Sandrine Salmon

Sylvie Rebuffat

Soizic Prado

Michel Sablier

' Haese

Jian-Sheng Sun

Jean-François Ponge
email: ponge@mnhn.fr

Cyrille D'haese

D ' Haese

François Ponge

Chemical communication in springtails: a review of facts and perspectives

Keywords: Chemical communication, Collembola, Pheromones, Aggregation, Phylogeny, Community

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Chemical communication among soil organisms is a still poorly explored scientific domain, despite recent focus on the functional importance of chemical signalling in plant-microbial [START_REF] Peters | Phenolic compounds as regulators of gene expression in plant-microbe interactions[END_REF][START_REF] Sanon | Rhizosphere microbiota interfers with plant-plant interactions[END_REF][START_REF] Combès | Chemical communication between the endophytic fungus Paraconiothyrium variabile and the phytopathogen Fusarium oxysporum[END_REF], plant-faunal [START_REF] Blouin | Belowground organism activities affect plant aboveground phenotype, inducing plant tolerance to parasites[END_REF][START_REF] Rosenstiel | Sex-specific volatile compounds influence microarthropod-mediated fertilization of moss[END_REF][START_REF] Leach | Communication in the phytobiome[END_REF]) and plant-faunal-microbial interactions [START_REF] Bonkowski | Protozoa and plant growth: the microbial loop in soil revisited[END_REF][START_REF] Bais | The role of root exudates in rhizosphere interactions with plants and other organisms[END_REF][START_REF] Jousset | Secondary metabolite production facilitates establishment of rhizobacteria by reducing both protozoan predation and the competitive effects of indigenous bacteria[END_REF]). However, better knowledge of these interactions might illuminate our still imperfect knowledge of soil food webs (DeAngelis 2016). Ignored for a long time, positive interactions [START_REF] Bertness | Positive interactions in communities[END_REF][START_REF] Leigh Eg | The evolution of mutualism and other forms of harmony at various levels of biological organization[END_REF][START_REF] Stachowicz | Mutualism, facilitation, and the structure of ecological communities[END_REF]) are now recognized as a key process in the building and stability of communities, involving plants, microbes and animals [START_REF] Wall | Interactions underground: soil biodiversity, mutualism, and ecosystem processes[END_REF][START_REF] Callaway | Positive interactions among alpine plants increase with stress[END_REF][START_REF] Wisz | The role of biotic interactions in shaping distributions and realised assemblages of species: implications for species distribution modelling[END_REF]. Positive interactions are also acknowledged as a driving force of evolution [START_REF] Laland | Niche construction, co-evolution and biodiversity[END_REF][START_REF] Rosenberg | Microbes drive evolution of animals and plants: the hologenome concept[END_REF][START_REF] Prinzing | Benefits from living together? Clades whose species use similar habitats may persist as a result of eco-evolutionary feedbacks[END_REF]. Negative and positive interactions among species are at least partly governed by chemical signals [START_REF] Callaway | The detection of neighbors by plants[END_REF][START_REF] Taga | Chemical communication among bacteria[END_REF]. This makes chemical communication a key factor of species coexistence (Vet 1999; [START_REF] Austin | There's no place like home ? An exploration of the mechanisms behind plant litter-decomposer affinity in terrestrial ecosystems[END_REF].

Chemical signals, based on volatile (olfactory) or contact molecules (O 'Connell 1986), are known to play a prominent role in positive as well as negative interactions between soil organisms [START_REF] Wenke | Belowground volatiles facilitate interactions between plant roots and soil organisms[END_REF][START_REF] Lavelle | Ecosystem engineers in a self-organized soil: a review of concepts and future research questions[END_REF]. In soil invertebrates, they play a direct role in mating [START_REF] Zizzari | Love at first sniff: a spermatophore-associated pheromone mediates partner attraction in a collembolan species[END_REF], search for food [START_REF] Brückner | Track the snaff: olfactory cues shape foraging behaviour of decomposing soil mites (Oribatida)[END_REF], search of safe sites for growth and reproduction [START_REF] Kuenen | Olfactory orientation in some land-isopods (Oniscoidea, Crustacea)[END_REF], aggregation [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF], avoidance of death risk (Nilsson and Bengtsson 2004a), prey location and catching [START_REF] Huber | Prey attraction and immobilization by allomone from nymphs of Womersia strandtmanni (Acarina: Trombiculidae)[END_REF]) and deterrence of predators [START_REF] Smolanoff | Polyzonimine: a novel terpenoid insect repellent produced by a millipede[END_REF]. Some chemical compounds excreted by soil animals have hormonal properties which stimulate the growth of plants and their resistance against stress factors and pathogens [START_REF] Puga-Freitas | A review of the effects of soil organisms on plant hormone signalling pathways[END_REF]. All these chemical signals are known as semiochemicals and are classified into two broad classes depending on their intra-or interspecific action. Pheromones are intraspecific compounds capable of acting outside the body of the secreting individual to impact the behaviour of a receiving individual of the same species (Karlson and Lüscher 1959). These chemical signals allow individuals of the same species to interact between them in the course of aggregation (aggregation pheromones), mating (sex pheromones) or by signalling a risk (alarm pheromones) [START_REF] Wilson | Chemical communication among animals[END_REF][START_REF] Shorey | Behavioural responses to insect pheromones[END_REF]. Conversely, allelochemicals are interspecific molecules that are grouped according to the benefits they provide to both producer and receiver. Allomones, kairomones and synomones are produced by a given species and act on another species [START_REF] Sbarbati | Allelochemical communication in vertebrates: kairomones, allomones and synomones[END_REF].

They are favourable either to the producer (allomones, e.g. repulsive odours) or to the receiver (kairomones, e.g. fungal odours), or to both of them (synomones). Springtails (Collembola) are among the most widespread and abundant groups of soil arthropods (20,000 to 400,000 individuals.m -²), and as such they impact ecosystem functioning [START_REF] Seastedt | The role of microarthropods in decomposition and mineralization process[END_REF][START_REF] Hopkin | Biology of the springtails (Insecta: Collembola)[END_REF]. They have long been considered as insects, but now belong to a separate taxonomic class. Collembola are phylogenetically sisters of other hexapods, which comprise mainly the insects. They are divided into four orders: Entomobryomorpha (~4000 known species), Poduromorpha (~3300 species), Symphypleona (~1200 species) and Neelipleona (~50 species). These microarthropods (0.25-3 mm in length) live mainly, but not exclusively, in soil and litter. The main role of Collembola is the decomposition of organic matter and the recycling of nutrients that can subsequently be taken up by plants [START_REF] Vries | Soil food web properties explain ecosystem services across European land use systems[END_REF]. They exert a direct action on organic matter decomposition and recycling of nutrients through the consumption of litter. Indirect effects occur through the regulation and dispersion of microorganisms (fungi and bacteria on which they feed), which are responsible for the mineralization of organic matter. As the role of Collembola varies according to species, the chemical communication regulating the interactions between Collembola species indirectly impacts soil functioning [START_REF] Bardgett | Belowground biodiversity and ecosystem functioning[END_REF].

Collembola are known for a long time for their aggregative behaviour [START_REF] Glasgow | A population study of subterranean soil Collembola[END_REF].

Aggregation is mediated by environmental heterogeneity [START_REF] Poole | An ecological study of the Collembola in a coniferous forest soil[END_REF][START_REF] Usher | Some properties of the aggregations of soil arthropods: Collembola[END_REF], mass dispersal [START_REF] Lyford | Overland migration of Collembola (Hypogastrura nivicola Fith) colonies[END_REF], selection of proper habitats [START_REF] Joosse | The formation and biological significance of aggregations in the distribution of Collembola[END_REF]Verhoef and Nagelkerke 1977;[START_REF] Salmon | Distribution of Heteromurus nitidus (Hexapoda, Collembola) according to soil acidity: interactions with earthworms and predator pressure[END_REF][START_REF] Auclerc | Experimental assessment of habitat preference and dispersal ability of soil springtails[END_REF]) and food resources [START_REF] Ponge | Vertical distribution of Collembola (Hexapoda) and their food resources in organic horizons of beech forests[END_REF][START_REF] Salmon | The impact of earthworms on the abundance of Collembola: improvement of food resources or of habitat?[END_REF][START_REF] Chauvat | Foraging patterns of soil springtails are impacted by food resources[END_REF].

Evidence for the existence of sex (Waldorf 1974a) and aggregation pheromones [START_REF] Mertens | Aggregation pheromone in Hypogastrura viatica (Collembola)[END_REF]Verhoef et al. 1977a, b) stimulated studies on olfactory signals involved in communication between congeners and mates in an environment and a zoological group such as soildwelling species where visual systems are rudimentary [START_REF] Leinaas | Synchronized moulting controlled by communication in group-living Collembola[END_REF][START_REF] Pfander | Chemical communication in Ceratophysella sigillata (Collembola: Hypogastruridae): intraspecific reaction to alarm substances[END_REF][START_REF] Porco | Sex pheromone in Tetrodontophora bielanensis (Waga, 1842) (Collembola: Onychiuridae): indirect reproduction mediated by cuticular compounds[END_REF]. Remarkable fossils preserved in Early Cretaceous amber allow describing a courtship behaviour in a Symphypleona and an aggregative behaviour in an Entomobryomorpha, thus suggesting the presence of pheromones more than 105 Ma ago [START_REF] Sánchez-García | Mating and aggregative behaviors among basal hexapods in the Early Cretaceous[END_REF]). However, information on the importance of chemical communication for coexistence or avoidance and the evolution of collembolan species is still scarce (Nilsson andBengtsson 2004b, Wertheim et al. 2005).

Given the accumulated data it is now time to review the present state of knowledge on chemical communication in Collembola, and make a prospect of ecological and evolutionary perspectives.

The use of pheromones, allomones and kairomones by Collembola

Collembola are endowed with a strong sensitivity to olfactory signals. Receptors are mostly localized on the antenna, in particular the two apical segments. This has been demonstrated directly by anatomical investigations [START_REF] Keil | Morphology and development of the peripheral olfactory organs[END_REF][START_REF] Kollmann | Brain organization in Collembola (springtails)[END_REF], total and partial antennal amputations [START_REF] Waldorf | Antennal amputation in Sinella curviseta (Collembola: Entomobryidae)[END_REF]Verhoef et al. 1977b), and indirectly by checking antennal cleaning behaviour (Waldorf 1974b). However, the chemosensitivity of other cephalic organs like postantennal organs cannot be excluded [START_REF] Karuhize | The structure of the postantennal organ in Onychiurus sp. (Insecta: Collembola) and its connection to the central nervous system[END_REF][START_REF] Altner | The postantennal organ: a specialized unicellular sensory input to the protocerebrum in apterygotan insects (Collembola)[END_REF]. Although this has not been demonstrated as yet on these tiny arthropods, it can be postulated that the double nature of the antennae allows them not only to detect chemical signals at distance, but also to evaluate the direction of the odour source like in insects [START_REF] Bell | Chemo-orientation[END_REF]. This might explain directional movements towards preferred food [START_REF] Auclerc | Assessment of movement patterns in Folsomia candida (Hexapoda: Collembola) in the presence of food[END_REF], as it has been observed in aboveground invertebrates [START_REF] Schooley | Finding habitat patches and directional connectivity[END_REF]. Furthermore, pheromones produced by collembolans can also be diverted from their purpose and used as kairomones by predators to detect their prey [START_REF] Kielty | Behavioral responses of three species of ground beetle (Coleoptera: Carabidae) to olfactory cues associated with prey and habitat[END_REF].

