Clément Trystram
email: clement.trystram@gmail.com

Karyne M Rogers

Marc M Soria

Sébastien Jaquemet

Feeding patterns of two sympatric shark predators in coastal ecosystems of an oceanic island

come

Introduction

Understanding the role of large predators in marine ecosystems is important, given that many of these populations are in steep decline, and therefore the critical ecological functions they provide are threatened [START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF][START_REF] Estes | Trophic downgrading of planet Earth[END_REF]. Both empirical and theoretical research supports the belief that large predatory sharks have a key role in marine ecosystem functioning via various processes, such as enhancing ecosystem connectivity due to their high mobility [START_REF] Mccauley | Assessing the effects of large mobile predators on ecosystem connectivity[END_REF]) and the top-down pressure they exert on their prey through direct predation [START_REF] Myers | Cascading effects of the loss of apex predatory sharks from a coastal ocean[END_REF][START_REF] Baum | Cascading top-down effects of changing oceanic predator abundances[END_REF][START_REF] Heupel | Sizing up the ecological role of sharks as predators[END_REF]. Moreover, the presence of predators leads to indirect behavioral effects such as risk avoidance, wherein herbivorous prey become more cautious, thus reducing herbivore intensity, even in the absence of effective predation [START_REF] Heithaus | Towards a predictive framework for predator risk effects: the interaction of landscape features and prey escape tactics[END_REF][START_REF] Rizzari | Not worth the risk: apex predators suppress herbivory on coral reefs[END_REF]. However, all of these effects involve a high plasticity of predator ecological niches, resulting in unpredictable consequences of predator loss on ecosystem dynamics and ecological interactions, including stock exploitation by human fisheries [START_REF] Newman | Structure of Caribbean coral reef communities across a large gradient of fish biomass[END_REF][START_REF] Myers | Cascading effects of the loss of apex predatory sharks from a coastal ocean[END_REF][START_REF] Rizzari | Not worth the risk: apex predators suppress herbivory on coral reefs[END_REF].

Tiger sharks (Galeocerdo cuvier) and bull sharks (Carcharhinus leucas) are two of the largest marine top-predators. Both species are widespread in tropical and subtropical coastal waters [START_REF] Compagno | Sharks of the World[END_REF].

Bull sharks are one of the few truly euryhaline elasmobranchs and can often be found in freshwater as well as marine environments [START_REF] Compagno | Sharks of the World[END_REF]. Although not fully euryhaline, tiger sharks nevertheless also exhibit a high level of tolerance for diverse habitats [START_REF] Compagno | Sharks of the World[END_REF]. While principally inhabitants of coastal environments, they are known to undertake large-scale oceanic migrations [START_REF] Werry | Reef-fidelity and migration of tiger sharks, Galeocerdo cuvier, across the Coral Sea[END_REF][START_REF] Lea | Repeated, long-distance migrations by a philopatric predator targeting highly contrasting ecosystems[END_REF]. As such, these species are suitable models to test hypotheses about the physical and biological drivers of ecological niche plasticity.

Historically, the trophic ecology of tiger sharks and bull sharks has been studied mainly via analysis of their stomach contents (e.g., [START_REF] Rancurel | Le requin tigre, Galeocerdo cuvier, Lacépède des eaux Néocalédoniennes examen des contenus stomacaux[END_REF][START_REF] Werry | Habitat ecology of the bull shark, Carcharhinus leucas, on urban coasts in eastern Queensland[END_REF][START_REF] Bornatowski | Feeding ecology and trophic comparisons of six shark species in a coastal ecosystem off southern Brazil[END_REF]) and, more recently, through the use of nonlethal stable isotope techniques [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF][START_REF] Daly | Comparative feeding ecology of bull sharks (Carcharhinus leucas) in the coastal waters of the southwest Indian Ocean inferred from stable isotope analysis[END_REF][START_REF] Heithaus | Apparent resource partitioning and trophic structure of large-bodied marine predators in a relatively pristine seagrass ecosystem[END_REF]. Carbon isotopic values (␦ 13 C) vary between organic matter sources (with higher values for benthic algae than for phytoplankton) and habitats (inshore-offshore gradient, with lower values in offshore environments), whereas the nitrogen isotopic value (␦ 15 N) increases through the food chain (with the relative abundance of 15 N higher in consumers than in prey), providing a tracer of organic matter sources and serving as a proxy for trophic level [START_REF] Peterson | Stable isotopes in ecosystem studies[END_REF]. As tissue turnover requires more time than digestion, stable isotopes represent a longer-term average of dietary habits (turnover rates of muscle and blood are approximately 1 year and 7.5 months, respectively, in large shark species; [START_REF] Macneil | Variable uptake and elimination of stable nitrogen isotopes between tissues in fish[END_REF][START_REF] Logan | Stable isotope dynamics in elasmobranch fishes[END_REF][START_REF] Malpica-Cruz | Tissuespecific isotope trophic discrimination factors and turnover rates in a marine elasmobranch: empirical and modeling results[END_REF]. Although stable isotopes can provide information about long-term dietary habits, they provide much less information about prey taxonomy. Using stable isotope analyses in combination with traditional stomach content analyses forms a robust tool for investigating the trophic ecology of animals (e.g., [START_REF] Fisk | Using anthropogenic contaminants and stable isotopes to assess the feeding ecology of greenland sharks[END_REF][START_REF] Polo-Silva | Trophic shift in the diet of the pelagic thresher shark based on stomach contents and stable isotope analyses[END_REF].

Despite both tiger and bull sharks being widely distributed, only a handful of studies have directly compared their trophic ecologies [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF], and to our knowledge, no study has made interspecies comparisons of the two sharks in the same location and at the same time. [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF] suggested contrasting patterns of individual specialization between these two species (studied in two distinct areas) that result in different functional roles in marine food webs. The presence of these two species in the waters around Reunion Island provides a unique opportunity to explore the relationships between these two apex predators.

Reunion Island is a young oceanic island of volcanic origin that lies to the east of Madagascar and within the Madagascar regional hotspot of biodiversity. The geomorphology of the island is characterized by the absence of an island shelf, with the exception of the leeward west coast, where it extends up to 5 km offshore. This isolated system is ideal to study tiger and bull shark ecology at a fine spatial scale.

Stomach contents and two tissue types (blood and muscle) with different turnover rates were analyzed using stable isotopes to ascertain the trophic ecology of these two shark species through the following questions: (1) Are they generalist or specialist foragers at the population level? (2) Is there overlap in their trophic niches? (3) Are their foraging habits dependent on individual length, body condition, capture season, or sex? (4) Do they exhibit trophic specialization at the individual level (i.e., are individuals homogeneous or heterogeneous)?

Methods

Sample collection and preparation

Muscle and (or) whole blood and (or) stomach content samples from 31 bull sharks (C. leucas) and 65 tiger sharks (G. cuvier) were collected from individuals caught almost exclusively by local fisherman along the island shelf (ϳ160 km 2 ; Fig. 1) between Saint-Paul and Saint-Gilles on the west coast of Reunion Island between August 2012 and December 2014.

Only individuals older than 2 years of age (i.e., body lengths >200 cm for tiger sharks and >110 cm for bull sharks; [START_REF] Branstetter | Age and growth estimates of the bull shark, Curcharhinus zeucas, from the northern Gulf of Mexico[END_REF]Branstetter et al. 1987) were included in the study to negate any artifacts of maternal enrichment influences due to slow muscle turnover rates [START_REF] Olin | Maternal meddling in neonatal sharks: implications for interpreting stable isotopes in young animals. Rapid Commun[END_REF]. Additional blood samples were collected from live animals during tagging programs [START_REF] Blaison | Seasonal variability of bull and tiger shark presence on the west coast of Reunion Island, western Indian Ocean[END_REF]. All samples were kept in a cool box following collection and thereafter frozen at -20 °C in the laboratory until further analysis.

