domain: sde sdv

text:
Claudia Gerard
email: claudia.gerard@univ-rennes1.fr

Thomas Trancart

Elsa Amilhat

Elisabeth Faliex

Laure Virag

Eric Feunteun

Anthony Acou

Claudia Ge ´rard

Influence of introduced vs. native parasites on the body condition of migrant silver eels

Keywords: Anguilla anguilla, Silver eels, Metazoan parasite communities, Introduced parasites, Body condition

Introduction

In spite of the perennial scientific mysteries about its biology and population genetic structure, the European eel Anguilla anguilla is considered as panmictic, therefore a breeder emigrating from a given river theoretically contributes to subsequent glass eel recruitment along the whole continental distribution range [START_REF] Als | All roads lead to home: panmixia of European eel in the Sargasso Sea[END_REF][START_REF] Dannewitz | Panmixia in the European eel: a matter of time[END_REF][START_REF] Feunteun | Management and restoration of European eel population (Anguilla anguilla): an impossible bargain?[END_REF]. Depending on subpopulations and habitat characteristics, eels spend about 3-30 years in fresh and brackish waters of Europe and North Africa, growing and accumulating fat reserves for an active swimming migration as silver eel stage across the Atlantic Ocean and reproduction once in the Sargasso Sea [START_REF] Kettle | Where once the eel and the elephant were together: decline of the European eel because of changing hydrology in southwest Europe and northwest Africa?[END_REF]. The probability that silver eels reach the spawning grounds and reproduce successfully is likely to vary greatly among continental growing sites [START_REF] Acou | Differential production and condition indices of premigrant eels (Anguilla anguilla) in two small Atlantic coastal catchments of France, in Eels at the Edge: Science, Status, and Conservation Concerns[END_REF][START_REF] Eelrep | Estimation of the reproduction capacity of European eel[END_REF]. A pan-European methodology to estimate the overall breeding potential of silver eels according to relevant criteria (e.g., fat composition, contamination by chemicals, parasitic load) is not ready for use because of the complexity in implementing such an approach. The use of silver eel condition indices may constitute a first step in this direction. Body condition has been demonstrated to indicate energy reserves of Salmo salar [START_REF] Sutton | Relationships among fat weight, body weight, water weight, and condition factors in wild Atlantic salmon parr[END_REF], Coregonus artedi [START_REF] Pangle | Temporal changes in the relationship between condition indices and proximate composition of juvenile Coregonus artedi[END_REF], and Gadus morhua [START_REF] Grant | Variation in condition of coastal Newfoundland 0-group Atlantic cod (Gadus morhua): field and laboratory studies using simple condition indices[END_REF][START_REF] Lambert | Can simple condition indices be used to monitor and quantify seasonal changes in the energy reserves of Atlantic cod (Gadus morhua)?[END_REF]. Considering the importance of energy reserves during both transoceanic migration and reproduction of the European eel [START_REF] Boe ¨tius | Experimental maturation of female silver eels, Anguilla anguilla. Estimates of fecundity and energy reserves for migration and spawning[END_REF], we hypothesize that body condition may represent a good proxy of silver eel fitness [START_REF] Acou | Differential production and condition indices of premigrant eels (Anguilla anguilla) in two small Atlantic coastal catchments of France, in Eels at the Edge: Science, Status, and Conservation Concerns[END_REF][START_REF] Jakob | Estimating fitness: a comparison of body condition indices[END_REF].

Because most parasitized organisms are generally in poorer condition than unparasitized ones [START_REF] Combes | Interactions durables. E ´cologie et e ´volution du parasitisme[END_REF][START_REF] Flegr | Frozen Evolution: Or, that's not the way it is, Mr. Darwin -Farewell to selfish gene[END_REF][START_REF] Thomas | E ´cologie et e ´volution des syste `mes parasite[END_REF] for reviews] and because parasites interact with natural and anthropogenic stressors to increase mortality and reduce animal health in myriad ways [START_REF] Marcogliese | Combined effects of parasites and contaminants on animal health: parasites do matter[END_REF] for review], parasitism is among the reasons invoked to explain the European eel collapse [START_REF] Brusle | L'anguille europe ´enne Anguilla anguilla, un poisson sensible aux stress environnementaux et vulne ´rable a diverses atteintes pathoge `nes[END_REF][START_REF] Fazio | Macroparasite community and asymmetry of the yellow eel Anguilla anguilla in Salses-Leucate lagoon, Southern France[END_REF][START_REF] Haenen | The health status of European silver eels, Anguilla anguilla, in the Dutch River Rhine watershed and Lake Ijsselmeer[END_REF][START_REF] Kennedy | The pathogenic helminth parasites of eels[END_REF]. Up to now, 161 species or taxa have been described in A. anguilla from fresh, brackish, and marine waters in 30 countries of Europe and North Africa [32 for review]. Among them, the introduced Anguillicoloides crassus (swimbladder nematode) and Pseudodactylogyrus sp. (gill monogenean) are considered as an important factor able to induce a stress that probably decreases the host fitness and seriously hampers the recovery of the European eel [for reviews : 9, 35]. The scientific research program EEL-IAD (2008)(2009)(2010)(2011)(2012) aimed to resolve some of the mysteries of eel biology in order to help conserving European eel stocks. EELIAD provided us the opportunity to investigate the parasitism patterns of some organs (gills, heart, intestine, and swimbladder) of silver eels sampled at the start of their transoceanic migration across five far-distant sites from northern Europe, i.e., Sweden, Denmark, Ireland (two sites), and France. Our objectives were (i) to describe the metazoan parasite community of these five silver eel subpopulations, and (ii) to evaluate in each site the potential parasitic constraint (on the whole and depending on parasite taxa) on the body condition of silver eels. Because a greater pathogenicity is often observed for recent (vs. long-term coevolved) host-parasite associations as for Pseudodactylogyrus sp. and A. crassus in A. anguilla [START_REF] Kania | Divergence between Asian, European and Canadian populations of the monogenean Pseudodactylogyrus bini indicated by ribosomal DNA patterns[END_REF][START_REF] Kennedy | The distribution and abundance of the nematode Anguillicola australiensis in eels Anguilla reinhardtii in Queensland, Australia[END_REF][START_REF] Moravec | Spreading of the nematode Anguillicola crassus (Dracunculoidea) among eel populations in Europe[END_REF], we suppose that the least parasitized silver eels by these introduced species could be the most susceptible to reach the spawning grounds and to reproduce in the Sargasso Sea compared to the most heavily infected ones.

Materials and methods

Study areas and silver eel sampling (Figure 1; Table 1) Silver eels (149) were caught during their seaward migration between October and February (i.e., at the beginning of their transoceanic migration) in five coastal water bodies from northern Europe: Stockholm Archipelago in Sweden The eel collection was focused on silver eel stage at the same maturation degree (see below the three criteria for silver eel assessment). Migrant silver eels were intercepted using pound nets (SWE-ARCH), coghill nets (IRE-COR), or eel-traps installed on dams (DEN-GUD, IRE-BUR, and FRA-FRE) depending on the site, these different methods of capture being adapted to both the size of the eels and the studied systems. Water bodies sampled differed in trophic status and level of anthropogenic pressure (Table 1). Among sites, IRE-BUR is considered as the pristine reference site due to the clean water and null anthropogenic pressure (no fishery, controls on movement of boat, water, or fishing gear).