Aggregation pheromones

Pheromones play a key role in the aggregation process of Collembola, such as in numerous other arthropods [START_REF] Ishii | An aggregation pheromone of the German cockroach Blatella germanica L. (Orthoptera: Blatellidae). I. Site of the pheromone production[END_REF][START_REF] Wertheim | Pheromone-mediated aggregation in nonsocial arthropods: an evolutionary ecological perspective[END_REF]. They have been first evidenced by their effects, e.g. the fact that a place previously conditioned by the presence of one or more individuals becomes attractive to conspecifics [START_REF] Mertens | Aggregation pheromone in Hypogastrura viatica (Collembola)[END_REF]Verhoef et al. 1977a, b;[START_REF] Mertens | Aggregation pheromone in Orchesella cincta (Collembola)[END_REF][START_REF] Joosse | Evidence for the presence of aggregation pheromones in Onychiurus armatus (Collembola), a pest insect in sugar beet[END_REF]. When extracted from a conditioned substrate with an appropriate solvent, pheromones were able to reproduce the same phenomenon [START_REF] Manica | An aggregation pheromone in the intertidal collembolan Anurida maritima[END_REF]). However, these experiments did not demonstrate that the molecules were remotely detected. In fact, erratic movements of an animal may favour random encounters with conditioned places, upon which the animal ceases to move: only contact (non-volatile) molecules are required in this type of chemical signalling. Other experiments showed that individuals attracted and induced the aggregation of conspecifics without any contact or visual signal, suggesting a volatile signal [START_REF] Leonard | Aggregative behaviour in Folsomia candida (Collembola: Isotomidae), with respect to previous conditioning[END_REF][START_REF] Krool | Reproduction, development, pheromone secretion in Heteromurus nitidus Templeton 1835 (Collembola, Entomobryidae)[END_REF]. Motility decreased then ceased totally at high concentration, i.e. at short distance from conspecifics [START_REF] Shorey | Behavioural responses to insect pheromones[END_REF][START_REF] Mertens | Aggregation pheromone in Hypogastrura viatica (Collembola)[END_REF].

Aggregation pheromones have been evidenced in a large number of collembolan species (Table 1), encompassing Poduromorpha and Entomobryomorpha living above or below ground in various environments. Surprisingly, Neelipleona and Symphypleona are absent from this list, when excluding short courtships displayed between sexual partners [START_REF] Mc | Les parades sexuelles primitives chez les Collemboles Symphypléones[END_REF][START_REF] Kozlowski | Ritual behaviors associated with spermatophore transfer in Deuterosminthurus bicinctus (Collembola: Bourletiellidae)[END_REF].

Aggregation pheromones allow encounters between individuals of the same species in places favourable for feeding [START_REF] Barra | Experimental study of aggregations during the development of Pseudosinella impediens (Collembola, Entomobryidae)[END_REF][START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF], mating [START_REF] Joosse | The formation and biological significance of aggregations in the distribution of Collembola[END_REF]Verhoef and Nagelkerke 1977), moulting [START_REF] Joosse | On the aggregational habits of surface dwelling Collembola[END_REF]Verhoef and Nagelkerke 1977) and ovipositing (Verhoef and Nagelkerke 1977;[START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]. Aggregation in favourable microniches is of paramount importance to limit water losses [START_REF] Joosse | The formation and biological significance of aggregations in the distribution of Collembola[END_REF][START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]. Limiting water loss is more particularly required during moulting periods which happen repetitively all along the life of Collembola (Verhoef and Nagelkerke 1977). Synchronized moulting controlled by chemical communication has been demonstrated in group-living hypogastrurids [START_REF] Leinaas | Synchronized moulting controlled by communication in group-living Collembola[END_REF]. It also appears that species that are most sensitive to desiccation would be also most prone to aggregation [START_REF] Joosse | The formation and biological significance of aggregations in the distribution of Collembola[END_REF][START_REF] Joosse | Ecological aspects of aggregation in Collembola[END_REF]. Awaiting further confirmation on a larger array of species, parthenogenetic species are less prone to aggregation than sexual species [START_REF] Chernova | Ecological significance of parthenogenesis in Collembola[END_REF]. Although aggregations comprising different species are commonplace in Collembola [START_REF] Poole | An ecological study of the Collembola in a coniferous forest soil[END_REF][START_REF] Hale | A population study of moorland Collembola[END_REF][START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF][START_REF] Joosse | The formation and biological significance of aggregations in the distribution of Collembola[END_REF], only two cross-species attractions by pheromones have been demonstrated (Tomocerus minor attracted to Orchesella villosa and O. cincta) out of the three interactions tested in two independent studies (Verhoef et al. 1977a[START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF].

Aggregation pheromones are produced by adults of both sexes (Verhoef et al. 1977a;[START_REF] Krool | Reproduction, development, pheromone secretion in Heteromurus nitidus Templeton 1835 (Collembola, Entomobryidae)[END_REF], even during starvation [START_REF] Verhoef | Releaser and primer pheromones in Collembola[END_REF] and moulting periods (Verhoef and Nagelkerke 1977;Verhoef et al. 1977a). Their production is negatively abundance-dependent [START_REF] Verhoef | Releaser and primer pheromones in Collembola[END_REF]. They are deposited on the substrate [START_REF] Mertens | Aggregation pheromone in Hypogastrura viatica (Collembola)[END_REF] and some of them are seemingly volatile, being attractive at distance (Verhoef et al. 1977a;[START_REF] Leonard | Aggregative behaviour in Folsomia candida (Collembola: Isotomidae), with respect to previous conditioning[END_REF][START_REF] Krool | Reproduction, development, pheromone secretion in Heteromurus nitidus Templeton 1835 (Collembola, Entomobryidae)[END_REF][START_REF] Manica | An aggregation pheromone in the intertidal collembolan Anurida maritima[END_REF]Nilsson and Bengtsson 2004b). They are emitted through the intestine, as being found in the faeces [START_REF] Verhoef | Releaser and primer pheromones in Collembola[END_REF], but probably also through the cuticle, as they were detected in extracts from entire uncrushed individuals [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF].

Sex pheromones

Sex pheromones are produced by females, conditioning the deposition of spermatophores by males (Waldorf 1974a) and attracting them [START_REF] Porco | Sex pheromone in Tetrodontophora bielanensis (Waga, 1842) (Collembola: Onychiuridae): indirect reproduction mediated by cuticular compounds[END_REF]) while spermatophores produced by males are also known to attract females [START_REF] Zizzari | Love at first sniff: a spermatophore-associated pheromone mediates partner attraction in a collembolan species[END_REF]. They are present in the cuticle [START_REF] Porco | Sexual discrimination with cuticular lipids in Schoetella ununguiculata (Tullberg, 1869) (Collembola: Hypogastruridae)[END_REF][START_REF] Porco | Sex pheromone in Tetrodontophora bielanensis (Waga, 1842) (Collembola: Onychiuridae): indirect reproduction mediated by cuticular compounds[END_REF] and in spermatophores [START_REF] Zizzari | Love at first sniff: a spermatophore-associated pheromone mediates partner attraction in a collembolan species[END_REF]. Sex pheromones are thought to mediate sexual selection [START_REF] Hedlund | Mate choice and male competition in Orchesella cincta (Collembola)[END_REF][START_REF] Stam | Distant lovers: spermatophore deposition and destruction behaviour by male springtails[END_REF][START_REF] Zizzari | Female preference and fitness benefits of mate choice in a species with dissociated sperm transfer[END_REF]. Although still largely speculative in the studied group, the idea of an evolutionary role of sex pheromones has been considered by entomologists [START_REF] Greenfield | Moth sex pheromones: an evolutionary perspective[END_REF][START_REF] Symonds | The evolution of pheromone diversity[END_REF], but deeper research in this direction is needed before clear hypotheses could emerge.

Alarm pheromones and allomones

Alarm pheromones have been recorded in a number of species, belonging to Poduromorpha, Entomobryomorpha and Symphypleona (Table 2). They are emitted after rupture of the cuticle when springtails die (necromones), or in the presence of predators. They elicit a dispersive behaviour of conspecifics [START_REF] Purrington | Alarm pheromone in a gregarious poduromorph collembolan (Collembola: Hypogastruridae)[END_REF]Nilsson and Bengtsson 2004a). Defence strategies vary according to the taxonomic group considered. Some collembolan species have an aggregative behaviour in the presence of a predator, while others move away from each other. This suggests various behavioural effects of alarm pheromones [START_REF] Negri | Spatial distribution of Collembola in presence and absence of a predator[END_REF]).

Predator-repulsive allomones have been shown to be emitted by podurid species, e.g.

Ceratophysella denticulata against a rove beetle [START_REF] Bitzer | Benzoic acid derivatives in a hypogastrurid collembolan: temperature-dependent formation and biological significance as deterrents[END_REF], or the giant springtail Tetradotonphora bielanensis against a ground beetle [START_REF] Dettner | Chemical defense of giant springtail Tetrodotonphora bielanensis (Waga) (Insecta: Collembola)[END_REF]. The emission of predatorrepulsive allomones occurs when predatory attack is simulated with a needle as in Protaphorura lata [START_REF] Usher | A defence mechanism in Onychiurus (Collembola, Onychiuridae)[END_REF]. However, the odour of springtails may also be used by some of their predators as a cue to catch them, such as in the ground beetle Nebria brevicollis [START_REF] Kielty | Behavioral responses of three species of ground beetle (Coleoptera: Carabidae) to olfactory cues associated with prey and habitat[END_REF].

Kairomones

Preferences for some food items are known for a long time [START_REF] Knight | A preliminary study of the dietary requirements of Tomocerus (Collembola)[END_REF]. Despite a wide array of food items potentially used by the same species, preferences may differ to a large extent among species [START_REF] Jørgensen | Selective foraging of fungi by collembolans in soil[END_REF][START_REF] Ferlian | Trophic niche differentiation and utilisation of food resources in collembolans based on complementary analyses of fatty acids and stable isotopes[END_REF]. It has been shown that selection of or attraction to preferred food can be mediated by olfaction. Indeed, it occurs in advance, and even in the absence of tasting [START_REF] Salmon | Earthworm excreta attract soil springtails: laboratory experiments on Heteromurus nitidus (Collembola: Entomobryidae)[END_REF][START_REF] Staaden | Olfactory cues associated with fungal grazing intensity and secondary metabolite pathway modulate Collembola foraging behaviour[END_REF][START_REF] Chauvat | Foraging patterns of soil springtails are impacted by food resources[END_REF]). Attraction to preferred food is associated with directional and rapid movements towards the odour source upon detection at a distance varying from 1 to 40 cm [START_REF] Bengtsson | Food-and density-dependent dispersal: evidence from a soil collembolan[END_REF][START_REF] Salmon | Earthworm excreta attract soil springtails: laboratory experiments on Heteromurus nitidus (Collembola: Entomobryidae)[END_REF][START_REF] Auclerc | Assessment of movement patterns in Folsomia candida (Hexapoda: Collembola) in the presence of food[END_REF]. Attraction is followed by cessation of movement in the presence of food [START_REF] Westerberg | The effect on dispersal from complex correlations in small-scale movement[END_REF].

