Jessica Kojadinovic
email: jessica.kojadinovic@univ-nantes.fr

Frédéric Ménard

Paco Bustamante

Richard P Cosson

Matthieu Le Corre

Trophic ecology of marine birds and pelagic fishes from Reunion Island as determined by stable isotope analysis

Keywords: Feeding behavior, Seabirds, Tunas, Dolphinfish, Top predators, Western Indian Ocean, δ 15 N, δ 13 C

Stable nitrogen and carbon isotopes were used to investigate trophic ecology in tropical marine bird and fish communities from Reunion Island, western Indian Ocean. Firstly, isotope signatures in the liver of Barau's petrels Pterodroma baraui, Audubon's shearwaters Puffinus lherminieri bailloni, and white-tailed tropicbirds Phaethon lepturus were used to compare their trophic levels and determine whether they forage in the same areas while breeding on Reunion Island. Spatial and trophic segregations were noted among these seabirds. Barau's petrels seem to feed on prey of higher trophic levels than Audubon's shearwaters. Different isotopic signatures in adults and juveniles of these species suggest that these chick-rearing Procellariiformes adopt a dual food-provisioning strategy, making separate foraging trips to feed their fledglings and for their own maintenance. Satellite tracking should be undertaken to verify this hypothesis. Furthermore, novel data were obtained on the seabirds' interbreeding period by analyzing feather signatures. White-tailed tropicbirds are thought to change foraging areas during this season, although none of the birds seemed to shift diets. Secondly, isotopic signatures in the muscle of yellowfin tuna Thunnus albacares, skipjack tuna Katsuwonus pelamis, and common dolphinfish Coryphaena hippurus were used to gather information on their feeding behaviors in Reunion Island waters. Spatial and trophic segregations were also observed, particularly between common dolphinfish and the tuna species, where the former fed more on low trophic level coastal organisms under fish aggregating devices than did the latter. Finally, trophic interactions in bird and fish communities were investigated. Seabirds appear to be trophically more structured than fish, foraging in a wider range of areas. Our results confirmed feeding associations between Audubon's shearwaters and yellowfin tuna.

INTRODUCTION

Due to the importance of predator-prey relationships and their dynamics in the structure and evolution of animal communities, ecologists continuously focus on better understanding trophic interactions. Investigations of trophic interactions between seabirds and surface predating fish is currently a major objective of the study of top predators in the western Indian Ocean. Different methods such as stomach content analysis and satellite tracking have previously been used to this end. An alternative and complementary approach to these methods is the measurement of naturally occurring stable isotopes in consumers and their prey. The principle underlying this approach is that stable isotope deviations of nitrogen and carbon in consumers reflect those of their prey as they are enriched in a predictable manner. Conventionally expressed as δ 15 N (‰), the deviation of 15 N to 14 N generally exhibits a stepwise enrichment from 2 to 5‰ relative to dietary nitrogen [START_REF] Kelly | Stable isotopes of carbon and nitrogen in the study of avian and mammalian trophic ecology[END_REF]. This discrepancy of δ 15 N is caused by a selective retention of the heavy isotope and excretion of the light one. It provides a tool for comparing the relative trophic level of various consumers living in the same environment. The deviation of 13 C to 12 C (denoted as δ 13 C) is also enriched with respect to dietary carbon, but to a much lesser degree than δ 15 N, on the order of 1‰ [START_REF] Deniro | Influence of diet on the distribution of carbon isotopes in animals[END_REF]. The most common use of δ 13 C in marine ecological studies is as a spatial tracer. Relative reliance of primary consumers on coastal and/or benthic versus oceanic and/or pelagic primary production determines δ 13 C values in food webs [START_REF] France | Carbon-13 enrichment in benthic compared to planktonic algae: foodweb implications[END_REF], which can also vary among water masses in the open ocean [START_REF] Deniro | Influence of diet on the distribution of carbon isotopes in animals[END_REF][START_REF] Francois | Changes in the δ 13 C of surface water particulate organic matter across the Subtropical Convergence in the SW Indian Ocean[END_REF]. Stable isotope deviations also have the advantage of offering information on a larger time scale than stomach content analysis, which, although more precise in prey determination, only allows a very narrow insight into the animal's global trophic habits. Given that tissues have different isotopic turnover rates, δ 15 N and δ 13 C measurements of multiple tissues provide dietary information on different temporal scales.

Here we focused on 6 top predators belonging to the tropical seabird and pelagic fish communities of the marine ecosystem surrounding Reunion Island. Our knowledge of the local feeding behavior of 3 avian species, viz. Barau's petrel Pterodroma baraui, which is endemic to Reunion Island, Audubon's shearwater Puffinus lherminieri bailloni, and white-tailed tropicbird Phaethon lepturus, is limited to general information on foraging by adults (basic temporal habits, foraging techniques, and interspecific associations) obtained by sighting around Reunion Island and diet composition (percentages of squid versus fish prey) acquired from stomach contents of adults and chicks during the breeding period. The specific diet composition of these birds and their fledglings, their foraging areas during and outside of the breeding season, and the extent of associations among seabird species and between seabirds and pelagic fishes remain unknown.

Three epipelagic fishes, viz. yellowfin tuna Thunnus albacares, skipjack tuna Katsuwonus pelamis, and common dolphinfish Coryphaena hippurus, are distributed worldwide and are thus more studied. Fairly detailed knowledge about their diets (species compositions and quantities) and foraging behaviors (e.g. geographical locations, depths, diurnal and seasonal cycles, associations) has been acquired for these fishes from numerous studies conducted on populations from various locations in the world. Additional information specific to their feeding behaviors in the western Indian Ocean is needed, as these species are subject to increasing commercial fishing pressure in this area where data are still laking. These 3 fishes gather in Reunion Island waters in the austral summer [START_REF] Roos | Les métiers de la pêche à La Réunion: description et évolution des techniques de pêche sur les DCP[END_REF]. During this season, 6 species of seabirds breed on the island, including Barau's petrels, Audubon's shearwaters, and white-tailed tropicbirds [START_REF] Barré | Tuna food habits related to the micronekton distribution in French Polynesia[END_REF]. The combined study of these 2 communities is of much interest, as their foraging behaviors are closely linked [START_REF] Jaquemet | Seabird community structure in a coastal tropical environment: importance of natural factors and fish aggregating devices (FADs)[END_REF]. Subsurface predators, such as tunas and dolphinfish, drive prey to the surface to facilitate their capture, thereby making these prey accessible to seabirds from the air. Using stable isotope analyses, we investigated and compared the trophic position and foraging range of each bird and fish species in light of data previously obtained by transects at sea and stomach content studies in an attempt to better understand the trophic position of each species, with respect to the others, when simultaneously present around Reunion Island.

MATERIALS AND METHODS

Sampling.

The marine birds analyzed in this study were collected between 2001 and 2004 on Reunion Island (21°07'S, 55°33'E) situated 700 km east of Madagascar (

). The sampling of chicks and adults took place during the breeding period from October to April for Barau's petrels and Audubon's shearwaters, and throughout the year for white-tailed tropicbirds. The fish were caught between January and May 2004 within 40 km of the Réunion Island coastline (Fig. 1).

Seabirds. The seabird community on Reunion Island comprises 6 species: Barau's petrel (3000 to 5000 pairs), Audubon's shearwater (5000 pairs), white-tailed tropicbird (2000 to 3000 pairs), brown noddy Anous stolidus (500 pairs), wedge-tailed shearwater Puffinus pacificus (several hundred pairs), and Mascarene black petrel Pseudobulweria aterrima (50 pairs at the most) [START_REF] Bretagnolle | Audubon's shearwaters Puffinus lherminieri on Reunion Island, Indian Ocean: behaviour, census, distribution, biometrics and breeding biology[END_REF], Le Corre et al. 2002, authors' unpubl. data). The species forming the most numerous colonies were sampled for this study. We collected 51 Barau's petrels, 59

Audubon's shearwaters, and 49 white-tailed tropicbirds on land following their collision with urban lights or from poaching seizures. Two age classes were determined (juvenile and adult) using characteristic features of the beak and the feathers (Barré et al. 1996, authors' unpubl. data).

Only adults were sexed during dissection, as the gonads were not developed enough in fledglings to differentiate males from females. The nutritional condition of each bird was also evaluated using the body condition index (BCI) proposed by [START_REF] Wenzel | The suitability of oiled guillemots (Uria aalge) as monitoring organisms for geographical comparisons of trace element contaminants[END_REF] that corresponds to the ratio of liver to kidney masses. BCI is significantly negatively correlated to the degree of emaciation, such that a smaller index indicates a more emaciated bird. The liver was removed and frozen at -20°C. Breast feathers were cut at their base and placed in plastic bags.

We chose to analyze δ 13 C and δ 15 N in the liver and feathers of birds, as these tissues reflect their dietary habits in different periods of their reproductive cycle. The feather provides a record of their diet during the time of its formation [START_REF] Thompson | Stable-isotope ratios of carbon and nitrogen in feathers indicate seasonal dietary shifts in Northern Fulmars[END_REF]. Most seabirds molt at sea during the interbreeding season [START_REF] Warham | The behavior, population biology and physiology of petrels[END_REF]. As such, the analysis of feathers collected on adults gives information on their dietary intake and foraging areas during the interbreeding season, whereas the analysis of feathers collected on juveniles is informative of their trophic ecology during the chick-rearing period. Furthermore, because carbon and nitrogen turnover rates are high in liver (2.6 d half-life for carbon in Japanese Quail; Hobson & Clark 1992ab), the isotopic deviation analysis in this tissue is an indicator of the trophic ecology of these seabirds at the time they were sampled, i.e. during their breeding season on Reunion Island.

Marine fishes. We sampled 20 yellowfin tuna, 40 skipjack tuna, and 45 common dolphinfish from sportfishing vessels. Many individuals were caught in the vicinities of fish aggregating devices (FADs) anchored within 23 km of Reunion Island, where these fish reside and feed while seasonally present in Reunion Island waters [START_REF] Roos | Les métiers de la pêche à La Réunion: description et évolution des techniques de pêche sur les DCP[END_REF]. Each fish was measured using the fork length (FL, from the tip of the snout to the fork of the caudal fin) and weighed when possible. Individuals were sexed during dissection by examination of the gonads. White muscle was sampled in the abdominal area above the vent of the fish and frozen at -20°C. In fish, we chose to analyze δ 13 C and δ 15 N in muscle because (1) it is the most commonly used tissue in trophic studies owing to the smaller amount of lipid and inorganic carbonates and to the low intrasample variability of δ 13 C and δ 15 N measurements [START_REF] Sweeting | Variance in isotopic signatures as a descriptor of tissue turnover and degree of omnivory[END_REF], and (2) the isotopic turnover rate in this tissue (half-life around 50 d in yellowfin tuna, Graham unpubl. data) is likely to reflect fish diet in Reunion waters.