Silver eels: Development stage and sex determination, aging by otoliths Each fish collected was anesthetized with Benzocaine at a concentration of 0.15 g L À1 . Then, total weight (TW, g) and total length (TL, mm) were measured respectively to the nearest g and mm. Silver eels were identified by three criteria [START_REF] Acou | Towards a standardized characterization of the potentially migrating silver European eel (Anguilla anguilla L.)[END_REF][START_REF] Acou | Errors in ocular index measurements in European eel Anguilla Anguilla (L.)[END_REF]: color of the back and belly, presence of a well-defined lateral line, as well as ocular index (OI ! 6.5, Pankhurst's 1982).

The sex was assigned by macroscopic observation of the gonads using the criteria described by Colombo et al. [START_REF] Colombo | Gonad differentiation and body growth in Anguilla anguilla L[END_REF]. Because most silver eels sampled were females (132 vs. 17 males), we only selected female eels to study the parasitic impact on the eel body condition.

As a proxy of fitness, we used index developed by Le Cren [START_REF] Cren | The length-weight relationship and seasonal cycle in gonad weight and condition in the perch (Perca fluviatilis)[END_REF] by calculating the body condition (BC) as the relative total weight TWr = 100 TW/TWstd, where TWstd is the predicted standard total weight of a fish at the same TL, as calculated according to the log 10 TW À log 10 TL regression equation (least squares means fit) for the whole sample of 132 female silver eels.

A sample of 58 female silver eels (SWE-ARCH, n = 4; DEN-GUD, n = 10; IRE-BUR, n = 17; IRE-COR, n = 11; FRA-FRE, n = 16) was aged by examination of the sagittal otoliths. The extracted otoliths were glued dorsal side up with crystal bond, grounded along the longitudinal plane until the nucleus was reached. Following the method described by the EIFAC ICES working group on eel [START_REF] Ices | Annex 4: manual for the ageing of Atlantic eel[END_REF], the fish continental age (number of years in fresh water) was determined through a stereomicroscope by counting the number of annuli from the first growth check outside the so-called zero band. This band is commonly assumed as the beginning of eel's continental growth.

Parasitological research in sampled Anguilla anguilla

All the 149 silver eels sampled were frozen before the research of metazoan parasites made on the gills, the heart, the digestive tract, and the swimbladder (the other organs were not available for parasitic investigation). Each organ was dissected using binocular microscope and all the metazoan parasites were numbered per organ and per eel (except for Myxozoa for which the presence or absence of cysts was recorded), identified and preserved in alcohol 70°. The parasitological parameter used to describe the community structure of parasites was species richness (number of parasite species in a sample of hosts); those to describe parasite populations (for a given parasite species) were prevalence (number of hosts infected with a particular parasite species/number of examined hosts), mean abundance (average abundance of a parasite species among all members of a host sample), and mean intensity (total number of parasites of a particular species found in a sample divided by the number of hosts infected with that parasite) [START_REF] Bush | Parasitology meets ecology on its own terms: Margolis et al. revisited[END_REF].

Statistical analyses

All the statistical analyses were made with the R-Cran project free software (http://www.r-project.org/). Differences were considered statistically significant at p < 0.05. Mean values of data are reported as means ± standard error (SE) except for prevalences ± 95% confidence limits (CL).

Inter-site comparison of the communities of parasites

Taxa with low prevalence ( 2%) were excluded of the analysis (Acanthocephalus lucii, Echinorhynchus truttae, Eustrongylides sp., Pomphorhynchus laevis, and Raphidascaris acus) because of their rarity and probable limited impact on silver eel subpopulations.

First, the structure of the component parasite communities in silver eels (n = 149) among the five sites was detected and represented by a Multiple Correspondence Analysis (MCA, Ade4 package) performed on the presence/absence of eight parasite taxa (Myxidium giardi, Pseudodactylogyrus sp., Bothriocephalus claviceps, A. crassus, Paraquimperia tenerrima, Camallanus lacustris, Acanthocephalus clavula, and Acanthocephalus anguillae). Second, similar tests (MCA) were focused on the abundances of the three major parasite taxa (i.e., the two introduced A. crassus and Pseudodactylogyrus sp., and the native A. clavula) that could exert a significant constraint on silver eels, potentially compromising the migration and spawning success [START_REF] Haenen | The health status of European silver eels, Anguilla anguilla, in the Dutch River Rhine watershed and Lake Ijsselmeer[END_REF][START_REF] Kennedy | The pathogenic helminth parasites of eels[END_REF]. In factorial maps (Figures 2 and3) realized from MCA results, ellipses are only visual summary. The prob-ability to be located in the ellipse is p = 1 À exp(Àk 2 /2) with k = 1.5 showing that around 67% of observations were located in the ellipses.

Factors influencing the body condition of female silver eels

Five variables which could explain BC were chosen: the age of female silver eels, the study-site, and the number of each of the three major parasite taxa (A. crassus, A. clavula, and Pseudodactylogyrus sp.). Their influence was analyzed using generalized linear models (glm, base package). We tested all possible combinations with these variables including interactions. Nineteen different models were tested and compared using classical selection by both step-akaike information criterion corrected from small sample sizes (AICc) and deviance explained (DE) [START_REF] Hurvich | Model selection for extended quasi-likelihood models in small samples[END_REF]. Then, the effect of parameters was analyzed by analysis of variance (ANOVA) from the selected model. As the sample size (58) used for the glm analyses was small according to the number of study-sites and variables tested, a power test using pwr package (pwr.f2.test) was performed on the selected model in order to give confidence in the acceptation of the null hypothesis (the fitness loss was not affected in some localities).

Results

Composition of the metazoan parasite community

A total of 149 European silver eels were dissected among them 89 ± 5% (132) were infected by one to six metazoan parasite taxa among the 13 identified in the whole sampling (Table 2). No trematodes were found. The nematode A. crassus, the only species recorded in the swimbladder, was the most common (prevalence of 56 ± 8%, mean intensity of 9.01 ± 1.19, and mean abundance of 4.98 ± 0.76), followed by A. clavula (30 ± 7%, 28.16 ± 4.17, and 8.45 ± 1.64) in the intestine, and then Pseudodactylogyrus sp. (17 ± 6%, 18.78 ± 4.73, and 2.47 ± 0.78) on the gills.

