominantly data for adults with a westernized lifestyle. Individuals under the age of 19 years accounted for less than 8% of this dataset [newborns (2.9%), children (3.4%), and school-age (1.4%)] [Table 1 and Supplementary Figure 1]. Given the multiple differences in analytical procedures between the studies included in the cMD, we first investigated whether our analysis of the cMD database could reproduce published findings for Bifidobacterium in humans, such as differences according to age, lifestyle, antibiotic use, and health status.

We found that subjects with a westernized lifestyle had higher relative abundances of Bifidobacterium (Mann-Whitney, P < 0.001) [Supplementary Figure 2] and a higher prevalence of B. animalis and B. longum [Supplementary Figure 3]. The gut microbiome of newborns was more enriched in Bifidobacterium (median of 7.76 %, IQR 0.48%-41.3%) than that of the other age categories (median 2%, IQR 0.01%-9.8%) (Kruskal Wallis, P < 0.001) [Supplementary Figure 2], with a higher prevalence of B. breve in children under the age of three years (50%), decreasing to < 10% thereafter. B. adolescentis was more prevalent in adults (71%) than in younger subjects (35%-50%) and seniors (60%). By contrast, B. longum was highly prevalent at all ages (> 80%) [Supplementary Figure 3]. The relative abundance of Bifidobacterium was significantly lower in most of the disease groups, especially Bowel (IBD) (median 0.03%, IQR 0%-0.2%) than in healthy individuals (median 3.4%, IQR 0.6%-10.7%) (Kruskal Wallis, P < 0.001) [Supplementary Figure 2]. In particular, B. adolescentis was more prevalent in healthy individuals (70%) than in those with the disease, particularly for IBD and metabolic diseases (40%-50%), whereas the opposite pattern was found for B. dentium [Supplementary Figure 3]. Finally, the relative abundance of Bifidobacterium was lower in subjects with recent antibiotic intake (Mann-Whitney, P < 0.001) [Supplementary Figure 2]. Despite the considerable analytical differences between studies, this dataset reproduced the major associations previously reported between Bifidobacterium and age [START_REF] Schnorr | Gut microbiome of the Hadza hunter-gatherers[END_REF] , lifestyle [START_REF] Gupta | A predictive index for health status using species-level gut microbiome profiling[END_REF] , health status [START_REF] Pokusaeva | Carbohydrate metabolism in Bifidobacteria[END_REF][START_REF] Gacesa | Environmental factors shaping the gut microbiome in a Dutch population[END_REF][START_REF] Fitzgerald | Improved gut microbiome recovery following drug therapy is linked to abundance and replication of probiotic strains[END_REF] , and antibiotic use [START_REF] Palleja | Recovery of gut microbiota of healthy adults following antibiotic exposure[END_REF][START_REF] Mangin | Long-term changes in human colonic Bifidobacterium populations induced by a 5day oral amoxicillin-clavulanic acid treatment[END_REF][START_REF] Liu | Microbiome-based stratification to guide dietary interventions to improve human health[END_REF] in individual cohorts.

Bifidobacterium-based partitioning of the human gut microbiome

We further explored the ecology and variation of Bifidobacterium species between subjects. We used the Dirichlet multinomial mixtures (DMM) partitioning method [START_REF] Huerta-Cepas | EggNOG 5.0: a hierarchical, functionally and phylogenetically annotated orthology resource based on 5090 organisms and 2502 viruses[END_REF] , which is commonly used to identify partitions of the human gut microbiome; however, in this case, we applied it exclusively to Bifidobacterium species. This made it possible to focus specifically on the association of the variable within-Bifidobacterium community distribution with the ecological features of the gut microbiome, lifestyle, and health. We applied DMM to 5.329 subjects (see methods). On the basis of BIC minimization, k = 6 was chosen for individuals for whom Bifidobacterium was detected [Figure 1A]. We added a group (k = 7) corresponding to subjects for whom no Bifidobacterium reads were detected (n = 380). All partitions had a median number of reads above 30 M [Supplementary Table 1].

Partitions #1 and #2 accounted for 48% of subjects, whereas the partition corresponding to the detection of no Bifidobacterium species (partition #7) accounted for the smallest number of subjects (17%) [Figure 1B]. We extracted the scaled contribution of each Bifidobacterium species to each partition, which reflects the relative abundance of these species within Bifidobacterium. Some of the dominant Bifidobacterium species (shown in yellow) differed in abundance between partitions [Figure 1C].

In partitions #1, #

Association between Bifidobacterium-based partitions and the gut microbiome and health

We then investigated the distribution of Bifidobacterium-based partitions as a function of age category, lifestyle (westernized/non-westernized), and health conditions [Figure 2A-C and Supplementary Figure 4]. The associations of Bifidobacterium partitions with health status (healthy or with one of the health conditions considered), lifestyle (westernized/ non-westernized), and age category were significant (chisquared, P-value < 0.05).

The prevalence of seniors and subjects with a non-western lifestyle was higher in the lon and no_Bif partitions, respectively [Figure 2C]. The Bifidobacterium partitions lon_pse, lon, ado_pse, and no_Bif included ~70% healthy subjects, whereas more than 90% of the subjects in the Bifidobacterium partitions ado_lon, ado_lon_bif, and lon_ado_cat were healthy. We also found that 64% of the healthy subjects belonged to the Bifidobacterium partitions ado_lon, ado_lon_bif, and lon_ado_cat, whereas 74% of the subjects with health conditions belonged to the Bifidobacterium partitions lon_pse, lon, ado_pse, and no_Bif.

We then investigated the association of the Bifidobacterium partitions with the gut microbiome. We found that Bifidobacterium partitions were significantly associated with Bifidobacterium abundance [Figure 3A]. and gut microbiome Shannon diversity [Figure 3B] (Kruskal-Wallis, P < 0.001). Bifidobacterium partitions ado_lon, ado_lon_bif, and lon_ado_cat had a higher relative abundance of Bifidobacterium, and a higher gut microbiome α-diversity (Shannon index) than the other partitions [Supplementary Table 1] (Mann-Whitney test, P < 0.001). Given that cMD also includes infants less than one-year-old, which may influence alpha-diversity results, we further investigated whether partitions could be detected in single adult cohorts in which samples were processed with the same analytical procedure. We selected three large cohorts from the cMD (> 1.000 adults) and extracted their Bifidobacterium partitions: (1) 1.098 individuals from the UK enrolled in the Personalised Responses to Dietary Composition Trial (PREDICT 1) study [START_REF] Nayfach | New insights from uncultivated genomes of the global human gut microbiome[END_REF] ; (2) 1.135 participants from the Dutch population-based cohort LifeLines-DEEP [START_REF] Filippis | Newly explored faecalibacterium diversity is connected to age, lifestyle, geography, and disease[END_REF] ; and (3) 800 individuals from an Israeli cohort. All the identified partitions were detected in the three cohorts with different prevalences [Supplementary Figure 5]. As for the cMD, the ado_lon and ado_lon_bif partitions were the most prevalent (> 50%), and the no_Bif partition was the least prevalent (< 10%). We identified several partitions related to Bifidobacterium composition in the adult gut microbiome. These differences were also observed when only adults from the cMD were selected [Supplementary Table 1].

We then used DESeq2 to identify bacterial species for which abundance differed between partitions (FDR < 0. Overall, the partitions enriched in both B. longum and B. adolescentis were associated with higher gut microbiome diversity and abundance of butyrate producers and were more prevalent in healthy individuals (i.e., health-associated Bifidobacterium communities).