The nutritional and reproductive states of each animal were assessed, as they may affect δ 13 C and/or δ 15 N values [START_REF] Doucett | Effects of the spawning migration on the nutritional status of anadromous Atlantic salmon (Salmo salar): insights from stable-isotope analysis[END_REF]. The hepatosomatic index (HSI) served as an indicator of the individual's body condition: HSI = (liver weight/body weight) × 100

(1)

The gonadosomatic index (GSI) was calculated, as it is generally considered a useful measure of gonad maturation and spawning readiness and is based on the broad assumption that proportionally larger gonads indicate greater development [START_REF] West | Methods of assessing ovarian development in fishes -a review[END_REF]. GSI was calculated as follows:

GSI = (gonad weight/body weight] × 100

(2)

Due to the limited data on the body weight of the sampled fish, this parameter was estimated from the length using weight-length relationships presented in .

Isotopic determination. To prepare for isotopic composition determination, bird liver and fish muscle were freeze-dried and ground to fine powder. As lipids are highly depleted in 13 C with regard to other af components due to fractionation occurring during lipid synthesis [START_REF] Deniro | Mechanism of carbon fractionation associated with lipid synthesis[END_REF], isotopic analyses were performed on lipid-extracted samples. Lipid extraction was performed using 20 ml of cyclohexane on powder aliquots of about 1 g. An ultra Turax was used to homogenize the mixture (2 × 15 s). The sample was then centrifuged for 2 min at 804 × g. The supernatant containing lipids was discarded, whereas the pellet was dried on an aluminum plate at 60°C for 12 h. All utensils were washed with detergent, rinsed with tap water and deionized water (Milli-Q quality), followed by absolute ethanol, and dried in an oven at 60°C before use.

Pectoral feathers were washed vigorously in triple baths of 0.25 N sodium hydroxide solution alternated with triple baths of deionized water in order to remove adherent external contamination as well as the external lipidic layer resulting from preening. Feathers were then dried in an oven

for 24 h at 50°C.

Stable carbon and nitrogen isotope assays were carried out on 1 ± 0.02 mg subsamples of powder loaded into tin cups. The 13 C, 12 C, 15 N, and 14 N abundances in the samples were determined using continuous-flow isotope-ratio mass spectrometry (CF-IRMS). Analyses were conducted using a Europa Scientific ANCA-NT 20-20 Stable Isotope Analyzer with ANCA-NT Solid/Liquid Preparation Module (Europa Scientific).The CF-IRMS was operated in the dual isotope mode, allowing δ 15 N and δ 13 C to be measured on the same sample. Every 10 samples were separated by 2 laboratory standards (leucine), which were calibrated against 'Europa flour' and IAEA standards N1 and N2. Experimental precision (based on the standard deviation of replicate measurements of the internal standards) was 0.10‰ for δ 13 C and 0.14‰ for δ 15 N.

Stable isotope deviations are expressed in delta notation (δ), defined as the part per thousand (‰) deviation from a standard material:

δ = (R sample /R standard -1) × 1000 (3)
where R sample and R standard are the fractions of heavy to light isotopes in the sample and standard, respectively. The international standards were the Pee Dee Belemnite (PDB) marine fossil limestone formation from South Carolina for δ 13 C and atmospheric nitrogen for δ 15 N.

Data analysis.

Statistical analyses were performed using the GNU R statistical system. All statistical samples submitted to tests were first checked for normality and homogeneity of the variances by Shapiro-Wilk and Bartlett tests, respectively. In the case of non-departure from normality, parametric tests were used in the subsequent analyses; otherwise, non-parametric analogs were used. The significance of differences of stable isotope signatures between tissues was tested by t-tests or Wilcoxon tests for paired samples using Do the-value correction. The influence of species on δ 15 N and δ 13 C measurements was tested by means of analyses of variance (ANOVA) or Kruskal-Wallis tests followed, in case of significant differences, by multiple pairwise comparison t-tests or Wilcoxon tests, respectively, for independent samples using Bonferroni's P-value correction. The influence of age and sex was tested by means of t-tests or Wilcoxon tests. Furthermore, the potential influence of length, reproductive status (GSI), and nutritional condition (HSI) of fish, as well as nutritional condition of birds (BCI) on isotope values were investigated for each species by means of Pearson's linear correlation coefficient.

Analysis of covariance (ANCOVA) was used to assess the effect of body length on δ 15 N values of the 3 fish species. Residuals were checked for normality by means of Shapiro tests, and for homoscedasticity by plotting fitted values versus residuals. Differences in coefficients of variation were tested between the bird and fish groups, and between species within groups, using the t-test based approach described in Sokal & Rohlf (l995).

RESULTS

δ 13 C and δ 15 N values in liver and feathers of juvenile and adult birds, and in fish muscle are presented in . Results are given as mean ± standard deviation (SD) except when otherwise specified.

Isotope measurements in seabirds

The δ 15 N values of feather samples were up to 1.3‰ depleted in 15 N compared to liver samples (Table 2). Only in juvenile white-tailed tropicbirds did these values not differ significantly (

). The differences between values obtained from both tissues were even higher for δ 13 C, as feathers were up to 2.5‰ enriched in 13 C in comparison to liver (Table 3).

The sex and the nutritional state of the bird influenced stable isotope signatures only in very few cases (data not shown), allowing a non-discriminant use of isotope results obtained from birds with different sexes or BCIs.

In both liver and feathers, δ 13 C and δ 15 N measurements differed significantly among species for juveniles and for adults (all Kruskal-Wallis tests were significant, ). Irrespective of the subgroup considered (Fig. 2), Barau's petrels displayed the highest δ 15 N values, and Audubon's shearwaters showed the lowest. The δ 15 N differences between Barau's petrels and white-tailed tropicbirds were not significant. In juveniles and in the livers of adults, Barau's petrels were the most depleted in 13 C, followed by Audubon's shearwaters and white-tailed tropicbirds. In the feathers of adults, δ 13 C values were lowest in Audubon's shearwaters (Fig. 2).

The differences in isotope deviations between juveniles and adults were also tested. Isotope signatures in the liver of young Barau's petrels were significantly lower than in adults (δ 13 C: P Wilcoxon = 0.001; δ 15 N: P Wilcoxon = 0.020). Hepatic δ 13 C measurements were also significantly lower in juvenile Audubon's shearwaters than in adults (P t-test < 0.001), although their hepatic δ 15 N values were not (P Wilcoxon = 0.072). Conversely, hepatic δ 13 C and δ 15 N values did not differ significantly in white-tailed tropicbirds (P t-test = 0.088 and P Wilcoxon = 0.366 respectively); feather δ 15 N values also did not differ (P t-test = 0.377). However, feathers of juveniles were significantly depleted in 13 C compared to adult feathers (Table 2; P t-test = 0.002).

Isotope measurements in fish

There was no influence of sex or nutritional or reproductive states on isotope deviations in yellowfin tuna, skipjack tuna, and common dolphinfish (data not shown).

We noted significant positive correlations between δ 15 N and the length of yellowfin tuna (r = 0.69; p < 0.001; n = 17) and common dolphinfish (r = 0.65; p < 0.001; n = 42;

). For skipjack tuna, the range of sizes was narrow (41 to 85 cm), and the significance of the correlation results was borderline (r = 0.35; p = 0.067; n = 28; Fig. 3).

Differences in isotope signatures were found among species. Muscular δ 13 C measurements differed significantly among all 3 fishes (P Kruskal-Wallis < 0.001). Skipjack tuna was the most depleted in 13 C, followed in increasing order by yellowfin tuna and common dolphinfish ( ).

Significant differences in δ 15 N measurements were observed between common dolphinfish and both yellowfin and skipjack tunas (P Wilcoxon = 0.019 and 0.001, respectively), but not between the 2 tuna species (Fig. 4). Common dolphinfish showed the lowest δ 15 N values. The result of the ANCOVA model showed separate slopes and intercepts for each species (P ANCOVA < 0.001, regression coefficients are displayed in Fig. 3). Fish body length influences muscular δ 15 N values differently in the 3 species. The effect of body length was the strongest for common dolphinfish, while the slopes for yellowfin and skipjack tunas were close (Fig. 3). With respect to the length ranges, δ 15 N values differed among individuals of equivalent length belonging to different species. Common dolphinfish exhibited the lowest predicted δ 15 N values. Skipjack tuna showed greater δ 15 N values than yellowfin tuna of the same body length, while large yellowfin tuna (length > 120 cm) exhibited the greatest predicted δ 15 N values among the 3 fish species (Fig. 4).

Furthermore, the highest coefficient of variation for δ 15 N values was observed in common dolphinfish. The difference was significant between common dolphinfish and skipjack tuna (p < 0.001) and between yellowfin and skipjack tunas (p < 0.030).

Comparing bird and fish isotope measurements

The isotope signatures of both taxonomic groups are presented jointly in Fig. 5. δ 13 C and δ 15 N values measured in the 3 birds and 3 fishes were compared by means of multiple pairwise comparison tests. In all but one case, isotope measurements were significantly different between bird and fish species (P Wilcoxon < 0.001). δ 15 N values were not significantly different between Audubon's shearwater and yellowfin tuna (P Wilcoxon = 0.145). Furthermore, a higher variance in δ 13 C signatures in birds was expressed by significantly higher coefficients of variation in birds than in fish (p = 0.020; Fig. 5).

DISCUSSION

Seabird trophic ecology

Because of the fast turnover of stable carbon and nitrogen isotopes in the liver, hepatic isotope signatures could be considered as indicators of adult foraging area and trophic level during the breeding season. Furthermore, isotope signatures of the feathers provide information about their trophic situation during the interbreeding period. For their nutrition, chicks exclusively depend on food brought back by their parents. Hence, isotope analyses of their liver and feathers provide information on their trophic level and on the geographic origin of prey delivered by their parents.