Component communities of metazoan parasites in each subpopulation of Anguilla anguilla

According to the MCAs (Figures 2 and3), the structure of the parasite component communities (Table 3) varied among the study-sites both on the presence/absence of the total parasite taxa and on the abundance of the three most prevalent taxa (A. crassus, A. clavula, and Pseudodactylogyrus sp.). IRE-BUR strongly differed from the four other sites. First, it was the only site where A. crassus was completely absent. Second, the intestine was heavily infected with a total prevalence of 90% (CL = 78-97%), mostly by the acanthocephalan A. clavula that was highly occurrent in terms of prevalence (84% with CL = 71-93%), mean intensity (30.51 ± 4.40), and mean abundance (25.53 ± 4.02). A second site, SWE-ARC, was also significantly different from the others (even with the small sample size of 10 fish compared to other sites) due to an overall prevalence less than half (40% with CL = 12-74% vs. 90-95% with CL = 77-100% in other sites) and to a total absence of parasites on the gills (Table 3). The three other sites: DEN-GUD, IRE-COR, and FRA-FRE, were characterized by high prevalences of A. crassus varying from 89% (CL = 67-104%) to 93% (80-99%). Among all sites, DEN-GUD and IRE-COR were the most similar because of the highest species richness (eight parasite taxa vs. four and five in other sites) and of the highest prevalences of Pseudodactylogyrus sp.; no monogenean was found in silver eels from FRA-FRE (Table 3).

Influence of parasitism on the body condition of female silver eels

Results from the 19 linear models are summarized in Table 4. According to the two criterions of selection (AICc and DE), the model #18 was selected. Results of ANOVA performed on this model are summarized in Table 5. The selected model explained 42.43% of the total deviance, and 33.10% of this explained deviance was explained only by the site (or 14.04% of total deviance, p = 0.013). The combined effect of female age and study-site was responsible of 27.70% of the explained deviance (p = 0.061). Among the three major parasite taxa, the number of Pseudodactylogyrus sp. had a strong negative effect on the BC of female silver eels (p = 0.003) and explained 27.10% of the explained deviance. The number of A. crassus tends to have a positive effect on the BC (p = 0.048). Model selection using step-AICc process showed that A. clavula had no significant effect on the BC. Indeed, AICc and explained deviance were not different between models #18 and #19, this latter being similar to #18 but with A. clavula as additional variable (Table 5).

Discussion

Whatever the host-parasite combination, parasites and their host compete for resources in such a way that both survival and fecundity of the host could be affected, even if no pathology is obvious and even if this effect may be drowned in the background noise of all other factors that affect survival and fecundity [START_REF] Combes | Interactions durables. E ´cologie et e ´volution du parasitisme[END_REF][START_REF] Flegr | Frozen Evolution: Or, that's not the way it is, Mr. Darwin -Farewell to selfish gene[END_REF][START_REF] O ¨stlund-Nilsson | Parasitic isopod Anilocra apogonae, a drag for the cardinal fish Cheilodipterus quinquelineatus[END_REF][START_REF] Thomas | E ´cologie et e ´volution des syste `mes parasite[END_REF] for reviews]. Other environmental stressors (e.g., pollutants, pathogens) that can also influence the health status of the eels [START_REF] Acou | Evidence of silver eels contamination by microcystin-LR at the onset of their seaward migration: What consequences for their breeding potential[END_REF][START_REF] Geeraerts | The effects of contaminants in European eel: a review[END_REF][START_REF] Gravato | Comparative study about the effects of pollution on glass and yellow eels (Anguilla anguilla) from the estuaries of Minho, Lima and Douro Rivers (NW Portugal)[END_REF][START_REF] Robinet | Sublethal effects of exposure to chemical compounds: a cause for the decline in Atlantic eels?[END_REF] could be considered as a potential confounding factor, but in most cases, contaminants and parasites occurring together are shown to exacerbate the detrimental effects on individuals (synergistic effects), suggesting Table 3. Component communities of metazoan parasites in Anguilla anguilla (149) from the five sampling sites: species richness, prevalence (P%), mean intensity (I ± SE) and mean abundance (A ± SE). See abbreviations in Tables 1 and2. Myxidium giardii cysts (Mgia) were not numbered in the gills. CL95% was indicated in parentheses for total prevalence per site.

that parasitized fish in polluted environments are in a poorer condition than unparasitized fish [47 for review]. Thus, to consider parasitism and its impact can help to understand the collapse of the European eel. In our study, 89 ± 5% of all silver eels were parasitized and 13 metazoan taxa were identified including Myxozoa, Monogenea, Cestoda, Nematoda, and Acanthocephala commonly recorded in A. anguilla [32 for review]. No Trematoda were found from any site in spite of 39 trematode species described in A. anguilla as definitive host [START_REF] Walter | A checklist of the protozoan and metazoan parasites of European eel (Anguilla anguilla): checklist of Anguilla anguilla parasites[END_REF]. One possible explanation is that our silver eels were originated from waters with salinity 5 psu and that trematodes infecting eels are shown to be significantly less frequent in fresh vs. marine waters [START_REF] Køie | Parasites in European eel Anguilla anguilla (L.) from Danish freshwater, brackish and marine localities[END_REF][START_REF] Kristmundsson | Parasite communities of eels Anguilla anguilla in freshwater and marine habitats in Iceland in comparison with other parasite communities of eels in Europe[END_REF].

The body condition of silver eels was used as a proxy of fitness that can reveal the impact of parasite infections. Among the three major parasite taxa recorded in the female silver eels studied here (A. crassus, A. clavula, and Pseudodactylogyrus sp.), only Pseudodactylogyrus sp. decreased the host body condition in relation with increasing abundance (up to 80 worms) on the gills. Pseudodactylogyrus sp. is the third more prevalent taxon in this study (17 ± 6%) and a specific gill monogenean of the genus Anguilla transferred from its native host Anguilla japonica to A. anguilla after introduction in Europe in 1977 [START_REF] Kania | Divergence between Asian, European and Canadian populations of the monogenean Pseudodactylogyrus bini indicated by ribosomal DNA patterns[END_REF][START_REF] Kennedy | The pathogenic helminth parasites of eels[END_REF][START_REF] Woo | Fish diseases and disorders[END_REF]. Pseudodactylogyrus sp. is browsing the host gill surface and induces epithelial lesions potentially leading to lethal hypoxia of the highly susceptible European eel and facilitating infections by various opportunist pathogens (virus, bacteria, and fungi) [START_REF] Kania | Divergence between Asian, European and Canadian populations of the monogenean Pseudodactylogyrus bini indicated by ribosomal DNA patterns[END_REF][START_REF] Kennedy | The pathogenic helminth parasites of eels[END_REF][START_REF] Woo | Fish diseases and disorders[END_REF].