Microcosm experiments demonstrated that fungal strains are preferred or avoided when their odour is attractive or repellent, respectively [START_REF] Bengtsson | Fungal odour attracts soil Collembola[END_REF][START_REF] Hedlund | Fungal odour discrimination in two sympatric species of fungivorous collembolans[END_REF][START_REF] Sadaka-Laulan | Feeding preferences of the collembolan Onychiurus sinensis for fungi colonizing holm oak litter (Quercus rotundifolia Lam.)[END_REF]. They also demonstrated that discrimination between preferred and discarded fungal colonies is mediated by secondary metabolites [START_REF] Staaden | Olfactory cues associated with fungal grazing intensity and secondary metabolite pathway modulate Collembola foraging behaviour[END_REF]. Preferred fungal strains maximize fitness parameters, with the exception of a few erroneous choices [START_REF] Sadaka-Laulan | Feeding preferences of the collembolan Onychiurus sinensis for fungi colonizing holm oak litter (Quercus rotundifolia Lam.)[END_REF][START_REF] Heděnec | Grazing preference and utilization of soil fungi by Folsomia candida (Isotomidae: Collembola)[END_REF]. The volatile compounds present in the odour of preferred fungal strains were isolated and tested. An attractive effect of some specific compounds was demonstrated on the onychiurid Protaphorura armata [START_REF] Bengtsson | Selective odor perception in the soil Collembola Onychiurus armatus[END_REF]. A strong influence of fungal defence chemicals on the aggregative behaviour of the fungus-feeding Folsomia candida has also been shown by using wild-type fungi and modified strains lacking the ability to secrete secondary metabolites [START_REF] Stötefeld | Fungal chemical defence alters density-dependent foraging behaviour and success in a fungivorous soil arthropod[END_REF].

Collembolan olfactory cues are also provided by mosses. A win-win process is mediated by moss volatile compounds which act as kairomones. They increase moss fertilization success by favouring collembolan search for preferred habitat [START_REF] Gerson | Moss-arthropod associations[END_REF]) and even for preferred food, although it has not been proven that sperm cells can act as a reward [START_REF] Rosenstiel | Sex-specific volatile compounds influence microarthropod-mediated fertilization of moss[END_REF].

Kairomones are also emitted by leaf litter, repellence to the collembolan Sinella coeca being induced by isoprene (2-methylbuta-1,3-diene) [START_REF] Michelozzi | Eco-ethological analysis of the interaction between isoprene and the behaviour of Collembola[END_REF]. Repellence mediated by the smell of artificial compounds, such as the herbicide 2,4,5-T (2,4,5-trichlorophenoxyacetic acid), has also been demonstrated in Onychiurus quadriocellatus [START_REF] Eijsackers | Side effects of the herbicide 2,4,5-T affecting mobility and mortality of the springtail Onychiurus quadriocellatus Gisin (Collembola)[END_REF]. The odour of soil animals, e.g. earthworm themselves and their excreta, can be perceived at distance and used as a cue to find proper food and habitat, including protection from predators [START_REF] Salmon | Earthworm excreta attract soil springtails: laboratory experiments on Heteromurus nitidus (Collembola: Entomobryidae)[END_REF][START_REF] Salmon | Earthworm excreta (mucus and urine) affect the distribution of springtails in forest soils[END_REF][START_REF] Salmon | The impact of earthworms on the abundance of Collembola: improvement of food resources or of habitat?[END_REF][START_REF] Salmon | Earthworms and Collembola relationships: effects of predatory centipedes and humus forms[END_REF].

All these results point to a fundamental role of chemical communication in the functional relationships between soil organisms (DeAngelis 2016) and co-adaptation experiments even suggest that they could play a key role in the co-evolution between insects and fungi [START_REF] Biedermann | Evolutionary chemo-ecology of insect-fungus interactions: still in its infancy but advancing[END_REF].

It is here suggested that attractive, neutral and repellent odours could explain the so-called 'home-field advantage hypothesis', i.e. the optimization of litter decomposition when decomposers are processing litter in their own habitat [START_REF] Perez | Home-Field Advantage: a matter of interaction between litter biochemistry and decomposer biota[END_REF][START_REF] Austin | There's no place like home ? An exploration of the mechanisms behind plant litter-decomposer affinity in terrestrial ecosystems[END_REF][START_REF] Veen | Litter quality and environmental controls of home-field advantage effects on litter decomposition[END_REF]. They could also explain 'natal habitat preference induction', i.e. the propensity of animal species to disperse in an environment that contains stimuli similar to those found in their original environment [START_REF] Stamps | Someplace like home: experience, habitat selection and conservation biology[END_REF].

Composition of collembolan semiochemicals

Most insect pheromones are made of mixtures of various molecules, the proportion of which varies among species and within and among populations of the same species [START_REF] Wertheim | Pheromone-mediated aggregation in nonsocial arthropods: an evolutionary ecological perspective[END_REF][START_REF] Symonds | The evolution of pheromone diversity[END_REF]. In various species, aggregation and sex pheromones are alcohols, esters, amino acids, fatty acids, aldehydes and ketones commonly linked to a hydrophobic alkyl chain. They are often derived from fatty acids, or from phenol (O 'Connell 1986;[START_REF] Leal | Evolution of sex pheromone communication in plant-feeding scarab beetles[END_REF][START_REF] Tillman | Insect pheromones: an overview of biosynthesis and endocrine regulation[END_REF].

Recognition mechanisms based on insect pheromones are highly species-specific [START_REF] Roelofs | Pheromones and evolutionary relationships of Tortricidae[END_REF]De Bruyne and Baker 2008;[START_REF] Kaissling | Pheromone reception in insects: the example of silk moths[END_REF]. This suggests that in collembolan too, the ratio of chemical compounds found in pheromones allows a fine tuning between emitters and receptors [START_REF] Bruce | Insect host location: a volatile situation[END_REF][START_REF] Lei | Responses of protocerebral neurons in Manduca sexta to sexpheromone mixtures[END_REF].

In Collembola, various compounds have been isolated among alarm pheromones and necromones (chemicals arising from dead individuals). Most of them are aromatic compounds and fatty acids [START_REF] Purrington | Alarm pheromone in a gregarious poduromorph collembolan (Collembola: Hypogastruridae)[END_REF][START_REF] Messer | Phenolic compounds in Neanura muscorum (Collembola, Neanuridae) and the role of 1,3-dimethoxybenzene as an alarm substance[END_REF]Nilsson and Bengtsson 2004b). Some compounds commonly isolated from collembolan alarm pheromones, such as linoleic acid, have a wide spectrum of action, being accordingly repulsive to ants, aphids, beetles and woodlice [START_REF] Greenway | Response of Myzus persicae to components of aphid extracts and to carboxylic acids[END_REF][START_REF] Nijholt | Pine oil and oleic acid delay and reduce attacks on logs by ambrosia beetles (Coleoptera: Scolytidae)[END_REF][START_REF] Howard | Chemical ecology and biochemistry of insect hydrocarbons[END_REF][START_REF] Rollo | Fatty acid necromones for cockroaches[END_REF]). Three compounds have been isolated from collembolan repulsive allomones (Table 3). The onychiurids T. bielanensis, Deuteraphorura scotaria and Onychiurus circulans were shown to excrete 2,3-dimethoxy-pyrido[2,3b] pyrazine, a heterocyclic aromatic compound. However, the repellent effect of this molecule has only been observed in T. bielanensis. Furthermore, all onychiurids do not produce this molecule, since it has not been found in P. armata [START_REF] Dettner | Chemical defense of giant springtail Tetrodotonphora bielanensis (Waga) (Insecta: Collembola)[END_REF]Nilsson and Bengtsson 2004b). These compounds were identified in fluids excreted by secretory organs (pseudocella) and were not found in food and faeces [START_REF] Dettner | Chemical defense of giant springtail Tetrodotonphora bielanensis (Waga) (Insecta: Collembola)[END_REF]. The hypogastrurid Ceratophysella denticulata was shown to excrete 3-hydroxy-4,5-dimethoxybenzoic acid (DmHbA) and 4-hydroxy-3,5-dimethoxybenzoic acid, better known as syringic acid [START_REF] Bitzer | Benzoic acid derivatives in a hypogastrurid collembolan: temperature-dependent formation and biological significance as deterrents[END_REF]). These phenolic acids are known for their strongly irritating properties [START_REF] Hayashi | A quantitative structureactivity relationship study of the skin irritation potential of phenols[END_REF]. The strong resemblance between repellent substances emitted by arthropods and allelochemicals produced by plants against herbivores (and other plants, i.e. allelopathy) has been highlighted a long time ago [START_REF] Smolanoff | Polyzonimine: a novel terpenoid insect repellent produced by a millipede[END_REF]. It can be explained by the use of plant semiochemicals as precursors of insect pheromones [START_REF] Reddy | Interactions of insect pheromones and plant semiochemicals[END_REF][START_REF] Stökl | Evolutionary origin of insect pheromones[END_REF][START_REF] Blomquist | Pheromone production in bark beetles[END_REF]). Nevertheless, [START_REF] Malcicka | De novo synthesis of linoleic acid in multiple Collembola species[END_REF][START_REF] Malcicka | An evolutionary perspective on linoleic acid synthesis in animals[END_REF] showed that, similar to some other soil invertebrates, some collembolan species were able to synthesize de novo linoleic acid, a common component of alarm pheromones, while other arthropods needed to get it from their food. The attraction of males of T. bielanensis by the odour of females was shown to be mediated by the polar fraction of cuticular extracts, e.g. fatty acids, ketones, alcohols and esters [START_REF] Porco | Sex pheromone in Tetrodontophora bielanensis (Waga, 1842) (Collembola: Onychiuridae): indirect reproduction mediated by cuticular compounds[END_REF].

The (Z)-14-tricosenol hydrocarbon was shown to be the main volatile compound emitted by spermatophores (spermatic capsules deposited on the substrate) of the entomobryid Orchesella cincta [START_REF] Zizzari | Love at first sniff: a spermatophore-associated pheromone mediates partner attraction in a collembolan species[END_REF]). We did not find any study describing unambiguously the identification and biological activity of aggregation pheromones isolated from collembolan extracts. The only exception is a recent study in which potentially attractive cuticular extracts were isolated and fatty acids proposed as the pheromones [START_REF] Liu | Chemical attraction of conspecifics in Folsomia candida (Collembola)[END_REF]. However, the authors neither demonstrated unambiguously the role of the extracts in aggregation nor the emission or excretion of the compounds in the environment. Hence in the conducted experiments, the extracted fatty acids could have been dissolved from the animal body [START_REF] Malcicka | De novo synthesis of linoleic acid in multiple Collembola species[END_REF], making the results questionable. In the search of necromones (repellent molecules), synthetic palmitic acid was shown to be attractive to P. armata and was present in P. armata, Folsomia candida, D. scotaria and O. circulans (Nilsson and Bengtsson 2004a, b). However, these authors could not demonstrate that P. armata was attractive to O. circulans and D. scotaria. It suggests that the sensitivity to a given concentration of palmitic acid can vary according to species. Furthermore, fatty acids and fatty acid-related compounds (e.g. n-alkanes, nalcohols) are well-known as persistent components of soil organic matter [START_REF] Marseille | n-Alkanes and free fatty acids in humus and A1 horizons of soils under beech, spruce and grass in the Massif-Central (Mont-Lozère), France[END_REF].

Consequently, distinguishing their potential role in aggregation phenomena is very challenging [START_REF] Baldock | Soil organic matter[END_REF]Nelson 2000, Reiffarth et al. 2016).

Despite their lack of chemical identification, diverse solubility and polarity properties of some collembolan aggregation pheromones have been established as yet. Those produced by Hypogastrura viatica and O. cincta [START_REF] Mertens | Aggregation pheromone in Orchesella cincta (Collembola)[END_REF]) are soluble in water while those emitted by the semiaquatic Anurida maritima are insoluble in water but soluble in methanol [START_REF] Manica | An aggregation pheromone in the intertidal collembolan Anurida maritima[END_REF].