Feeding habits during the breeding period

Species trophic segregation

Particle organic matter δ 13 C signatures vary from coastal to oceanic environments as well as latitudinally, with relatively well-defined shifts across frontal boundaries [START_REF] Rau | Plankton 13C:12C ratio changes with latitude: differences between northern and southern oceans[END_REF][START_REF] Francois | Changes in the δ 13 C of surface water particulate organic matter across the Subtropical Convergence in the SW Indian Ocean[END_REF]). The very slight increase in δ 13 C with increasing trophic levels [START_REF] Post | Using stable isotopes to estimate trophic position: models, methods, and assumptions[END_REF] enables δ 13 C values in predators to serve as indicators of the water masses in which they feed [START_REF] Cherel | Geographical variation in carbon stable isotope signatures of marine predators: a tool to investigate their foraging areas in the Southern Ocean[END_REF]. However, a recent study over a broad latitudinal zone has evidenced weak δ 13 C variations in tropical predatory fish of the western tropical Indian Ocean (Ménard et al. 2007). Overall, latitudinal patterns in δ 13 C and δ 15 N at the base of the food chain remain poorly documented in the tropical Indian Ocean. Because of these uncertainties on the isotopic patterns in this area, our isotope results cannot be interpreted in terms of well-defined foraging zones but will instead be used to delineate general trends. The δ 13 C results (Fig. 2) obtained for juveniles (liver and feathers) and adults (liver) indicate that during the breeding period, the 3 birds prospect for food in different areas, hence avoiding competition. These conclusions are in agreement with observations made at sea. Barau's petrels have been sighted more then 1800 km south of Reunion Island during the breeding season [START_REF] Stahl | Distribution, abundance and aspects of the pelagic ecology of Barau's Petrel (Pterodroma baraui) in the south-west Indian Ocean[END_REF], whereas Audubon's shearwaters and white-tailed tropicbirds seem to feed within 80 km and hundreds of km from their colonies, respectively (Bailey 1967). None of these birds has been followed by satellite tracking. This technique would not only be very useful in completing our understanding of the dispersion of these birds at sea during and outside the breeding period, but would also indirectly allow us to study whether water particulate organic mater δ 13 C signatures follow a decreasing gradient with increasing latitudes throughout the tropical zone of the western Indian Ocean as they do south of 30° S [START_REF] Francois | Changes in the δ 13 C of surface water particulate organic matter across the Subtropical Convergence in the SW Indian Ocean[END_REF][START_REF] Cherel | Geographical variation in carbon stable isotope signatures of marine predators: a tool to investigate their foraging areas in the Southern Ocean[END_REF]. If this were asserted, it would be possible to use δ 13 C signatures in birds to describe their geographic dispersion according to latitude.

δ 15 N results show trophic differences between Audubon's shearwaters and the other species in both juveniles and adults. Lower δ 15 N measurements in Audubon's shearwaters, the smallest species, indicate that they probably feed on prey of a lower trophic level than the other 2 seabirds.

They are suspected to consume equal amounts of fish and cephalopods, whereas Barau's petrels and white-tailed tropicbirds appear to feed almost exclusively on cephalopods (98 and 80% respectively, authors' unpubl. data), which can have fairly high δ 15 N signatures [START_REF] Froese | Trophic signatures of marine organisms in the Mediterranean as compared with other ecosystems[END_REF]). These δ 15 N results substantiate predictions of a previous study of these birds in which lower mercury levels in Audubon's shearwater were an argument for the lower trophic level of its prey [START_REF] Kojadinovic | Trace elements in three marine birds breeding on Reunion Island (Western Indian Ocean). Part 1. Factors influencing their bioaccumulation[END_REF]. This is also coherent with what is known of the foraging behavior of white-tailed tropicbird and Barau's petrel. The former covers large distances to find isolated prey of large size (Ballance & Pitman 1999), and the latter shows scavenging behavior (Ballance & Pitman 1999, authors' unpubl. data), giving it the opportunity to feed on large prey of high trophic levels such as dead marine mammals or large fishes or squid. These assumptions should be regarded cautiously, since these seabirds feed in different areas that may potentially be characterized by different nitrogen baseline levels impeding a rigorous interpretation of the data.

Food provisioning strategies

Young and adult Barau's petrels showed differences in δ 13 C and δ 15 N values in liver. Chicks may differ from adults in their metabolism such that stable isotope fractionation during liver or feather synthesis is not consistent between chicks and adults. However, no growth-related difference has been found in the stable isotope ratios for various species [START_REF] Minagawa | Stepwise enrichment of 15 N along food chains: further evidence and the relation between δ15N and animal age[END_REF][START_REF] Hobson | Stable isotope analyses of tooth annuli reveal temporal dietary records: an example using Steller sea lions[END_REF]. The most parsimonious explanation for relative differences in the signatures of both isotopes in the liver of young and adult Barau's petrels is therefore that it reflects different stable isotope values in foods of young and adults rather than a systematic age-related difference in isotopic fractionation [START_REF] Hobson | Trophic relationships among high Arctic seabirds: insights from tissuedependent stable-isotope models[END_REF]). Such differences indicate that fledglings and adults do not rely on the same prey. A probable explanation is that parents forage in separate areas when fishing for their fledglings and when fishing for themselves (lower δ 13 C values in chicks) at which time they possibly feed on larger prey (higher δ 15 N values in adults). Such dual foodprovisioning strategies have been shown in other Procellariiformes [START_REF] Chaurand | The regular alternation of short and long foraging trips in the blue petrel Halobaena caerulea: a previously undescribed strategy of food provisioning in a pelagic seabird[END_REF][START_REF] Cherel | Using stable isotopes to study resource acquisition and allocation in procellariiform seabirds[END_REF][START_REF] Congdon | Dual-foraging and co-ordinated provisioning in a tropical Procellariiform, the wedge-tailed shearwater[END_REF]) and usually consist of parents making short foraging trips during which most of the food is brought back to their young and long trips for their own maintenance. The long trips are profitable for adults through a build up of energy reserves enabling them to maintain themselves during the short trips, which are profitable for chicks through an increase in their feeding frequency [START_REF] Cherel | Using stable isotopes to study resource acquisition and allocation in procellariiform seabirds[END_REF]. The results obtained for Audubon's shearwaters, the other procellariiform species studied here, also support the hypothesis of an alternation of feeding areas when the food was destined to be eaten by the parent or by the chick; however, differences in δ 15 N values were not observed. This first evidence of dual foodprovisioning strategies in Barau's petrels and Audubon's shearwaters relies on fairly small differences in isotope values between chicks and adults (1.1‰ for δ 13 C and 0.7‰ for δ 15 N in Barau's petrel, and 0.6‰ for δ 13 C in Audubon's shearwater) and should thus be substantiated by further investigations combining satellite tracking and isotope measurements of adults returning to the colony.

In contrast to what is observed for the procellariiform species, white-tailed tropicbird parents do not seem to discriminate their food from that fed to their young, since hepatic δ 15 N and δ 13 C values did not differ between age classes.

Feeding habits during the interbreeding period

Although very little precise information is available, the 3 seabird species are believed to spread out over larger oceanic areas during the interbreeding season (Bailey 1967[START_REF] Stahl | Distribution, abundance and aspects of the pelagic ecology of Barau's Petrel (Pterodroma baraui) in the south-west Indian Ocean[END_REF].

During the breeding season on Reunion Island, juvenile and adult white-tailed tropicbirds had equivalent hepatic isotopic signatures implying similar diets. Consequently, isotopic deviations in feathers of juveniles were considered to be representative of isotope deviations in feathers of breeding adults.

Feather δ 13 C values of both age classes were thus compared to investigate whether adults visit the same foraging zones during the breeding and interbreeding seasons. Although the difference in δ 13 C values was small (0.4‰), its statistical significance, added to the weak δ 13 C latitudinal variations in this part of the Indian Ocean (Ménard et al. 2007), suggests a shift in white-tailed tropicbird foraging areas between seasons. The extent and direction of this interseasonal shift remains unknown.

Feather δ 15 N values of both age classes were compared to investigate whether adults maintain the same diet during the breeding and interbreeding seasons. δ 15 N values did not differ significantly between feathers of juvenile and adult tropicbirds, indicating that the trophic levels of the latter do not change between seasons, implying that they probably maintain the same diet year-round.

Moreover, when considering all 3 seabirds, we noted that the trophic distinction between Audubon's shearwaters and the other 2 species described for the breeding season is conserved during the interbreeding season. This is another element indicating the absence of large shifts in feeding behaviors of these 3 seabirds from one season to another, although it should be kept in mind that baseline δ 15 N values may vary between areas [START_REF] Popp | Insight into the trophic ecology of yellowfin tuna, Thunnus albacares, from compound-specific nitrogen isotope analysis of proteinaceous amino acids[END_REF], thus affecting the above interpretations if the seabirds feed in very distinct environments.

Fish trophic ecology

Similarly to seabirds, tunas and common dolphinfish are primarily visual predators and opportunistic feeders [START_REF] Roger | Relationships among yellowfin and skipjack tuna, their prey-fish and plankton in the tropical western Indian Ocean[END_REF][START_REF] Massuti | Diet and feeding of dolphin (Coryphaena hippurus) in western Mediterranean waters[END_REF][START_REF] Ménard | Opportunistic predation in tuna: a size-based approach[END_REF]. They are mainly piscivorous, although they also feed on cephalopods and crustaceans [START_REF] Roger | Relationships among yellowfin and skipjack tuna, their prey-fish and plankton in the tropical western Indian Ocean[END_REF][START_REF] Massuti | Diet and feeding of dolphin (Coryphaena hippurus) in western Mediterranean waters[END_REF], Potier et al. 2004). Most fish prey belong to the epipelagic fauna (small Carangidae, juvenile Scombridae) and reef fishes (Holocentridae, Tetraodontidae, Stromateidae, Mullidae; [START_REF] Potier | Feeding partitioning among tuna taken in surface and mid-water layers: the case of yellowfin (Thunnus albacares) and bigeye (T. obesus) in the Western Tropical Indian Ocean[END_REF]. Differences in δ 15 N measurements between common dolphinfish and both yellowfin and skipjack tunas (Fig. 4) suggest that the composition of common dolphinfish diet varies significantly from that of the tunas in terms of prey species and/or proportions. Yellowfin and skipjack tunas seem to occupy significantly higher trophic positions than common dolphinfish, with yellowfin tuna exhibiting the highest δ 15 N values. This is not surprising, as yellowfin tuna tend to consume progressively larger prey as they grow [START_REF] Ménard | Opportunistic predation in tuna: a size-based approach[END_REF]. For instance, large yellowfin tuna feed on other Scombridae, including skipjack tuna (Ménard et al. 2000a). The difference in diet between the dolphinfish and the tunas may also be attributable to differences in foraging behaviors (in terms of e.g. depth, feeding hours) giving common dolphinfish access to prey that differ from those available to yellowfin and skipjack tunas [START_REF] Moteki | Composition of piscine prey in the diet of large pelagic fish in the eastern tropical Pacific Ocean[END_REF]. Adult yellowfin tuna usually feed during daylight hours in the surface mixed layer above the thermocline, although they can dive to at least 1160 m [START_REF] Roger | Relationships among yellowfin and skipjack tuna, their prey-fish and plankton in the tropical western Indian Ocean[END_REF][START_REF] Dagorn | Deep diving behavior observed in yellowfin tuna (Thunnus albacares)[END_REF].