In contrast with Pseudodactylogyrus sp., the infection by the introduced hematophagous A. crassus, the most prevalent species (56 ± 8%), had no significant effect on the condition factor of A. anguilla, as also shown by other studies [58 for review], and even tended to have a positive effect with increasing nematode abundance in the swimbladder. But this criterion is not the best reflecting the eel pathogenicity due to the short life cycle of A. crassus, and severely damaged swimbladders are shown to harbor very few or even no living nematodes [START_REF] Lefebvre | The health state of the eel swimbladder as a measure of parasite pressure by Anguillicola crassus[END_REF][START_REF] Zimmerman | Prevalence of Anguillicoloides crassus and growth variation in migrant yellow-phase American eels of the upper Potomac River drainage[END_REF]. Nevertheless, using the Swimbladder Degenerative Index, Lefebvre et al. [START_REF] Lefebvre | Is the continental life of the European eel Anguilla anguilla affected by the parasitic invader Anguillicoloides crassus[END_REF] recently demonstrated that the most affected eels had greater body length and mass (+11% and +41% respectively) than unaffected eels of the same age. Despite these surprising counterintuitive results, high virulence and severe impacts of A. crassus are expected because A. anguilla lacks an adaptive immune response, and various pathogenic potentially lethal effects (e.g., anemia, energy drain, swimming performance decrease) have been demonstrated, threatening the success of spawning migration in the Sargasso Sea [START_REF] Kennedy | The pathogenic helminth parasites of eels[END_REF][START_REF] Lefebvre | An evaluation of indices of gross pathology associated with the nematode Anguillicoloides crassus in eels[END_REF][START_REF] Lefebvre | Is the continental life of the European eel Anguilla anguilla affected by the parasitic invader Anguillicoloides crassus[END_REF][START_REF] Palstra | Swimming performance of silver eels is severely impaired by the swimbladder parasite Anguillicola crassus[END_REF][START_REF] Sjo ¨berg | Effects of the swimbladder parasite Anguillicola crassus on the migration of European silver eels Anguilla anguilla in the Baltic Sea[END_REF].

The body condition of female silver eels was also not influenced by the second most prevalent parasite of our study, A. clavula (30 ± 7%), a native generalist acanthocephalan with A. anguilla as preferred definitive host [START_REF] Brown | A key to the species of Acanthocephala parasitic in British freshwater fishes[END_REF][START_REF] Byrne | Utilization of brown trout Salmo trutta by Acanthocephalus clavula in an Irish lake: is this evidence of a host shift[END_REF]. The apparent unchanged fitness of A. clavula-infected eels is in accordance with the absence of pronounced symptoms of disease generally observed for most fish infected by acanthocephalans (including those with high parasite intensities) [60 for review].

Component communities of metazoan parasites and body condition of silver eels widely varied between sites, suggesting differences in biocenosis (e.g., host species occurrence) and environmental constraints or stressors (including parasitism), with probable consequences on the spawning migration and reproduction success. Based on the inter-site differences and the virulence of Pseudodactylogyrus sp. and A. crassus, the silver eels from Stockholm Archipelago (Sweden) and from Burrishoole (Ireland) could be the most successful, whereas the migration success may be affected by the pathogenic effects of introduced parasites for the silver eels from the other sites studied, i.e., Fre ´mur (France), Corrib (Ireland), and Gudenaa (Denmark). Indeed, only four out of the ten Swedish silver eels were infected and always with a very low parasite intensity. Moreover, they harbored no parasites on their gills and had both low A. crassus intensity (one helminth per swimbladder) and prevalence (three out of ten fish) suggesting low parasite pathogeny. Therefore, despite the low sample size, one can suppose that metazoan parasites have probably a limited influence on the condition of Swedish silver eels and on their migration success.

Burrishoole is considered as a pristine site with oligotrophic acid waters, no anthropogenic influence, and low levels of organic contaminants [START_REF] Machugh | The occurrence of persistent chlorinated and brominated organic contaminants in the European eel (Anguilla anguilla) in Irish waters[END_REF]. The growth rate of eels from this site is known to be extremely low [START_REF] Poole | Age and growth of yellow eel, Anguilla anguilla (L.), determined by two different methods[END_REF], thus we consider that the one-year sooner sampling date (compared to the four other sites) had no significant impact on the eel parasite community and did not introduce a bias in our inter-site comparison. In our study, 90 ± 8% of the silver eels from Burrishoole were parasitized but this high total prevalence was not synonym of a strong parasitic constraint since they were mainly infected by the native intestinal acanthocephalan A. clavula (prevalence of 84 ± 10%, intensity of 31 ± 4) for which we demonstrated no influence on the host body condition. A. crassus was not recorded suggesting that the nematode may not complete its entire life cycle in Burrishoole despite its numerous intermediate crustacean (ostracods, copepods) and paratenic fish (and even snails) hosts [START_REF] Haenen | Detection of larvae of Anguillicola crassus (an eel swimbladder nematode) in freshwater fish[END_REF][START_REF] Moravec | Aquatic invertebrates (snails) as new paratenic hosts of Anguillicola crassus (Nematoda: Dracunculoidea) and the role of paratenic hosts in the life cycle of this parasite[END_REF]. Moreover, no parasitic pathology probably occurred in the gills of Burrishoole silver eels because of the rarity of Pseudodactylogyrus sp. (only two infected out of the 49 specimens, harboring only one or three monogeneans).

Despite differences in the anthropogenic influence and the trophic status of Fre ´mur (France), Corrib (Ireland), and Gudenaa (Denmark), A. crassus was omnipresent in A. anguilla from these sites with a prevalence varying from 89% (67/104) to 93% (80/99) probably inducing pathogenic effects [35 for review]. In addition, for 60% (36/81) and 71% (49/87) of the Corrib and Gudenaa silver eels, hypoxia due to the impaired gills was probably a major symptom as they were severely infected by Pseudodactylogyrus sp. with a mean intensity of 23 ± 12 and 19 ± 6 respectively. The parasite constraint appeared more substantial for the silver eels from Corrib and Gudenaa having both damaged swimbladder and gills than for those from Fre ´mur since Pseudodactylogyrus sp. was absent.

Finally, based on the potential fitness loss induced by parasitism, we suppose that the migrant silver eels from Stockholm Archipelago (Sweden) and Burrishoole (Ireland) are able to contribute to the recruitment and gene pool of A. anguilla pop-ulation, whereas those from our other study-sites [in particular from Corrib (Ireland) and Gudenaa (Denmark)] have a lower probability to reach the spawning grounds in Sargasso Sea.

Numerous aspects remain to be explored to explain the decline of the European eel population, probably induced by various interacting abiotic and biotic factors (e.g., habitat loss or fragmentation, changing hydrology, overfishing, pollution, and pathogens [START_REF] Brusle | L'anguille europe ´enne Anguilla anguilla, un poisson sensible aux stress environnementaux et vulne ´rable a diverses atteintes pathoge `nes[END_REF][START_REF] Fazio | Macroparasite community and asymmetry of the yellow eel Anguilla anguilla in Salses-Leucate lagoon, Southern France[END_REF][START_REF] Feunteun | Management and restoration of European eel population (Anguilla anguilla): an impossible bargain?[END_REF][START_REF] Haenen | The health status of European silver eels, Anguilla anguilla, in the Dutch River Rhine watershed and Lake Ijsselmeer[END_REF][START_REF] Kettle | Where once the eel and the elephant were together: decline of the European eel because of changing hydrology in southwest Europe and northwest Africa?[END_REF][START_REF] Robinet | Sublethal effects of exposure to chemical compounds: a cause for the decline in Atlantic eels?[END_REF]) and resulting in difficult and complex measures of preservation of this species. Enhancing the production of viable silver eels by water system represents the current target of the conservation strategy of the European eel [START_REF] Eu | Council Regulation (EC) No 1100/2007 of 18 September 2007 establishing measures for the recovery of the stock of European eel[END_REF]. However, we believe that the question of animal quality among river systems, which is presumed to influence the reproductive success, is also a key issue that must be urgently pursued for European eel conservation. Every potential impact on silver eel body condition warrants examination, more especially as synergistic effects can occur between environmental stressors such as parasites and contaminants, increasing the mortality of exposed organisms [47 for review].