Aggregation pheromones extracted from Cryptopygus antarcticus and Friesea grisea were more soluble in methanol than in water and acetone, and were insoluble in hexane [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF].

Those extracted from P. armata and F. candida were soluble in cyclohexane, hexane or dichloromethane (Nilsson and Bengtsson 2004a;[START_REF] Liu | Chemical attraction of conspecifics in Folsomia candida (Collembola)[END_REF]. This wide range of polarities, while pointing to a common poor solubility in water, highlights the diversity of chemical compounds that can play a role as aggregation pheromones among collembolan species, as well as the difficulty to establish an efficient and common protocol for their isolation. The chemical characterization of such pheromones is still also challenging, most probably because there are produced at very low concentration. This implies a high number of individuals necessary for pheromone collection [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]. Olfactory attraction tests allowed deducing that aggregation pheromones are at least composed of some volatile molecules, while the effect of non-volatile molecules acting by contact cannot be excluded [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]. Given our knowledge of the chemical compositions of insect pheromones, it can be reasonably hypothesized that collembolan pheromones should be also complex mixtures of a broad range of chemical compounds [START_REF] Wertheim | Pheromone-mediated aggregation in nonsocial arthropods: an evolutionary ecological perspective[END_REF].

Ecological and evolutionary perspectives in chemical communication among Collembola

Given that our knowledge on chemical communication among arthropods is mainly based on insect pheromones, we will present some 'insect' facts which should be of interest for future studies of Collembola semiochemicals. Among the most attractive research directions we can identify (i) the evolutionary point of view, and (ii) the roles played by the microbiota associated with springtails. The metabolic potential of endo-and ectomicrobiota opens new tracks. Furthermore, some of the molecules previously identified as springtail pheromones may possibly originate from the microorganisms they host. First, we question at which phylogenetic level olfactory signals vary, and whether they intervene in speciation mechanisms. It is known that scarab beetles of the same genus share the same sex pheromones [START_REF] Leal | Evolution of sex pheromone communication in plant-feeding scarab beetles[END_REF]. This suggests that in this zoological group, speciation was allopatric and mediated by natural geographic or temporal discontinuities. Inter-species pheromone interactions even occur between species belonging to distinct insect orders, e.g. male wine flies responding to honey bee queen sex pheromones [START_REF] Croft | Sexual response of male Drosophila to honey bee queen mandibular pheromone: implications for generic studies of social insects[END_REF]. In butterflies there were strong discrepancies between phylogenies based on morphological characters and those based on synthetic pathways of sex pheromones [START_REF] Dugdale | Pheromone and morphology-based phylogenies in New Zealand tortricid moths[END_REF]. Most of these data nevertheless suggest a strong phylogenetic conservatism of sexual olfactory signals, as otherwise demonstrated in insect-plant interactions [START_REF] Futuyma | Insect-plant interactions: the evolution of component communities[END_REF].

Contrary to sex pheromones, variegated results were obtained with insect aggregation pheromones. Despite preferences for conspecifics, cross-attraction between species belonging to various families is known as an adaptive mechanism thought to enhance species coexistence (Vet 1999; [START_REF] Wertheim | Pheromone-mediated aggregation in nonsocial arthropods: an evolutionary ecological perspective[END_REF]. Working with phylogenetic distances, [START_REF] Symonds | The mode of pheromone evolution: evidence from bark beetles[END_REF] showed that no links between morphologically-and chemically-based scolytid phylogenies could be evidenced. On the contrary, a close relationship between both phylogenies was shown in drosophilids [START_REF] Symonds | The mode of evolution of aggregation pheromones in Drosophila species[END_REF]. In the former case, random saltational shifts in chemical communication systems, a typical case of punctuated equilibrium [START_REF] Gould | Punctuated equilibrium comes of age[END_REF], are expected to favour sympatric speciation. In this case, like geographical or ecological isolation, aggregation pheromones create behavioural barriers against gene flow [START_REF] Baker | Mechanism for saltational shifts in pheromone communication systems[END_REF][START_REF] Wertheim | Pheromone-mediated aggregation in nonsocial arthropods: an evolutionary ecological perspective[END_REF][START_REF] Mendelson | Mutation-order divergence by sexual selection: diversification of sexual signals in similar environments as a first step in speciation[END_REF]). In the latter case, phyletic gradualism is expected because of the absence of selection pressure on aggregation pheromones, non-chemical cues being used in mate assortment [START_REF] Symonds | The mode of evolution of aggregation pheromones in Drosophila species[END_REF].

Despite the lack of large datasets allowing discerning evolutionary trends in Collembola, some multi-species experiments deserve our attention. Positive as well as negative results of cross-repulsion by necromones of another species were obtained [START_REF] Purrington | Alarm pheromone in a gregarious poduromorph collembolan (Collembola: Hypogastruridae)[END_REF]Nilsson and Bengtsson 2004b). These results are not in favour of the existence of a phylogenetic signal related to necromones.

Thus, further research is needed on this question, especially on sex or aggregation pheromones, before drawing straightforward conclusions. Multi-species attraction experiments in the order Entomobryomorpha showed that Tomocerus minor (family Tomoceridae) was attracted by aggregation pheromones of two species of Orchesella (family Entomobryidae), while Orchesella species were only attracted by their conspecifics (Verhoef et al. 1977a). On the contrary, two co-occurring Antarctic species, Cryptopygus antarcticus (Entomobryomorpha, family Isotomidae) and Friesea grisea (Poduromorpha, family Neanuridae) did not reveal any cross-attraction [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]). The same was observed for P. armata versus O. circulans and D. scotaria, all belonging to the same Onychiuridae family in Poduromorpha (Nilsson and Bengtsson 2004b). To date, no research of crossattraction has been reported in Symphypleona and Neelipleona, the former displaying elaborate mating courtships in many species. Clearly, all these results are insufficient to discern a pattern of attraction as a function of the phylogenetic distance between species, since only one cross-attraction was observed between species belonging to different families. Consequently inferring a general trend in the role of chemical communication in collembolan species assembly rules and speciation is premature.

Nevertheless, the partial results so far obtained point to the existence of asymmetrical effects of semiochemicals across species. Some species repel or attract other species, while the reverse does not apply (species "1" repels or attracts species "2", but species "2" does not repel or attract species "1"). This might contribute to explain dominance effects in multi-species laboratory cultures [START_REF] Christiansen | Competition between collembolan species in culture jars[END_REF][START_REF] Culver | Competition between Collembola in a patchy environment[END_REF][START_REF] Christiansen | Interspecific interactions between collembolan populations in culture[END_REF][START_REF] Walsh | Effects of diet on the interactions between Hypogastrura denticulata Bagnall and Onychiurus furcifer Börner in laboratory cultures[END_REF]. Such mechanisms might also explain the field co-occurrence [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF] or on the contrary segregation [START_REF] Ponge | Application de l'analyse factorielle des correspondances à l'étude des variations annuelles dans les populations de microarthropodes[END_REF][START_REF] Ponge | Spatial and taxonomic correlates of species and species trait assemblages in soil invertebrate communities[END_REF] of species sharing similar ecological requirements. Some aspects of chemical communication, so far unquestioned within soil ecosystems, should now deserve the attention of researchers. When several species are attracted to the same food or habitat [START_REF] Maraun | Adding to the 'enigma of soil animal diversity': fungal feeders and saprophagous soil invertebrates prefer similar food substrates[END_REF][START_REF] Staaden | Olfactory cues associated with fungal grazing intensity and secondary metabolite pathway modulate Collembola foraging behaviour[END_REF][START_REF] Boulay | Mixed-species aggregations in arthropods[END_REF], several events can occur following the encounter instar. Both species can compete by using the same resource (exploitative competition), or be expelled from each other (interference competition), or even one of them may kill the other (egg predation, cannibalism). The sign (negative, positive, or nil) of pheromone interactions between species might determine the fate of their encounters. Indifference, avoidance or attraction might possibly act in advance of any contact or agonistic behaviour. Such a phenomenon cannot be revealed in laboratory culture conditions where species are constrained to cohabit in a restricted space.

Positive or nil chemical interactions (between-species attraction or indifference, respectively) will result in multi-species aggregations around the same food or habitat, in the former case with a stabilizing mechanism according to [START_REF] Barot | Mechanisms promoting plant coexistence: can all the proposed processes be reconciled?[END_REF]. Conversely, negative interactions, whether symmetrical or asymmetrical, will result in either niche shifts or checkerboard distributions, according to different or similar ecological species requirements, respectively. A better knowledge of chemical communication systems might thus contribute to explain fine-scale niche segregation [START_REF] Anderson | Competition between two unrelated species of soil Cryptostigmata (Acari) in experimental microcosms[END_REF][START_REF] Zhao | Predatory beetles facilitate plant growth by driving earthworms to lower soil layers[END_REF]) and competition-mediated over-dispersion of functional traits [START_REF] Widenfalk | Small-scale Collembola community composition in a pine forest soil: overdispersion in functional traits indicates the importance of species interactions[END_REF], as observed in soil samples, and then to finally resolve the famous "enigma of soil animal species diversity" [START_REF] Anderson | The enigma of soil animal species diversity[END_REF][START_REF] Maraun | Adding to the 'enigma of soil animal diversity': fungal feeders and saprophagous soil invertebrates prefer similar food substrates[END_REF][START_REF] Wehner | Specialization of Oribatid mites to forest microhabitats: the enigmatic role of litter[END_REF].

Finally, the development of our knowledge on the roles played by symbiotic and more largely associated microorganisms in the biology, physiology and biochemistry of any organism, from plants to mammals, thanks to metagenomics and metabolomics methods, opens avenues for further research on springtail communication molecules. Furthermore, many of the molecules previously identified as springtail pheromones are possibly originating from the microorganisms they host, rather than from the springtails themselves.

Undoubtedly, the role of the Collembola-associated microbiota in intra-and interspecies communication should be a major research field in the next decade. Until now, this role has been scarcely explored in insect communication since the 1970's, and more actively since 2000 [START_REF] Dillon | Mutualism between the desert locust Schistocerca gregaria and its gut microbiota[END_REF][START_REF] Sharon | Commensal bacteria play a role in mating preference of Drosophila melanogaster[END_REF]. Gut microbiota appears as the most important insect-associated microbial community, contributing to the production of sex or aggregation pheromones, or to the insect defence. As an example, the gut microbiota of the desert locust Schistocerca gregaria was shown to produce guaiacol and phenols having both antimicrobial and aggregation pheromone functions [START_REF] Dillon | Mutualism between the desert locust Schistocerca gregaria and its gut microbiota[END_REF]. Molecular defence in insects through their associated microorganisms has been largely explored (for a review, see [START_REF] Van Arnam | Defense contracts: molecular protection in insect-microbe symbioses[END_REF]). On another side, chemical studies point to the production of pyrazines as common communication molecules in many organisms, including Collembola [START_REF] Dettner | Chemical defense of giant springtail Tetrodotonphora bielanensis (Waga) (Insecta: Collembola)[END_REF]Nilsson and Bengtsson 2004b). This suggests the possible involvement of the microbial compartment in animal chemical communication because pyrazine molecules are synthesized by several bacterial species [START_REF] Dickschat | Novel pyrazines from the myxobacterium Chondromyces crocatus and marine bacteria[END_REF].

However, in springtails, these orientations are still nascent. [START_REF] Bahrndorff | Diversity and metabolic potential of the microbiota associated with a soil arthropod[END_REF] described the composition of the microbial communities associated with Orchesella cincta (Entomobryidae).

The results supported a functional role of the microbiota in the enzymatic digestion of food, in the metabolism of aromatic amino acids that act as growth factors, and in soil metabolism including decomposition processes and biodegradation. However, the search for a role in communication was not a focus of the study.