The prey items retrieved from stomachs were counted, weighed to the nearest 0.01 g, identified to the lowest possible taxonomic level using identification keys adapted to taxonomic groups and (or) to anatomical parts (cephalopod beaks, teleost otoliths, and so forth; [START_REF] Clarke | A handbook for the identification of cephalopods' beaks[END_REF][START_REF] Smith | Smith's sea fishes[END_REF][START_REF] Smale | Otolith atlas of southern African marine fishes[END_REF], and then compared with species lists for the area around Reunion [START_REF] Letourneur | An updated checklist of the marine fish fauna of Reuin Island, South-Western Indian Ocean[END_REF][START_REF] Durville | Inventaire des poissons récoltés lors de l'éruption volcanique d'avril 2007 du piton de la Fournaise (île de La Réunion)[END_REF]) and our own reference collection. Paired and unpaired otoliths, beaks, and exoskeleton fragments (cephalothorax and abdomen) were used to assess the number of digested fish, cephalopods, and crustaceans, respectively.

To characterize the trophic habitat of Reunion Island sharks, 36 samples of sedimentary organic matter (SOM) were collected between November 2012 and August 2014 at depths ranging from 10 m to 100 m along six inshore-offshore transects. Sixty samples of particulate organic matter (POM) were also collected at the same time. POM samples were obtained by filtering 5 L of subsurface seawater through precombusted (4 h at 450 °C) Whatman GF/F filters (25 mm). Filters were then stored in a cool box and brought back to the laboratory, where they were oven-dried at 50 °C for 24 h [START_REF] Lorrain | Decarbonation and preservation method for the analysis of organic C and N contents and stable isotope ratios of low-carbonated suspended particulate material[END_REF]. Prior to carbon and nitrogen isotope analysis of the organic matter, subsamples of POM and SOM were treated with 1 mol•L -1 HCl to remove inorganic carbon [START_REF] Kolasinski | Sources of particulate organic matter at the ecosystem scale: A stable isotope and trace element study in a tropical coral reef[END_REF]. Published POM and SOM values collected by the same method on the fringing reef of Reunion Island in 2006 and 2007 were used to complement the baseline values of the study area [START_REF] Kolasinski | Sources of particulate organic matter at the ecosystem scale: A stable isotope and trace element study in a tropical coral reef[END_REF].

All samples intended for stable isotope analyses, except POM samples, were freeze-dried at <50 Pa and < -40 °C for 48 h and then ground into a fine, homogeneous powder using an automated grinder. Approximately 0.5 mg of muscle and blood were weighed and packed into tin capsules. Isotopic composition (with a precision of 0.1‰ for ␦ 13 C and 0.1‰ for ␦ 15 N) and carbon percentage and nitrogen percentage content were measured at the Stable Isotope Laboratory, GNS Science, Lower Hutt, New Zealand, using an Isoprime isotope ratio mass spectrometer interfaced to an EuroEA elemental analyzer in continuous-flow mode (EA-IRMS). The results were expressed in conventional delta notation (␦), according to the following equation [START_REF] Peterson | Stable isotopes in ecosystem studies[END_REF]:

(1) X ϭ [(R sample /R standard ) Ϫ 1] 1000
where R sample and R standard are the fractions of heavy to light isotopes in the sample and standard, respectively. The ␦ 13 C and ␦ 15 N isotopes were measured relative to the international standards of Vienna PeeDee Belemnite (VPDB) for carbon and atmospheric air for nitrogen. Some chemical effects (lipid and urea concentrations) could affect isotopic values, leading several researchers to propose chemical extraction or mathematical correction (Hussey et al. 2012a). However, the higher variability in results and interpretations [START_REF] Logan | Stable isotope dynamics in elasmobranch fishes[END_REF]Hussey et al. 2012a) has led others to recommend that no chemical treatment be applied when the C/N ratio (proxy of lipid contents) is low [START_REF] Post | Getting to the fat of the matter: models, methods and assumptions for dealing with lipids in stable isotope analyses[END_REF]). In our case, the C/N ratio was low (<3.5) and constant between tissue types; therefore, no chemical extraction was undertaken [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF][START_REF] Vaudo | Dietary niche overlap in a nearshore elasmobranch mesopredator community[END_REF][START_REF] Malpica-Cruz | Tissue-specific stable isotope ratios of shortfin mako (Isurus oxyrinchus) and white (Carcharodon carcharias) sharks as indicators of size-based differences in foraging habitat and trophic level[END_REF].

Diet and trophic niche widths

Prey importance was expressed as the relative numerical abundance (%N, number of individual prey divided by the total number of consumed prey), relative mass abundance (%W), and occurrence (%O, number of stomachs containing the given prey divided by the total number of stomachs, expressed as a percentage). The trophic niche width (TNW) of each population was calculated using the Shannon diversity index [START_REF] Bolnick | Measuring individual-level resource specialization[END_REF]):

(2)

TNW ϭ Ϫ ͚ kϭ1 S p k ln(p k )
where S is the number of prey families, and p is the proportional numerical abundance of each prey family (%N). This index varies from 0 to ln(S). The Piélou equitability was then calculated as TNW/ln(S), with values varying between 0 (diet dominated by one specific prey type) and 1 (all prey were of equal importance).

Stable isotopes were used to calculate several metrics based on the adaptation of a community index (Layman et al. 2007a) applied at the species scale using each individual as a sampling unit. The mean distance to centroid (CD) provides a measure of the average degree of trophic diversity, which can then be divided into exploited trophic-level diversity (␦ 15 N range: NR) and organic matter source diversity (␦ 13 C range: CR). The mean nearest neighbor distance (MNND) provides a measure of trophic similarity between individuals. Total isotopic niche assessments have previously been determined using the minimum convex hull area (from ␦ 15 N-␦ 13 C isotopic plots) that contain all individuals, although it is highly biased by outliers (i.e., individuals with extreme positions; Layman et al. 2007b). Kernel-based estimators are robust for small sample sizes and are less sensitive to outliers but are still capable of considering outliers as part of the overall distribution [START_REF] Fieberg | Kernel density estimators of home range: smoothing and the autocorrelation red herring[END_REF]. Analogous to home range, 95% kernel area was used to assess total isotopic niche (e.g., [START_REF] Franco-Trecu | Individual trophic specialisation and niche segregation explain the contrasting population trends of two sympatric otariids[END_REF]. The 95% kernel area was generated using the "ks" package of the statistical software R [START_REF] Duong | ks: Kernel density estimation and kernel discriminant analysis for multivariate data in R[END_REF].

Moreover, the meaningful trophic niche width (i.e., that is likely to be important for a randomly chosen individual) was represented by the standard ellipse corrected for small sample size (SEA c ), introduced by [START_REF] Jackson | Comparing isotopic niche widths among and within communities: SIBER -Stable Isotope Bayesian Ellipses in R[END_REF] as a bidimensional equivalent of unidimensional mean and variance. Applied SEA c on non-Gaussian distribution could lead to misinterpretation because this representation is based on the bivariate normal distribution hypothesis. If required, outliers were removed one by one until an isotopic normal distribution was obtained. SEA c is independent from sample size, allowing comparison between species with different sample sizes [START_REF] Jackson | Comparing isotopic niche widths among and within communities: SIBER -Stable Isotope Bayesian Ellipses in R[END_REF]. Except for kernel area, all other isotopic metrics were calculated using "SIAR", a statistical software package using R [START_REF] Parnell | SIAR: Stable isotope analysis in R. R Package[END_REF]. As isotopic values are expressed in ‰, the kernel area and SEA c are expressed in ‰ 2 because these indices represent an area extent on the isotopic biplot.

Evaluation of sampling effort

Sampling effort was evaluated using cumulative curves generated by plotting cumulative observations (number of prey families or the 95% isotopic kernel area) against sampling effort (number of preycontaining stomachs or isotopic value). For stomach contents, individual ranking was randomized 100 times to eliminate bias [START_REF] Cortés | A critical review of methods of studying fish feeding based on analysis of stomach contents: application to elasmobranch fishes[END_REF]. For stable isotopes, bootstrap analyses (100 random selections) were performed to examine the mean 95% isotopic kernel area across varying sample sizes. Asymptotic curves were then fitted to the observed cumulative prey curves to assess their theoretical diet diversity and breadth [START_REF] Dengler | Which function describes the species-area relationship best? A review and empirical evaluation[END_REF]. To determine if a curve had reached an asymptote, the slope of a linear regression based on the last four endpoints was statistically compared with zero [START_REF] Bizzarro | Comparative feeding ecology of four sympatric skate species off central California[END_REF]). If the slope was not zero, the asymptote was not reached, whereas if the slope was zero, the cumulative curve was considered to reach an asymptote.