  (SWE-ARCH, n = 10, Oct 2009), River Gudenaa in Denmark (DEN-GUD, n = 21, Dec 2009), Burrishoole (IRE-BUR, n = 49, Nov 2008), and Corrib (IRE-COR, n = 26, Nov 2009) in Ireland, and River Fre ´mur in France (FRA-FRE, n = 43, Feb 2010).

Figure 1 .

 1 Figure 1. Geographical position of the sample sites of Anguilla anguilla in northern Europe: Burrishoole (IRE-BUR) and Corrib (IRE-COR) in Ireland, Fre ´mur in France (FRA-FRE), Gudenaa in Denmark (DEN-GUD), and Stockholm Archipelago in Sweden (SWE-ARC).

Table 1 .

 1 Characteristics of the five coastal water bodies sampled, i.e., Stockholm Archipelago (SWE-ARC), Gudenaa (DEN-GUD), Burrishoole (IRE-BUR), Corrib (IRE-COR), and Fre ´mur (FRA-FRE). Trophic status according to[START_REF] Carlson | A trophic state index for lakes[END_REF] [START_REF] Carlson | A trophic state index for lakes[END_REF]. Sampling dates from October 2009 to February 2010 correspond to the same seaward migration of the silver eels whereas sampling in Burrishoole occurred one-year sooner.

Figure 2 .

 2 Figure 2. Factorial map according to the Multiple Correspondence Analysis on the presence/absence of the eight parasite taxa [Acanthocephalus anguillae (Aang), Acanthocephalus clavula (Acla), Anguillicoloides crassus (Acra), Bothriocephalus claviceps (Bcla), Camallanus lacustris (Clac), Myxidium giardia (Mgia), Paraquimperia tenerrima (Pten), and Pseudodactylogyrus sp. (Pseu)] in silver eels from the five study-sites (DEN-GUD, FRA-FRE, IRE-BUR, IRE-COR, and SWE-ARC).

Figure 3 .

 3 Figure 3. Factorial map according to the Multiple Correspondence Analysis on the abundance of the three major parasite taxa (Acanthocephalus clavula, Anguillicoloides crassus, and Pseudodactylogyrus sp.) in silver eels from the five study-sites (DEN-GUD, FRA-FRE, IRE-BUR, IRE-COR, and SWE-ARC)

SWE

Table 2 .

 2 Metazoan parasite species and their ecological parameters in Anguilla anguilla (149) from the five study-sites (CL 95% was calculated for each prevalence).

	Species

Table 4 .

 4 List of all the 18 models tested with AICc (akaike information criterion corrected from small sample sizes) and DE (deviance explained, %); BC = body condition.

	No.	Model	AIC	DE
	1	BC ~site	393.13	14.04
	2	BC ~age	390.25	12.23
	3	B C Pseudodactylogyrus sp.	394.03	5.86
	4	B C Ã. clavula	396.10	2.19
	5	B C Ã. crassus	397.04	0.47
	6	BC ~site + age	389.05	23.19
	7	BC ~site + age + Pseudodactylogyrus sp.	384.96	31.38
	8	BC ~site + age:site + Pseudodactylogyrus sp. + A. clavula	387.90	37.51
	9	BC ~site + age + Pseudodactylogyrus sp. + A. clavula + A. crassus	384.83	36.42
	10	BC ~site + age + Pseudodactylogyrus sp.:site + A. clavula:site + A. crassus:site	389.27	40.49
	11	BC ~site + age:site + Pseudodactylogyrus sp.:site + A. clavula:site + A. crassus:site	389.27	46.74
	12	BC ~age:site + Pseudodactylogyrus sp.:site + A. clavula:site + A. crassus:site	388.67	41.14
	13	BC ~site + site:age + Pseudodactylogyrus sp.:site + A. crassus:site	385.78	46.24
	14	BC ~site + age:site + Pseudodactylogyrus sp.:site	387.90	37.51
	15	BC ~site + age:site + Pseudodactylogyrus sp.:site + A. clavula:site	391.39	38.10
	16	BC ~site + age + Pseudodactylogyrus sp.:site + A. clavula:site	390.44	32.04
	17	BC ~site + age + Pseudodactylogyrus sp. + A. crassus	382.84	36.42
	18	BC ~site + age:site + Pseudodactylogyrus sp. + A. crassus	382.44	42.43
	19	BC ~site + age:site + Pseudodactylogyrus sp. + A. crassus + A. clavula	385.47	42.43

Table 5 .

 5 Deviance explained (%) by each significant variable (studysite, Pseudodactylogyrus sp., Anguillicoloides crassus and site:age) of the selected model.

	Variables	Deviance	P-value
	Study-site	14.04	0.013
	Pseudodactylogyrus sp.	11.50	0.003
	A. crassus	5.12	0.048
	Site:age	11.77	0.061
	Total deviance explained	42.43	0.002

C. Ge ´rard et al.: Parasite 2013, 20, 38

Acknowledgements. We would like to thank Dave Righton that supervised the EELIAD program and all European collaborators for providing the fish used in this study (Ha ˚kan Wickstro ¨m, Niklas Sjoberg, Kim Aarestrup, Russel Poole, Michael Ingemann Pedersen, Gustavo Becerra-Jurado, Paddy Gargan, Alan Walker, Fish Pass, and Bretagne Grands Migrateurs). This study was funded by Grant Agreement GOCE-2008212133 (EELIAD) of the European Union FP7 research program.


domain: chim sde sdv

text:
Chloé Boutillez

Aurélie Anne Perez

Francois Guyot

Stéphanie Rossano

Atmo

Atmospheric alteration of stained glass: influence of Mn-oxidising bacteria (P seudomona putida)

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Atmospheric alteration of stained glass: influence of Mn-oxidising bacteria (Pseudomona putida)

Chloé Boutillez 1 , Anne Perez 1 , Aurélie Verney-Carron 2 , François Guyot 3 and Stéphanie Rossano 1 For several centuries, the external and internal surfaces of mediaeval stained glass windows are continuously exposed to water (rain events, humidity, condensation/dry cycle), which causes the alteration of their external/internal surfaces. These conditions of high humidity also favours the development of microorganisms. Due to their composition, medieval glasses can be attractive substrates for the development of biofilms, as they represent a source of nutrients for microorganisms [1]. However, due to the multitude of parameters involved, these interactions are not well understood and are still poorly documented.

To test the influence of glass substrates on their colonisation and alteration processes, a panel of 5 model medieval glasses was subjected to short term (closed system, 7 days) and medium term (opened system in bioreactor, 1 month) alteration experiments in an aqueous environment. Five glasses having identical potash-lime-silica base and variable Mn and Fe contents (between 0 and 2 wt% of oxide) were chosen, in order to target specifically the role of Mn and Fe in the development of bacteria/biofilm formation, and investigate the potential role of bacteria in the browning pathology known to develop at the surface stained glass windows containing Mn. The Pseudomonas putida strain, known for its affinity for Fe and its Mn oxidation properties, was selected in a first step. In a second step, experiments were carried out with Arthrobacter Crystallopoietes strain, which was collected from exposed stained glass windows and is known for its altering capacity.