Skipjack tuna are also predominantly diurnal feeders [START_REF] Roger | Relationships among yellowfin and skipjack tuna, their prey-fish and plankton in the tropical western Indian Ocean[END_REF]. They are limited to surface temperatures of about 17 to 30°C [START_REF] Wild | A review of the biology and fisheries for skipjack tuna, Katsuwonus pelamis, in the Pacific Ocean. Tech Rep 2[END_REF] and inhabit mostly the mixed layer, although they can make repetitive dives below the thermocline to depths greater than 300 m [START_REF] Shaeffer | Vertical movement patterns of skipjack tuna (Katsuwonus pelamis) in the eastern equatorial Pacific Ocean, as revealed with archival tags[END_REF]. Both species also feed on nyctemeral migrating communities. Given their feeding behavior, these 2 species have access to the same prey and therefore exhibited close δ 15 N values. Common dolphinfish are generally present from the surface up to 30 m in depth, and although they dive down to 75 m when moving in non-aggregated schools, they spend 95% of the time within 5 m from the surface [START_REF] Massuti | Diet and feeding of dolphin (Coryphaena hippurus) in western Mediterranean waters[END_REF][START_REF] Taquet | Le comportement agrégatif de la dorade coryphène (Coryphaena hippurus) autour des objets flottants[END_REF]). Although common dolphinfish are primarily visual predators, they can also feed at nighttime [START_REF] Shcherbachev | The biology and distribution of the dolphins (Pisces, Coryphaenidae)[END_REF].

Furthermore, common dolphinfish are more associated with the FADs anchored around Reunion Island than the tunas and reside under these floating objects for much longer periods of time [START_REF] Taquet | Le comportement agrégatif de la dorade coryphène (Coryphaena hippurus) autour des objets flottants[END_REF]. Consequently, common dolphinfish potentially rely in large part on reef fishes drawn offshore that take shelter under these FADs (Bertrand et al. 2002). This idea is also supported by the significantly higher δ 13 C values measured in this species, since coastal fauna that use benthic carbon sources, such as reef fish, are enriched in 13 C compared to fauna dependent on oceanic sources [START_REF] France | Carbon-13 enrichment in benthic compared to planktonic algae: foodweb implications[END_REF]).

The non-selective feeding behavior of the 3 fishes is confirmed by relatively high intraspecific coefficients of variation for δ 15 N values that reflect a large range of prey species and/or prey sizes in their diet. The diet of tunas feeding under FADs is usually more diversified compared to the diet of unassociated individuals (Ménard et al. 2000b). In the present case, the highest coefficients of variation were observed in common dolphinfish, which suggests that their diet is relatively more varied than that of the other 2 species. This observation agrees with the idea that common dolphinfish are more closely linked to floating objects than the tunas. Part of the δ 15 N variability might also originate from ontogenetic shifts in diet, since individuals of different sizes (and thus most probably different ages) were sampled within the same species. Larger fish are able to eat larger prey that are likely to be situated at a higher position in the food web and thus exhibit higher δ 15 N values [START_REF] Kelly | Stable isotopes of carbon and nitrogen in the study of avian and mammalian trophic ecology[END_REF][START_REF] Scharf | Predator size-prey size relationships of marine fish predators: interspecific variation and effects of ontogeny and body size on trophic-niche breadth[END_REF], Ménard et al. 2007).

Trophic interactions between birds and fishes

Reunion Island is located in the South Indian tropical gyre and is considered an oligotrophic environment characterized by low productivity, which is associated with low abundance in prey, mostly distributed in patches (Ballance & Pitman 1999). Prey patches are usually associated with frontal boundaries or particular topographic features. However, these physical characteristics have little influence on tropical seabird foraging distributions, which seem to be more tightly linked to the presence of subsurface predators such as pelagic fishes [START_REF] Weimerskirch | Foraging strategy of a top predator in tropical waters: great frigatebirds in the Mozambique Channel[END_REF]).

Subsurface predators drive prey to the surface because the air-water interface acts as a boundary that prey cannot escape. Under these circumstances, seabirds can access these same prey from the air. Presumably as a result of this pressure from below, some prey species (exocoetid flying fish and flying squids, mostly ommastrephids) have evolved a flight escape response (Ballance & Pitman 1999). While this has made them less vulnerable to predation from below, it has made them more vulnerable to predation from above. Flying fish and squid comprise a substantial portion of the diet of many seabirds including the species studied here. A majority of Audubon's shearwater feeding events around Reunion Island occur in association with subsurface predators, and flocks are largest when they feed in association with skipjack and yellowfin tunas [START_REF] Jaquemet | Seabird community structure in a coastal tropical environment: importance of natural factors and fish aggregating devices (FADs)[END_REF]). Barau's petrels are also commonly observed feeding in association with subsurface predators. Among the studied animals, white-tailed tropicbirds are the only solitary feeders that seldom associate with other birds or subsurface predators (Ballance & Pitman 1999).

In our study, Audubon's shearwaters and yellowfin tuna were characterized by similar nitrogen isotopic signatures, indicating that they feed on prey of similar trophic levels. There is probably an overlap in their diet, given that these species form the most frequently observed feeding associations in the vicinity of Reunion Island [START_REF] Jaquemet | Seabird community structure in a coastal tropical environment: importance of natural factors and fish aggregating devices (FADs)[END_REF]) and that Audubon's shearwaters, contrary to the other 2 seabirds that mainly have access to flying fish and squid, dive and feed on prey under water. The other nitrogen results show that the birds generally feed at higher trophic levels than the fishes (mean δ 15 N up to 4.1‰ higher in birds, which corresponds to about 1 trophic level; Table 2). These results are difficult to compare to other ecosystems because of the scarcity of joint isotopic studies on seabirds and pelagic fish. However, when considering separate studies on marine birds and fish conducted in the same area during the same year, it appears as though our results illustrate a situation that occurs in other parts of the world. For example in the Mediterranean Sea, δ 15 N values for Cory's shearwater Calonectris diomedea diomedea and Auduin's gull Larus andouinii feathers were higher than those measured in the muscle of bluefin tuna Thunnus thynnus by 0.84 to 3.4‰ [START_REF] Gómez-Díaz | Geographic assignment of seabirds to their origin: combining morphologic, genetic, and biogeochemical analyses[END_REF][START_REF] Sanpera | Audouin's gull chicks as bioindicators of mercury pollution at different breeding locations in the western Mediterranean[END_REF][START_REF] Sarà | Feeding habits and trophic levels of bluefin tuna Thunnus thynnus of different size classes in the Mediterranean Sea[END_REF]. Another example can be found in the northwestern Atlantic Sea off the Massachusetts coast, where seabird δ 15 N values were higher than those measured in tunas: δ 15 N values were higher by 2.3, 4.3, and 5.3‰ in common tern Sterna hirundo feathers than in the muscle of bluefin, yellowfin, and albacore Thunnus alalunga tunas respectively [START_REF] Nisbet | Use of stable isotopes to investigate individual differences in diets and mercury exposures among common terns Sterna hirundo in breeding and wintering grounds[END_REF][START_REF] Estrada | Diet and trophic position of Atlantic bluefin tuna (Thunnus thynnus) inferred from stable carbon and nitrogen isotope analysis[END_REF]. Consequently, although most of these 2 types of predators associate to feed, their diets do not seem to be identical. Previous studies have indeed shown that these seabirds tend to rely primarily on squid, whereas these fishes are dominantly piscivorous [START_REF] Roger | Relationships among yellowfin and skipjack tuna, their prey-fish and plankton in the tropical western Indian Ocean[END_REF], Moteki et al. 2001, authors' unpubl. data). The differences in nitrogen signatures might be amplified with respect to the true differences in diets due to shifts in isotopic fractionation and/or turnover rates, which might occur between tissues or taxonomic groups [START_REF] Deniro | Influence of diet on the distribution of carbon isotopes in animals[END_REF][START_REF] Deniro | Influence of diet on the distribution of nitrogen isotopes in animals[END_REF]. For example, the nitrogen diet-tissue fractionation factor determined in ring-billed gulls L.delawarensis is higher for liver (+2.7‰) than for muscle (+1.4‰; Hobson & Clark 1992ab), which in turn is different from that observed for Atlantic salmon Salmo salar muscle (+2.3‰; [START_REF] Persson | Fractionation of δ15N and δ13C for Atlantic salmon and its intestinal cestode Eubothrium crassum[END_REF].

Although no significant difference has been reported in carbon diet-tissue fractionation factors between liver and muscle in birds, a bias linked to differences in fractionation factors or turnover rates between taxonomic groups might also have affected the carbon results that show differences in foraging areas between birds and fish. The largest differences were noted between common dolphinfish and the seabirds. This result may seem surprising at first, as this fish is more epipelagic than the other 2 and could be expected to have closer isotopic signatures to the birds than the tunas. Once again, this observation may be explained by the particular foraging behaviors of common dolphinfish in relation to floating objects, which tend to increase its δ 13 C signatures.

More generally, a wider δ 13 C range in birds (4.1‰ in birds versus 3.3‰ in fish) and higher variances in δ 13 C values of birds indicate greater spatial distinctions in foraging areas among birds than among fish. Seabirds are flying homeothermic animals capable of covering very large distances and are much less restricted by temperatures than tunas and common dolphinfish. Their thermal regulation capacity enables them to spread to latitudes where temperatures preclude the presence of the fish species considered here. For example, Barau's petrels have been sighted more then 1800 km south of Reunion (Bailey 1967[START_REF] Stahl | Distribution, abundance and aspects of the pelagic ecology of Barau's Petrel (Pterodroma baraui) in the south-west Indian Ocean[END_REF], which roughly corresponds to 38° S, whereas skipjack tuna, which of the 3 fishes are adapted to the lowest temperatures, are limited by the 17°C isotherm situated around 30° S in the western part of the Indian Ocean.

According to the known feeding behaviors of these animals and the above isotope results, we suggest that these 2 taxonomic groups forage somewhat in different water masses but feed in association to reduce foraging effort when present in the same areas.

CONCLUSION

This study is the first to focus on the trophic ecology of an assemblage of marine pelagic fishes and seabirds from Reunion Island, including the endemic Barau's petrel. Our results revealed that during the breeding period, Barau's petrels, Audubon's shearwaters, and white-tailed tropicbirds show differences in the ecological niches they occupy and suggest that both Procellariiformes might adopt a dual food-provisioning strategy, making separate foraging trips to feed their fledglings and for their own maintenance. Furthermore, evidence suggests that white-tailed tropicbirds change foraging areas between breeding and molting seasons, although no diet shift seems to occur.

Our results also support the idea that differences in feeding strategies exist among the 3 fishes close to Reunion Island. Common dolphinfish presumably prey more on low trophic level coastal organisms than yellowfin and skipjack tunas, which are more associated with birds, namely Audubon's shearwaters, during feeding events. These results on species associations confirm prior reports of at-sea observations. Additionally, our study corroborates the previously established strong aggregative behavior of common dolphinfish to FADs floating or anchored around Reunion Island.

To further investigate the diet and spatial distribution of these top predators, isotopic baselines of δ 13 C and δ 15 N in marine pelagic food webs, stable isotope values in potential prey, and satellite tracking data are needed.