Niche overlap

The mean carbon and nitrogen isotope values in muscle and blood samples from both tiger sharks and bull sharks were determined. Differences between species were tested with a Student t test for independent samples for normally distributed data or a Mann-Whitney-Wilcoxon signed rank test for non-normally distributed data. Finally, the area of overlap between the two species' SEA c was calculated to assess the extent of isotopic niche overlap between the two species and obtain an interpretable value similar to previous indices.

Similarities between predator species diets were quantified with Czekanowski index (C xy ;[START_REF] Bolnick | Measuring individual-level resource specialization[END_REF]):

(3)

C xy ϭ 1 Ϫ 1 2 ͚ iϭ1 S |p xi Ϫ p yi |
where S is the number of prey families, and p x and p y are the proportional numerical abundances of prey families consumed by species x and y, respectively. C xy varies between 0 (complete partition) and 1 (complete overlap). Analogously with Morisita-Horn index, a C xy greater than 0.6 will be considered as significant overlap [START_REF] Zaret | Competition in tropical stream fishes: support for the Competitive Exclusion Principle[END_REF].

Length, body condition, year, capture season, and sex effect

The body condition was determined using a ratio based on a measurement of each individual shark's circumference at the base of the first dorsal fin divided by fork length. The capture season was defined based on [START_REF] Conand | A ten-year period of daily sea surface temperature at a coastal station in Reunion Island, Indian Ocean (July 1993 -April 2004): patterns of variability and biological responses[END_REF] monitoring of sea-surface temperatures (winter: May-October; summer: November-April). The data were tested for normality using a Jarque-Bera test [START_REF] Thadewald | Jarque-Bera test and its competitors for testing normality: a power comparison[END_REF]. The correlation between two quantitative variables (effect of body condition and fork length on carbon and nitrogen isotopic values) was tested using a Pearson or Kendall test depending on normality of the variables. The effects of year, capture season, and sex on carbon and nitrogen isotopic values were tested using Student or Mann-Whitney tests depending on variable normality. All statistical analyses were carried out using R 3.1.2 for Windows with a significant p level set at 0.05.

Individual specialization

Isotopic values can be used to determine individual feeding stability over time. When more than one isotopic value per individual is available (e.g., two tissue types), it is possible to separate the total variation into the following two components: the mean sum of square within (MSW) and between (MSB) individuals [START_REF] Bolnick | Measuring individual-level resource specialization[END_REF][START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF]). These indices were calculated using the following equations:

Pagination not final (cite DOI) / Pagination provisoire (citer le DOI)

(4) MSW ϭ 1 N(K Ϫ 1) ͚ kϭ1 K ͚ jϭ1 N (x jk Ϫ x ¯j) 2 (5) MSB ϭ 1 N Ϫ 1 ͚ jϭ1 N (x ¯j Ϫ x ¯)2
where N is the number of individuals, K is the number of tissues, x jk is the isotopic value (␦ 13 C or ␦ 15 N) of tissue k measured on the jth individual, x ¯j is the individual isotopic mean, and x ¯is the population isotopic mean. Therefore, MSW + MSB represents the total variation, and MSW MSWϩMSB (hereinafter IS) measures the relative degree of individual stability, ranging from 0 (constant diet) to 1 (diet change through time). The calculation was performed using two tissues with high turnover rates: muscle (turnover ϳ1 year) and whole blood (turnover ϳ7.5 months; Malpica-Cruz et al. 2012), so the assessed stability only covered a short time frame (ϳ5 months).

An adaptation of the Czekanowski index allows a dietary overlap calculation between each individual and the overall population. The mean proportional similarity between individuals and the population (PSI) provided a measure of individual similarity [START_REF] Bolnick | Measuring individual-level resource specialization[END_REF]:

(6) PSI ϭ 1 N ͚ iϭ1 N ͩ 1 Ϫ 1 2 ͚ jϭ1 S |p ij Ϫ p j | ͪ
where N is the number of individuals (stomach number), S is the number of prey categories, p ij is the number of jth prey category individuals found in the ith stomach, and p j is the proportion of the jth resource category in the population's niche. Calculated at the individual level, PSI (ranging from 0, a specialist, to 1, a generalist) allowed a statistical comparison between G. cuvier and C. leucas.

Results

Dietary composition

The stomach contents of 24 bull sharks and 58 tiger sharks were analyzed; 16 (67%) and 30 (52%) stomachs were found to contain prey, respectively. The prey items that were identified to the family level represented 55% and 70% of all prey found in bull and tiger shark stomachs, respectively.

The bull sharks we analyzed fed on three major prey groups, representing 13 prey families, with the most common prey consisting of teleosts, represented by 10 identified families, followed by cephalopods and elasmobranchs (Table 1). Indigestible items, such as plastic bags, were found in 25% of bull shark stomachs.

Tiger sharks consumed a wider variety of prey that consisted of eight major prey categories represented by 22 families. The most common prey items were teleosts and cephalopods, followed by birds. Marine turtles, crustaceans, elasmobranchs, land mammals, and sea stars were also recorded in rare instances (Table 1). Indigestible items, such as plastic bags, hooks, sand, and vegetables were found in 27% of tiger shark stomachs.

Cumulative curve analyses suggested that C. leucas could consume up to 15 prey families and G. cuvier up to 26 prey families, as assessed by asymptote ordinates. The slope of linear regressions performed on the four last points of cumulative curves were significantly different from 0 (p < 0.05 in both cases), while being very low (0.08 for G. cuvier and 0.07 for C. leucas), demonstrating that the number of C. leucas and G. cuvier stomachs included in the analyses were close to the optimal number necessary for describing the diet (Fig. 2A).

Overall, the stomach content compositions of both species varied greatly among individuals, as indicated by the large standard deviations for each parameter compared with the mean values (Table 2). Thus, the mean number of prey per stomach, the mean mass of the stomach contents, and the mean individual mass of the prey items identified from the stomach contents did not differ significantly between species (Mann-Whitney test, all p > 0.05; Table 2). The number of prey families found per tiger shark stomach (2.4 ± 1.1) was significantly higher than those found in bull shark stomachs (1.8 ± 1.5, Mann-Whitney test, p = 0.043); however, Shannon indices based on numerical abundance (TNW) were quite similar between tiger sharks (2.25) and bull sharks (2.41).

Characterization of organic matter sources

In the study area, POM ␦ 13 C values were lower than SOM ␦ 13 C values (Welch's t test, p < 0.001), whereas POM displayed more similar ␦ 15 N values than SOM (Welsh's t test, p = 0.057; Fig. 3A). Organic matter previously sampled from the fringing reef exhibited more positive ␦ 13 C values and lower ␦ 15 N values than organic matter sampled from outside the reef in this study (Fig. 3A). Thus, the organic matter sources exhibited an overall ␦ 13 C range of 8.3‰ between nearshore waters and deeper waters.

Shark isotopic niche

Cumulative curve analyses indicated that the maximal isotopic niche (assessed using the kernel approach) for bull sharks would be 4.83‰ 2 for muscle and 3.97‰ 2 for blood, whereas for tiger sharks, the isotopic niche would extend to 5.93‰ 2 for muscle and 6.76‰ 2 for blood. Moreover, these analyses indicated that the numbers of individuals included in this study were not sufficient to describe the isotopic diversity for both species given that their cumulative prey curves did not reach an asymptote (all slopes of linear regressions performed on the four last points were significantly different from 0, p < 0.05 for all test; Fig. 2B). Thus, their isotopic niche sizes appear to be slightly underestimated.

All isotopic niche metrics calculated for both muscle and blood (CR, NR, CD, and 95% kernel) were found to be larger for tiger sharks than for bull sharks (Table 3), whereas CD did not statistically differ between species or tissues (Kruskal test, p = 0.42). The slightly higher mean trophic diversity (assessed by the mean distance to centroid, CD) of tiger sharks compared with bull sharks could be due to the larger range of values for both carbon isotopes (CR) and nitrogen isotopes (NR). Tiger shark CR represented 63% of the total range of ␦ 13 C values for the organic matter sources originating from coral reef and coastal waters, whereas bull shark CR covered 36% of the ␦ 13 C value range of organic matter sources (Fig. 3A). The isotopic niche estimates were higher for G. cuvier than for C. leucas using both a kernel approach and unbiased standard ellipse, SEA c . Outliers not included in the calculation of the SEA c (see Materials and methods) had more negative ␦ 13 C values for tiger sharks and less negative ␦ 13 C values for bull sharks (Figs. 3B and3C). Finally, the tiger shark population was more homogeneous than the bull shark population and exhibited lower MNNDs (Table 3).