A pangenomic analysis of Bifidobacterium reveals functions associated with gut ecology and health

Finally, we investigated whether intra-species functions were associated with Bifidobacterium partitions as a surrogate for a more diverse gut microbiome and a higher prevalence of healthy subjects. We selected 2,263 MAGs constructed from an extensive dataset [START_REF] Leviatan | An expanded reference map of the human gut microbiome reveals hundreds of previously unknown species[END_REF] included in the cMD database. Therefore, we could pair participants' Bifidobacterium partitions with their Bifidobacterium MAG content. An analysis of 11,673 unique OGs functionally distinguished MAGs from different Bifidobacterium species on the basis of OGs prevalence (chi-squared test, FDR < 0.05 within species) [Supplementary Table 2]. The most significant OGs were that for asparagine synthase (COG0367), which was detected in 99% of the MAGs assigned to B. adolescentis and 0.25% of those from other species. We confirmed known functional differences, relating, for example, to glycoside hydrolases (GH) involved in the metabolism of host carbohydrates (mucin/milk), such as GH 20, GH 29, GH 33, and GH 95, which were specific to B. bifidum and had prevalences ranging from 90 to 97%, vs. 0.1% in other species. Similarly, we detected a high prevalence of alpha-Larabinofuranosidase (COG3534) for the MAGs of B. longum (99% vs. 0.7% in other species). For B. pseudocatenulatum, we detected OGs assigned to the GH 43 family (xylosidase). We then studied the functions of Bifidobacterium associated with previously identified Bifidobacterium partitions (health-associated ado_lon, ado_lon_bif, and lon_ado_cat or others (lon_pse, lon, ado_pse, and no_Bif) by pairing MAG content to a Bifidobacterium partition for each subject. This analysis compared the gene content of MAGs from specific Bifidobacterium species regardless of their differential abundance between health groups. We found 38 OGs significantly associated with these two types of Bifidobacterium partitions (chi-squared, FDR < 0.1). Fifteen of these OGs were less prevalent in health-associated Bifidobacterium partitions (< or > 25% prevalence in health-associated and others, respectively) [Supplementary Table 3]. These OGs were assigned to phages and included integrases, transposases, and helicases. We then investigated the phylogenetic relationships between the MAGs harboring these 15 OGs in B. bifidum. We computed Mash distances between 198 B. bifidum MAGs and visualized the prevalence of 15 significant OGs in the 198 B. bifidum MAGs. These 15 OGs had a higher prevalence in a cluster of MAGs associated with the Bifidobacterium partitions most frequently detected in individuals with diseases [Figure 4]. This finding suggests that a B. bifidum subspecies or strain may be enriched in phage-related genes in subjects with more altered gut microbiomes.

DISCUSSION

In this study, we performed a large-scale analysis of the Bifidobacterium community in the human gut microbiome, with data from a large adult population, including individuals with various health conditions. By combining ecological and functional analyses of the Bifidobacterium community, we identified variable associations of partitions of Bifidobacterium species and functions associated with gut microbiome features and human health. Overall, our results confirm and extend previous findings on the ecological and functional relevance of the Bifidobacterium community for the gut microbiome and human health.

The human gut microbiome varies significantly between subjects, and this variation may obscure the effect of diet or treatment. Stratification of the gut microbiome on the basis of its composition has been used to identify the subjects most likely to respond to dietary interventions [START_REF] Feng | Targeting gut microbiota for precision medicine: focusing on the efficacy and toxicity of drugs[END_REF] or medical treatments [START_REF] Odamaki | Age-related changes in gut microbiota composition from newborn to centenarian: a crosssectional study[END_REF] . Bifidobacterium is a common member of the human gut microbiome, with different species co-existing in the host at different ages [START_REF] Costea | Enterotypes in the landscape of gut microbial community composition[END_REF] . A number of studies have shown that various metabolic, immune, and intestinal disease states coincide with the depletion of Bifidobacterium from the gut microbiota [START_REF] Pokusaeva | Carbohydrate metabolism in Bifidobacteria[END_REF] . Here, we studied the variability of the gut microbiome as a function of the resident Bifidobacterium community, using a public database compiling curated metagenomics-based studies, mostly performed in adults from Western countries. We first checked that we could reproduce the previously reported findings of a lower abundance of Bifidobacterium species in antibiotic users [START_REF] Palleja | Recovery of gut microbiota of healthy adults following antibiotic exposure[END_REF][START_REF] Mangin | Long-term changes in human colonic Bifidobacterium populations induced by a 5day oral amoxicillin-clavulanic acid treatment[END_REF][START_REF] Liu | Microbiome-based stratification to guide dietary interventions to improve human health[END_REF] and individuals with diseases [START_REF] Pokusaeva | Carbohydrate metabolism in Bifidobacteria[END_REF] or adopting a nonwesternized lifestyle [START_REF] Gupta | A predictive index for health status using species-level gut microbiome profiling[END_REF] , the differential prevalence of most Bifidobacterium species depending on age [START_REF] Bolte | Long-term dietary patterns are associated with pro-inflammatory and anti-inflammatory features of the gut microbiome[END_REF] , the high prevalence of B. longum throughout the human lifespan [START_REF] Schnorr | Gut microbiome of the Hadza hunter-gatherers[END_REF] , and the higher prevalence of B. adolescentis in healthy subjects than in those with diseases [START_REF] Gacesa | Environmental factors shaping the gut microbiome in a Dutch population[END_REF][START_REF] Fitzgerald | Improved gut microbiome recovery following drug therapy is linked to abundance and replication of probiotic strains[END_REF] .

We then studied the variability of Bifidobacterium community composition with the Dirichlet multinomial mixtures (DMM) method, which has been used for gut microbiome clustering on the basis of composition in many studies [START_REF] Huerta-Cepas | EggNOG 5.0: a hierarchical, functionally and phylogenetically annotated orthology resource based on 5090 organisms and 2502 viruses[END_REF][START_REF] Stewart | Temporal development of the gut microbiome in early childhood from the TEDDY study[END_REF][START_REF] Beller | Successional stages in infant gut microbiota maturation[END_REF][START_REF] Vieira-Silva | Statin therapy is associated with lower prevalence of gut microbiota dysbiosis[END_REF][START_REF] Ding | Dynamics and associations of microbial community types across the human body[END_REF][START_REF] Roswall | Developmental trajectory of the healthy human gut microbiota during the first 5 years of life[END_REF][START_REF] Das | In vitro co-cultures of human gut bacterial species as predicted from cooccurrence network analysis[END_REF] . We identified partitions enriched in different combinations of Bifidobacterium species in a database containing predominantly adult data. The partitions that were more prevalent were characterized by a higher abundance of B. longum and B. adolescentis, whereas the partition corresponding to the non-detection of Bifidobacterium was the least prevalent. A specific analysis of three individual cohorts of adults (~1,000 subjects) confirmed the detection of several partitions, indicating an effect of between-subject variability rather than technical differences between studies. Those associated with a healthier state were dominated by B. longum and B. adolescentis. In previous studies, the species-level analysis revealed a positive correlation or covariation between multiple Bifidobacterium species [START_REF] Kato | Age-related changes in the composition of gut bifidobacterium species[END_REF][START_REF] Abu-Ali | Metatranscriptome of human faecal microbial communities in a cohort of adult men[END_REF] or between specific species, such as B. adolescentis and B. longum [START_REF] Hansen | A low-gluten diet induces changes in the intestinal microbiome of healthy Danish adults[END_REF][START_REF] Nagara | Selective induction of human gut-associated acetogenic/ butyrogenic microbiota based on specific microbial colonization of indigestible starch granules[END_REF][START_REF] Avershina | Bifidobacterial succession and correlation networks in a large unselected cohort of mothers and their children[END_REF] or B. adolescentis and B. bifidum [START_REF] Hansen | A low-gluten diet induces changes in the intestinal microbiome of healthy Danish adults[END_REF] . However, another study reported a negative correlation between B. adolescentis and B. longum [START_REF] Gotoh | Sharing of human milk oligosaccharides degradants within bifidobacterial communities in faecal cultures supplemented with Bifidobacterium bifidum[END_REF] , suggesting variability between studies or study subjects. It remains unclear whether positive correlations indicate metabolic cross-feeding or similar niches, and this aspect requires further investigation in vitro.

Cross-feeding between Bifidobacterium species on human milk oligosaccharides has been studied for the species prevalent in infants [START_REF] Walsh | Human milk oligosaccharide-sharing by a consortium of infant derived Bifidobacterium species[END_REF][START_REF] Lawson | Breast milk-derived human milk oligosaccharides promote Bifidobacterium interactions within a single ecosystem[END_REF][START_REF] Belenguer | Two routes of metabolic cross-feeding between Bifidobacterium adolescentis and butyrate-producing anaerobes from the human gut[END_REF] , but, to our knowledge, there have been no studies investigating crossfeeding on complex dietary fibers between B. longum, B. adolescentis and B. pseudocatenulatum, which are more common in adults.