Acknowledgements. This research was supported by the Conseil Régional de La Réunion and is part of the program REMIGE (ANR Biodiv 2005-011). J.K. also benefited from support of the Conseil Régional de La Réunion and the European Social Fund through a PhD grant. We thank the Société d'Etudes Ornithologiques de La Réunion (SEOR), J. Bourjea and M. Taquet (Ifremer, La Réunion), as well as Maeva Pêche au Gros, Réunion Fishing Club, L. and P. Berthier, and other anonymous Reunionese fishermen for their assistance in the sampling effort. We are also grateful to E. Robert, M. Rouquette, P. Grondin, N. Ghanem (University of La Réunion) and P.

Richard (CRELA, La Rochelle) for their help in sample preparation and analysis. We would also like to thank the referees for very helpful comments on the manuscript.

LITERATURE CITED

Bailey RS (1967) The pelagic distribution of seabirds in the Western Indian Ocean.

Figure captions Fig. 1 .

 captions1 Figure captions

Table 2 Table 1 .

 21 Thannus albcares, Katsuwonus pelamis and Coryphaena hippurus. Relationships between weight and length for 3 tropical pelagic fishes: data from studies in the Indian Ocean. These relationships are presented as exponential regressions on the following model: W = a × L b where W is the weight and L the length

	Species	n	a	b	r²	Source
	Yellowfin tuna (females)	194	54.10 -6	2.72	0.97	Tantivala (2000)
	Yellowfin tuna (males)	174	41.10 -6	2.79	0.97	Tantivala (2000)
	Skipjack tuna	22	6.65.10 -6	3.28	0.98	Present study
	Common dolphinfish	51	5.94.10 -6	3.04	0.93	Present study, Bourjea pers. comm.

Table 2 .

 2 13 C and 15 N values ( ) of seabirds and pelagic fishes from Reunion Island. CV: coefficient of variation. Shaded areas correspond to data representing the trophic behavior of adult birds at sea

	during the inter-
	breeding season
	White-tailed
	tropicbird	Juvenile	Liver	17	-17.9 ± 0.8	14.2 ± 2.2
					4.5 (-19.7 / -17.5)	15.3 (11.4 / 20.4)
			Feathers	17	-15.6 ± 0.45	13.4 ± 1.1
					2.9 (-16.3 / -14.9)	8.5 (11.6 / 15.0)
		Adult	Liver	32	-17.5 ± 0.6	14.4 ± 1.4
					3.5 (-19.4 / -16.2)	9.8 (12.1 / 18.1)
			Feathers	31	-15.2 ± 0.3	13.7 ± 1.0
					2.0 (-15.7 / -14.5)	7.5 (10.7 / 15.5)
	Fish
	Yellowfin tuna	107 ± 34	Muscle	20	-16.3 ± 0.3	12.0 ± 1.0
		63 / 170			1.9 (-16.7 / -15.5)	8.5 (10.0 / 13.8)
	Skipjack tuna	68 ± 15	Muscle	37	-16.8 ± 0.3	11.9 ± 0.6
		41 / 85			2.1 (-17.8 / -16.0)	5.2 (10.7 / 13.2)
	Common dolphinfish	87 ±15	Muscle	45	-15.8 ± 0.6	11.1 ± 1.1
		61 / 112			3.8 (-17.3 / -14.5)	9.7 (9.5 / 13.7)

Table 3 .

 3 Comparison of stable isotopic signatures in liver and feathers of seabirds and pelagic fishes from Reunion Island. Full species names in Table 2. Significant results are in bold

		n	δ 13 C	δ 15 N
	Juveniles
	Barau's petrel	25	P t-test < 0.001	P t-test = 0.005
	Audubon's shearwater	37	P t-test < 0.001	P t-test < 0.001
	White-tailed tropicbird	17	P t-test < 0.001	P Wilcoxon = 0.517
	Adults
	Barau's petrel	14	P t-test < 0.001	P t-test = 0.007
	Audubon's shearwater	21	P t-test < 0.001	P t-test = 0.007
	White-tailed tropicbird	31	P t-test < 0.002	P t-test = 0.009


Arnaud Bourdin

Nicolas Molinari

Gary Ferguson

Mohd Barinder Singh

Kashif Siddiqui

Ulf Holmgren

Mario Ouwens

Martin Jenkins

Enrico De Nigris

Efficacy and Safety of Budesonide/Glycopyrronium/Formoterol Fumarate versus Other Triple Combinations in COPD: A Systematic Literature Review and Network Meta-analysis

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


Claude Manté
email: claude.mante@mio.osupytheas.fr

Keywords: Riemannian manifold, geodesics, cut locus, bers, Poisson Approximation, Multidimensional Scaling

Seeking for shortest paths between negative binomial distributions. Application to the statistical analysis of counts data.

Notations

Consider a Riemannian manifold M, and a parametric curve α : [a, b] → M; its rst derivative will be denoted α. A geodesic curve γ connecting two points p and q of M will be denoted p q, and p s ⊕ s q will denote the broken geodesic [1] connecting p to q with a stopover at s. We will also consider for any θ ∈ M the local norm V g (θ) associated with the metrics g on the tangent space T θ M :

∀ V ∈ T θ M, V g (θ) := V t .g(θ).V .

(

) 1
The length of a curve α traced on M will be denoted Λ (α). In addition, R + * := ]0, +∞[, and M F will denote the Frobenius norm of the matrix M ; logical propositions will be combined by using the classical connectors ∨ (or) and ∧

(and).

A parametric probability distribution L i will be identied with its coordinates with respect to some chosen parametrization; for instance, we will write L i ≡ φ i , µ i for some negative binomial distribution.

Introduction

The statistical analysis of counts of living organisms brings information about the collective behavior of species (schooling, habitat preference, etc), possibly associated with their socio-biological characteristics (aggregation, growth rate, reproductive power, survival rate, etc). In the spirit of Manté et al. [2], we propose an original exploratory method, consisting in measuring the dissimilarity between species through the probability distribution of some characteristic, and analyzing the obtained dissimilarity table through MDS. In [2], this characteristic was the dispersion of each species while here it will be its abundance.

There is a wide range of statistical methods to deal with distributional data, fundamentally depending on the chosen metrics on the probabilities set. Recently, multivariate methods with a geometric dominance appeared in the literature, based on Riemannian structures equipping spaces of probability densities:

non-parametric Fisher-Rao metrics [START_REF] Srivastava | Riemannian Analysis of Probability Density Functions with Applications in Vision[END_REF] or Wasserstein metrics [START_REF] Seguy | Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric[END_REF]; see also [START_REF] Petersen | Functional data analysis for density functions by transformation to a Hilbert space[END_REF][START_REF] Cazelles | Geodesic PCA versus Log-PCA of Histograms in the Wasserstein Space[END_REF].

But all these methods were designed in a non-parametric setting, for absolutely continuous distributions, while our data are discrete. Furthermore, even if the parametric approach is quite sound from the ecological point of view (see [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF] and the references therein), it is ill-suited for Exploratory Data Analysis (EDA): the visual distance between parameters of several distributions is misleading, because on the one hand it depends on the chosen parametrization and, on the other hand, these parameters are not commensurable in general (dierent ecological meaning, dierent ranges, ...).

In a seminal paper, Rao [START_REF] Rao | Information and the Accuracy Attainable in the Estimation of Statistical Parameters[END_REF] noticed that, equipped with the Fisher information metrics denoted g (•), a family of probabilities depending on p parameters can be considered as a p-dimensional Riemannian manifold. The associated Riemannian (Rao's) distance between the distributions θ 1 and θ 2 is

D R θ 1 , θ 2 := ¢ 1 0 γt (t) .g (γ (t)) . γ (t)dt (2)
where γ is the segment (minimal length curve) connecting θ 1 = γ (0) to θ 2 = γ (1). Naturally, Rao [START_REF] Rao | Information and the Accuracy Attainable in the Estimation of Statistical Parameters[END_REF][START_REF] Rao | Comment to Kass' paper[END_REF] proposed to use (2) as a distance between populations or for Goodness-Of-Fit (GOF) testing, followed by a number of authors [START_REF] Carter | FINE: Fisher Information Nonparametric Embedding[END_REF][START_REF] Galanis | Wave height characteristics in the north Atlantic ocean: a new approach based on statistical and geometrical techniques[END_REF][START_REF] Dodson | Some illustrations of information geometry in biology and physics[END_REF][START_REF] Cubedo | A dissimilarity based on relevant population features[END_REF][START_REF] Ilea | Statistical Hypothesis Test for Maritime Pine Forest Sar Images Classication 20 Based on the Geodesic Distance[END_REF][START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF][START_REF] Kass | The geometry of asymptotic inference[END_REF][START_REF] Menendez | Statistical Tests Based on Geodesic Distances[END_REF][START_REF] Cubedo | Hypothesis testing: a model selection approach[END_REF]. The Rao's distance between members of a common family of distributions has been calculated in a number of classical cases [START_REF] Burbea | Informative geometry of probality spaces[END_REF] but it cannot be obtained in a closed form, generally. In such cases, like the NB distributions (when both parameters are undened), D R must be obtained by numerically solving a second-order nonlinear dierential equation, frequently hard to integrate. Manté and Kidé [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF] proposed numerical remedies to this issue, which are completed here by Poisson Approximation combined with Dierential Geometry techniques.

Few elements of Riemannian geometry

According to the fundamental theorem of Riemannian geometry [1], there is a unique symmetric connection ∇ compatible with a given metrics g, giving in our case the Rao's distance. It is noteworthy that other statistically sound (but not Riemannian) connections can be fruitfully considered (Amari et al. [START_REF] S.-I. Amari | Methods of information geometry[END_REF]). Geodesics with respect to ∇ are solutions of the Euler-Lagrange equation [START_REF] Gray | Modern dierential geometry of curves and surfaces with Mathematica[END_REF]1,[START_REF] Burbea | Informative geometry of probality spaces[END_REF]:

∀ 1 ≤ k ≤ p, γk (t) + p i,j=1 Γ k i,j γi (t) γj (t) = 0 (3)
where each Christoel symbol Γ k i,j only depends on g and is dened in coordinates by:

Γ k i,j := p m=1 g km 2 ∂g jm ∂θ i + ∂g im ∂θ j - ∂g ij ∂θ m (4)
and g im (resp. g mk ) is some entry of g -1 (resp. g). The segment connecting L 1 to L 2 (if it exists) is necessarily a geodesic, but building it is not straightforward: a geodesic is not necessarily a segment, due to the possible existence of cut points.

Theorem 1. [1,[START_REF] Carmo | Riemannian geometry[END_REF] Let p = α (0) be the initial point of a geodesic. Then there is some 0 < t 0 ≤ +∞ such that α is a segment from p to α (t) for every t ≤ t 0 and for t > t 0 thereafter never again a segment from p to any α (t) for t > t 0 .

This number t 0 is called the cut value of α and α (t 0 ) is called the cut point of α. There are only two possible reasons (which can occur simultaneously) for α (t 0 ) to be to be the cut point of α:

there is a segment from p to α (t 0 ) dierent from α α (t 0 ) is the rst conjugate point on α to p (i.e. t 0 α (0) is a critical point of the exponential map.