Niche overlap

Bull sharks exhibited carbon isotopic values that were significantly 13 C-enriched compared with tiger sharks (Mann-Whitney-Wilcoxon test, p < 0.001 in both tissues; Table 3). Bull shark muscle was slightly 15 N-enriched compared with tiger shark muscle (Student test, p < 0.001), whereas the two species had similar blood ␦ 15 N values (Student test, p = 0.06). These differences suggested a niche partitioning between the two species, with no SEA c overlap (Figs. 3B and3C).

This niche partition is also observed with the Czekanowski dietary overlap index calculated using only the identified prey families found in the stomach contents, which was under the 0.6 threshold (C xy = 0.34).

Length, body condition, year, capture season, and sex effect

Tiger sharks sampled in this study were significantly longer than bull sharks (total length = 333 ± 34 cm and 268 ± 39 cm, respectively; Mann-Whitney test, p < 0.001). For tiger sharks, carbon and nitrogen stable isotopes measured in muscle and blood showed no fork length, body condition, seasonal, annual, or sex effects (all tests p > 0.05). In contrast, bull shark muscle ␦ 13 C was affected by fork length ␦ 13 C (Pearson test, r = -0.54; p = 0.002). A seasonal effect was also observed in whole blood ␦ 13 C (Student test, p = 0.016), switching from -15.43‰ ± 0.29‰ for individuals caught in the summer to -14.97‰ ± 0.49‰ for animals caught in the winter. Year, body condition, and sex did not affect C. leucas carbon and nitrogen isotopic values (all tests p > 0.05). The ␦ 13 C difference between blood and muscle (⌬␦ 13 C blood-muscle ) was positively correlated with fork length Note: "Total" summarizes the number of prey identified, the cumulative mass of stomach contents, and the number of stomachs containing identifiable prey. NI, not identified; IRI, %O(%N + %W), expressed as percentage.

Pagination not final (cite DOI) / Pagination provisoire (citer le DOI) (r = 0.66; p < 0.001), with longer individuals exhibiting larger intertissue isotopic differences.

Individual specialization

Individual prey specialization was assessed by comparing carbon and nitrogen isotopes of paired muscle and blood from each of the 22 bull and 49 tiger sharks. Both bull and tiger sharks showed significant isotopic differences between blood and muscle (⌬␦ 13 C blood-muscle = 0.77‰ ± 0.24‰, Student test, p < 0.001, ⌬␦ 15 N blood-muscle = 0.21‰ ± 0.24‰, Student test, p < 0.001 for bull sharks; ⌬␦ 13 C blood-muscle = 0.80‰ ± 0.40‰, Student test, p < 0.001, ⌬␦ 15 N blood-muscle = 1.05‰ ± 0.39‰, Mann-Whitney test, p < 0.001 for tiger sharks).

Individual specialization was evaluated using both stable isotopes and stomach contents through the indices IS and PSI, respectively, with higher values indicating greater similarity between individuals. Both methods provided consistent results, with the bull shark population being more heterogeneous than the tiger shark population. Actually, bull shark IS ranged from 14% (using ␦ 15 N values) to 57% (using ␦ 13 C values), whereas tiger shark IS ranged from 59% (using ␦ 15 N values) to 38% (using ␦ 13 C values). Moreover, bull shark PSI (0.15 ± 0.13) was significantly lower than tiger shark PSI (0.29 ± 0.18) (Mann-Whitney test, U = 90, p = 0.02).

Discussion

Although bull sharks and tiger sharks have a long history of interaction with humans [START_REF] Burgess | ISAF statistics on attacking species of shark[END_REF] and are thought to have important roles in ecosystem functioning [START_REF] Heithaus | Predicting ecological consequences of marine top predator declines[END_REF]), only a handful of studies have explored their foraging ecology to any great extent (a compilation of studies focusing on stomach content analyses is presented in Table 4). The limited amount of research on these species is mainly due to the difficulties of study-ing large and scarce predators, particularly in marine systems. The goal of this study was thus to improve our understanding of the factors influencing trophic niche plasticity. Considering the difficulty to investigate the trophic ecology of these predators, the novelty lies in the use of both stomach content and stable isotope analyses to concomitantly compare two sympatric populations. The description of tiger and bull shark trophic niches was achieved by investigating their pattern of individual specialization. Difficulties related to sampling large, mobile, and scarce animals coupled with their opportunistic feeding behavior led to a lower level of statistic robustness as demonstrated by cumulative curves even though our sample size in this study was comparable to those of previous studies.

Dietary habits at Reunion Island confirms the opportunistic feeding behaviour of tiger and bull sharks

In this study, both tiger and bull sharks were found to be more piscivorous than previously reported (Table 4). Similarly, the dietary contribution of cephalopods was high for both species (based on stomach content found in 47% of tiger shark stomachs and 31% of bull shark stomachs). Such a high proportion of cephalopod consumption by sharks has previously only been reported once, in tiger sharks from South Africa [START_REF] Smale | Cephalopods in the diets of four shark species (Galeocerdo cuvier, Sphyrna lewini, S. zygaena and S. mokarran) from Kwazulu-Natal, South Africa[END_REF]), but cephalopods have never been known to exceed 5.8%O of the bull shark diet (Table 4). In contrast with previous studies, turtles, crustaceans, other elasmobranchs, sea snakes, and marine mammals were less abundant or absent in the sharks we analyzed (Table 4). The key difference between the diets of these two species at Reunion Island compared with other localities is likely to be related to the local composition of available prey communities rather than predator selectivity; indeed, both species were considered opportunist because their diets differed regionally. The main prey of tiger sharks have been shown to vary considerably based on location, from te-

Note:

The values are given as the mean ± standard deviation. N stomach, sample size; N prey, number of prey; TL, predator total length; TNW, total niche width; PSI, individual similarity index. Mean taxonomic richness per stomach is expressed in terms of number of prey families. leost fish to sea snakes (in New Caledonia and Eastern Shark Bay), marine turtles (Western Shark Bay), and crustaceans (Northern Hawaiian Island) (Table 4). Interestingly, some tiger sharks have been shown to travel long distances between abovementioned locations [START_REF] Lowe | Using acoustic telemetry monitoring techniques to quantify movement patterns and site fidelity of sharks and giant trevally around French Frigate shoals and Midway atoll[END_REF][START_REF] Heithaus | Long-term movements of tiger sharks satellite-tagged in Shark Bay, Western Australia[END_REF][START_REF] Werry | Reef-fidelity and migration of tiger sharks, Galeocerdo cuvier, across the Coral Sea[END_REF], and thus their diets differ between different locations. These large-scale movements between locations highlight the tiger shark's adaptability to local resource availability. Euryhaline bull sharks can switch between freshwater piscivorous diets and estuary-and marine-based diets [START_REF] Werry | Habitat ecology of the bull shark, Carcharhinus leucas, on urban coasts in eastern Queensland[END_REF]) and demonstrate opportunistic foraging strategies, inspecting boats and shore anglers to feed on bait or hooked fish (McCord and Lamberth 2009) or associating with scombrids to feed on anchovy [START_REF] Pinault | What appears to be a defensive behavior in the anchovy could also be a double-edged sword[END_REF].