Conclusions and future research needs

This review shows that the existence of aggregation, alarm and sex pheromones is unquestionable and verified experimentally. However, the studies suffered from several difficulties in collecting enough material for the identification of the molecules eliciting a signal. The small size of these animals, and difficulties in rearing them, notably but not only for Symphypleona, may explain why collembolan communication science does not make progress as fast as for other insects. Nothing is known on the receptors involved although strong clues indicate that antennal organs would play a role [START_REF] Kollmann | Brain organization in Collembola (springtails)[END_REF]). It appears highly probable that given their phylogenetic position at the base of Hexapoda [START_REF] Giribet | Is Ellipura monophyletic? A combined analysis of basal hexapod relationships with emphasis on the origin of inscts[END_REF]), Collembola would share with bristletails the absence of specialized olfactory receptors [START_REF] Missbach | Evolution of insect olfactory receptors[END_REF] to the benefit of generalized ionotropic receptors. Indeed, basal hexapods are proposed to have ionotropic receptors in common with Crustacea [START_REF] Corey | Ionotropic crustacean olfactory receptors[END_REF].

Studies on antennal transcriptomes might solve this question.

The origin and kind of molecules involved in collembolan chemical communication needs more attention to the methods used to extract them and experimentally reconstruct the processes in play in aggregation, mating, foraging, and other behavioural responses. The origin of extracted molecules should be clearly determined. Especially, it is crucial to know whether they are excreted in the environment by the animal or not, and from which organ or compartment in the animal they arise.

The role of extracted and identified molecules should be unambiguously demonstrated. A simple and instantaneous attraction does not mean that a given molecule enters mandatorily in the composition of aggregation pheromones. These molecules have also to induce a decrease in motility and a clustering of individuals in the course of time. As an example, fatty acids extracted from the body of Folsomia candida and attracting F. candida [START_REF] Liu | Chemical attraction of conspecifics in Folsomia candida (Collembola)[END_REF] are known to attract also other microarthropods towards food sources such as bacteria [START_REF] Brückner | Track the snaff: olfactory cues shape foraging behaviour of decomposing soil mites (Oribatida)[END_REF]. Thus, all attractive molecules are not necessarily pheromone components. Rather than yes/no sensitivity to a single compound, there are clues in favour of the ability of Collembola to react quantitatively to chemical 'fingerprints', like this has been shown to occur in other arthropods [START_REF] Barata | Encoding of host and non-host plant odours by receptor neurones in the eucalyptus woodborer, Phoracantha semipunctata (Coleoptera: Cerambycidae)[END_REF].

We also opened avenues in the evolutionary and functional importance of collembolan chemical communication. More research on multi-species aggregation (versus segregation) might throw light on the reasons why so many species contribute to similar or complementary functions in the same soil sample [START_REF] Bardgett | Causes and consequences of biological diversity in soil[END_REF] while repeated sampling shows so much unexplained microscale variation [START_REF] Ims | Spatial and temporal variation in patch occupancy and population density in a model system of an arctic Collembola species assemblage[END_REF]). Moreover, the roles that the associated microbial communities could play in the production of collembolan semiochemicals may explain several discrepancies observed in the results acquired until now, such as variations in the chemical nature of the main pheromone components identified according to the location of collected individuals or their number [START_REF] Benoit | Antarctic collembolans use chemical signals to promote aggregation and egg laying[END_REF]. It also appears that many of the results accumulated for the last 40 years would need being reexamined in the light of analysing the composition and metabolic potential of the endo-and global associated microbiota, using metagenomics and metabolomics. In this context it is clear that, taking into-account the current methodological advances in analytical chemistry, massively parallel sequencing technology (NGS) and the potential applications of signalling molecules in various Fatty acids [START_REF] Liu | Chemical attraction of conspecifics in Folsomia candida (Collembola)[END_REF]

Table 3 .

 3 Communication molecules in springtails, localization and proposed functions 809

domains and particularly in agricultural ecology to reduce pest incidence (Beck et al 2018), deciphering springtail communication strategies offers an extraordinary opportunity for future multidisciplinary research.


Mourad Chamekh
email: amourad.chamekh@enit.utm.tn

Nabiha Brik
email: bnabiha.brik@enit.utm.tn

Malek Abid
email: cabid@irphe.univ-mrs.fr

Mathematical Study of the concentration effect of red cells in blood

Keywords: Bingham model, advection-diffusion-reaction equation, Abdominal Aortic Aneurysm, Papanastasiou regularization

In this paper, a mathematical and numerical study of the effect of red blood cell concentration on a circular cross section tube is presented. The considered PDE system use the Bingham model describing blood as a non-Newtonian fluid. This system consists of Navier-Stokes equations describing the behavior of fluid and advection-reaction-diffusion equations that take into account the influence of chemical reactions on transient flow behavior in the arteries. Finally, numerical tests are presented to treat blood flow in the case of fusiform aneurysms caused by abdominal aortic aneurysm disease.

Introduction

The study of the behavior of fluids according to their viscosity is one of the interesting physical problems. Viscosity is a very important parameter that characterizes the behavior of the fluid since it links the stress of a fluid in motion with the rate of deformation of their constituent elements. For this reason, researchers are increasingly using generalized fluid models to meet the specific needs of modeling, such as the non-Newtonian fluids, which the vis-cosity cannot be defined as a constant value because of the non-linear relation between the strain rate and the shear stress [Marrero et al. 2014]. If the non-Newtonian character is manifested by the decreasing of the viscosity with increasing shear rate, then it is in the case of rheofluidifiant fluids (or pseudoplastic) [Kim 2002, Chamekh and[START_REF] Chamekh | [END_REF]. These are the fluids most commonly found in nature, where a constraint applied to this type of fluids must exceed a critical value, called yield stress for the flow to begin to occur.

Therefore, the yield stress is a free limit between two different behaviors (rigid and viscoplastic) of the material.

Among these fluids, we can characterize the Bingham fluid that is a theoretical model that describes the flow of a rigid viscoplastic and incompressible fluid with a yield stress [Messelmi 2014]. It is of great interest because of its ever-growing industry applications. It has been used in various publications to model the flow of metals, plastics, and various polymers. If the stress applied to the fluid is less than the yield stress, no deformation occurs and the fluid does not flow, then, we obtain a phenomenon of a rigid zone in the flow is transformed into a winder because of the growth of the elastic limit which can contribute to a blockage. This is have been related to the increase of the concentration of the particles constituting this fluid which are in contact with the effect of interaction forces. This behavior is similar to the behavior of the blood, which is considered among the fluids that are characterized by this blocking property since it flows under cardiac activity and its flow can block under the coagulation phenomenon that is quite sensitive to a number of factors such as plasma temperature, viscosity and the concentration of blood cells [ Messelmi 2011]. This blocking situation may help to understand the abdominal aortic aneurysm disease (AAA). This disease can cause fusiform aneurysms and/or saccular aneurysms. In the both cases, the aneurysm is dangerous; they can evolve into rupture, allowing the blood to flow from the vessel. When this situation is reached, blood accumulates and is life-threatening.

In this study, we treat the equations modeling the blood behavior in the abdominal aorta in case of an aneurysm involving the non-Newtonian model of Bingham. This modeling is based on the biochemical problem of blood flow in AAA diseases and is described by the convection-diffusion-reaction phenomenon, whose blood is considered as an incompressible and rigid viscoplastic fluid with a yield stress of Bingham taking into account the concentration effect of red cells. Then, a result of existence and uniqueness of a maximum principle is proposed for the concentration corresponding to this problem in a three-dimensional delimited domain, also some preliminary and notations found in [Merouani andMesselmi 2015, Messelmi 2014]. In the last part, we simulate the blood flow in AAA regularized by Papanastasiou and some numerical results on velocity field and concentration of red cells in AAA are presented.

Biological background

Blood is probably the most important biological fluid and its rheology is interesting from both a theoretical and applied point of view. It is a concentrated suspension of several species of particles in the plasma. It contains almost 45% of figurative elements in volume and 55% plasma [Brujan 2010[START_REF] Marieb | [END_REF], Toungara 2011, Robertson et al. 2008]. In the 45%, we find 97% of red blood cells or erythrocytes, and the other 3% are distributed between white blood cells or leucocytes and platelets or thrombocytes [START_REF] Marieb | [END_REF], Toungara 2011, Robertson et al. 2008]. Therefore, the red blood cells give a rheological behavior of blood [Brujan 2010, Robertson et al. 2009]. It is well known that blood is an incompressible Newtonian fluid [Paramasivam et al. 2010, Janela et al. 2010, Marrero et al. 2014]. However, in times and places, we can notice that there are non-Newtonian properties that can be important [Robertson et al. 2009, Khanafar et al. 2006 ]. For example, in larger blood vessels such as the aorta, the apparent viscosity is constant and the blood was treated as an incompressible Newtonian fluid since the shear rate is high. However, in certain pathological conditions, it is clear to find non-Newtonian effects, because of the presence of circular particles, the concentration of the recirculation zones increases and the particles touch, which induces a low shear rate, the viscos-ity high due to the aggregation of red blood cells. Since, the change in blood viscosity generally depends on the temperature, the deformation of red blood cells concentration and is related to the change in the percentage of hematocrit. Many studies have been conducted to better understand blood flow in the arterial system, with the idea to find trustworthy answers to these cardiovascular pathologies and complications. By the way, cardiovascular disease, which refers to a group of diseases of the heart and arteries increase, represent a major cause of death. Every year more people die from cardiovascular disease than from any other cause. The AAA disease is one of the most common cardiovascular diseases that usually leads to death in patients. It is a pathology of the aortic wall that affects the main artery of the human body that carries blood from the heart to all other organs. The AAA disease is a dilatation of the wall of the abdominal aorta, mainly caused by inflammation of the endothelium [Toungara 2011, Paramasivam et al. 2010, Marrero et al. 2014]. When the yield strength of the wall is reached, the AAA ruptures causing internal bleeding that is usually fatal in the patient. As a result, the work done on this pathology is a field of investigation of paramount importance in terms of public health since their rupture leads to the death of patients from 80% to 90% cite a52.

Due to hemodynamic forces of pressure and shear, the aneurysm continues to expand.

In general, the natural evolution of AAA differs between humans [Mofidi 2006, DeRubertis 2007]. The risk of re-offending is 25% for aneurysms between 4.1cm and 7cm in diameter and 9.5% for aneurysms less than 4cm [Paramasivam et al. 2010, Khanafar et al. 2006 ]. There are many constitutive models that are more complex used to study blood flow as the law of power or Cross [Robertson et al. 2008, Johnstona et al. 2004, Messelmi 2011]. More recently, the Bingham model has been used in the modeling of blood flow to describe its behavior through threshold stress and blocking phenomena that describe blood clotting, formation, and lysis of blood clots. We can find in several works on the phenomena of convection-diffusion-reaction which make the study of the coagulation and the formation of blood clots [Merouani and Messelmi 2015, Messelmi 2011, Anand et al. 2003, Hansen et al. 2015]. To do this study, we consider a mathematical model that describes the flow of blood involving the non-Newtonian model of Bingham.

Problem description

The purpose of our analysis is to determine the effect of red blood cell concentration on blood behavior. In order to guarantee this, we consider an incompressible flow inside a domain Ω; Ω ⊂ R n (n = 2, 3) an open bounded Lipschitzian boundary Γ, S n is the space of the symmetric tensors of order n -1 on R n . The blood temperature is assumed constant. We have formulated a nonlinear advection-diffusion-reaction problem for the Bingham fluid in a bounded domain of R 3 . For this problem, viscosity, yield strength, and diffusivity are supposed to depend on the concentration of red blood cells. The modeling is carried out thanks to a system with partial derivatives whose existence and the uniqueness of the solution are studied locally.