In addition, the distance function D R (p, •) is not dierentiable at α (t 0 ) [START_REF] Itoh | Cut loci and distance functions[END_REF]1].

Remark 1. No matter the cause of the phenomenon, the main point for us is

that if t 0 is a cut value of the unit-speed geodesic α, ∀ t ≤ t 0 , D R (p, α (t)) = t while ∀ t > t 0 , D R (α (0) , α (t)) < t.
This is the basis of the method proposed by Manté and Kidé [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF] for detecting cut points (see the Supplementary Material).

Remark 2. If α := p q is a segment and V 0 := α (0), because of uniqueness of geodesics, exp p (V 0

) := α B(V0) (1) = q; reciprocally, if V 1 := -α (1), we have also that exp q (V 1 ) := α B(V1) (1) = p.

The special case of N B(D R )

There is a large number of parametrizations for the NB distribution, and the most classical one is probably

P (X = j; (φ, p)) =   φ + j -1 φ -1   p j (1 -p) φ j ≥ 0 (5)
with (φ, p) ∈ R + ×]0, 1[. Nevertheless, because of its orthogonality, we chose instead the parametrization used by Chua and Ong [START_REF] Chua | Test of misspecication with application to negative binomial distribution[END_REF]:

P (X = j; (φ, µ)) =   φ + j -1 j   µ µ + φ j 1 - µ µ + φ φ , j ≥ 0 (6)
(φ, µ) ∈ R + × R + ; here, µ is the mean of the distribution and φ is the so-called index parameter". In these coordinates, the information matrix is:

g(φ, µ) =   G φφ 0 0 G µµ  
where G µµ = φ µ(µ+φ) , while the expression of G φφ is more complicated:

G φφ = - µ + φ (µ + φ) ( φ /µ+φ) φ -1 ψ 1 (φ) φ (µ + φ) (7)
where ψ 1 is the Trigamma function [START_REF] Abramowicz | Handbook of mathematical functions with formulas, graphs and mathematical tables[END_REF]. The reader will nd in Burbea and Rao [START_REF] Burbea | Informative geometry of probality spaces[END_REF] the closed-form expression of the Rao's distance for a number of probability families; the Rao's distance between the Poisson distributions P(λ 1 ) and P(λ 2 )

is D P (λ 1 , λ 2 ) := 2 λ 1 -λ 2 . ( 8
)
We will denote P(D P ) the Riemannian manifold of Poisson distributions equipped with this distance. These authors also reported that if the index parameter φ of two NB distributions is the same,the Rao's distance is given by

D N B(p) φ, p 1 , φ, p 2 := 2 φ cosh -1 1 -p 1 p 2 (1 -p 1 ) (1 -p 2 ) (9)
in the parametrization [START_REF] Petersen | Functional data analysis for density functions by transformation to a Hilbert space[END_REF]. Of course, if L 1 = N B φ, p 1 and L 2 = N B φ, p 2 , we have necessarily:

D R L 1 , L 2 ≤ D N B(p) L 1 , L 2 . ( 10
)
Due to the complexity of ( 7), D R L 1 , L 2 cannot be obtained in a closed-form.

It must be computed by nding the numerical solution of (3) completed in the parametrization (6) by the conditions (boundary value problem)

γ (0) = φ 1 , µ 1 , γ (1) = φ 2 , µ 2 . (11) 3.1. Numerical approximation of D R L 1 , L 2 [15]
From now, L i ≡ φ i , µ i will denote some NB distribution parametrized in the system [START_REF] Cazelles | Geodesic PCA versus Log-PCA of Histograms in the Wasserstein Space[END_REF], but notice that our purpose could be extended to any parametric family of probabilities.

Firstly, all the Christoel symbols (4) were calculated from the expression [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF] of G φφ , with the help of Mathematica [START_REF] Inc | Mathematica[END_REF] . Then, the dierential equation [START_REF] Srivastava | Riemannian Analysis of Probability Density Functions with Applications in Vision[END_REF] was numerically solved under the the boundary conditions [START_REF] Galanis | Wave height characteristics in the north Atlantic ocean: a new approach based on statistical and geometrical techniques[END_REF], for a number of distributions of counts of marine species whose parameters had been estimated in [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF]. In most cases a solution could be found in an acceptable time (four CPU minutes), with a good numerical precision (20 digits), but was each one of the geodesics found a segment? And what about failures met in computation?

We indeed had to face various problems detailed in [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF], where numerical remedies were proposed. The main one consisted in inserting a well-placed stopover S between each pair of problematic distributions A and B, in such a way that D R (A, S) and D R (S, B) could be computed in a reasonable time, while D R (A, B) could not. Furthermore, S was placed in order that D R (A, S)+ D R (S, B) should by a good approximation of D R (A, B). For sake of brevity, we moved to the Supplementary Material useful information and illustrations about this previous work. All references to this supplement will be preceded by an S.

Making computations easier through Dierential Geometry techniques and Poisson Approximation

From the numerical side, it is noteworthy that the index parameter φ often takes large values, causing diculties in the evaluation of quantities associated with Γ (φ), like formulas ( 6) and ( 7) or Christoel's symbols [START_REF] Seguy | Principal Geodesic Analysis for Probability Measures under the Optimal Transport Metric[END_REF].

From the statistical side, the convergence in distribution of some L ≡ (φ, µ)

towards a Poisson distribution P when φ → ∞ is well-known. Majsnerowska [START_REF] Majsnerowska | A note on Poisson approximation by w-functions[END_REF] proved the following result:

d T V (L, P (λ)) ≤ ∆ (φ, µ) := 1 -e -µ µ φ (12)
where λ = ω (φ, µ) := φ µ φ+µ and d T V denotes the total variation distance. Consequently, we can claim that (φ µ) ∨ (µ small) ⇒ ∆ (φ, µ) small and conclude that in such cases it may be quite impossible to nd a dierence between L and P (λ), even when the index parameter is small or moderate! This fact suggests to replace the NB model by the Poisson one when both distributions are very close to each other. This is also biologically sound, since the former is well-suited for aggregative species, while the latter is associated to species with a random behavior (see [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF]2] and the references therein). But since P(D P ) is not a sub-manifold of N B(D R ), there is no clear relationship between the associated Rao's distances: we cannot mix both types of distributions. To avoid this conceptual diculty, instead of superseding the original distribution L ≡ (φ, µ) by P (ω (φ, µ)) when both distributions are very close, we propose to supersede L by another "equivalent" NB distribution, more easily manageable.

Let's focus now on the application

     ω : R + * × R + * → R + * (φ, µ) → λ
associating to any NB distribution (φ, µ) the corresponding P(λ).

Lemma 2. The tangent application

T (φ,µ) ω = 1 (φ+µ) 2 µ 2 φ 2
is surjective.

Proof. Let us x some ρ ∈ R + * ; one can easily show that the set of solutions of the equation

T (φ,µ) ω (x, y) = ρ is the line of equation y = -µ φ 2 x+ρ 1 + µ φ 2
As a consequence, ω is a surjective submersion and the ber F λ := ω -1 (λ)

associated with any λ ∈ R + * is a sub-manifold of N B(D R ).
Proposition 3. F λ is dened by either equation:

µ (λ; φ) = λ 1-λ /φ : φ > λ φ (λ; µ) = λ 1-λ /µ : µ > λ . (13)
Proof. F λ := (φ, µ) : φ µ φ+µ = λ ; thus, the strictly positive parameters λ, φ and µ are linked by the relationship φµ = φλ+λµ, which proves that φ = λ+λ φ µ and

µ = λ + λ µ φ . Consequently, λ < min (φ, µ) and lim φ→+∞ µ (λ; φ) = lim µ→+∞ φ (λ; µ) = λ Lemma 4.
One can easily verify that :

∀ λ ∈ R + * , F λ = ∅ ∀ (φ, µ) ∈ R + * × R + * , (φ, µ) ∈ F ω(φ,µ) ∀ (λ 1 , λ 2 ) ∈ R + * × R + * , λ 1 = λ 2 ⇒ F λ1 ∩ F λ2 = ∅.
Theorem 5. Suppose L ≡ (φ, µ) ∈ F λ and ∆ (φ, µ) ≤ δ, where δ is some xed threshold chosen for deciding whether L can be identied with P (λ). Then, if L ≡ (φ , µ ) ∈ F λ is another distribution, such that φ > φ, ∆ (φ , µ ) < δ and L cannot be practically distinguished from P (λ) too.

Proof. See Appendix 5 Corollary 6. Consider L ≡ (φ, µ), such that ∆ (φ, µ) ≤ δ. Then L ∈ F λ , with λ = ω (φ, µ), and we can determine the initial distribution L * (δ) := (φ * , µ * ) (δ) of F λ dened by:

   φ * = arg φ:(φ,µ)∈F λ (∆ (φ, µ) = δ) = arg φ (∆ (φ, µ (λ; φ)) = δ) µ * = λ 1-λ /φ * . ( 14
)
Let us now x L 0 ≡ φ 0 , µ 0 , such that ∆ φ 0 , µ 0 ≤ δ. L 0 can be identied with P λ 0 , as well as any distribution of the ber whose index parameter is greater than φ 0 * , due to the propositions above. Consider now the following part of F λ 0 : P λ 0 , δ := (φ, µ) ∈ F λ 0 : φ ≥ φ 0 * . ≡ L 2 and L 3 δ ≡ L 2 ; there exists λ 1,2 : L i ∈ P (λ 1,2 , δ) , i = 1, 2 and λ 2,3 : Proof. We just have to compute, thanks to formulas ( 14) and [START_REF] Cubedo | A dissimilarity based on relevant population features[END_REF] φ := max φ * λ 1 , δ , φ * λ 2 , δ , μ1 := µ λ 1 ; φ and μ2 := µ λ 2 ; φ

L 1 δ ≡ L 2 ⇔ ∃λ : L i ∈ P (λ, δ) , i = 1, 2.
L i ∈ P (λ 2,3 , δ) , i = 2, 3. Consequently, L 2 ∈ F λ1,2 ∩ F λ2,3 which is empty if λ 1,2 = λ 2,
Under the same conditions as in the preceding corollary, the following alternative strategy is always possible too.

Corollary 10. Suppose φ 1 ≤ φ 2 ; thanks to formula [START_REF] Cubedo | A dissimilarity based on relevant population features[END_REF] we can determine μ1 := µ (λ 1 ; φ 2 ), such that Ľ1 := (φ 2 , μ1 ) ∈ F λ1 is Poisson-like too (because of ( 5)) and belongs to the same class as L 1 ; D R L 1 , L 2 can be superseded by D R Ľ1 , L 2 . In addition, we can easily compute D N B(p) Ľ1 , L 2 , which is an upper bound for D R Ľ1 , L 2 .