The partitions with a higher abundance of Bifidobacterium, and particularly those dominated by both B. longum and B. adolescentis, were associated with a higher gut microbiome diversity and a higher abundance of butyrate-producing species, including Roseburia faecis, Coprococcus catus, C. eutactus, C. comes, and Eubacterium hallii. Covariation between Bifidobacterium species and other resident species has been detected for butyrate producers in the metagenomic analysis [START_REF] Kato | Age-related changes in the composition of gut bifidobacterium species[END_REF][START_REF] Nagara | Selective induction of human gut-associated acetogenic/ butyrogenic microbiota based on specific microbial colonization of indigestible starch granules[END_REF] , and metabolic interactions between B. adolescentis, B. longum, and butyrate producers have been observed in vitro in the presence of complex dietary substrates [START_REF] Rivière | Mutual cross-feeding interactions between Bifidobacterium longum subsp. longum NCC2705 and Eubacterium rectale ATCC 33656 explain the bifidogenic and butyrogenic effects of arabinoxylan oligosaccharides[END_REF][START_REF] Falony | Cross-feeding between Bifidobacterium longum BB536 and acetate-converting, butyrate-producing colon bacteria during growth on oligofructose[END_REF][START_REF] Roy | Yoghurt consumption is associated with changes in the composition of the human gut microbiome and metabolome[END_REF] . These partitions were more frequently found in healthy subjects, suggesting that individual stratification exclusively on the basis of gut Bifidobacterium species abundance is associated with differential gut microbiome structure and state of health. Notably, the partition corresponding to an absence of Bifidobacterium detection, which contained a larger number of subjects with a non-westernized lifestyle than the other partitions, was depleted of Streptococcus salivarius, which is detected in consumers of yogurts, including yogurts supplemented with B. animalis subsp. lactis [START_REF] Derrien | Fate, activity, and impact of ingested bacteria within the human gut microbiota[END_REF] . Overall, our results extend previous findings on associations with the gut microbiome by differentiating gut microbiomes enriched in specific Bifidobacterium types. It would be interesting to determine whether these partitions are associated with differential gut microbiome permissivity to distinct exogenous Bifidobacterium species/strains (during and/or after the cessation of consumption) [START_REF] Maldonado-Gómez | Stable engraftment of bifidobacterium longum AH1206 in the human gut depends on individualized features of the resident microbiome[END_REF][START_REF] Zhang | Ecological robustness of the gut microbiota in response to ingestion of transient food-borne microbes[END_REF][START_REF] Blanco | Revisiting the metabolic capabilities of bifidobacterium longum susbp. longum and bifidobacterium longum subsp. infantis from a glycoside hydrolase perspective[END_REF] . Following our exploratory analysis of the ecology of the Bifidobacterium community, we evaluated the functions of Bifidobacterium associated with partitions in a pangenomic analysis (i.e., functional variability within species). We studied Bifidobacterium-assigned MAGs retrieved from an extensive dataset for the human gut microbiome [START_REF] Leviatan | An expanded reference map of the human gut microbiome reveals hundreds of previously unknown species[END_REF] , which could be assigned to metadata and gut microbiome features. EggNOG analysis confirmed known functional differences between the most prevalent Bifidobacterium species, such as the specificity of alpha-L-arabinofuranosidase to B. longum, involved in the metabolism of arabinans, arabinoxylans, and arabinogalactans [START_REF] Schnorr | Gut microbiome of the Hadza hunter-gatherers[END_REF][START_REF] Katoh | Enzymatic adaptation of bifidobacterium bifidum to host glycans, viewed from glycoside hydrolyases and carbohydrate-binding modules[END_REF] , glycoside hydrolases of host carbohydrate metabolism (mucin, human milk oligosaccharide) for B. bifidum [START_REF] Watanabe | Xylan utilisation promotes adaptation of Bifidobacterium pseudocatenulatum to the human gastrointestinal tract[END_REF] , and glycoside hydrolase GH 43 for B. pseudocatenulatum [START_REF] Schöpping | Genome-wide assessment of stress-associated genes in bifidobacteria[END_REF] . Asparagine synthetase was found to be highly specific to B. adolescentis [62] .

The association between species function and Bifidobacterium partitions revealed a difference in the functional features of B. bifidum MAGs across Bifidobacterium partitions in association with health status. Specifically, B. bifidum MAGs harboring a set of genes potentially related to phages were more prevalent in partitions associated with a lower gut microbiome diversity and were genetically more closely related. This potentially highlights the existence of a B. bifidum subspecies with a selective advantage for the colonization of gut microbiomes with a particular composition. Interest in the possible contribution of phages to gut microbiome ecology has increased significantly over the last decade, and one recent study [START_REF] Schmidt | Strain-level analysis of bifidobacterium spp. from gut microbiomes of adults with differing lactase persistence genotypes[END_REF] showed the phages of Bifidobacterium to be rather specific. Overall, our pangenomic analysis revealed several functional features of B. bifidum differing between Bifidobacterium partitions as a function of health status [Figure 5].

This study has several limitations. First, it is based on the pooling of studies, an approach that is increasingly used to increase the sample size for ecological analysis. However, there are inherent differences in technical parameters between studies. Second, only a small amount of metadata is included. Diet is a major factor underlying gut microbiome variation between subjects. Carbohydrates are the dietary component most frequently reported to be positively associated with Bifidobacterium [START_REF] Bolte | Long-term dietary patterns are associated with pro-inflammatory and anti-inflammatory features of the gut microbiome[END_REF] . In previous metagenomics-based studies with species-level analysis, B. adolescentis was identified as the bifidobacterial species most significantly associated with dietary habits [START_REF] Asnicar | Microbiome connections with host metabolism and habitual diet from 1,098 deeply phenotyped individuals[END_REF] , whereas both common and different associations between different Bifidobacterium species and food scores were identified [START_REF] Nayfach | New insights from uncultivated genomes of the global human gut microbiome[END_REF] . The associations between dietary habits, partitions, variation of the LCT gene (lactase persistence), and other parameters should, therefore, be studied specifically in future studies. Another limitation of this study is that only cross-sectional analysis was performed. However, a previous metagenomic analysis revealed that the B. longum, B. adolescentis, and B. bifidum communities remained stable within individuals over a period of several years [START_REF] Rajilić-Stojanović | Long-term monitoring of the human intestinal microbiota composition[END_REF] , consistent with the stability of the genus Bifidobacterium reported in a 10-year study [65] .

This study is novel in the stratification of the gut microbiome according to specific resident bacterial species and the association with ecological features of the gut microbiome and health. These features could be applied to other genera. This first such study may provide insights for further investigations of the association between partitions and more exhaustive analyses of the host and environmental factors, including dietary habits. This exploratory study constitutes a first step towards understanding the ecology and variability of Bifidobacterium, with a view to guiding the selection of specific Bifidobacterium strains for use in subjects as a function of the partition present.

Figure 1 .

 1 Figure 1. Bifidobacterium-based gut microbiome partitions. (A) Model fit according to BIC; (B) distribution of subjects across the 7 partitions. Partition #7 consists of subjects for whom no Bifidobacterium was detected (no-Bif); (C) scaled contribution of each Bifidobacterium species to each partition. Higher DMM model contributions are associated with a higher relative abundance of a particular species. Species are ordered according to hierarchical clustering based on Euclidean distance. Gray indicates alpha parameters below 10 -5 . More abundant species: B. longum to B. dentium and sub-dominant species: B. moukalabense to B. pseudolongum.

Figure 2 .

 2 Figure 2. Distribution of subjects between Bifidobacterium-based gut microbiome partitions according to metadata (without recent antibiotic intake). (A) Age category; (B) westernized lifestyle (adults); (C) health status (adults).

Figure 3 .

 3 Figure 3. Variation of the gut microbiome between Bifidobacterium-based partitions. (A) Relative abundance of Bifidobacterium across partitions; (B) species-based Shannon index for the gut microbiome; (C) differential analysis of the gut microbiome across partitions. The top 30 most abundant species are depicted. Red indicates a higher species abundance in the first partition tested (significant when FDR < 0.05, trend when FDR < 0.1). Species and partitions are ranked according to hierarchical clustering based on Euclidean distance, with the exclusion of Bifidobacterium species from the graph.

Figure 4 .

 4 Figure 4. Heatmap of Mash distances between the B. bifidum MAGs. The heatmap is annotated with the prevalence of the 15 OGs related to phages. The gradient from blue to red indicates increasing Mash genetic distances between the MAGs. Hierarchical clustering was performed by the Ward2 method. MAGs derived from subjects without Bifidobacterium partition information are shown in white.

Figure 5 .