Although the Shannon indices were quite comparable between the two shark species (2.25 for tiger sharks and 2.41 for bull sharks), tiger sharks consumed a wider range of prey (22 prey families belonging to eight major prey groups) than bull sharks (13 prey families belonging to three major prey groups). The higher number of prey families consumed by tiger sharks was shown to be independent from sample size (16 versus 30 stomachs containing food for bull and tiger sharks, respectively) because tiger shark's cumulative prey curve was above that for bull sharks for any given sampling effort. Therefore, the number of prey families potentially consumed by bull sharks (estimated at 15 prey families) is predicted to be less than the number of prey families potentially consumed by tiger sharks (estimated at 26 prey families). At the population level, tiger sharks from Fig. 3. Biplot representing individual carbon and nitrogen stable isotope ratios measured in (A) tiger sharks, bull sharks, and organic matter sources. POM and SOM are presented as the mean ± standard deviation; reef values were obtained from [START_REF] Kolasinski | Sources of particulate organic matter at the ecosystem scale: A stable isotope and trace element study in a tropical coral reef[END_REF]. (B) Shark muscle and (C) shark blood, illustrating the 95% kernel contour and standard ellipse area (SEA c ). Outliers were included in the kernel calculation but not in the SEA c representation.

Pagination not final (cite DOI) / Pagination provisoire (citer le DOI)

Reunion Island would have a larger dietary range than bull sharks in accordance with previous studies (Table 4).

The consumption of a wider range of prey coupled with the fact that C. leucas and G. cuvier did not feed on the same fish families (mostly Fistularidae and Diodontidae found in G. cuvier stomachs and Chanidae and Mullidae in C. leucas stomach) led to a clear trophic niche segregation of these two species, as demonstrated by the low overlap index (equal to 0.34) well below the 0.6 threshold of a significant overlap.

In limited coastal environments, tiger and bull sharks reduce the competition for resources

Both stomach contents and stable isotope tiger shark metrics had slightly higher values than bull shark metrics. This observation suggests a higher mobility of tiger sharks than bull sharks and is supported by evidence that tiger sharks undertake largescale offshore migration [START_REF] Lowe | Using acoustic telemetry monitoring techniques to quantify movement patterns and site fidelity of sharks and giant trevally around French Frigate shoals and Midway atoll[END_REF][START_REF] Heithaus | Long-term movements of tiger sharks satellite-tagged in Shark Bay, Western Australia[END_REF][START_REF] Werry | Reef-fidelity and migration of tiger sharks, Galeocerdo cuvier, across the Coral Sea[END_REF]) while bull sharks seem less mobile (but see [START_REF] Heupel | Conservation challenges of sharks with continental scale migrations[END_REF]. Tiger sharks appeared to exploit both a greater diversity of organic matter sources and trophic levels. Moreover, bull sharks exhibited higher ␦ 13 C values in both muscle and blood than tiger sharks. As carbon isotope values vary between pelagic and benthic organic matter sources, carbon isotopic ratios could be considered as a proxy of feeding distance to shore [START_REF] France | Carbon-13 enrichment in benthic compared to planktonic algae: foodweb implications[END_REF][START_REF] Cresson | Spatiotemporal variation of suspended and sedimentary organic matter quality in the Bay of Marseilles (NW Mediterranean) assessed by biochemical and isotopic analyses[END_REF]. The difference in ␦ 13 C values suggests a segregation of trophic habitats between the two species, with bull sharks depending more on neritic production. This trophic niche partitioning agreed with the higher presence rate of tagged bull sharks (30% ± 9%) than tiger sharks (3.2% ± 0.7%) tracked by the network of listening stations deployed along the west coast of Reunion Island [START_REF] Blaison | Seasonal variability of bull and tiger shark presence on the west coast of Reunion Island, western Indian Ocean[END_REF], as well as with published data (coastal-pelagic tiger sharks and coastal, estuarine, and riverine bull sharks; [START_REF] Werry | A multi-faceted approach for quantifying the estuarine-nearshore transition in the life cycle of the bull shark, Carcharhinus leucas[END_REF][START_REF] Lea | Repeated, long-distance migrations by a philopatric predator targeting highly contrasting ecosystems[END_REF].

Both populations exhibited ranges of isotope values comparable to the enrichment between a prey and its consumer (i.e., the difference between the individual presenting the lowest values and the individual presenting the highest values was about one trophic level: ϳ1‰ ␦ 13 C, ϳ2‰ ␦ 15 N; [START_REF] Hussey | ␦ 15 N and ␦ 13 C diet-tissue discrimination factors for large sharks under semi-controlled conditions[END_REF][START_REF] Olin | Variable ␦ 15 N diet-tissue discrimination factors among sharks: implications for trophic position, diet and food web models[END_REF].

However, both tiger and bull sharks analyzed in this particular study presented a narrower isotopic niche width than has been previously found [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF][START_REF] Daly | Comparative feeding ecology of bull sharks (Carcharhinus leucas) in the coastal waters of the southwest Indian Ocean inferred from stable isotope analysis[END_REF][START_REF] Heithaus | Apparent resource partitioning and trophic structure of large-bodied marine predators in a relatively pristine seagrass ecosystem[END_REF]; for instance, 18 bull sharks sampled in Mozambique had a ␦ 13 C range of 6‰ [START_REF] Daly | Comparative feeding ecology of bull sharks (Carcharhinus leucas) in the coastal waters of the southwest Indian Ocean inferred from stable isotope analysis[END_REF], whereas our study of 27 bull sharks sampled in Reunion Island had a ␦ 13 C range of only 2.5‰. Nevertheless, the extrapolation from the isotopic niche to the trophic niche depends on the isotopic range of food and nutrient sources [START_REF] Newsome | A niche for isotopic ecology[END_REF]). In fact, the ␦ 13 C values of both bull sharks and tiger sharks covered 30% and 63% of the range of organic matter sources, respectively, demonstrating a high diversity of resource use in the coastal ecosystems of Reunion Island. Thus, the ␦ 13 C range difference between that found in this study and the scientific literature is probably due to the lower diversity of organic matter sources around Reunion Island compared with the large continental shelves offshore of other locations, such as Mozambique, Australia, and Florida, which experience strong fluvial (containing freshwater and terrestrial) inputs.

Bull sharks do not exhibit the same foraging strategies over time

At the intraspecific level, bull shark ␦ 13 C values were influenced by individual fork length (longer individuals had more negative ␦ 13 C muscle values) and capture season (␦ 13 C blood values were greater between November and May than between April and October). The seasonal influence on ␦ 13 C may result either from a baseline seasonal variation or small-scale seasonal movement along the inshoreoffshore gradient or a baseline seasonal variation. The body length effect on ␦ 13 C suggested that smaller bull sharks derived their energy from resources that depend on coastal primary production, as was found in another empirical study from eastern Australia [START_REF] Werry | A multi-faceted approach for quantifying the estuarine-nearshore transition in the life cycle of the bull shark, Carcharhinus leucas[END_REF]. It is important to note, however, that this length effect was detected only in individuals longer than 205 cm (length at maturity is approximately 204 cm; [START_REF] Cruz-Martínez | Age and growth of the Bull Shark, Carcharhinus leucas, from southern Gulf of Mexico[END_REF]). Consequently, this result demonstrated that mature individuals are not homogeneous and that foraging shifts occur not only between immature and mature stages but also within the mature stage. Further sampling of juveniles and subadults would be necessary to explore ontogenetic trophic shifts and confirm this trend. The absence of length and seasonal effects on the blood and muscle isotope values for tiger sharks suggests that dietary shifts in this species are limited across adulthood and that individuals exploit the coastal environment of Reunion Island similarly all year round.