Since P(D P ) is not a sub-manifold of N B(D R ), there is no clear relationship between the associated Rao's distances, in general (for instance, if

L 1 = L 2 belong to the same ber F λ , D R L 1 , L 2 > D P ω L 1 , ω L 2 = 0). Never-
theless, one can easily prove, with the same notation as in Proposition 10, that

     D N B(p) L1 , L2 = C (μ 1 , μ2 ) D P P(λ 1 ), P(λ 2 ) D N B(p) Ľ1 , L 2 = C (μ 1 , µ 2 ) D P P(λ 1 ), P(λ 2 ) ( 15
)
where C (μ 1 , µ 2 ) and C (μ 1 , μ2 ) ≥ 1 are given by the function dened hereunder (up to a simple change of parametrization).

Lemma 11. The function C is dened in the parametrization (5) by Proof. In the parametrization (5), the mean µ = K p 1-p and thus p = µ φ+µ , while φ = K. Consequently, λ i = K p i and D P P(λ 1 ),

C p 1 , p 2 = cosh -1 1- √ p 1 p 2 √ (p 1 -1)(p 2 -1) -2 p 1 p 2 + p 1 + p 2 ≥ 1 if p 1 = p 2 ,
P(λ 2 ) = 2 √ K p 1 -p 2 .
Then, because of formula ( 9), we have:

D N B(p) Ľ1 , Ľ2 D P (P(λ 1 ), P(λ 2 )) = C p 1 , p 2

A detailed example

Lets us x δ := 0.01, and consider L 1 = (172.236, 0.974793) and L 2 = (6, 0.05). Both these distributions are

Poisson-like, with L 1 ∈ F 0.0495868 and L 2 ∈ F 0.969307 . We plotted on Figure 2 interesting portions of these bers. On each one of the panels, the big gray point (of coordinates (λ, λ)) corresponds to the lower bound of φ and µ, while

EDA of eld data: representation of counts distributions of marine species

The Mauritanian coast, situated on the Atlantic side of the northwestern African continent, embeds a wide long continental shelf of about 750 km and 36000km 2 , with an Exclusive Economic Zone (the MEEZ) of 230000km 2 . Manté et al. [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF] considered the abundance of species of sh and invertebrates collected in the MEEZ during annual scientic trawl surveys since 1997 to now. Because the spatial distribution of groundsh species is strongly inuenced by the physical environment, we split this set into an optimal number (four) of subsets (typical habitats) associated with homogeneous physical conditions determined by available environmental variables. The counts associated with each species found in each one of the four habitats were then gathered, and tted by a NB distribution (for further information, see [START_REF] Manté | Fitting the truncated negative binomial distribution to count data. A comparison of estimators, with an application to groundshes from the Mauritanian Exclusive Economic Zone[END_REF]).

Benets of the proposed method: speeding up computations

Processing these data, Manté and Kidé [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF] found an overwhelming proportion of simple congurations (more than 70%), while numerical cut points were quite rare. In the intricate cases, the rough solution was generally accepted (more than 90% of occurrences). We display on Figure 3 statistics about the computational cost of the 26 × 26 distance matrix corresponding to C4. Among the 325 distances computed, 287 were simple cases, with a median computation cost of 25"; the remaining 38 cases were intricate, with a median cost of 826". Superseding each Poisson-like distribution L by either L * (which always exists), or L, or Ľ (depending on the situation -see Section 3.2) , we found that the proportion of simple congurations was greater than 90%, excepted for the 1). Furthermore, in the intricate cases, the rough solution was always accepted. In our previous study, numerical cut points were rare (less than a pair per class), but now we detect a single numerical cut point (see Table 1)! Poisson-like distributions were quite rare, but they were generally so pathological that their replacement by equivalent better-suited NB distributions changed a lot the results. This is shown by the statistics displayed on Figure 4. Among the 325 distances computed, 305 were simple cases, with a median computation cost of 20"; the remaining 20 cases were intricate, with a median cost of 404".

                   

Parametric representation of species from the habitat C4

We display on Figure 5 the estimated parameters of the counts distribution of species sampled in the zone 4 of the MEEZ. This region is of paramount importance: it is a high plankton productivity area, supporting a large variety of sh communities, with many commercial species that sustain shing activities.

We distinguished two categories of species on Figure 5, according to the index parameter: φ > 1 or φ ≤ 1. Species belonging to the second category being very MDS of the table of distances. We also plotted on Figure 5 theoretical condence ellipses centered on the spatial medians [START_REF] Sering | Nonparametric multivariate descriptive measures based on spatial quantiles[END_REF] of both species categories.

                                305

Representation of the Rao's distance table

Only one of the 26 species retained in C4 was Poisson-like: HISP00 (Hippocampus sp). Consequently, the distance table was dened this way: ∆ i,j = D R L i , L j , excepted when one of these species was HISP00 (index h, say).

In these cases, we used Poisson Approximation, as in Section 3.2.1; MDS of the 26 species is thus represented on Figure 6. Clearly, the counts of the species HISP00 and PHHU76 (Physiculus huloti, a type of cod) are distributed in a very special manner, which was not so obvious on Figure 5. In the inset region we represented the other 24 species separately analyzed by MDS of the restricted table ∆ AC (no point from the rst MDS is hidden).

Results and discussion

Motivated by the analysis of a large data set of marine species counts collected in the MEEZ, we developed a parameter-free method to compare species counts distributions in the setting of the Riemannian manifold N B(D R ) of negative binomial distributions, equipped with D R .

We focused rst [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF] on numerical problems met in computing D R L 1 , L 2 : lengthy computations could result from the presence of a cut point on the geodesic L 1 L 2 , requiring to determine a stopover S somewhere between

these distributions. D R L 1 , L 2 was then bounded by D R L 1 , S +D R S, L 2 .
In this work, we have essentially shown that Poisson Approximation, combined with Dierential Geometry techniques, can be used to evaluate more eciently D R L 1 , L 2 when one (at least) of the distributions involved is "Poisson- like". Superseding original NB distributions by equivalent ones (in the sense of Denition 7), we could obtain lower upper bounds of the distances than with the former strategy, with a lower computational cost. More precisely, this renement enabled us to get around computation issues: the number of intricate congurations has been approximately divided by two, the computational cost of the corresponding distances has been approximately divided by two, and numerical cut points were nearly eliminated.

Appendix

Proof of Theorem 5

Notice rst that on the ber F λ , because of ( 13), the expression of ∆ dened in [START_REF] Dodson | Some illustrations of information geometry in biology and physics[END_REF] is

λ e λφ λ-φ -1 λ-φ , giving: ∂∆ ∂φ (φ) = λ e λφ λ-φ (λ 2 +λ-φ)-λ+φ (λ-φ) 3
. Since φ > λ, the denominator of this expression is always negative while the numerator is clearly positive, excepted

potentially if λ 2 + λ -φ < 0. Substituting λ 2 + λ + ζ to φ (with ζ > 0) in
the equation, we get a simpler expression for ∂∆ ∂φ :

- ≥ 0. Consequently, ∂∆ ∂φ (φ) is negative and ∆ (φ, µ) is a decreasing function of φ on a ber.

λ -ζ e ζ ζ+λ 2 -λ-1 + ζ + λ 2 (ζ + λ 2 )

Numerical cut points and broken geodesics

Consider a Riemannian manifold M; we have the following plain proposition. Proposition 1. Let γ : I → M be a geodesic with respect to the metric connection ∇. Then γ has constant speed in the local norm M1

γ g := γ (•) g (γ (•)) = γt (•) .g (γ (•)) . γ (•)
and, for any Suppose that a satisfactory solution γ = L 1 ≡ φ 1 , µ 1 φ 2 , µ 2 ≡ L 2 of problem (M3) under the boundary condition (M11) has been found; according to Proposition 1 above, D R L 1 , L 2 is naturally approximated by γ g . But notice that γ g is an upper bound which is attained only when there is no cut point in γ ([0, 1]) (Theorem M1). That is why we needed some test to detect a possible cut point on some geodesic curve. Suppose a cut point φ c (1,2) , µ c (1,2) has been detected on γ. Then, it is natural [1] to supersede γ by the broken geodesic

φ 1 , µ 1 φ c(1,2) , µ c(1,2) ⊕ φ c(1,2) , µ c(1,2) φ 2 , µ 2
whose length is shorter than Λ (γ), provided that φ c (1,2) , µ c(1,2) φ 2 , µ 2 is also a segment. Since in our case geodesics are obtained by numerically solving a dierential equation, we are only able to locate numerical cut points. Such points were called (J, ε)-cut points, since their detection depends on two parameters: the chosen number J of sub-intervals of [0, 1] determining the accuracy of computations, and some xed threshold ε ∈ ]0, 1[.

Locating some (J, )-cut point on a geodesic γ

For that purpose, the unit interval is rst divided into J intervals: [0, 1] = J i=1 δ i , with δ i := [ i-1 J , i J [. Suppose there exists a cut point γ (t c ) on γ, such that t c ∈ δ ic . Consider the set 1

C J (γ) := γ 1 := γ 1 J , • • • , γ k := γ k J , • • • , γ J-1 := γ J -1 J ⊂ M
and, for each 1 ≤ i ≤ J the geodesic α i := γ i-1 γ i obtained by solving (M3)

under the constraints

{α i (0) = γ i-1 , α i (1) = γ i } .
Because of the uniqueness of segments (Proposition 1 and Remark M1), ∀ i < i c , γ g J = Λ (α i ) = αi g . On the contrary, when i ≥ i c , the distance between γ i-1 and γ i along γ is γ g J yet, while αi g should be smaller. More precisely, if the resolution 1 J is small enough (for instance, smaller than the injectivity radius of M [1]), γ i-1 γ i is a segment and we may write:

           ∀ i < i c , γ g J -αi g = 0 γ g J -αic g > 0 ∀ i > i c , γ g J -αi g ≥ 0.
Thus, after xing some (small) , we can locate possible cut points, with a precision depending on (J, ) and lay a denition for numerical cut points.

Denition 2. We will say that γ ic ∈ C J (γ) is a (J, )cut point on γ if i c = arg min 1≤i≤J-1 γ g J -αi g > .

Possible numerical issues

Various numerical problems were met in computing φ 1 , µ 1 φ 2 , µ 2 :

(P1) no solution was found (due to time limit) (P2) an unsuitable solution was found: for some t ∈ [0, 1], (φ (t) , µ (t)) / ∈ R + × R + (P3) the boundary condition (M11) was not fullled with a satisfactory precision.

Thus, two kinds of conguration were distinguished: simple, when none of the above issues is met, or intricate in the alternative. For further information, see [2].