 5 Figure 5. Graphical summary.

Table 1 . Datasets used in the study All individuals No antibiotic intake No antibiotic intake and adults only Characteristic N= 9.515 N= 5.728 N= 4.921

 1

	Antibiotic intake	216 (3.6%)
	Unknown	3.571
	No		5.728 (100%)	4.921 (100%)
	Health status
	Control	7.016 (79%)	4.876 (85%)	4.348 (88%)
	Adenoma	153 (1.7%)	39 (0.7%)	29 (0.6%)
	Colorectal	439 (4.9%)	110 (1.9%)	74 (1.5%)
	Metabolic	656 (7.4%)	495 (8.6%)	320 (6.5%)
	Bowel	535 (6.0%)	99 (1.7%)	45 (0.9%)
	Arthritis	94 (1.1%)	89 (1.6%)	85 (1.7%)
	Behçet's disease (BD)	20 (0.2%)	20 (0.3%)	20 (0.4%)
	Unknown	602
	Age category
	Newborn	278 (2.9%)	137 (2.4%)
	Child	322 (3.4%)	160 (2.8%)
	School-age	135 (1.4%)	88 (1.5%)
	Adult	7.745 (81%)	4.921 (86%)	4.921 (100%)
	Senior	1,035 (11%)	422 (7.4%)
	Westernized lifestyle	8.701 (91%)	5.577 (97%)	4.774 (97%)
	n (%)

Acknowledgments

The authors wish to thank Aurélie Cotillard and Matthieu Pichaud for their helpful discussions.

Financial support and sponsorship

This work was funded by Danone Nutricia Research.

Availability of data and materials

All the data used in this study are publicly available. CuratedMetagenomicData is available from https:// github.com/waldronlab/curatedMetagenomicData.

DECLARATIONS

Authors' contributions

Designed the research: Tap J, Derrien M Analyzed the data: Tap J, Ladeira R Interpreted the results: Ladeira R, Tap J, Derrien M Wrote the manuscript: Ladeira R, Tap J, Derrien M

Conflicts of interest

Julien Tap and Muriel Derrien are employees of Danone Nutricia Research. Ruben Ladeira was an intern at Danone Nutricia Research at the time of the analysis.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.


Mostafa Abaali
email: mostafaabaali@gmail.com

Salih Ouchtout
email: salih.ouchtout@uit.ac.ma

Asymptotic behavior analysis of a biodenitrification process model *

Keywords: Bioprocesses, Biodenitrification, Ordinary differential equations, Asymptotic behaviour

Day Mth. Year) Revised (Day Mth

Introduction

Food safety and environmental protection are two key issues in nowadays societies. Due to a remarkable increase of the world population and in order to increase the yield of agricultural production, massive and excessive use of nitrogen fertilizers and sustained irrigation have been widely practised in recent years. This strategy leads to a large transfer of nitrates to groundwater, leading to a major environmental problem of nitrate contamination in water, intended for drinking water consumption. However, the natural environment can offer a possibility of remediation. One of the most effective solutions is the degradation of nitrites and nitrates, into gaseous nitrogen, is the use of heterotrophic bacteria during the denitrification process. This process is characterized by the fact that when the oxygen concentration is negligible, it will be replaced by oxidized nitrogen (NO2 and/or NO3) which is the best electron acceptor after oxygen so the biodenitrification process can be described as a respiratory process in which denitrifying bacteria use nitrates instead of oxygen as an acceptor of electrons [START_REF] Chevron | Dénitrification biologique d'une nappe phréatique polluée par des composés azotés d'origine industrielle: expérimentations en laboratoire sur les cinétiques, le métabolisme et les apports de nutriments[END_REF][START_REF] Chapelle | Ground-water microbiology and geochemistry[END_REF][START_REF] Xi | Study the catalyzing mechanism of dissolved redox mediators on bio-denitrification by metabolic inhibitors[END_REF]. It generally takes place under conditions, called anoxic, intended to provide energy for cell activity and the synthesis of new cells [START_REF] Mccarty | Energetics of organic matter degradation[END_REF][START_REF] Payne | Reduction of nitrogenous oxides by microorganisms[END_REF][START_REF] Dahab | Nitrate reduction by in-situ bio-denitrification in groundwater[END_REF]. Successful biodenitrification process requires that the nitrate should be considered in the modeling as a limiting compound [START_REF] Abaali | Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media[END_REF]. In [START_REF] Badge | Finite element model to study potassium dynamics in the rhizosphere of a wheat root due to presence of bio-physical source[END_REF], the authors studied the dynamics of potassium in the rhizosphere of the wheat root in the presence and in the absence of a biophysical source and they showed that the presence of bio-fertilizer leads to significant changes in potassium concentration in the rhizosphere of the wheat root. In [START_REF] Badge | Two-dimensional finite element model to study the effect of water flux on the nitrate dynamics in the rhizosphere of a maize root[END_REF], the authors proposed a mathematical model to study the effect of water flux on nitrate dynamics in the rhizosphere of a single maize root taking into account important mechanisms such as diffusion and mass flux of solutes in the rhizosphere. Charpentier et al. [START_REF] Chen-Charpentier | Mathematical modeling of bioremediation of trichloroethylene in aquifers[END_REF] modeled the Trichloroethylene (TCE) biodegradation process where they considered two types of bacteria, the first one degrades the contaminant while the second forms a biobarrier to prevents the contamination from infiltration into groundwater. They gave four spatialized models according to the consumption of the nutrient by the two types of bacteria. In [START_REF] Chen-Charpentier | Numerical simulation of multi-species biofilms in porous media for different kinetics[END_REF] Charpentier et al. gave numerical simulations of several types of collaboration of two types of bacteria to control pollutants are given. Recent theoretical and experimental studies on biofilms [START_REF] Christensen | Physical and chemical properties of biofilms[END_REF][START_REF] Costerton | Microbial biofilms[END_REF][START_REF] Ouchtout | Analysis of an anaerobic digestion model in landfill with mortality term[END_REF] confirm that the growth of bacteria on the walls can be a serious problem for bioreactors and fermenters and have fundamental implications for natural environments. But in biodegradation processes, colonization of the reactor by bacteria can be useful for the results obtained. Freter et al. [START_REF] Freter | Survival and implantation of Escherichia coli in the intestinal tract[END_REF][START_REF] Freter | An understanding of colonization resistance of the mammalian large intestine requires mathematical analysis[END_REF] confirmed that adherent bacteria play an important role in the observed stability of the microflora of the mammalian large intestine to colonization by invading organisms. For biodegradation processes, adherent bacteria also play an important role for the evolution of bacteria in the reactor because free bacteria can be influenced by the flow rate. F. Chevron [START_REF] Chevron | Dénitrification biologique d'une nappe phréatique polluée par des composés azotés d'origine industrielle: expérimentations en laboratoire sur les cinétiques, le métabolisme et les apports de nutriments[END_REF], during the experiments, has confirmed the existence of free bacteria in the reactors which accelerated the biodenitrification. M. Ballyk et al. [START_REF] Ballyk | A model of microbial growth in a plug flow reactor with wall attachment[END_REF] analyzed the case of a single strain of bacteria for the general model given by Ballyk et al. [START_REF] Ballyk | A flow reactor with wall growth[END_REF] they have shown that there are two possible steady state regimes, the first is the total leaching of bacteria from the reactor(washout) and the second is the successful colonization of the reactor by the bacterial strain. They have shown that one of these regimes is stable under conditions related to system parameters. The main objective of the present paper is to give deeper insights and predictions on the evolution of bacteria and substrates involved in the biodenitrification process with the help of a mathematical model, and consequently their exploitation in heuristic and/or optimal control algorithms.

The paper is organized as follows. In the next section, model and assumptions are presented and discussed. Then, the dynamics is mathematically analyzed in the following section. In particular, we show that there exists a value of the injected carbon concentration from which we ensure the success of the biodenitrification process. Finally, we perform some numerical simulations in agreement with the theoretical results and we propose a heuristic algorithm to control the biodenitrification process.