Individual specialization is higher in bull sharks compared with tiger sharks

Tiger and bull shark populations displayed almost identical isotopic niches (assessed by SEA c ) despite tiger sharks consuming a higher diversity of prey families. The two following hypotheses could explain this finding: (i) a greater isotopic redundancy of tiger shark prey (interpretable in terms of functional redundancy by extrapolation) and (or) (ii) a greater dietary similarity among tiger sharks than among bull sharks. It is not possible to delineate between these hypotheses without sampling prey for stable isotope analysis, but other clues derived from both stomach contents and stable isotopes indicated that there was a higher degree of similarity in prey items among tiger shark individuals than among bull shark individuals. The feeding niche of bull sharks was more equitably partitioned between individuals who specialized on different prey than that of tiger sharks, as suggested by a lower taxonomic richness per stomach coupled with higher equitability at the population scale. In a theoretical case, if each individual feeds on unique prey, with key differences from one individual to another, then pooling all individuals would result in a good equitability at the population level, with very low taxonomic diversity per stomach. Furthermore, the bull shark mean overlap between each individual and the population average diet (expressed as PSI) was significantly lower than in tiger sharks. This indicated that tiger shark individuals' diet was more similar to each other than diet within bull shark individuals, which were composed of heterogeneous individuals. Isotopically, smaller isotope nearest neighbor distances and a greater ratio of intertissue to interindividual difference (the IS index) also suggested a greater similarity between generalist tiger shark individuals [START_REF] Bolnick | Measuring individual-level resource specialization[END_REF]Layman et al. 2007a). Based on both stable isotopes and stomach contents, the IS based on two high turnover rates tissues (i.e., measuring diet stability between 1 year and several months before catch) to the PSI based on the individual's last meals were consistent. Dietary resources (in term of species number and biomass) are an important parameter affecting individual specialization; the higher the diversity of resources (number of prey species), the higher the individual predator specialization [START_REF] Araújo | The ecological causes of individual specialisation[END_REF]. Moreover, in the case of resource scarcity (low biomass), individuals who specialize on a narrow range of prey demonstrated higher foraging efficiency than those exploiting more diverse resources [START_REF] Bolnick | The ecology of individuals: incidence and implications of individual specialization[END_REF]. In Reunion Island, a high number of species and low biomass compared with other localities in the Indian Ocean [START_REF] Chabanet | Coral reef monitoring at Reunion island (Western Indian Ocean) using the GCRMN method[END_REF][START_REF] Chabanet | Coral reef monitoring in the Iles Eparses, Mozambique Channel (2011-2013[END_REF]Pinault et al. 2013) would be expected to promote individual specialization, especially for neritic bull sharks that forage chiefly on coastal fish in these small coastal ecosystems. This pattern of heterogeneous bull sharks and homogeneous tiger sharks in terms of diet has been observed in geographically distinct populations (Florida and Australia, respectively), which could thus be interpreted as being context-dependent [START_REF] Matich | Contrasting patterns of individual specialization and trophic coupling in two marine apex predators[END_REF]). Our results demonstrated different feeding patterns between two sympatric apex predator populations in a restricted habitat area, suggesting that difference in individual specialization is not only contextdependent but also species-dependent. Further assessment of bull and tiger shark relative abundance in the region is needed to estimate intensity of inter-and intraspecific competition and thus their importance in coastal ecosystems. Moreover, sampling potential prey for stable isotopes would help to both quantify long-term shark diet through mixing models and the trophic environment of the studied shark population.

Caution should be taken when interpreting the results of multitissue stable isotope analyses, as physiological (tissue-specific discrimination factor) and chemical (lipid and urea concentration) effects could lead to intertissue isotopic differences even with a constant diet (Hussey et al. 2012b). Indeed, trophic discrimination factors (i.e., differences between prey and consumer) depend on consumer species, diet quality, and tissue type [START_REF] Malpica-Cruz | Tissuespecific isotope trophic discrimination factors and turnover rates in a marine elasmobranch: empirical and modeling results[END_REF][START_REF] Caut | Diet-and tissuespecific incorporation of isotopes in the shark Scyliorhinus stellaris, a North Sea mesopredator[END_REF]. No controlled feeding experiments have ever been performed on tiger and bull sharks, and published studies on other species do not provide any applicable generalizations (such as correction factors between muscle and blood; [START_REF] Malpica-Cruz | Tissuespecific isotope trophic discrimination factors and turnover rates in a marine elasmobranch: empirical and modeling results[END_REF][START_REF] Caut | Diet-and tissuespecific incorporation of isotopes in the shark Scyliorhinus stellaris, a North Sea mesopredator[END_REF]. The trend that emerges from these studies suggests a reduction in ␦ 13 C blood-muscle values with treatment, with no effect on nitrogen isotopes, which would exacerbate the observed degree of individual dissimilarity without affecting differences between tiger shark and bull shark populations.

This study was the first to examine the trophic ecology of the sympatric tiger shark and bull shark populations in an isolated oceanic island using both stomach content and multitissue stable isotope analyses. The results of this study demonstrated the clear segregation in the foraging niches of these two apex predators, even for individuals caught in the same area during the same time period, underlining the existence of mechanisms that moderate trophic competition between the two shark species in coastal habitats with limited food resources. This further suggests that the two species have different functions in these coastal habitats and, as such, must be considered independently in terms of conservation and management. Further research on other coastal and oceanic predators (such as giant trevally (Caranx ignobilis) and yellowfin tuna (Thunnus albacares)), prey availability, and relative abundance of tiger and bull sharks in the region would help to better understand their ecological functions.

Fig. 1 .

 1 Fig. 1. Map of the fringing reef and particulate organic matter (POM) and sedimentary organic matter (SOM) sampling sites (solid stars) along Reunion Island's western leeward coast. Inside reef results (open star) were obtained from Kolasinski et al. (2011).

Fig. 2 .

 2 Fig. 2. Cumulative prey curves based on (A) stomach contents and (B) stable isotopes. The number of consumed prey families was used as the taxonomic richness. The dashed lines in panel A represent asymptotic model and associated asymptote ordinate.

Table 1 .

 1 Stomach content composition of bull sharks (Carcharhinus leucas) and tiger sharks (Galeocerdo cuvier) caught along the western coast of Reunion Island, expressed in terms of percentage of number (%N), mass (%W), and occurrence (%O) of prey categories.

	Bull sharks	Tiger sharks

Table 2 .

 2 Stomach-based trophic niche metrics.

		C. leucas	G. cuvier	Statistics
	N stomach	16	30
	N prey	56	148
	Mean TL (min.-max.)	227.1 (178-267)	279.8 (206-336)	U = 44.5; p = 0.0001
	Mean prey number per stomach	3.5±2.9	4.9±3.2	U = 168; p = 0.096
	Mean prey mass (g)	496.5±1170.4	618.8±1231.2	U = 2949; p = 0.33
	Mean stomach content mass (g)	1719.1±3528.1	2208.3±2681.4	U = 117; p = 0.17
	Taxonomic richness per stomach	1.8±1.5	2.4±1.1	U = 104; p = 0.043
	Prey family equitability	0.87	0.70
	TNW	2.41	2.25
	PSI	0.15±0.13	0.29±0.18

U = 90; p = 0.02

Table 3 .

 3 Metrics quantifying isotopic niche. N, number of samples; CR and NR, ranges of ␦ 13 C and ␦ 15 N values, respectively; TA, convex hull area; CD, mean distance to centroid; MNND, mean nearest neighbor distance; 95 K, 95% kernel area; SEA c , standard ellipse area. Letters indicate statistically significant differences by a Kruskal-Wallis test coupled with Mann-Whitney multiple comparisons.

		Tiger shark (G. cuvier)	Bull shark (C. leucas)
	Tissue	Muscle	Blood	Muscle	Blood
	N
	Mean fork length (min.-max.; cm) 333 (246-402) 333 (246-402) 268 (205-325) 272 (205-325)
	Mean ␦ 13 C (‰)	-16.91±0.68	-16.13±0.82	-15.79±0.55	-15.15±0.51
	CR (‰)	4.10	5.21	2.58	2.53
	Mean ␦ 15 N (‰)	11.96±0.71	12.97±0.62	12.54±0.60	12.69±0.57
	NR (‰)	2.96	2.47	2.2	2.04
	C/N	2.90±0.13a	2.84±0.26a	3.02±0.09b	2.97±0.18b
	TA (‰ 2 )	5.84	6.33	3.90	3.35
	CD (‰)	0.78±0.59a	0.84±0.72a	0.67±0.44a	0.60±0.42a
	MNND ± SD (‰)	0.16±0.17	0.20±0.27	0.27±0.34	0.24±0.28
	95 K (‰ 2 )	4.18	4.64	2.82	2.51
	SEA c (‰ 2 )	0.63	0.64	0.48	0.61

Note:

Table 4 .

 4 A global review of the dietary composition of tiger sharks (Galeocerdo cuvier) and bull sharks (Carcharhinus leucas) from previous studies expressed as a percentage of occurrence (%O) and number (%N).