Simple congurations

When none of the above issues is met, we rst check that there is no (J, ε)-

cut point on γ = φ 1 , µ 1 φ 2 , µ 2 .
Then, the solution found is accepted, and we can write:

D R L 1 , L 2 = Λ (γ) = γ g . (1)
If a (J, ε)-cut point φ c (1,2) , µ c(1,2) is detected on γ, and if φ c(1,2) , µ c(1,2) φ 2 , µ 2 is free of (J, ε)-cut point, we adopt as an upper bound for

D R L 1 , L 2 Λ φ 1 , µ 1 φ c(1,2) , µ c(1,2) + Λ φ c(1,2) , µ c(1,2) φ 2 , µ 2 .

Intricate congurations

When (P1) or (P2) is met, we consider that the best achievable solution consists in breaking γ = φ 1 , µ 1 φ 2 , µ 2 by inserting a well-placed stopover.

But since γ is undetermined, how should this stopover φ S(1,2) , µ S(1,2) be placed? We proposed two heuristics for that purpose:

1. compute a rough solution γ R to the original problem M3, contenting ourselves with low-precision (5 digits), and substitute γ R for γ to search for φ S(1,2) , µ S(1,2)

2. when γ R cannot be obtained, merely use instead

γ L (t) := t φ 1 , µ 1 + (1 -t) φ 2 , µ 2 .
In the second case, the stopover S naturally corresponds to the shortest broken geodesic determined by:

     φ S(1,2) , µ S(1,2) = γ L k L J with k L := arg min 1≤k≤J-1 Λ φ 1 , µ 1 γ L k J + Λ γ L k J φ 2 , µ 2 .
(

) 2
In the rst case, two eventualities must be considered: 2) , and the length of the broken geodesic is computed in full precision 2. if no (J, ε)-cut point is detected, proceed like in (2)

1. a (J, ε)-cut point φ c(1,2) , µ c(1,2) is detected on γ R ([0, 1]): then φ S(1,2) , µ S(1,2) = φ c(1,2) , µ c(1,
     φ S(1,2) , µ S(1,2) = γ R k R J with k R := arg min 1≤k≤J-1 Λ φ 1 , µ 1 γ R k J + Λ γ R k J φ 2 , µ 2 .
(3)

Boundary issues

(P3) is easy to solve: we just have to complete (1) by the corrective boundary error term:

BE (γ) := γ(0) -L 1 g L 1 + γ(1) -L 2 g L 2 . (4)
Finally, whatever the selected geodesic (broken, or not) may be, we obtain the upper bound:

D R L 1 , L 2 ≤ Λ (γ) + BE (γ) . (5)

A bestiary of computational problems

Before processing data, it was necessary to tune the accuracy parameters;

we xed: (J, ) = (10, 0.05).

We will display in this section typical cases of intricate congurations encountered in computing D R between marine species count distributions. Each illustration will be composed of three panels. On the left one we superimposed to the rough solution of (3), γ R (suits), the nal solution: a broken geodesic (black curve). On both the right panels, we featured the structure of each component of the broken geodesic in the neighborhood of the stopover S, with the help of the exponential map: remember that geodesics can be computed by solving (M3) under the alternative constraints (initial value problem) Here V is the initial velocity of the geodesic, and this solution is associated with the exponential map at φ 1 , µ 1 . We rst determined this way γ 1 = A S

γ (0) = φ 1 , µ 1 , γ (0) = V ∈ R 2 . (6)
where the angle of the rotation ρ acting on the initial direction B ( γi (0)) is θ k ∈ {0, ±0.1, ±0.2, ±0.3} (in degrees). Equation (M3) was then solved under the constraints (6) with V = V i (θ k ), giving rise to two bundles of seven geodesics.

In all these plots, the red (color gure online) point will be A and the black one will be B, while the stopover is represented by the big gray point; exponential maps corresponding to θ = 0 (unrotated) are plotted in red (color gure online).

On Figures S1 and S2

Figure 1 :

 1 Figure 1: Four instances of Poisson-like distributions; ∆ is given by Formula (12) and vertical bars are associated with NB probabilities while continuous curves are associated with Poisson ones.

Corollary 8 .

 8 The relation δ ≡ is an equivalence relation between Poisson-like distributions. Proof. Reexive and symmetric properties are straightforward. Suppose now L 1 δ

  3 (see Lemma 4), and these three Poisson-like distributions belong to the same ber. Thus, L 1 δ ≡ L 3 and δ ≡ is transitive Suppose L 1 δ ≡ L 2 belong to a common ber, F λ1,2 . Being indiscernible, these distributions should be necessarily close to each other, and it would be sound to merely supersede D R L 1 , L 2 by δ. The following corollaries about distributions belonging to dierent bers also stem from Theorem 5. Corollary 9. Consider two Poisson-like distributions L 1 and L 2 belonging to dierent bers, F λ1 and F λ2 . One can always determine a pair of Poissonlike distributions L1 := φ, μ1 ∈ F λ1 and L2 := φ, μ2 ∈ F λ2 such that D R L 1 , L 2 can be superseded by D R L1 , L2 . In addition, thanks to for- mula (9), we can straightforwardly compute D N B(p) L1 , L2 , which is an upper bound for D R L1 , L2 .

  and C (p, p) := 1.

L≡ L 2 (

 2 * := (φ * , µ * ) (λ, δ) is the "initial" distribution given by[START_REF] Ilea | Statistical Hypothesis Test for Maritime Pine Forest Sar Images Classication 20 Based on the Geodesic Distance[END_REF].All the distributions situated on the right of L * are Poisson-like. It is the case of L 1 and L 2 , represented on Figure 2 by small gray points. We found that φ 1 * = 63.2096 < φ 1 and φ 2 * = 0.412537; thus, φ = φ 1 * and L1 = L 1 * , while L2 = L 2 * . Next, in accordance with Corollary 9, we com- puted D N B(p) L1 , L2 ≈ 1.53375, which is rather close to D P (P(λ 1 ), P(λ 2 )) ≈ 1.52371. Since φ 2 < φ 1 , it is also possible to determine the distribution Ľ2 := (φ 1 , μ2 ) δ black point on the lower panel) and, thanks to Corollary 10, to compute D N B(p) Ľ2 , L 1 ≈ 1.52737 (very close to D P (P(λ 1 ), P(λ 2 )) too).3.2.1. Application to EDALet L 1 and L 2 ∈ N B(D R ); three cases may be met: both of them are Poisson-like, only one of them is Poisson-like, or none of them is so.

Figure 2 :

 2 Figure 2: Portions of bers associated with two Poisson-like distributions (δ = 0.01): F 0.0495868 (lower panel) and F 0.969307 (upper panel).

Figure 3 :

 3 Figure 3: Statistics of the computational burden for processing (without Poisson approximation) species collected in zone 4 of the MEEZ: number of distances of each type, box-plots of computation length.

Figure 4 :

 4 Figure 4: Statistics of the computational cost for processing the same species as in Figure 3, with Poisson approximation.

Figure 5 :

 5 Figure 5: Species collected in zone 4 of the MEEZ; the green dotted (resp. orange dashed) closed curve corresponds to the condence region of 0.99 level associated with the spatial median of the rst (resp. second) category species .

Figure 6 :

 6 Figure6: MDS of the Rao's distance between the 26 selected species (big points); the inset graph corresponds to the same analysis, performed after removing both the species HISP00 and PHHU76. .

3 whose sign depends on the sign of 1 -e ζ ζ+λ 2 -λ- 1 .

 321 Since the only solutions of ζ ζ+λ 2 -λ -1 = 0 are λ = 0 ∧ ζ = 0 and ζ = -λ + λ 2 ∧ λ = 0, this expression is negative, and 1 -e ζ ζ+λ 2 -λ-1

  [a, b] ⊆ I, we have: ¢ b a γt (t) .g (γ (t)) . γ (t)dt = (b -a) γ g .

Figure S1 :

 S1 Figure S1: Numerical cut point ( γ R satisfactory). Left panel: rst guess (suits) and nal solution (black curve). Right panels: plot of the two bundles of geodesics issued from A or B. Red curve (color gure online): θ = 0 in equation (7); dashed curves: θ = 0. The header corresponds to the parameters of the distributions in the system M6.

(resp. γ 2

 2 = B S) by solving equation (M3) under the constraints (M11). We afterward considered {V i (θ k ) := ρ (θ k ) B ( γi (0)) : i = 1, 2} ,

  are represented typical cases of intricate situations where γ R was a satisfactory rst guess for the geodesic. The nal solution A S ⊕ S B is similar to γ R . The structure of A S and B S is investigated on both right panels of each sub-gure. Notice they look like genuine cut points (arriving at the cut locus means some kind of catastrophe [1, p. 279]; cf. also the last statement of Theorem M1).On Figures S3 and S4 are represented worse cases, where the rst guess γ R

Figure S2 :

 S2 Figure S2: Numerical cut point ( γ R satisfactory). Same structure as in S1.

Figure

  Figure S3: A numerical cut point with an unsatisfactory γ R . Same structure as S1.

Figure S4 :

 S4 Figure S4: Linear rst guess. Same structure as S1.

Table 1 :

 1 Global results obtained in the four habitats of the MEEZ (δ = 0.01).

	Habitat	Number of species	Simple	Intricate	Cut points	Poisson-like
		(well-tted)	congurations	(Rough, Linear)		distributions
	C1	30	400	(35,0)	1	3
	C2	19	147	(24,0)	0	3
	C3	26	309	(16,0)	0	5
	C4	26	304	(21,0)	0	1

Acknowledgments

We thank the Mauritanian Institute of Oceanographic Research and Fisheries (especially S. O. Kidé) and the Department of Cooperation and Cultural Action of the Embassy of France in Mauritania for their support for this study.

We also thank all scientists who contributed to eld surveys and data collection.

Suppose rst L 1 and L 2 belong to distinct bers F λ 1 and F λ 2 and each L i ∈ P λ i , δ . Then µ i ≈ λ i and we can use each one of the Corollaries 6, 9 or 10 to determine a pair of equivalent distributions, whose distance would be easier to compute. So, one will successively (try to

as soon as possible in order to alter the data as little as possible.

Suppose now L 1 is Poisson-like while L 2 is not; if φ 1 ≤ φ 2 (or even if φ 1 * ≤ φ 2 ), we can again consider Ľ1 := φ 2 , μ1 . One will successively (try to) compute (intricate conguration with a cut point) was 45.1321 (denitions of simple and intricate congurations are reminded in the supplementary materialfor further details, see [START_REF] Manté | Approximating the Rao's distance between negative binomial distributions. Application to counts of marine organisms[END_REF]).

In the next case (Figure S3) L 1 = (0.00996246, 0.121282), while the second distribution is the same as in the previous case. The original distance corresponded to an intricate conguration with a cut point, and to the upper bound 43.1519. We found instead D R L 2 * , L 1 = 3.48809.

In the last case (Figure S4), L Ľ2 , L 1 , but we found that D R L 2 * , L 1 = 12.5294 (intricate conguration with an acceptable rough solution and a stopover), while the original D R L 1 , L 2 (intricate conguration with linear interpolation) gave rise to the upper distance 21.351.