Mathematical model

In order to describe the mathematical model that governs the water biodenitrification process, we consider that the reactor contains nitrified water, denitrifying bacteria and nutrients. The bacteria in the reactor are divided into two categories: a first type which adhere to the walls of the reactor with a maximum value denoted by ω ∞ , forming a biofilm, denoted by b f , and a second type which remain mobile and free floating, called planktonic cells and denoted by b m . The evolution of each type of bacteria is given by the following equation (see [START_REF] Abaali | Mathematical modelling of biodegradation in situ application to biodenitrification[END_REF])

db i dt = µ i (S) -K i b i , for i = m, f, (2.1)
where b i (t) and S(t) represent respectively the concentrations of bacteria and substrate at time t. Symbols K i and µ i are successively the loss coefficient (mortality, change of category, ...) and the specific growth rate which will be described later. These denitrifying bacteria react with the contaminant nitrate, considered as an electron acceptor, its concentration will be denoted by S N . Moreover, bacteria also need carbon for its growth, its concentration will be denoted by S. The input substrate concentrations of S and S N are respectively S in and S in N , and we consider that denitrified water is extracted at the outlet of the reactor. Based on the fundamental relations of biological kinetic (that are the growth rate of bacteria and the use rate of the substrate), the evolution of the concentrations S and S N , is governed respectively by the following equations [START_REF] Abaali | Mathematical modelling of biodegradation in situ application to biodenitrification[END_REF]

dS dt = (S in -S)d S - 1 Y m b m µ m (•) - 1 Y f γ -1 b f µ f (•), (2.2)
dS N dt = (S in N -S N )d S N - R Y m b m µ m (•) - R Y f γ -1 b f µ f (•), (2.3)
where Y m and Y f are successively the yield coefficient of mobile bacteria adherent bacteria. Parameters R, d S and d S N represent respectively the degradation rate of nitrates, the substrate dilution rate and the nitrate dilution rate. Symbol γ represents the conversion coefficient from volume density to areal density. It can be considered as the ratio between the surface and the perimeter of a section of the domain in the flow direction [START_REF] Ballyk | A model of microbial growth in a plug flow reactor with wall attachment[END_REF].

Generally, the evolution of the bacteria b f and b m depend on each other. In what follow, we describe and introduce this interdependence.

• We take firstly the equation (2.1) governing the fixed bacteria i.e. "i=f", and we consider that the loss coefficient K f is the sum of both the mortality rate , denoted k f , and the term corresponding to the detachment rate of these bacteria from surfaces, denoted by β, then we have

K f = k f + β, (2.4)
In agreement with [START_REF] Freter | Survival and implantation of Escherichia coli in the intestinal tract[END_REF] and [START_REF] Freter | An understanding of colonization resistance of the mammalian large intestine requires mathematical analysis[END_REF], only a proportion of the daughter cells of the fixed bacteria find place on the surface of the reactor, the rest of cells will be evacuated into the liquid. Let us consider the variable bf defined by

bf := b f ω ∞ . (2.5)
The first proportion of daughter cells is a function of bf (see [START_REF] Freter | Survival and implantation of Escherichia coli in the intestinal tract[END_REF]), denoted by G( bf ). In what follows we use the Freter's formula (see [START_REF] Freter | An understanding of colonization resistance of the mammalian large intestine requires mathematical analysis[END_REF]), given by

G(X) = 1 -X a + 1 -X , (2.6)
where a is a very small constant. Therefore, the growth rate of the fixed bacteria in the evolution equation (2.1) will be multiplied by the proportion G( bf ). Taking into account these improvements, from (2.1)-(2.6)-(2.4) we get

db f dt = µ f (•)G( bf ) -k f -β b f . (2.7)
In addition, a portion of planktonic bacteria can be attached to surfaces with a rate α. Therefore, the term αb m will be added to the equation (2.7) as source term. We consider (1 -bf ) as the availability of the adhesion surface and we introduce γ the conversion coefficient of the volume density into a surface density. Taking into account these progress, the equation (2.7) becomes

db f dt = µ f (•)G( bf ) -k f -β b f + α 1 -bf γb m . (2.8)
• Moving to the mobile bacteria, similarly to the previous case, we consider that the loss coefficient K m is the sum of both the mortality coefficient, denoted by k m , the dilution coefficient denoted by d and the attachment coefficient of mobile bacteria denoted by α i.e.

K m = k m + α + d. The evolution equation (2.1) of b m becomes db m dt = µ m (•) -k m -α -d b m .
In addition, the following modelling improvements will be introduced to this equation. The source term corresponding to the surface concentration of the detached fixed bacteria, multiplied by the conversion coefficient, will April 5, 2023 12:34 output Asymptotic behavior analysis of a biodenitrification process Model 5 be considered and denoted by γ -1 βb f . The concentration of bacteria which do not find a location to attach themselves will be taken into account and denoted by the term:

b f γ -1 µ f (•)(1 -G( bf )). We obtain db m dt = µ m (•) -k m -α -d b m + γ -1 b f µ f (•)(1 -G( bf )) + β . (2.9)
Finally, from the equations (2.8), (2.9), (2.2) and (2.3) , we obtain the following system (2.10) describing the evolution of the quantities (b f , b m , S, S N ) of the biodenitrification model.

                       db f dt = µ f (•)G( bf ) -k f -β b f + αγ(1 -bf )b m , db m dt = µ m (•) -k m -d -α(1 -bf ) b m + µ f (•)(1 -G( bf )) + β γ -1 b f , dS dt = (S in -S)d S - µ m (•) Y m b m - µ f (•) Y f γ -1 b f , dS N dt = (S in N -S N )d S N - Rµ m (•) Y m b m - Rµ f (•) Y f γ -1 b f .
(2.10) At time t = 0, we assume that the reactor contains nitrified water with an initial density S 0 N , mobile bacteria with an initial density b 0 m , fixed bacteria with an initial density b 0 f and substrate with an initial density S 0 .

       b f (0) = b 0 f , b m (0) = b 0 m , S(0) = S 0 , S N (0) = S 0 N . (2.11)
It is natural to assume that the initial quantities of the variables are positive

b 0 f ≥ 0, b 0 m ≥ 0, S ≥ 0, S N ≥ 0.
In what follows, we will study the existence, characteristics, asymptotic behaviour of the solution and stability of the steady states for the non-spatialized system (2.10), where we assume that the concentrations are uniformly distributed over the whole domain, given by [START_REF] Abaali | Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media[END_REF] for the purpose of generalizing the results to the spacialized problem.

In order to simplify the presentation of the system, we consider the following notations

b f = c 1 , b m = c 2 , S = c 3 , S N = c 4 and C = (c 1 , c 2 , c 3 , c 4 ).
The system (2.10) becomes:

                               dc 1 dt = F 1 (t, C), dc 2 dt = F 2 (t, C), dc 3 dt = F 3 (t, C), dc 4 dt = F 4 (t, C), (2.12)
where

                           F 1 (t, C) = µ f (c 3 (t), c 4 (t))G( c1 ω∞ (t)) -k f -β c 1 (t) + αγ(1 -c1 ω∞ (t))c 2 (t), F 2 (t, C) = µ m (c 3 (t), c 4 (t)) -k m -d -α(1 -c1 ω∞ (t)) c 2 (t) + µ f (c 3 (t), c 4 (t))(1 -G( c1 ω∞ (t))) + β γ -1 c 1 (t), F 3 (t, C) = (c in 3 -c 3 )d S -µm(c3(t),c4(t)) Ym c 2 (t) - µ f (c3(t),c4(t)) Y f γ -1 c 1 (t), F 4 (t, C) = (c in 4 -c 4 )d S N -Rµm(c3(t),c4(t)) Ym c 2 (t) - Rµ f (c3(t),c4(t)) Y f γ -1 c 1 (t), (2.13
) and

       c 1 (0) = c 0 1 > 0, c 2 (0) = c 0 2 > 0, c 3 (0) = c 0 3 > 0, c 4 (0) = c 0 4 > 0.
(2.14)

As we have mentioned before, µ f and µ m represent the growth function of adherent and free bacteria, respectively, and can obey the multiplicative formula (see [START_REF] Abaali | Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media[END_REF])

µ i (c 3 , c 4 ) = µ i max c 3 c 3 + K i 1 c 4 c 4 + K i 2 , i = f, m (2.15)
where the indices i = m and i = f designate respectively free and adherent bacteria. µ i max is maximum growth rate. The constants K i j , for i = m, f and j = 1, 2, ("j = 1" relates to consumption of nutrient and "j = 2" relates to consumption of nitrates) are in fact the Monod's law constants, also called the half-saturation constants. (see the profile of µ i in Figure 1, for example for µ i max = 0.7, K i 1 = 60 and K i 2 = 50). According to [START_REF] Abaali | Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media[END_REF], the functions G and µ i (•, •) satisfy the following properties. hypothesis H1) The specific growth rate function of bacteria µ i (x, y), for i = f, m, satisfies

µ i ∈ C 1 , µ i (0, y) = µ i (x, 0) = 0.
F := (F 1 , F 2 , .., F p ) ∈ C 1 (R + ×R p , R p ) with p ∈ N * is quasi-positive if F i (t, y) ≥ 0 whenever (t, y) ∈ (0, +∞) × R p + is such that y i = 0.
Proposition 3.2. Under Hypothesis 2.1 the system (2.13) admits a nonnegative maximal solution.