The five glasses were immersed in a minimalist liquid medium at 25°C in the presence of microorganisms and the (bio)altered glasses (7 days, 1 month) were characterised by SEM. In the shortterm experiments, the filtered cultures were periodically analysed by ICP-OES in order to study the dissolution kinetics of the different glasses in the presence of bacteria. The results of show that the presence of P. putida has no real effect on the short-term hydrolysis rate of glasses, but significantly modifies the dissolution and the speciation of certain elements (Mn, Fe, P) from the glass. The bacterial growth is also strongly linked to the composition of the glass: in the presence of Mn-bearing glass, consistent biofilms are observed, whereas in the presence of Fe-bearing glass, bacteria mostly develop in a planktonic form. Moreover, the absence of Fe in the glass triggers the production of siderophores. In all experiments with Mn and without Fe, the production of siderophore is followed by the formation of Mn oxides in solution and in biofilms attached on the glass surface.


Stef Graillat
email: stef.graillat@lip6.fr

Philippe Langlois
email: langlois@univ-perp.fr

REAL AND COMPLEX PSEUDOZERO SETS FOR POLYNOMIALS WITH APPLICATIONS

Keywords: Mathematics Subject Classification. 65F35, 68W30 Polynomial root, pseudozero set, uncertainty, perturbation, stability

Pseudozeros are useful to describe how perturbations of polynomial coefficients affect its zeros. We compare two types of pseudozero sets: the complex and the real pseudozero sets. These sets differ with respect to the type of perturbations. The first set -complex perturbations of a complex polynomial -has been intensively studied while the second one -real perturbations of a real polynomial -seems to have received little attention. We present a computable formula for the real pseudozero set and a comparison between these two pseudozero sets. We conclude that the complex pseudozero sets have to be preferred except when the perturbed real polynomials admit non-real zeros. We also give some applications of pseudozero set in control theory.

Introduction

The computation of polynomial roots is commonly used in numerous applications in scientific computing or engineering. In practice, the real or complex polynomial coefficients are often approximate values. Two well known sources of such approximation are data uncertainties and rounding errors. The sensitivity of the roots with respect to these perturbations of the polynomial coefficients has been studied with several kinds of methods.

Analytical sensitivity analysis introduces a condition number that bounds the magnitudes of the (first order) changes of the roots with respect to the coefficient perturbations. Numerous results in this direction are available, see for example Gautschi [START_REF] Gautschi | On the condition of algebraic equations[END_REF] or Wilkinson [START_REF] Wilkinson | Rounding errors in algebraic processes[END_REF]. Representing coefficient uncertainty with intervals and computing with interval arithmetic yield over-sets that enclose (sometimes pessimistically) the perturbed roots [START_REF]WWW resources about Interval Arithmetic[END_REF]. When the coefficient uncertainty is assumed to satisfy a given probabilistic behavior, stochastic arithmetic of Vignes and Chesneaux and its associated library CADNA provide a probabilistic estimate of perturbed roots [START_REF] Chesneaux | La bibliothèque CADNA : présentation et utilisation[END_REF][START_REF] Vignes | A stochastic arithmetic for reliable scientific computation[END_REF]. Continuous sensitivity analysis, introduced by Ostrowski [START_REF] Ostrowski | Solution of equations and systems of equations[END_REF], considers the uncertainty of the coefficients as a continuity problem. The most powerful tool of this last type of methods seems to be the pseudozero set of a polynomial we focus hereafter. Roughly speaking, it is the set of the roots of polynomials being near to a given polynomial. This set was first introduced by Mosier [START_REF] Mosier | Root neighborhoods of a polynomial[END_REF]. A very recent survey about pseudozero set is proposed by Stetter [START_REF] Stetter | Numerical Polynomial Algebra[END_REF].

In this paper, we study two kinds of pseudozero sets. The first one is the complex pseudozero set : it is the set of complex numbers that are the roots of polynomials with complex coefficients being near to a given polynomial p with complex coefficients. The second one is the real pseudozero set, that is the set of complex numbers that are the roots of polynomials with real coefficients being near to a given polynomial p with real coefficients.

For a given polynomial with real coefficients, it makes sense to compute both complex and real pseudozero sets even if the latter may be closer to the physical problem the polynomial represent. This is the case when the polynomial coefficients describe non-complex physical values as for example in transfer function for control theory. Previous works of the authors illustrate how the complex pseudozero set solves stability problems in this area [START_REF] Graillat | Testing polynomial primality with pseudozeros[END_REF][START_REF] Graillat | Pseudozero set decides on polynomial stability[END_REF]. Another motivation to constrain the pseudozero set to real perturbations comes from finite precision computation since the rounding error in real coefficients represented by fixed or floating numbers is always a non-complex perturbation. The aim of this paper is to compare these two pseudozero sets and evaluate which one is the more convenient and the easiest to compute.

Up to our knowledge, published results often consider only complex pseudozero sets. The real pseudozero set has not been studied a lot before. Hereafter we provide a computable formula for the real pseudozero set and we justify that the complex pseudozero set is more convenient than the real one.

The paper is organized as follows. In Section 2, we recall some definitions and known results about the complex pseudozero set. In Section 3, we introduce the real pseudozero set and we propose a computable expression for this set. In Section 4, we present some numerical simulations and we compare the real and complex pseudozero sets. The Section 5 is devoted to applications of pseudozero set in control theory and to robustness problems in particular.

Complex pseudozero set of polynomials

Let P n (C) be the linear space of polynomials of degree at most n with complex coefficients (n ≥ 1). Let p be a polynomial of P n (C) with complex coefficients p i

(0 ≤ i ≤ n) such that p(z) = n i=0 p i z i .
Representing p by the vector (p 0 , p 1 , . . . , p n ) T of its coefficients, we identify the norm on P n (C) to the 2-norm of the corresponding vector in C n+1 . Throughout this paper, we will only consider this 2-norm we denote • . It means that

p = n i=0 |p i | 2 1/2
.

(

) 1
The following real parameter ε bounds the uncertainty of the coefficients of p.

Given such an ε > 0, a complex ε-neighborhood of p is the set of all polynomials of P n (C), close enough to p, that is,

N ε (p) = { p ∈ P n (C) : p -p ≤ ε} .
The complex ε-pseudozero set of p is defined to include all the zeros of the ε-neighborhood of p. A definition of this set is

Z ε (p) = {z ∈ C : p(z) = 0 for some p ∈ N ε (p)} .
Theorem 2.1 below provides a computable counterpart of this definition.

Theorem 2.1 (Trefethen and Toh [START_REF] Toh | Pseudozeros of polynomials and pseudospectra of companion matrices[END_REF]). The complex ε-pseudozero set of p verifies

Z ε (p) = z ∈ C : g(z) := |p(z)| z ≤ ε , where z = (1, z, . . . , z n ) T .
This theorem was proved in [START_REF] Toh | Pseudozeros of polynomials and pseudospectra of companion matrices[END_REF] for the 2-norm and in [START_REF] Stetter | Polynomials with coefficients of limited accuracy[END_REF] for an arbitrary norm. We recall the proof of [START_REF] Toh | Pseudozeros of polynomials and pseudospectra of companion matrices[END_REF].