Let the space R 3 and S 3 are equipped by their the scalar product and the Euclidean norm, respectively. For a velocity field u ∈ H 1 (Ω) 3 , we consider the deformation rate tensor defined by

D(u) = (D ij (u)) 1≤i,j≤n , D ij (u) = 1 2 (u i,j + u j,i ), ∀ u ∈ H 1 (Ω) 3 .
The equation of fluid motion is given by

u∇u = div(σ) + f in Ω, (1)
div(u) = 0 in Ω, ( 2
)
with σ is the field of constraints and where the density is assumed to be 1 and f represents the density by volume forces acting on the fluid, under the condition of incompressibility (3) because the density is constant all along the fluid path.

The model of Bingham is characterized by

     σ D = 2µ(C)D(u) + g(C) D(u) |D(u)| if |D(u)| = 0 |σ D | ≤ g(C) if |D(u)| = 0, in Ω (3)
with σ D is the deviator of the constraint σ µ > 0 and g ≥ 0, which gives the variation of the velocity with the variation of the shear stress, whose viscosity µ and the yield g depends on the concentration C of the red cells.

Further, if we assume that the advection convection phenomena of the fluid undergo a chemical reaction under isothermal conditions, then the equation governing the concentration C is given by

u∇C -div(η(C)∇C) = R(C) in Ω. ( 4
)
This equation describes the variation of the concentration of the particles constituting the fluid where η is the diffusivity of the fluid and R is the reaction function (interaction and collision) between the different particles constituting the fluid. In a particular case, we have R(C) = -K C of first-order chemical reaction with K is a constant chemical reaction rate. We assume that the boundary conditions have been taken by:

u = 0 on Γ, C = 0 on Γ. (5)
The nonlinear advection -diffusion -reaction problem for stationary Bingham fluid is modeled by the partial differential system below which consists of to find:

• the velocity field u = (u i ) : Ω -→ R n , • the field of constraints σ = (σ ij ) : Ω -→ S n , • the concentration C : Ω -→ R,
that verify the equations ((1)-( 5)).

Variational formulation

To determine the variational formulation that corresponds to the problem ((1)-( 5)). We consider the following space H defined by

H = v ∈ H 1 0 (Ω) 3 \ divv = 0 ,
Where H is a Hilbert space with a scalar product and an induced norm, respectively,

(u, v) H = (u i , v i ) H 1 (Ω) , u H = u H 1 (Ω) ∀u, v ∈ H.
The set H is Banach space. We introduce the following operators

B : H 3 → R, B(u, v, w) = Ω u • ∇v • w dx E : H 1 (Ω) 2 × H → R, E(θ, τ, v) = Ω θ∇τ • v dx.
We assume the following hypothesis

∀x ∈ Ω, µ(•, x) ∈ C 0 (R) and ∃µ 1 , µ 2 > 0 : µ 1 ≤ µ(y, x) ≤ µ 2 ∀y ∈ R, ∀x ∈ Ω. (6) ∀x ∈ Ω, g(•, x) ∈ C 0 (R) and ∃g 0 > 0 : 0 ≤ g(y, x) ≤ g 0 ∀y ∈ R, ∀x ∈ Ω. (7) η > η * > 0. (8) R(•, x) : Ω × R → R is L Lipschitzian and R(•, x) ∈ L 2 (Ω × R) (9)
Multiplying (1) by a test function (v -u) ∈ H, we get

Ω u∇u(v -u) = Ω div(σ)(v -u) + Ω f (v -u). ( 10
)
Using a generalized Green formula, we obtain

Ω u∇u(v -u) = Γ σ(v -u) - Ω σ D(v -u) + Ω f (v -u) (11)
Multiplying (1) by D(v -u), we obtain

σ D D(v -u) = 2µ(C)D(u)D(v -u) + g(C) D(u) |D(u)| D(v -u) = 2µ(C)D(u)D(v -u) + g(C) D(u) |D(u)| D(v) -g(C)|D(u)| ≤ 2µ(C)D(u)D(v -u) + g(C)|D(v)| -g(C)|D(u)|
In addition,

σ D D(v -u) = (σ - 1 n T r(σ)I D )D(v -u).
Then,

ID(v) = n i=1 nD ii (v i ) = n i=1 ∂v i ∂x i = divv = 0, ID(v -u) = 0 =⇒ σ D D(v -u) = σD(v -u), σD(v -u) ≤ 2µ(C)D(u)D(v -u) + g(C)|D(v)| -g(C)|D(u)|.
We replace this result in the equation ( 11), we obtain the following variational formulation

Ω u∇u(v -u) + 2 Ω µ(C)D(u)D(v -u)+g(C) (|D(v)| -|D(u)|) ≥ Ω f (v -u), ∀ v ∈ H, (12)
Ω ∇Cϕu + Ω η(C)∇C∇ϕ = Ω R(C)ϕ , ∀ϕ ∈ H 1 0 (Ω). ( 13
)

Existence of the solution

The proof of existence is based on the application of Schauder's fixed point theorem, using in this framework two auxiliary existence results. The first result was obtained in [Kim 1987, Kim 2002], which consists of the existence of a local solution u λ ∈ H of problem (12)for each λ ∈ H 1 0 , satisfying u λ H ≤ c 1 , where c 1 is a positive constant does not depend on λ. The second is a result which shows that for u λ ∈ H is a solution of the problem (12), then there exists a unique solution C λ ∈ H 1 0 of the problem (13), satisfying even C λ H 1 0 ≤ c 2 , where c 2 is a positive constant does not depend on λ.

To begin, for the operator B and E, we have the following proprieties Lemma 5.1. See [START_REF] Lions | [END_REF] (i) The operator B is trilinear, continuous on H 3 and for all (u, v, w) ∈ H 3 we have

B(u, v, w) = -B(u, w, v). ( 14
) (ii)
The operator E is trilinear, continuous on H 1 0 (Ω) 3 and

E(θ, τ, v) = -E(τ, θ, v) for all (θ, τ, v). ( 15
)
We have a first result of existence Lemma 5.2. For λ ∈ H 1 0 (Ω), it exists a unique solution u λ of the problem

B(u λ , u λ , v) + 2 Ω µ(λ)ε(u λ )ε(v -u λ )+ g(λ)|ε(v)| -g(λ)|ε(u λ )| ≥ Ω f (v -u λ ) ∀v ∈ H, (16)
satisfy

u λ H ≤ c 1 , (17)
where c 1 is a positive constant does not depend on λ.

Theorem 5.1. Let u λ ∈ H is the solution of problem ( 16). Then, it exists a

unique solution C λ ∈ H of E(C λ , φ, u λ ) + Ω η(λ)∇C λ ∇ϕdx = Ω R(C λ )ϕdx ∀ϕ ∈ H 1 0 (Ω). ( 18
)
and satisfy

C λ H 1 0 (Ω) ≤ c 2 , ( 19
)
where c 2 is a positive constant does not depend on λ.

Proof. Let E the following bilinear form

E : H 1 0 (Ω) × H 1 0 (Ω) → R, (ψ, ϕ) → E(ψ, ϕ, u λ ) + Ω η(λ)∇ψ∇ϕdx
The trilinear operator E is continuous on H 1 0 (Ω) × H 1 0 (Ω) × H. In fact, using the Hölder inequality and Poincaré theorem, we have

| Ω ∇ψϕ • vdx| ≤ ∇ψϕ L 2 (Ω) v L 2 (Ω) ≤ ∇ψ L 2 (Ω) ϕ L 2 (Ω) v L 2 (Ω) ≤ ψ H 1 0 (Ω) ϕ H 1 0 (Ω) v H .
Then, we obtain

| Ω ∇ψϕ • vdx| ≤ ψ H 1 0 (Ω) ϕ H 1 0 (Ω) v H . ( 20
)
Consider r ∈ L 2 (Ω) and the following linear problem

E(C λr , ϕ) = Ω rϕdx ∀ϕ ∈ H 1 0 (Ω). ( 21
)
Using the continuity of E, we can simply show that E is continuous.

|E(C λr , ϕ)| = | Ω ∇C λr ϕ • u λ + Ω η(λ)∇C λr • ∇ϕdx| ≤| Ω ∇C λr ϕ • u λ |+ | Ω η(λ)∇C λr • ∇ϕdx|.
Using Hölder inequality we get

≤ ∇C λr ϕ L 2 (Ω) u λ L 2 (Ω) + η(λ) ∇C λr L 2 (Ω) ∇ϕ L 2 (Ω) ≤ ∇C λr L 2 (Ω) ϕ L 2 (Ω) u λ L 2 (Ω) + η(λ) ∇C λr L 2 (Ω) ∇ϕ L 2 (Ω) .
Using Poincaré theorem, then

≤ C p C λr H 1 0 (Ω) ϕ H 1 0 (Ω) u λ H + η(λ) C λr H 1 0 (Ω) ϕ H 1 0 (Ω) .
We have also

u λ H ≤ c 1 , ≤ c 1 C p C λr H 1 0 (Ω) ϕ H 1 0 (Ω) +η(λ) C λr H 1 0 (Ω) ϕ H 1 0 (Ω) .
Then, we obtain

|E(C λr , ϕ)| ≤ K C λr H 1 0 (Ω) ϕ H 1 0 (Ω) , with K = (c 1 C p + η(λ)).
In addition, by ( 15), we remark that E(u, u, v) = 0. Then,

E(C λr , C λr ) = Ω η(λ) | ∇C λr | 2 dx
, using in addition that (8), we have the

coercivity of E. E(C λr , C λr ) ≥ η * C λr 2 H 1 0 (Ω) . Adding that l(ϕ) = Ω rϕdx is lineair continuous by Poincaré formula, we have | l(ϕ)| =| Ω rϕdx| ≤ r L 2 (Ω) ϕ L 2 (Ω) ≤ C 2 p r H 1 0 (Ω) ϕ H 1 0 (Ω) .
Using Lax-Milgrame theorem on H 1 (Ω) × H 1 (Ω), then, the problem (23) have a unique solution C λr ∈ H 1 0 (Ω). To get auxiliary solution of problem (18), we consider the following operator

   L : L 2 (Ω) → L 2 (Ω), Lr = R(C λr ). ( 22
)
Since η verify (8), for r 1 , r 2 ∈ L 2 (Ω), we have

|Lr 1 -Lr 2 | =| R(C λr1 ) -R(C λr2 ) |≤ L | C λr1 -C λr2 |.
We denote C 1 = C λr1 and C 2 = C λr2 , respectively, solutions of the following problems

E(C λr1 , ϕ) = Ω r 1 ϕdx ∀ ϕ ∈ H 1 0 (Ω), (23)
E(C λr2 , ϕ) = Ω r 2 ϕdx ∀ ϕ ∈ H 1 0 (Ω), (24)
Subtracting the equations ( 23) et (24), and using ϕ = C λr1 -C λr2 as a test function, we get