Proof. The functions F i for i = 1, 2, 3, 4 are Lipschitz and then according to the Cauchy-Lipschitz theorem, the system (2.13) admits a maximal solution. In addition, the function F = (F 1 , F 2 , F 3 , F 4 ) is quasi-positive and consequently the solutions are nonnegative (see [START_REF] Bothe | Global wellposedness for a class of reaction-advection-anisotropic-diffusion systems[END_REF] page 3).

Proposition 3.3. Let (c 1 , c 2 , c 3 , c 4 ) be the solution of (2.13), where c 0 i ≤ c in i for i = 3, 4. We consider the following quantities

M 1 = max µ f (c in 3 , c in 4 ) -k f ; µ m (c in 3 , c in 4 ) -k m -d , M 2 = min µ f (c 3 , c 4 ) -k f ; µ m (c 3 , c 4 ) -k m -d .
Thus,

• if M 1 ≤ 0, then γ -1 c 1 (t) + c 2 (t) ≤ γ -1 c 0 1 + c 0 2 . • if M 2 ≥ 0, then γ -1 c 1 (t) + c 2 (t) ≥ γ -1 c 0 1 + c 0 2 .

Proof.

Multiplying the first equation of the system (2.13) by γ -1 and then making the sum of the first and the second equaion of the system (2.13), we get

d(γ -1 c 1 + c 2 ) dt = γ -1 µ f (c 3 , c 4 ) -k f c 1 + µ m (c 3 , c 4 ) -k m -d c 2 .
Or the functions µ f and µ m are increasing, for each variable, so

M 2 γ -1 c 1 + c 2 ≤ d(γ -1 c 1 + c 2 ) dt ≤ M 1 γ -1 c 1 + c 2 .
If M 1 ≤ 0 then the function (γ -1 c 1 + c 2 )(t) decrease from where

∀t ≥ t 0 , (γ -1 c 1 + c 2 )(t) ≤ γ -1 c 0 1 + c 0 2 . If M 2 ≥ 0 then the function (γ -1 c 1 + c 2 )(t) increase from where ∀t ≥ t 0 , (γ -1 c 1 + c 2 )(t) ≥ γ -1 c 0 1 + c 0 2 .
Remark 3.4.

• The first point of Theorem 3.3 confirms that the concentration of bacteria will be lower than the initial data and therefore the bacteria will die out and the biodenitrification process will be unsuccessful. • The second point of Theorem 3.3 confirms that the concentration of bacteria will be greater than the initial data, and consequently the bacteria will succeed in colonizing the reactor and the biodenitrification process will be successful.

Corollary 3.5. Let (c 1 , c 2 , c 3 , c 4 ) be the solution of (2.13). Under the Hypothesis 2.1, for i = 3 or 4, we get

c i (t) ≤ c in i ∀t > 0.
Proof. Thanks to Theorem 3.2, for i = 1 or 2 we have c i ≥ 0 then

     dc 3 dt ≤ (c in 3 -c 3 )d S , dc 4 dt ≤ (c in 4 -c 4 )d S N , which implies      d(c 3 -c in 3 ) dt ≤ -(c 3 -c in 3 )d S , d(c 4 -c in 4 ) dt ≤ -(c 4 -c in 4 )d S N , equivalent to      d(c 3 -c in 3 ) dt ≤ (c 3 -c in 3 )(-d S ), d(c 4 -c in 4 ) dt ≤ (c 4 -c in 4 )(-d S N ).
By using the Gronwall's lemma we get

c 3 -c in 3 ≤ (c 0 3 -c in 3 ) exp(-d S t), c 4 -c in 4 ≤ (c 0 4 -c in 4 ) exp(-d S N t),
and by the fact that c 0 3 ≤ c in 3 and c 0 4 ≤ c in 4 we obtain

c 3 -c in 3 ≤ 0, c 4 -c in 4 ≤ 0,
Finally we get

c 3 ≤ c in 3 , c 4 ≤ c in 4 .
Proposition 3.6. There exists a value c * 3 such that for any c in 3 verifying the inequality c in 3 < c * 3 we get M 1 < 0, where

c * 3 = min (K f 2 + c in 4 )k f (K f 1 + µ f max ) c in 4 µ f max ; (K m 2 + c in 4 )(k m + d)(K m 1 + µ m max ) c in 4 µ m max . (3.1)
Proof. . The conditions for that M 1 < 0 are

µ f (c in 3 , c in 4 ) -k f < 0, (3.2)
and

µ m (c in 3 , c in 4 ) -k m -d < 0. (3.3)
According to (2.15), the first condition eq. (3.2) implies

µ f max c in 3 K f 1 + c in 3 × c in 4 K f 2 + c in 4 -k f < 0 ⇔ c in 3 K f 1 + µ f max < (K f 2 + c in 4 )k f c in 4 µ f max ⇔ c in 3 < (K f 2 + c in 4 )k f (K f 1 + µ f max ) c in 4 µ f max . (3.4)
The second condition eq. ( 3.3) implies

µ m max c in 3 K m 1 + c in 3 × c in 4 K m 2 + c in 4 -k m -d < 0 ⇔ c in 3 K m 1 + µ m max < (K m 2 + c in 4 )(k m + d) c in 4 µ m max ⇔ c in 3 < (K m 2 + c in 4 )(k m + d)(K m 1 + µ m max ) c in 4 µ m max (3.5)
From (3.4) and (3.5), we obtain

c in 3 < min( (K f 2 + c in 4 )k f (K f 1 + µ f max ) c in 4 µ f max ; (K m 2 + c in 4 )(k m + d)(K m 1 + µ m max ) c in 4 µ m max ) = c * 3 .
Proposition 3.7. Let (c 1 , c 2 , c 3 , c 4 ) be the solution of (2.13). The following properties hold true:

• If M 1 < 0, then lim t→∞ (γ -1 c 1 (t) + c 2 (t)) = 0. • If M 2 > 0, then lim t→∞ (γ -1 c 1 (t) + c 2 (t)) = c * > γ -1 c 0 1 + c 0 2 . • lim t→∞ c 3 (t) ≤ c in 3 . • lim t→∞ c 4 (t) ≤ c in 4 .
Proof.

• By multiplying the first equation of (2.12) by γ -1 and making the sum of the first and the second equation of the system (2.12), we get

d(γ -1 c 1 + c 2 ) dt = γ -1 µ f (c 3 , c 4 ) -k f c 1 + µ m (c 3 , c 4 ) -k m -d c 2 which implies M 2 γ -1 c 1 + c 2 ≤ d(γ -1 c 1 + c 2 ) dt ≤ M 1 γ -1 c 1 + c 2 .
By using the Gronwall's lemma we have 0

≤ (γ -1 c 1 + c 2 ) ≤ (γ -1 c 0 1 + c 0
2 )e M1t since M 1 < 0 and for t → ∞ then we obtain the result. • For the adherent bacteria and from eq. (2.5), we have c 1 ≤ ω ∞ . For the free bacteria and according to the growth function proprieties there exists a maximal concentration c max