				Prey item
								Land				Sea	Marine
	Location	Indicator N	Teleosts Cephalopods Birds Elasmobranchs	mammals Crustaceans Turtles Echinoderms	snakes	mammals Gastropods Reference
	Tiger sharks
	Southern	%O	NS = 198	58	13	23	22	19	33	11			5	Lowe et al. 1996
	Hawaiian
	Islands
	Northern	%O	NS = 57	46.8	14.4	53.7 27.9		64.2	26.7			17.4	Taylor and Naftel,
	Hawaiian												unpublished data, and
	Islands												De Crosta et al. 1984,
													cited in Lowe et al. 1996
	West Shark Bay %O	NS = 84	31	11	14.3 10.7		3.6	45.2		21.4	18.9	6	Simpfendorfer et al. 2001
	East Shark Bay %O	NS = 15	7	7	7	20			27		60	47	Heithaus 2001
	Queensland	%O	NS = 553	37.2	0.4	9.3			10.3	9.4		29.3	Simpfendorfer 1992
	New Caledonia %O	NS = 30	21.3	6.5	43.7 4.3		13.0	20.0		61.7	14.8	Rancurel and Intes 1982
	Brazil	%N	NS = 22	75.4	2	6.1 10.1		6.1				Bornatowski et al. 2014
	Reunion Island %O	NS = 30	90	47	33	10	6.7	10	10	3.3		This study
	Bull sharks
	Florida	%N	NS = 50; NP = 73 71			19		10				Snelson et al. 1984
	Costa Rica	%O	NS = 42	86			2	3	10	5		2	Tuma 1976
	Brazil	%N	NS = 41; NP = 46 89			11						Sadowsky 1971
	East Australia %O	NS = 485	52			9			3.4			Werry 2010
	South Africa	%O	-	71.1	5.8		57.5		5.3				7.7	Olin et al. 2013
	South Africa	%O	NS = 309	57.0	4.3	0.7 50.7	2.9	4.0	2.0			4.3	0.5	Cliff and Dudley 1991
	Reunion Island %O	NS = 16	94	31		13						This study
	Note: NS, number of stomachs; NP, number of prey.

Pagination not final (cite DOI) / Pagination provisoire (citer le DOI) Trystram et al.

Can. J. Fish. Aquat. Sci. Vol. 00, 0000 Published by NRC Research Press Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by Couperin on 09/14/16For personal use only.

Published by NRC Research PressCan. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by Couperin on 09/14/16For personal use only.

Acknowledgements

This study was carried out under the scientific program CHARC (Connaissances de l'écologie et de l'habitat de deux espèces de requins côtiers a `La Réunion) with financial support from the Commission of the European Communities (FEDER fund), the French Government, and the Regional Council of Reunion Island. This work was also supported in part by the Western Indian Ocean Marine Science Association (WIOMSA), under a MARG ii Grant (contract 4/2012). The views expressed herein are those of the authors and do not necessarily reflect the views of WIOMSA and Swedish International Development Cooperation Agency. CT was also supported by the Europe Social Fund through a PhD grant delivered by Regional Council of Reunion Island. The authors are grateful to the fishermen of Reunion Island, A. Blaison, B. Reche (veterinarian), D. Guyomard (Comité Régional des Pêches Maritimes et des Elevages Marins de La Réunion), and the numerous other individuals who participated in sample collection.


D Chablat
email: damien.chablat@irccyn.ec-nantes.frangeles@cim.mcgill.ca

J Angeles

STRATEGIES FOR THE DESIGN OF A SLIDE-O-CAM TRANSMISSION

The optimization of the pressure angle in a cam-follower transmission is reported in this paper. This transmission is based on Slide-o-Cam, a cam mechanism with multiple rollers mounted on a common translating follower. The design of Slide-o-Cam, a transmission intended to produce a sliding motion from a turning drive, or vice versa, was reported elsewhere. This transmission provides pure-rolling motion, thereby reducing the friction of rack-and-pinions and linear drives. The pressure angle is a suitable performance index for this transmission because it determines the amount of force transmitted to the load vs. that transmitted to the machine frame. Two alternative design strategies are studied, namely, (i) increase the number of lobes on each cam or (ii) increase the number of cams. This device is intended to replace the current ball-screws in Orthoglide, a three-DOF parallel robot for the production of translational motions, currently under development at Ecole Centrale de Nantes for machining applications.

Introduction

In robotics and mechatronics applications, whereby motion is controlled using a piece of software, the conversion of motion from rotational to translational is usually done by either ball screws or linear actuators. Of these alternatives, ball screws are gaining popularity, one of their drawbacks being the high number of moving parts that they comprise, for their functioning relies on a number of balls rolling on grooves machined on a shaft; one more drawback of ball screws is their low load-carrying capacity, stemming from the punctual form of contact by means of which loads are transmitted. Linear bearings solve these drawbacks to some extent, for they can be fabricated with roller bearings; however, these devices rely on a form of direct-drive motor, which makes them expensive to produce and to maintain.

A novel transmission, called Slide-o-Cam, was introduced in [START_REF] González-Palacios | The design of a novel pure-rolling transmission to convert rotational into translational motion[END_REF] as depicted in Fig. 1, to transform a rotation into a translation. Slide-o-Cam is composed of four major elements: (i) the frame, (ii) the cam, (iii) the follower and (iv) the rollers. The input axis on which the cam is mounted, the camshaft, is driven at a time-varying angular velocity, by an actuator under computer-control. Power is transmitted to the output, the translating follower, which is the roller-carrying slider, by means of pure-rolling contact between cam and roller. The roller, in turn, comprises two components, the pin and the bearing. The bearing is mounted at one end of the pin, while the other end is press-fit into the roller-carrying slider. Contact between cam and roller thus takes place at the outer surface of the bearing. The mechanism uses two conjugate cam-follower pairs, which alternately take over the motion transmission to ensure a positive action; the rollers are thus driven by the cams throughout a complete cycle. The main advantage of using a cam-follower mechanism instead of an alternative transmission to transform rotation into translation is that contact through a roller reduces friction, contact stress and wear.

This transmission, once fully optimized, will replace the three ball screws used by the Orthoglide prototype [START_REF] Chablat | Architecture Optimization of a 3-DOF Parallel Mechanism for Machining Applications, the Orthoglide[END_REF]. Orthoglide features three prismatic joints mounted orthogonaly, three identical legs and a mobile platform, which moves in Cartesian space with fixed orientation, as shown in Fig. 2. The three motors are SANYO DENKI (ref. P30B08075D) with a constant torque of 1.2 Nm from 0 to 3000 rpm. This property enables the mechanism to move throughout the workspace a 4-kg load with an acceleration of 17 ms -2 and a velocity of 1.3 ms -1 . Furthermore, the pitch is 50 mm per cam turn, while the minimum radius of the camshaft is 8.5 mm. A new arrangement of camshaft, rollers and follower is proposed to reduce the inertial load when more than two cams are needed.

Roller

Follower

Conjugate cams

Figure 1: Layout of Slide-o-Cam Figure 2: The Orthoglide Unlike Lampinen [START_REF] Lampinen | Cam shape optimisation by genetic algorithm[END_REF], who used a genetic algorithm, or Zhang and Shin [START_REF] Zhang | A Computational Approach to Profile Generation of Planar Cam Mechanisms[END_REF], who used what they called "the relative-motion method," where the relative velocity and acceleration of the the follower with respect to the cam is prescribed, we use a deterministic method that takes into account geometric and machining constraints [START_REF] Bouzakis | Computer-Aided Optimum Design and NC Milling of Planar Cam Mechanims[END_REF].

The pressure angle µ (or its complement, the transmission angle) is a key performance index of cam-follower transmissions. One definition of pressure angle is the acute angle between the direction of the output force and the direction of the velocity of the point where the output force is applied [START_REF] Shigley | Theory of Machines and Mechanisms[END_REF].

Moreover, unlike Carra, Garziera and Pellegrini [START_REF] Carra | Synthesis of cams with negative radius follower and evaluation of the pressure angles[END_REF], who used a negative radius-follower to reduce the pressure angle, we use a positive radius-follower that permits us to assemble several followers on the same roller-carrier. To optimize the pressure angle, two alternative design strategies are studied, namely, (i) increase the number of lobes on each cam or (ii) increase the number of cams. The relations defined in [START_REF] Renotte | The Design of a Novel Prismatic Drive for a Three-DOF Parallel-Kinematics Machine[END_REF] for cam profiles with one lobe are extended to several lobes.