Proof. If z ∈ Z ε (p) then there exists p ∈ P n (C) such that p(z) = 0 and p-p ≤ ε. From Hölder's inequality |x T y| ≤ x y , we get |p(z)| = |p(z) -p(z)| = n i=0 (p i -p i )z i ≤ p -p z . It follows that |p(z)| ≤ ε z . Conversely, let u ∈ C be such that |p(u)| ≤ ε u . If u = 0, we can write u = |u|e iθ , θ ∈ [0, 2π) with |u| > 0.
Let us introduce the polynomials r and p u defined by

r(z) = n k=0 r k z k , with r k = |u| k e -ikθ , and p u (z) = p(z) - p(u) r(u) r(z).
It is clear that r(u) = u 2 = r 2 , and p u (u) = 0. So we have

p -p u = |p(u)| |r(u)| r ≤ ε.
Hence we obtain that u belongs to

Z ε (p). If u = 0, let us define p u (z) = p(z) -p(u). It is clear that p u (u) = 0. Besides, we have p -p u = |p(u)| ≤ ε by hypothesis.
In the same way, we get that u belongs to Z ε (p).

This theorem gives us an efficient way to compute the complex pseudozero set. The ε-pseudozeros of p belong to the interior of the area defined by the contour level (of value ε) of the normalized residual |p(z)|/ z . Using for example MATLAB primitives, following Algorithm 1 easily plots the ε-pseudozeros of a polynomial p for a given coefficient uncertainty ε.

Algorithm 1 Computation of complex ε-pseudozero set (MATLAB version)

Require: polynomial p and uncertainty ε Ensure: pseudozero set layout in the complex plane 1: We grid a square containing all the roots of p with the MATLAB command meshgrid. 2: We compute g(z) at the grid nodes z.

Real pseudozero set of polynomials

Now, we introduce the real pseudozero set and a computable expression for this set.

The notations are similar to the complex case. For n ≥ 1, let P n (R) be the linear space of polynomials of at most degree n with real coefficients. A polynomial p in P n (R) is represented by the vector (p 0 , p 1 , . . . , p n ) T of its coefficients, so p(z) = n i=0 p i z i . Again we identify the norm • on P n (R) to the 2-norm on R n+1 of the corresponding vector of its coefficients. Relation (1) is still valid to define this norm.

Let ε be a given bound of the polynomial coefficient uncertainties. A real ε-neighborhood of p is the set of all polynomials of P n (R), close enough to p, that is,

N R ε (p) = { p ∈ P n (R) : p -p ≤ ε} . ( 2
)
Then the real ε-pseudozero set of p is defined to include all the zeros of the real ε-neighborhood of p. A definition of this set is

Z R ε (p) = z ∈ C : p(z) = 0 for some p ∈ N R ε (p) . ( 3
)
For ε = 0, the pseudozero set Z R 0 (p) is the set of the roots of p we denote Z(p). One can easily notice that the real ε-pseudozero set Z R ε (p) is symmetric with respect to the real axis. Proof. Let z ∈ Z R ε (p). It means that there exists q ∈ N R ε (p) such that q(z) = 0. As the polynomial q has real coefficients, this implies that q(z) = 0. So z ∈ Z R ε (p).

Following Theorem 3.2 provides a computable counterpart of this definition. It can be considered as a special case of spectral value sets for the companion matrix using structured perturbations [START_REF] Hinrichsen | Spectral value sets: a graphical tool for robustness analysis[END_REF] when the polynomial is monic. We prove it hereafter using arguments developed in [START_REF] Hinrichsen | Robustness measures for linear systems with application to stability radii of Hurwitz and Schur polynomials[END_REF] but staying in the field of polynomials. The proof also deals with non-monic polynomials. We define for x, y ∈ R n+1 , d(x, Ry) = inf α∈R xαy , the distance of a point x ∈ R n+1 from the linear subspace Ry = {αy, α ∈ R}.

Theorem 3.2. The real ε-pseudozero set of p verifies

Z R ε (p) = Z(p) ∪ z ∈ C\Z(p) : h(z) := d(G R (z), RG I (z)) ≥ 1 ε , ( 4
)
where G R (z) and G I (z) are the real and imaginary parts of

G(z) = 1 p(z) (1, z, . . . , z n ) T , z ∈ C\Z(p). Proof. Let z ∈ Z R ε (p). If p(z) = 0 then z ∈ Z(p) else there exists q ∈ N R ε (p) such that q(z) = 0. In this case, we have p(z) = p(z) -q(z) = (p -q) T z, where z = (1, z, . . . , z n ) T . It follows that 1 = (p -q) T G(z). Hence we have 1 = (p -q) T G R (u) + i(p -q) T G I (u) and so (p -q) T G R (u) = 1, (p -q) T G I (u) = 0.
As a consequence, we have pq G R (u) -αG I (u) ≥ 1, for all α ∈ R. We conclude that

d(G R (u), RG I (u)) ≥ 1 p -q ≥ 1 ε • Conversely, let z ∈ Z(p) ∪ z ∈ C\Z(p) : d(G R (z), RG I (z)) ≥ 1 ε . If z belongs to Z(p) then it belongs to Z R ε (p).
Otherwise z satisfies G(z) ≥ 1/ε. From a duality theorem (see [START_REF] Luenberger | Optimization by vector space methods[END_REF], p. 119), there exists a vector u ∈ R n+1 with u = 1 satisfying

u T G R (z) = d(G R (z), RG I (z)) and u T G I (z) = 0.
Let us consider the real polynomial

q = p - u d(G R (z), RG I (z)) •
We have

q(z) = p(z) - u T z d(G R (z), RG I (z)) = p(z) - p(z)u T G(z) d(G R (z), RG I (z)) = 0. Furthermore we have q -p = 1/d(G R (z), RG I (z)). It follows p -q ≤ ε.
To compute the real ε-pseudozero set Z R ε (p), we only have to evaluate the distance d(G R (z), RG I (z)). We recall that • is the 2-norm and we denote with

•, • the associated inner product. In this case we have

d(x, Ry) = x 2 -x,y 2 y 2 if y = 0, x if y = 0.
Then the real pseudozero set is also the interior of the area defined by the level contour of function h defined by Relation 4. We can compute the real ε-pseudozero set with following Algorithm 2.

Algorithm 2 Computation of real ε-pseudozero set (MATLAB version)

Require: polynomial p and uncertainty ε Ensure: pseudozero set layout in the complex plane 1: We grid a square containing all the roots of p with the MATLAB command meshgrid.

2:

We compute h(z) at the grid nodes z.

Numerical simulations and comparisons

In this section, we present some numerical simulations and we compare the real and complex pseudozero sets. This comparison is motivated since it is more difficult to compute the real pseudozero set than the complex pseudozero set. It is clear that we always have Z R ε (p) ⊂ Z ε (p). Proposition 4.1. Let p be a polynomial of P n (R). Then the real pseudozero set is included in the complex pseudozero set, i.e.,

Z R ε (p) ⊂ Z ε (p).
Proof. Let z belonging to Z R ε (p). It means that there exists a polynomial q in P n (R) such that q(z) = 0. Since P n (R) is included in P n (C), it means that q is in P n (C). Consequently, z belongs to Z ε (p).