2 such that c 2 ≤ c max 2
, which gives

γ -1 c 1 (t) + c 2 (t) ≤ γ -1 ω ∞ + c max 2 .
Or M 2 > 0, then the concentration (γ -1 c 1 + c 2 ) is increasing, respecting time, and bounded. Then there exists a value c * such that

lim t→∞ (γ -1 c 1 (t) + c 2 (t)) = c * > γ -1 c 0 1 + c 0 2 .
• According to Theorem 3.2, we have

c i ≥ 0, i = 1, 2 then      dc 3 dt ≤ (c in 3 -c 3 )d S , dc 4 dt ≤ (c in 4 -c 4 )d S N , (3.6)
which implies

     d(c 3 -c in 3 ) dt ≤ -(c 3 -c in 3 )d S , d(c 4 -c in 4 ) dt ≤ -(c 4 -c in 4 )d S N , (3.7)
equivalent to

     d(c 3 -c in 3 ) dt ≤ (c 3 -c in 3 )(-d S ), d(c 4 -c in 4 ) dt ≤ (c 4 -c in 4 )(-d S N ). (3.8)
Using the Gronwall's lemma, we get

c 3 -c in 3 ≤ (c 0 3 -c in 3 ) exp(-d S t), c 4 -c in 4 ≤ (c 0 4 -c in 4 ) exp(-d S N t), (3.9)
which implies

c 3 ≤ (c 0 3 -c in 3 ) exp(-d S t) + c in 3 , c 4 ≤ (c 0 4 -c in 4 ) exp(-d S N t) + c in 4 . (3.10)
Finally we obtain lim

t-→+∞ c 3 ≤ c in 3 and lim t-→+∞ c 4 ≤ c in 4
In what follows, we study the existence and stability of the steady state. By combining the first and the second equation of (2.12) in a single equation, the system (2.12) can be writing in the following form

             d(γ -1 c 1 + c 2 ) dt = γ -1 µ f (c 3 , c 4 ) -k f c 1 + µ m (c 3 , c 4 ) -k m -d c 2 , d(c 3 ) dt = (c in 3 -c 3 )d S -µm(c3,c4) Ym c 2 - µ f (c3,c4) Y f γ -1 c 1 , d(c 4 ) dt = (c in 4 -c 4 )d S N -Rµm(c3,c4) Ym c 2 - Rµ f (c3,c4) Y f γ -1 c 1 . (3.11)
We make a change of variable

• c = γ -1 c 1 + c 2 , • µ(c 3 , c 4 )c = µ f (c 3 , c 4 )γ -1 c 1 + µ m (c 3 , c 4 )c 2 , • Kc = k f γ -1 c 1 + (k m + d)c 2 , • µ(c 3 , c 4 ) Y c = µ m (c 3 , c 4 ) Y m c 2 + µ f (c 3 , c 4 ) Y f γ -1 c 1 .
The system (3.11) then can be written as follows with three variables

           dc dt = (µ(c 3 , c 4 ) -K)c, dc 3 dt = (c in 3 -c 3 )d S - µ(c 3 , c 4 ) Y c, dc 4 dt = (c in 4 -c 4 )d S N - Rµ(c 3 , c 4 ) Y c. (3.12)
The steady states, denoted by (c ⋆ , c ⋆ 3 , c ⋆ 4 ), of the system (3.12) verify

         0 = (µ(c ⋆ 3 , c ⋆ 4 ) -K)c ⋆ , 0 = (c in 3 -c ⋆ 3 )d S - µ(c ⋆ 3 , c ⋆ 4 ) Y c⋆ , 0 = (c in 4 -c ⋆ 4 )d S N - Rµ(c ⋆ 3 , c ⋆ 4 ) Y c⋆ . (3.13)
We remark that the first steady state is the washout steady state and is (0, c in 3 , c in 4 ). In addition, the other steady states satisfy

     0 = (c in 3 -c ⋆ 3 )d S - K Y c⋆ , 0 = (c in 4 -c ⋆ 4 )d S N - RK Y c⋆ , (3.14)
which implies

     0 = (c in 3 -c ⋆ 3 )d S - K Y c⋆ , c ⋆ 4 = c in 4 d S N -Rd S (c in 3 -c ⋆ 3 ) d S N , (3.15)
hence

     c⋆ = Y d S (c in 3 -c ⋆ 3 ) K , c ⋆ 4 = c in 4 d S N -Rd S (c in 3 -c ⋆ 3 ) d S N , (3.16)
From the system (3.15), we remark that the system (3.13) admits an infinity of steady states. The linearization of the system (3.13) near the washout steady state is given using the following Jacobian matrix:

A =      µ f (c in 3 , c in 4 ) -K 0 0 -µ f (c in 3 , c in 4 ) Y -d S 0 -µ f (c in 3 , c in 4 ) Y 0 -d S N      , Proposition 3.8.
The following properties hold true.

• If M 1 + M 2 < 0, then the washout steady state is stable. Proof.

• If M 1 + M 2 > 0,
• The condition M 1 + M 2 < 0 gives µ f (c in 3 , c in 4 ) -K < 0 and then the matrix A has three negative eigenvalues and therefore the washout steady state is stable.

• The condition M 1 + M 2 > 0 gives µ f (c in 3 , c in 4 ) -K > 0 and then the matrix A admits a non negative eigenvalue and therefore the washout steady state is unstable. Using the Theorem 3.3 and the fact that the system (3.13) admits a solution then we obtain the result.

Numerical algorithm for solving the system:

The functions of the second members of system 2.12 are given in the general form, by writing them as a function of t and of the unknown vector C(t) = (c 1 (t), c 2 (t), c 3 (t), c 4 (t)). However, in our case, the system can be written as an autonomous system, i.e. F i are only functions of C(t).

The numerical scheme for solving the problem (2.12) has been implemented in Python computational software through the fourth-order Runge-Kutta method (see Algorithm 1). It is an efficient method for solving problems of ordinary differential equations with initial values and it guarantees a stable computing time [START_REF] Lambert | Numerical methods for ordinary differential systems: the initial value problem[END_REF].

Algorithm 1 Numerical algorithm based on RK4 for solving the problem.

C ← C(t 0 ) (initial conditions). 5: define F i (C), i = 1, 2, 3, 4 6: N ←- t f -t0 ∆t + 1 7: t ← t 0 8: for k ← 1 to N do 9: K i 1 ← ∆t F i (C(t)), i = 1, 2, 3, 4 10: K i 2 ← ∆t F i C(t) + ∆t 2 K i 1 , i = 1, 2, 3, 4
11:

K i 3 ← ∆t F C(t) + ∆tK i 2 2 , i = 1, 2, 3, 4
12:

K i 4 ← ∆t F i C(t) + ∆tK i 3 , i = 1, 2, 3,
c i (t) ← c i (t) + 1 6 (K i 1 + 2 K i 2 + 2 K i 3 + K i 4 ), i = 1, 2, 3, 4 14:
Registration. Heuristic algorithm for improving the process:

d 0.1cm 2 /h µ f max 0.7h -1 d SN 0.1cm 2 /h Y m 0.5 d S 0.1cm 2 /h Y f 0.5 a 0.1 β 0.2h -1 α 0.2h -1 T 60 days w ∞ 60g/cm 2 R 1.2 γ 0.8ml/cm 2 K f 1 60 mg carbon/l k m 0.005h -1 K m 1 60 mg carbon/l k f 0.005h -1 K m 2 50 mg N O - 3 /l µ m max 0.7h -1 K f 2 50 mg N O - 3 /l
In the theoretical study of the problem as well as in the previous numerical tests, we showed the existence of a specific value of the injected carbon concentration from which we ensure the success of the biodenitrification process. In this part of numerical results, we propose a heuristic algorithm, based on previous results, to improve and control the biodenitrification process over time.

In what follow, we will present and compare 2 different schemes, to manage a total available quantity of c 3 , to improve the process (one of them using our proposed heuristic algorithm Algo. 2). We recall that c 3 is the nutrient that helps bacteria c 2 to increase, and thus the consumption of nitracts c 4 (contamination). In both schemes, as already shown, we consider the correct given path that gives the success of the biodenitrification process, ie c in 3 ≥ c ⋆ 3 , but in the heuristic algorithm scheme we make several injections of c i 3 until the disappearance of the total available quantity of c 3 to manage, or until the elimination of c 4 (i.e. con-

Conclusion

In this work we have developed a mathematical model dedicated to the biodenitrification process, taking into account the fixed and mobile bacteria. We studied asymptotic behaviour and provided some information for the solution of the problem. The numerical simulations in agreement with the theoretical results show the existence of a value of the injected carbon concentration from which we ensure the success of the biodenitrification process. This study also allowed us to provide a heuristic algorithm in order to improve the situation of biodegradation by minimizing as much as possible the nitrate (contaminant) over time in the environment and then improving the biodenitrification process. As a perspective, taking into account the obtained results, the model can be treated in a complete study of optimal control for choosing optimally the proportion of splitting and the corresponding re-introduction time(s) to obtain the best performances. This could be the matter of a future work. Moreover, the introduction of spatial heterogeneity in the model, in terms of a system of reaction diffusion "partial differential equations" instead of a system of "ordinary differential equations" would be a reasonable extension of the present work, taking into consideration that spatial heterogeneity is usually observed in real problem.