E(C 1 -C 2 , C 1 -C 2 ) = Ω (r 1 -r 2 )(C 1 -C 2 )dx, Ω ∇(C 1 -C 2 )(C 1 -C 2 )•u λ + Ω η(λ) | ∇(C 1 -C 2 )| 2 dx = Ω (r 1 -r 2 )(C 1 -C 2 )dx, As η verify (8), we obtain Ω ∇(C 1 -C 2 )(C 1 -C 2 )•u λ + Ω η * (λ) | ∇(C 1 -C 2 )| 2 dx ≤ Ω ∇(C 1 -C 2 )(C 1 -C 2 ) • u λ + Ω η(λ) | ∇(C 1 -C 2 )| 2 dx = Ω (r 1 -r 2 )(C 1 -C 2 )dx. Using E(u, u, v) = 0, then, Ω ∇(C 1 -C 2 )(C 1 -C 2 ) • u λ > 0. Therefore η * C 1 -C 2 2 H 1 0 (Ω) dx ≤ r 1 -r 2 L 2 (Ω) C 1 -C 2 L 2 (Ω) , η * C 1 -C 2 2 H 1 0 (Ω) ≤ r 1 -r 2 L 2 (Ω) C p C 1 -C 2 H 1 0 (Ω) , C 1 -C 2 H 1 0 (Ω) ≤ C p η * r 1 -r 2 L 2 (Ω) .
We have

|Lr 1 -Lr 2 | ≤ L | C 1 -C 2 |.
Then,

Lr 1 -Lr 2 2 L 2 (Ω) ≤ L 2 C 1 -C 2 2 L 2 (Ω) ≤ L 2 C 2 p C 1 -C 2 2 H 1 0 (Ω) ≤ L 2 C 4 p η 2 * r 1 -r 2 2 L 2 (Ω) Lr 1 -Lr 2 2 L 2 (Ω) ≤ L 2 c1 r 1 -r 2 2 L 2 (Ω) , with c 1 = η * C 2 p .
We can write

Lr 1 -Lr 2 2 L 2 (Ω) ≤ L 2 c 1 r 1 -r 2 2 L 2 (Ω) , L 2 r 1 -L 2 r 2 2 L 2 (Ω) ≤ L 2 c 1 r 1 -r 2 2 L 2 (Ω) , . . . L m r 1 -L m r 2 2 L 2 (Ω) ≤ L 2 c 1 r 1 -r 2 2 L 2 (Ω) .
The operator L is contracting in the space L 2 (Ω). So, according to Banach's fixed point theorem L m admits a fixed point that we denote r ∈ L 2 (Ω) which gives L m r = Lr.

Then, the fixed-point uniqueness gives Lr = r which also gives Lr = R(C r ) = r, which implies that the equation ( 18) has a unique solution C λ ∈ H 1 0 (Ω). Finding now the estimate (19). We choose C λ as a test function of the equation ( 18). So we have

Ω ∇C λ C λ • u λ + Ω η(λ) | ∇C λ | 2 dx = Ω R(C λ )C λ dx, ∀C λ ∈ H 1 0 (Ω)
Using the Hölder inequality, we obtain

Ω ∇C λ C λ • u λ + η * (λ) C λ 2 H 1 0 (Ω) ≤ R L 2 (Ω) C λ L 2 (Ω) η * (λ) C λ 2 H 1 0 (Ω) ≤ R L 2 (Ω) C λ L 2 (Ω) .
Using the Poincaré inequality, we obtain

η * (λ) C λ 2 H 1 0 (Ω) ≤ C p R L 2 (Ω) C λ H 1 0 (Ω) .
Therefore,

C λ H 1 0 (Ω) ≤ C p η * R L 2 (Ω) .
So, we obtain

C λ H 1 0 (Ω) ≤ c 2 .
Theorem 5.2.

The problem ( 16)-( 18) has a locale solution (u, C) having the following regular-

ity u ∈ H ∩ W 1,6 0 (Ω) 3 et C ∈ H 1 0 (Ω) ∩ L 2 (Ω) (25)
It is easy to check that the regularity (25) using in particular the Theorem of intermediate values and Theorem extensions of Sobolev, that after a possible modification on a set of zero measure, the solutions are:

u ∈ H ∩ C 0, 1 2 (Ω) 3 et C ∈ L 2 (Ω). (26)
Proof. We will use the fixed point Theorem of Schauder, by considering the the following closed convex ball:

B = {λ ∈ H 1 0 (Ω) / λ H 1 0 (Ω) ≤ c 2 } Soit Λ : B → B λ → Λ(λ) = C λ ∈ B,
our interest is to check the compactness of the Λ operator. Therefore, we consider λ n in B and consequently this sequence is limited, and since it is a Hilbert space, we can extract a subsequence that we note lambda n and C n = C lambdan such that

λ n λ dans H 1 (Ω) λ n * λ dans L 2 (Ω) C n C λ dans H 1 (Ω) 13 C n * C λ dans L 2 (Ω).
As C n is a solution of the following problem

E(C n , φ, u n ) + Ω η(λ n )∇C n ∇ϕdx = Ω R(C n )ϕdx ∀ϕ ∈ H 1 0 (Ω). ( 27
)
with u n verifies the following inequality

B(u n , u n , v) + 2 Ω µ(λ n )ε(u n )ε(v -u n )+ g(λ n )|ε(v)| -g(λ n )|ε(u n )| ≥ Ω f (v -u n ) ∀v ∈ H, (28)
The lemma (5.2) gets a unique strong solution u n ∈ H. Then we can extract a subsuite is also noted by u n satisfies

u n * u λ in H ∩ C 0, 1 2 (Ω) 3 .
Then, the injection W 1,6 0 (Ω) 3 ⊂ C 0 ( Ōmega) is compact, using the compactness theorem of Aubin, as we can extract suites λ n , C n , u n (see [Brezis 1968[START_REF] Lions | [END_REF] ) such that

λ n → λ dans L 2 (Ω) C n → C λ dans L 2 (Ω) u n → λ dans C 0 ( Ω) 3 B(u n , u n , v) + 2 Ω µ(λ n )ε(u n ) • ε(v)dx + g(λ n )|ε(v)| - Ω f (v)dx ≥ 2 Ω µ(λ n ) | ε(u n )| 2 dx + g(λ n )|ε(u n )|dx - Ω f (u n )dx ∀v ∈ H. (29)
Following [START_REF] Lions | [END_REF]], we have B is continuous, and by passing to the limit

B(u n , u n , v) → B(u λ , u λ , v), ∀v ∈ H. ( 30
)
Using the * convergence de u n u λ dans H ∩ C 0, 1 2 (Ω) 3 and and the fact that λ n → λin Ω, l'hypothse ( 6), ( 7) and according to the dominated convergence theorem, we obtain for v ∈ H the following result as

Ω µ(λ n )ε(u n ) • ε(v)dx → Ω µ(λ)ε(u λ ) • ε(v)dx Ω f (u n )dx → Ω f (u λ )dx. (31)
Adding, we have

2 Ω µ(λ n ) | ε(u n )| 2 dx + g(λ n )|ε(u n )|dx = 2 Ω µ(λ) | ε(u n )| 2 dx + 2 Ω (µ(λ n ) -µ(λ)) | ε(u n )| 2 dx + g(λ)|ε(u n )|dx + (g(λ n ) -g(λ))|ε(u n )|dx.
Then, λ → λ ∈ Ω, the functions µ et g are continuous, and due to the low semi-continuity in the space H, continuity and convexity of functional

Ω µ(λ) | ε(v))| 2 dx et Ω g(λ) | ε(v)|dx, we deduct using Fatou lemma that 2 lim inf Ω µ(λ n )|ε(u n )| 2 dx + lim inf Ω g(λ n )|ε(u n )|dx ≥ 2 Ω µ(λ)|ε(u λ )| 2 dx + Ω g(λ)|ε(u λ )|dx. ( 32
)
On the other hand, the fact that u n , u λ ∈ H and after (29) all lead to the conclusion that u is a solution of ( 16). Moreover, it is necessary to pass by the limit in the equation ( 27). To do this, for every φ ∈ H 1 (Ω), we have

|E(C n , φ, u n ) -E(C λ , φ, u λ )| = Ω (u n • ∇(C n -C λ )φ + (u n -u λ ) • (∇C λ )φ)dx|.
By the lemma (5.1), we have

|E(C n , φ, u n ) -E(C λ , φ, u λ )| ≤| Ω (C n -C λ )u n • ∇φdx | + | Ω C λ (u n -u λ ) • ∇φdx| (33)
Elsewhere, using the regularity (26), we obtain the following inequality

Ω | u n • ∇φ | 2 dx ≤ u n 2 C 0 (Ω) φ 2 H 1 (Ω) ∈ φ ∈ H 1 (Ω).
Then, using the result of the convergence

C n → C λ dansH 1 (Ω), que |E(C n , φ, u n ) -E(C λ , φ, u λ )| → 0.
In addition,

C λ ∈ H 1 0 (Ω) ∩ L 2 (Ω), gives that C λ ∈ L 4 (Ω). Therefore, C λ ∇φ ∈ L 4 3 (Ω) 3 , ∀φ ∈ H 1 (Ω) (34)
Since, de u n → λ in C 0 ( Ω) 3 , we deduce

u n → u λ dans L 4 (Ω) 3 . (35)
Combining ( 35) avec (34). Then,

|E(C n , φ, u n ) -E(C λ , φ, u λ )| → 0.
We obtain

|E(C n , φ, u n ) -E(C λ , φ, u λ )| → 0 ∀φ ∈ H 1 (Ω). ( 36
)
Then, λ n → λp. p ∈ Ω, the lower limit is easily obtained, using

C n C λ in H 1 (Ω),
the assumptions ( 6), ( 7) and the use of the dominated convergence theorem

Ω η(λ n )∇C n • ∇φdx → Ω η(λ)∇C λ • ∇φdx ∀φH 1 (Ω) (37)
So that the passing to the limit in the equation ( 29) remains to verify that

Ω R(C n )φdx → Ω R(C λ )φdx, ∀φH 1 (Ω) (38)
This is obtained taking into account that the R Lipschitzsian function and

C n → C λ in L 2 (Ω) p. p. in Ω.
Proving now the following propriety

C n → C λ dans H 1 0 (Ω) ∩ L 2 (Ω) (39)
To do this, subtracting the two equations ( 18) and ( 27) and using φ =

C n -C λ , as a test function in the equation obtained, we obtain

1 2 C n -C λ 2 H 1 (Ω) + E(C n , C n -C λ , u n ) -E(C λ , C n -C λ , u n )+ Ω (η(λ n )∇C n -η(λ)∇C λ ) • ∇(C n -C λ ))dx = Ω (R(C n ) -R(C λ )(C n -C λ )dx.
Using the hypotheses ( 6) and ( 7), we have

1 2 C n -C λ 2 H 1 (Ω) + Ω η(λ n )|∇(C n -C λ )| 2 dx ≤ Ω | η(λ n ) -η(λ) || ∇C λ | |∇(C n -C λ )|dx +E(C n , C n -C λ , u n ) -E(C λ , C n -C λ , u n ) = L Ω | C n -C λ | 2 dx. (40)
In addition, (36) shows that

E(C n , C n -C λ , u n ) -E(C λ , C n -C λ , u n ) → 0. ( 41
)
Therefore, with the equation ( 40), we get the assumptions ( 6), ( 7) and the convergence results

λ n → λ in L 2 (Ω), C n → C λ in L 2 (Ω),
and the fact that

| ∇C λ || ∇(C n -C λ ) | belongs to a bounded L 1 (Ω), seeing
Lebesgue's theorem of dominated convergence.