We are interested to know whether the real and the complex pseudozero sets are similar or not.

Let us first comment the following example considering p(z) = 1/2+z 2 . Figure 1 represents the real and complex pseudozero sets for this polynomial p with ε = 0.1. The real pseudozero set (Fig. 1b) is clearly enclosed in the complex pseudozero set (Fig. 1a). The shapes of the real and the complex pseudozero sets are quite similar. This example clearly exhibits that the real and the complex pseudozero sets look very similar Let us now consider polynomial q(z) = z -1 with ε = 0.1. Figure 2 again illustrates that the real pseudozero set (Fig. 2b) is included in the complex pseudozero set (Fig. 2a). Nevertheless, the real pseudozero set and the complex pseudozero set now have very different shapes. In fact, the real pseudozero set should be included in the real axis. Since it appears to not be the case here, we explain this phenomenon thanks to the following lemma. Lemma 4.2 (Hinrichsen and Kelb [START_REF] Hinrichsen | Spectral value sets: a graphical tool for robustness analysis[END_REF]). The function

d : R n+1 × R n+1 → R + , (x, y) → d(x, Ry)
is continuous at all pairs (x, y) with y = 0 or x = 0, and discontinuous at all pairs

(x, 0) ∈ R n+1 × R n+1 , x = 0.
This lemma states that a discontinuity problem arises when vector y vanishes. In our case the discontinuity arises when G I (z) = 0, where G I is the imaginary part G I (z) of

G(z) = 1 p(z) (1, z, . . . , z n ) T .
It follows that G I vanishes for z ∈ R, that is along the real axis. This explains why the contour function of MATLAB may fail and gives some bad results along the real axis. Of course, if none of the zeros of the polynomial is real, the real pseudozero set is reliable because we do not evaluate the function G on the real axis. The first presented example and Figure 1 illustrate this case for instance. A reliable computation of the real pseudozero set on the real axis can be performed with the complex pseudozero counterpart thanks to the following lemma.

Lemma 4.3. Being z ∈ R, z belongs to Z R ε (p) if and only if z belongs to Z ε (p) (complex

version of the pseudozero set).

Proof. This is true because the formula involved in Theorem 2.1 and in the proof stays in the real field if z is real.

We conclude this comparison stating that in general complex pseudozero sets have to be preferred to real ones even to represent polynomials with perturbed real coefficients. Nevertheless real pseudozero sets yield an accurate and reliable description of perturbed zeros when it can be proved that none of them is real.

Applications in control theory

In control theory, classical transfer functions associated with some systems are often written as H(z) = N (z)/D(z), where N and D are complex polynomials and z is a parameter of the system. The system described with the function H is stable (in a sense to be defined later) if the polynomial D is stable. Since uncertainties of the coefficients of the polynomials are unavoidable in most real problems (data uncertainty, rounding error), it is useful to know whether the system still remains stable while the coefficient uncertainty is bounded by a given ε > 0. In such cases, the system is said to be robustly stable. Similarly the polynomial D is robustly stable when it remains stable when its coefficients suffer from an uncertainty of ε > 0. The robust stability of polynomial has been studied intensively (see [START_REF] Jaulin | Applied interval analysis[END_REF] and the references therein) for polynomials with complex coefficients and with complex perturbations of those coefficients. Hereafter, we focus on stability for real polynomials only allowing real perturbations of the coefficients. We name this as the real stability problem.

Deciding on polynomial stability: a first criterion

The real ε-pseudozero set can be used to decide the real stability of systems described by polynomials. A classic stability criterion in control theory is to compare the modulus roots to 1 (Schur stability). For a given polynomial with coefficients known with a tolerance ε, it is difficult to compute every root of all polynomials being in its ε-neighborhood. Of course, a sensitivity analysis that uses the condition number of the polynomial with respect to its coefficients can be performed and yields a first order criterion for arbitrary small uncertainty ε. Pseudozero sets provide a more general answer to this question since it neither neglect higher order effects of coefficient perturbations nor restrict these perturbations only to arbitrary small values. It suffices to draw the real ε-pseudozero set and to verify if it is included in the unit circle.

Figure 3 shows the real ε-pseudozero set of polynomial p(z) = (z -0.3)(z -0.6)(z -0.7)(z + 0.5), with two coefficient uncertainties bounded by ε = 0.1 and ε = 0.05.

From this figure, we cannot decide the real stability for ε = 0.1 because some 0.1-pseudozeros have modulus larger than 1. On the other hand, the figure proves that all the real 0.05-pseudozeros have a modulus less than 1 and so we conclude for the real stability of p while ε ≤ 0.05.

Deciding on polynomial stability: a second criterion

Another kind of real stability is defined when the real part of all the roots of a polynomial are negative (Hurwitz stability). For testing this other real stability criterion, it suffices to draw the real ε-pseudozero set and to verify if it is included in the left half-plane. This is shown with Figure 4. We exhibit for example that polynomial p(z) = (z +1) 2 (z +0.2+0.2i)(z +0.2-0.2i) is real stable while ε ≤ 0.06 but is unstable if ε = 0.6.

Conclusion

In this paper, we have proposed a computable formula for the real pseudozero set. Identifying this set is motivated from data errors and rounding errors that corrupt the coefficients of real polynomials. Alas this formula is more difficult to compute than the corresponding complex one. Thanks to numerical simulations, we have shown that computing real pseudozero set may yield unreliable results. We justify this breakdown comes from the discontinuity of the level function. Then it is in general better to prefer the complex pseudozero set for real polynomial instead of the real pseudozero set.

Nevertheless, if the perturbations are sufficiently small to guarantee the zeros of the real polynomial stay far from the real axis then we may prefer to draw the real pseudozero set (since the discontinuity only appears near the real axis). In this case the real pseudozero set represents more accurately than before all the roots of the really perturbed polynomials.

3 :

 3 We draw the level line |g(z)| = ε with the MATLAB command contour.

Proposition 3 . 1 .

 31 The real ε-pseudozero set Z R ε (p) is symmetric with respect to the real axis.

3 :

 3 We draw the level line |h(z)| = 1/ε with the MATLAB command contour.

(Figure 1 .

 1 Figure 1. Complex and real pseudozero sets for p(z) = 1/2 + z 2 .

  (a) Complex Zε(q) with ε = 0.1 (b) Real Z R ε (q) with ε = 0.1(c) Both sets on the same figure

Figure 2 .

 2 Figure 2. Complex and real pseudozero sets for q(z) = z -1.

Figure 3 .

 3 Figure 3. Real ε-pseudozero set of the polynomial p(z) = (z -0.3)(z -0.6)(z -0.7)(z + 0.5), with ε = 0.1 and ε = 0.05.

Figure 4 .

 4 Figure 4. Real ε-pseudozero set of p(z) = (z + 1) 2 (z + 0.2 + 0.2i)(z + 0.2 -0.2i), with ε = 0.06 and ε = 0.6.