Figure 1 : 3 . 3 . 1 .

 1331 Figure 1: Graphs of multiplicative function µ i , i = f, m with µ i max = 0.7, K i 1 = 60 and K i 2 = 50.

4

 4

Figure 2 :

 2 Figure 2: Evolution of c 1 , c 2 , c 3 and c 4 with c in 3 = 104 ≥ c * 3 .

Figure 3 :

 3 Figure 3: Evolution of c, c 3 and c 4 with c in 3 = 104 ≥ c * 3 .

Figure 4 :

 4 Figure 4: Graphs of bacteria rate [(γ -1 )c 1 +c 2 ] and initial bacteria rate [(γ -1 )c 0 1 +c 0 2 ] with c in 3 = 104 ≥ c * 3 .

Figure 5 :

 5 Figure 5: Evolution of c 1 , c 2 , c 3 and c 4 with c in 3 = 0.5 < c * 3 .

Figure 6 :

 6 Figure 6: Evolution of c, c 3 and c 4 with c in 3 = 0.5 < c * 3 .

Figure 8 :

 8 Figure 8: Evolution of c 4 with scheme 1 by solving directly the system on [0,100] (i.e without using the heuristic algorithm). In this plot: c 4 (t 3 ) = 47.8190 and c 4 (100) = 49.5331

Figure 9 :

 9 Figure 9: Evolution of c 4 by using our proposed heuristic algorithm Algo. 2 (i.e scheme 2). Step1 is on[t 0 , t 1 ], Step2 is on [t 1 , t 2 ],Step3 is on [t 2 , t 3 ], and at t 3 : c 4 (t 3 ) = 47.6137.

  the washout steady state is unstable and there is another state of equilibrium ( ċ1 , ċ2 , ċ3 ) such that ċ1 ≥ γ -1 c 0 1 + c 0 2 and it is stable.

	April 5, 2023 12:34	output
		Asymptotic behavior analysis of a biodenitrification process Model 13

1 :

 1 Initialisation: t 0 (initial time),

	2:	t f (final time),
	3:	∆t (the step size),

4:

Table 2 :

 2 Parameters.

Numerical results

In this section we present some numerical results about the success of the biodenitrification process in agreement with the theoretical results the problem (2.13). The different parameters used are given in Table 2 and come from [START_REF] Abaali | Impact of Dual Substrate Limitation on Biodenitrification Modeling in Porous Media[END_REF]. From the same reference, we take the initial conditions of the system as indicated in Table 1.

In the theoretical study, we showed the existence of a value c * 3 of the injected carbon concentration from which we ensure the success of the biodenitrification process and the stability of the steady states. To verify that numerically, we perform two numerical experiences. Before that we start to compute the value of c * 3 . For example, with the parameter values given in the Table 2 and according to eq. (3.1), we get c * 3 = 0.6504mg/l. In the first test, we take a value of the amount of carbon injected c in 3 such that c in 3 ≥ c * 3 . For example with c in 3 = c 3 (0) = 104mg/l, we get Figure 2, Figure 3 and Figure 4 (evolution/behaviour of bacteria and substrates). Figure 2 shows the evolution of c 1 , c 2 , c 3 and c 4 . Figure 3 shows the evolution of c, c 3 and c 4 and then the stability of the system in agreement with Theorem 3.8. Figure 4 shows the evolution (γ -1 )c 1 + c 2 and (γ -1 )c 0 1 + c 0 2 . We remark in this figure that for c in 3 ≥ c * 3 , bacteria evolve in the reactor and then the success of the biodenitrification process which is consistent with the results obtained in the Theorem 3.3 and the Theorem 3.5. We also observe that from a certain time the four components stagnate which means that we obtain a stable steady state as is confirmed in the Theorem 3.7 and Theorem 3.8.

In the second test, we take a value of the amount of carbon injected c in 3 such that c in 3 ≤ c * 3 . For example with c in 3 = c 3 (0) = 0.5mg/l, we get Figure 6, Figure 5, and Figure 7. Figure 5 shows the evolution of c 1 , c 2 , c 3 and c 4 . Figure 6 shows the evolution of c, c 3 and c 4 and then the stability of the system in agreement with Theorem 3.8. Figure 7 shows the evolution (γ -1 )c 1 +c 2 and (γ -1 )c 0 1 +c 0 2 . We remark in this figure that for c in 3 ≤ c * 3 , bacteria disappear from the reactor and then the biodenitrification process fails which is consistent with the results obtained in the Theorem 3.3 and the Theorem 3.5,we note that in this case we have M 1 = -0.0011, the value given in the Theorem 3.3.We also observe that from a certain time bacteria concentration decreases towards 0 and the concentrations of carbon and nitrates respectively increase towards c in 3 and c in 4 which means that (0, 0, c in 3 , c in 4 ) is a stable steady state as is confirmed in the Theorem 3.7 and Theorem 3.8. Table 1: Initial conditions. As we have shown, this is related to the value of c 3 available in the biodegradable medium. We assume that the total available quantity to manage is denoted by

, where c added,i

3

, i = 1, 2, .. are the quantities that will be added during different injections over time. In this experiment, we consider for example that k f = 0.05 and c in 4 = 50 in order to have another value of c ⋆ 3 larger than the one we obtained previously. So in this case, we get c ⋆ 3 = 8.6714 (see line 6 in Algo. 2). In addition, we assume for example that c total 3 = 21.025. As we have shown, c in 3 must be greater than c ⋆ 3 and c (t0)

3 . Therefore, we start our heuristic algorithm by choosing c in 3 = c ⋆ 3 (see line 8 in Algo.2). Then, we solve the system until the equilibrium or when c 4 begins to grow. The stopping time of this resolution is denoted t ⋆ (see line 10 in Algo. 2). At this moment, we inject a quantity, denoted c added,1 3 such that we obtain c 3 (t 1 ) + c in 3 = 2c ⋆ (see line 13 in Algo. 2 ). The same step will be done until the consumption of the total available quantity of c 3 to manage (or when c 4 = 0).

Figure 8 and Figure 9 show the evolution of c 4 in the two schemes respectively, i.e without the use of the heuristic algorithm (scheme 1) and with the use of the heuristic algorithm (scheme 2), with the same total available quantity c total

3

. For scheme 1 the resolution was made on [0,100] and thus Figure 8 shows the evolution of c 4 on [0, 100], devised over two intervals in order to subsequently show the compared curves. However, with the heuristic algorithm Algo. 2, 3 steps were performed to manage the total available quantity c total 3 (see Figure 9). The first resolution was April 5, 2023 12:34 output Asymptotic behavior analysis of a biodenitrification process Model 19 performed on [t0 = 0, t1 = 39.9], and the quantity c added,1 3 = 2.6212 was injected (and at t 1 : c 4 (t 1 ) = 47.7757). The second resolution was performed on [t 1 , t 2 = 44.4], and the quantity c added,2 3 = 0.7933 was injected (and at t 2 : c 4 (t 2 ) = 47.6265). The third resolution was performed on [t 2 , t 3 = 45.8], and the quantity c added,3 3 = 0.2677 was injected (and at t 3 : c 4 (t 3 ) = 47.6137). At instant t 3 : all the quantity c total In terms of comparison: with scheme 1, we obtained at t = 45.8 = t 3 : c 4 (t 3 ) = 47.8190 =: v1, and then it increases to the value 49.5331 at t =100 (see Figure 8). However with the use of Algo. 2, we obtained c 4 (t 3 ) = 47.6137 = v2 < v1 and we can also follow our process of decreasing c 4 if the total quantity available c total 3 to manage is greater. Moreover, with scheme 2, at time t 3 , we have a quantity c 2 greater than that obtained in scheme 1: c 2 (t 3 ) = 1.0528 "scheme 1" Vs c 2 (t 3 ) = 1.1248 "scheme 2" (c 2 is the bacteria that helps to reduce nitrates c 4 ).

Algorithm 2 Heuristic algorithm for improving the process.

// c total 3 is the total quantity available to manage. 6: Calcul c ⋆ 3 by using eq. (3.1) and c in 4 .

7:

// We suppose that at t 0 : c 3 (0) > c ⋆ . . . 22: until the consumption of the total quantity available of c 3 to manage (or when c 4 = 0).