C n → C λ dans H 1 (Ω ∩ L 2 (Ω). ( 42
)
By subtracting, we deduce

Ω | u n •∇C n -u λ •∇C λ ) | 2 dx ≤ c Ω | u n | 2 | ∇(C n -C λ ) | 2 dx+c Ω | u n -u λ | 2 | ∇C λ | 2 dx.
The use of Hölder inequality and (26), we have

Ω | u n •∇C n -u λ •∇C λ ) | 2 dx ≤ c u n 2 C 0 ( Ω) C n -C λ H 1 (Ω) +c u n -u λ 2 C 0 ( Ω) C λ 2 H 1 (Ω) (43)
As a result, C n converges strongly to C λ in the space H 1 0 (Ω) ∩ L 2 (Ω). Then the Λ operator is compact. Using Schauder fixed point theory, we have at least one 16), ( 18)).

fixed point λ = C λ inH 1 0 (Ω) ∩ L 2 (Ω) . Finally, we conclude that the function (uλ, Cλ ∈ H ∩ W 1,6 0 (Ω) 3 × H 1 0 (Ω) ∩ L 2 (Ω) solve the problem ((

Numerical treatment

Problem regularized

The basic idea is to use one of the regularization methods to approximate the apparent viscosity µ e of the law of the original behavior in order to avoid the discontinuity of this viscosity by µ e,ε , so that we may overcome the instability of calculation. In literature, the Papanastasiou model of regularization has been widely used in numerical simulations of flows of viscoplastic fluids [Glowinski and Ciarlet 2010, Messelmi 2014, Paramasivam et al. 2010]. The Bingham's law (3) is equivalent to the following system

σ D = 2µ e D(u), (44)
µ e = µ + g 2|D(u)| , (45)
The above system is obviously well suited to the ceded regions, that is to say the regions where the deformation rate tensor D(u) is non-zero. However, if ||D(u)|| → 0, the exponential model of Papanastasiou is as follows:

µ e,ε = µ + g 2|D(u)| (1 -exp( -2 ε |D(u)|)).
So the Papanastasiou-Bingham model is as follows:

σ D = 2µ e,ε D(u) = 2µD(u) + g D(u) |D(u)| (1 -exp( -2 ε |D(u)|)). (46)
We will consider now the problem ( (1)-( 5) with replacing (3) by ( 46)). The goal is treated this problem numerically simulation. To do this, here and below, we suppose that fluid is stationary, laminar, viscoplastic and incompressible. We suppose that the volume forces are negligible in front of the forces of viscosity, the blood diffusion coefficient η, the yield stress g and the blood viscosity µ don't depend on the concentration, except the Reaction function that is non linear and generally has an unknown expression. According to the bio-mechanical problem ((1)-( 5) with replacing (3) by ( 46)), the variational formulation is as follows

Ω u∇uv + Ω 2µD(u) : D(v) - Ω p∇.v - Ω q∇.u + Ω g D(u) : D(v) (D(u) : D(u)) 1 2 (1 -exp( -2 ε (D(u) : D(u)) 1 2 )) = 0, ∀v ∈ V 1 (47) Ω u∇Cφ + Ω η∇C∇φ = Ω R(C)φ, ∀φ ∈ H 1 0 (Ω) . (48)
In order to solving numerically the non-linear equations (( 47),( 48)), we will use the Newton algorithm. For this, we need to linearize these equations. To do this, we calculate the differential of each weak forms at point u, since the differential in general is a linear application. Then, we solve them in this form:

F (u + δ) = F (u) + DF (u)δ + o(δ). (49)
For the variational problem, we obtain

                                                     F 1(u, p) = Ω u∇uv + Ω 2µD(u) : D(v) + Ω g D(u) : D(v) (D(u) : D(u)) 1 2 (1 -exp( -2 ε (D(u) : D(u)) 1 2 )) - Ω p∇.v - Ω q∇.u DF 1(u, p)(δu, δp) = Ω (δu∇uv + u∇δuv) + Ω 2µD(δu) : D(v) +g D(δu) : D(v) (D(u) : D(u)) 1 2 (1 -exp( -2 ε (D(u) : D(u)) 1 2 )) -g D(u) : D(v)D(u) : D(δu) (D(u) : D(u)) 3 2 1 -exp( -2 ε (D(u) : D(u)) 1 2 ) + 2 ε g D(u) : D(v)D(u) : D(δu) D(u) : D(u) exp( -2 ε (D(u) : D(u)) 1 2 )) - Ω δp∇.v - Ω q∇.δu (50)
and

     F 2(C) = Ω u∇Cφ + Ω η∇C∇φ - Ω (1 -C)Cφ, DF 2(C)δC = Ω u∇δCφ + Ω η∇δC∇φ - Ω (1 -δC)δCφ
(51)

Numerical results

This section focuses on the 2D numerical study of the effect of concentration of blood cell in fusiform aneurysmsin caused by th AAA diseases in 2D

model. Blood is considered a non-Newtonian laminar flow, incompressible and stationary, which can present itself with a rigid wall.

Two-dimensional geometry construction

We use in this study two types of geometries of an Abdominal Aortic Aneurysms (AAA), one presents a Fusiform aneurysms caused by the AAA diseases without bifurcation. The second is more complex than the other (Fig. 6.2.1). It represents a fusiform AAA with a bifurcation at the level of the abdominal aorta which is split into two iliac arteries. The AAA models have a non dilated aorta diameter d = 1, 8cm, the maximum diameter of the AAA, D = 2.75d, the aneurysm has a length, L = 4d and the diameter of each outputs of the right and left branches of the bifurcation is d 0 = 0.72d, while the order of the average velocity of blood is u 0 ≈ 0, 25 m/s, since it does not circulate at the same velocity throughout the body. The two meshes were built with Freefem++ software, using a sinc function as follows [ Khanafar et al. 2006 ]

f (x) = D -d 4 1 + sin( 2πx L - π 2 ) + d 2 , 0 ≤ x ≤ L(= 4d), ( 52
)
where d is a diameter Abdominal Aortic (AA), D is a maximum aneurysm diameter of the abdominal aortic and L is the length of the aneurysm.

The most important blood properties are: the Newtonian viscosity µ ∞ = 0.00345 P a.s and µ 0 = 0.056 P a.s , at high and low shear rates, respectively.

In effect, the erythrocytes rolls are formed at low shear rate. But, if this rate increases, the aggregates of these globules will start to deform without the blood flowing, which implies that there is a critical shear stress (yield stress) g ≈ 0.003 P a, below which the flow does not occur. As found in [Picart et al. 1998],

that the yield stress varies from 0.001 P a to 0.03 P a for a normal blood at hematocrit 40, in general it increases with the rate of hematocrit.

The non-Newtonian blood property has a very significant effect on the velocity profiles, especially on the longitudinal section of the aorta specifically in the aneurysmal sac, without forgetting the concentration and pressure that corresponds to this velocity.

Numerical Interpretation

Let u x and u y the velocity component with x and y are horizontal and vertical spatial coordinates. Numerically, we found that the u x component of the speed along x in[0, L] varies nonlinearly between 0, 015m/s and 0, 25m/s, while the u y component of the speed varies between 0.0034m/s and 0.18m/s for a high rate of shear that due to a very low viscosity mu = 0.00345P a.s. In this state, we can conclude that blood behaves like a Newtonian fluid. This shows that the main flow is relatively dominated by the u x component of the velocity. The resolution is done by varying the number of meshes to test the convergence using triangular finite elements. The velocity is computed throughout aneurysmal fusiform at a variable rate of shear which respectively to the deformation of the red blood cell rolls on both models of AAA, shown in Fig. 2. The solutions of the velocity for different numbers of meshes, (Fig. . (2) ,

(3)) are obtained in a compatible way, which makes it possible to conclude to the convergence. For both models with or without bifurcation of AAA disease, a remarkable change in velocity component u x with a relatively high viscosity (µ = 0.056P a.s) can be observed, than to the case of a shear rate due to a lower viscosity (µ = 0.00345P a.s). To testing some regularization values, we set the number of meshes at (n = 100) for. The velocity component u x for both models and in both cases low and high shear rates are confounded, see figures eqref ph3 and eqref ph4), which

shows the convergents even if we let's change the regularization values. We noted in the case of AAA without bifurcation, when the shear rate is relatively low the value of u x is almost constant in the first half of the aneurysm sac (x in[0, f racL2]), then this value increases remarkably in the second half. This increase does not appear in the same way when the viscosity is relatively low.

On the other hand, for the bifurcated model, there is a remarkably large change in speed for the case of a low shear rate compared with the case of a high rate.

We have noticed that the smallest regularization value we can take to get convergent solutions for the AAA model without bifurcation is 4 • 10 -2 and 3 • 10 -2 for the other. This can influence the reasonableness of the result obtained, since it takes the value of regularization to be small to obtain precise solutions.

In what follows,, we are interested in the study of blood flow in the aneurysmal fusiform with velocity distribution analysis in the aorta. That was not the case for a low shear rate.

Discussions

When we treat the flow of blood and its viscosity, we talk about the concentration of red blood cells. It varies between 130k/m 3 and 170kg/m 3 in a healthy man and between 110k/m 3 and 150kg/m 3 in a woman. From the figure (7), we notice that the concentrations corresponding to the previous velocities for a high shear rate of two models decrease. On the other hand, for a very high viscosity, the concentration of hemoglobin throughout the hump remains stable at very important values for the two models of AAA with or without a bifurcation. This increase can produce a blockage of blood flow at this hump. On the one hand, for blood pressure Fig. ( 8), it's almost the same situation for both models of AAA at low viscosity. On the other hand, it is obviously important in the presence of the bifurcated geometry of the aorta. We can conclude that when the velocity is in the stage of acceleration Fig. ( 6), the pressure will be strongly increasing (8). This is a phenomenon that characterizes blood modeled as Bingham fluid. This is shown in Fig. ( 8) for pressure in case of a low shear

Figure 1 :

 1 Figure 1: Fusiform aneurysms caused by the AAA diseases without and with bifurcation.

Figure 2 :

 2 Figure 2: The velocity profiles ux on the regime x ∈ [0, L] for the fusiform AAA model without bifurcation, the left figure for a low rate (µ = 0.056 P a.s) and the right figure for a high rate (µ = 0.00345 P a.s).

Figure 3 :

 3 Figure 3: The velocity profiles ux on the regime x ∈ [0, L] for the fusiform AAA model with bifurcation , the left figure for a low rate (µ = 0.056 P a.s) and the right figure for a high rate (µ = 0.00345 P a.s).

Figure 4 :

 4 Figure 4: The velocity profiles ux on the regime x ∈ [0, L] for the fusiform AAA model , the left figure for a low rate (µ = 0.056 P a.s) and the right figure for a high rate (µ = 0.00345 P a.s).

Figure 5 :

 5 Figure 5: The velocity profiles ux on the regime x ∈ [0, L] for the fusiform AAA model with bifurcation , the left figure for a low rate (µ = 0.056 P a.s) and the right figure for a high rate (µ = 0.00345 P a.s).

Figure 6 :

 6 Figure 6: The velocity profiles ux on the regime x ∈ [0, L] for a low and high rate, the left figure for the fusiform AAA model without bifurcation and the right figure for the fusiform AAA model with bifurcation.

Figure 7 :

 7 Figure 7: The concentration that corresponds to the velocity profiles ux on the regime x ∈ [0, L] for a low and high rate, the left figure for the fusiform AAA model without bifurcation and the right figure for the fusiform AAA model with bifurcation.

Figure 8 :

 8 Figure 8: The pressure that corresponds to the velocity profiles ux on the regime x ∈ [0, L] for a low and high rate, the left figure for the fusiform AAA model without bifurcation and the right figure for the fusiform AAA model with bifurcation.

Fig. ( 6

 6 Fig. (6) shows the velocity profiles of an AAA with and without bifurcation for a high and low shear rate. Comparing the results, we find that both models of the AAA have the same velocity profiles for a low viscosity µ = 0.00345P a • s.