Synthesis of the Planar Cam Mechanism

Let the x-y frame be fixed to the machine frame and the u-v frame be attached to the cam, as depicted in Fig. 3. O 1 is the origin of both frames, while O 2 is the center of the roller, and C is the contact point between cam and roller. The geometric parameters defining the cam mechanism, with n lobes, are illustrated in the same figure. The notation of this figure is based on the general notation introduced in [START_REF] González-Palacios | Cam Synthesis[END_REF] and complemented in [START_REF] Renotte | The Design of a Novel Prismatic Drive for a Three-DOF Parallel-Kinematics Machine[END_REF], namely,

• p, the pitch, i.e., the distance between the center of two rollers on the same side of the follower, in single-lobed cams (distance is p/n in n-lobbed cams);

• n, the number of lobes;

• e, the distance between the axis of the cam and the line of centers of the rollers;

• a 4 , the radius of the roller bearing, i.e., the radius of the roller;

• ψ, the angle of rotation of the cam, the input of the mechanism;

• s, the displacement of the follower, given by the the position of the center of the roller, namely, the output function of the mechanism;

• µ, the pressure angle;

• f , the force transmitted from cam to roller.

In our design, p is set to 50 mm, in order to meet the Orthoglide specifications.

The Input-Output Function

The above parameters as well as the contact surface on the cam are determined by the geometric relations dictated by the Aronhold-Kennedy Theorem in the plane [START_REF] Waldron | Kinematics, Dynamics, and Design of Machinery[END_REF]. When the cam makes a complete turn (∆ψ = 2π), the displacement of the roller is equal to p, i.e., the distance between n + 1 rollers on the same side of the rollercarrying slider (∆s = p). Furthermore, if we consider the home configuration of the roller, as depicted in Fig. 4, the roller is below the x-axis for ψ = 0, so that we have s(0) = -p/(2n). Hence, the expression for the input-output function s is

s(ψ) = p 2π ψ - p 2n (1)
The expressions for the first and second derivatives of s(ψ) with respect to ψ will be needed; these are readily derived, namely,

s ′ (ψ) = p 2π and s ′′ (ψ) = 0 (2)

Cam-Profile Determination

The cam profile is determined by the displacement of the contact point C around the cam. This contact between one lobe and one roller takes place within 0 ≤ ψ ≤ 2π/n. For this domain we find the profile of one lobe. The remaining n -1 lobes are found by rotation around O 1 . The Cartesian coordinates of C in the u-v frame take the form [START_REF] González-Palacios | Cam Synthesis[END_REF]

u c (ψ) = b 2 cos(ψ) + (b 3 -a 4 ) cos(δ -ψ) v c (ψ) = -b 2 sin(ψ) + (b 3 -a 4 ) sin(δ -ψ)
with coefficients b 2 , b 3 and δ given by

b 2 = -s ′ (ψ) sin α 1 (3a) b 3 = [e + s ′ (ψ) sin α 1 ] 2 + [s(ψ) sin α 1 ] 2 (3b) δ = arctan -s(ψ) sin α 1 e + s ′ (ψ) sin α 1 (3c)
where α 1 is the directed angle between the axis of the cam and the translating direction of the follower; α 1 is positive in the ccw direction. Considering the sign-convention adopted for the input angle ψ and for the output s, as depicted in Fig. 3, we have

α 1 = -π/2 (4)
We introduce now the non-dimensional design parameter η, which will be extensively used:

η = e p (5)
Thus, from Eqs.( 1), (2a), (3a-c), ( 4) and ( 5), we derive the expressions for coefficients b 2 , b 3 and δ:

b 2 = p 2π (6a) b 3 = p 2π (2πη -1) 2 + ψ - π n 2 (6b) δ = arctan nψ -π 2nπη -n (6c)
whence a first constraint on η, η = 1/(2π), is derived. With such a definition, however, the cam profile does not close. An extended angle ∆ is introduced [START_REF] Lee | Design for Manufacturability of Speed-Reduction Cam Mechanisms[END_REF], so that the cam profile closes. Angle ∆ is obtained as the root of the equation v c (ψ) = 0. In the case of Slide-o-Cam, ∆ is negative, as shown in Fig. 5 for a cam with one lob. The contact between the cam profile and one roller is made (resp. is lost) for ψ = ∆ or ψ = 2π -∆ as depicted in Fig. 5 (c&d).

O 1 O 1 O 1 O 1 x x x x y y y y u u v v v C C C ψ ψ -u -v (a) (b) (c) (d)

Pitch-curve Determination

The pitch curve is the trajectory of the center O 2 of the roller, distinct from the trajectory of the contact point C, which produces the cam profile. The Cartesian coordinates of points O 2 in the x-y frame are (e, s), as depicted in Fig. 3. Hence, the Cartesian coordinates of the pitch curve in the u-v frame are

O 1 O 1 O 1 O 1 x x x x y y y y y (a) (b) (c) (c)
u p (ψ) = e cos(ψ) + s(ψ) sin(ψ) (7a) v p (ψ) = -e sin(ψ) + s(ψ) cos(ψ) (7b)
Figure 6 shows a plot of the cam profile and its pitch curve, within ∆ ≤ ψ ≤ 2π/n -∆, throughout all n lobes.

Geometric Constraints on the Mechanism

In order to lead to a feasible mechanism, the radius a 4 of the roller must satisfy two conditions, as shown in Fig. 7:

• Two consecutive rollers on the same side of the roller-carrying slider must not interfere with each other. Since p/n is the distance between the center of two consecutive rollers, we have the constraint 2a 4 < p/n. Hence the first condition on a 4 :

a 4 < p 2n (8)
• The radius b of the shaft on which the cams are mounted must be taken into consideration. Hence, we have the constraint a 4 + b ≤ e , the second constraint on a 4 in terms of the parameter η thus being1

a 4 ≤ ηp -b (9)
Considering the initial configuration of the roller, as depicted in Fig. 4, the v-component of the Cartesian coordinate of the contact point C is negative in this configuration, i.e., v c (0) ≤ 0. However, from the expression for v c (ψ) and for parameters b 3 and δ given in Eqs. (6b & c), respectively, the above relation leads to the condition:

p 2πn (2nπη -n) 2 + π 2 -a 4 sin arctan -π 2nπη -n ≤ 0 (10)
Further, we define parameters A and B as

A = p 2nπ (2nπη -n) 2 + π 2 -a 4 and B = sin arctan -π 2nπη -n (11)
Now we derive a constraint A. Since (2πnη -n)2 > 0, we have

(2nπη -n) 2 + π 2 > π Hence, A > p 2n -a 4
Furthermore, from the constraint on a 4 established in Eq. ( 8), we have p/(2n) -a 4 > 0, whence A > 0.

Consequently, the constraint v c (0) ≤ 0 leads to the constraint B ≤ 0.

We transform now the expression for B by using the trigonometric relation:

∀x ∈ R, sin(arctan x) = x √ 1 + x 2
Hence, the constraint v c (0) ≤ 0 becomes

-π n(2πη -1) 1 + π 2 /(2nπη -n) 2 ≤ 0
which holds only if 2πη -1 > 0. Finally, the constraint v c (0) ≤ 0 leads to a constraint on η, namely 2 ,

η > 1 2π (12)
Equations 8, 9 and 12 permit us to reduce the design parameter space. Figure 8 depicts the design parameter space for P = 50mm and b = 4.25mm. We can notice that when the number of lobes by cam increases, the maximum value of a 4 decreases.

Pressure Angle

The pressure angle of cam-roller-follower mechanisms is defined as the angle between the common normal at the cam-roller contact point C and the velocity of C as a point of the follower [START_REF] López-Cajún | Optimization of Cam Mechanisms[END_REF], as depicted in Fig. 3, where the pressure angle is denoted by µ. This angle plays an important role in cam design. The smaller |µ|, the better the force transmission. In the case of high-speed operations, i.e., angular velocities of cams exceeding 50 rpm, the pressure-angle is recommended to lie within 30 • .

For the case at hand, the expression for the pressure-angle µ is given in [START_REF] López-Cajún | Optimization of Cam Mechanisms[END_REF] as

tan µ = s ′ (ψ) -e s(ψ)
Considering the expressions for s and s ′ , and using the parameter η given in Eqs.( 1), (2a) and ( 6), respectively, the expression for the pressure angle becomes

tan µ = n -2nπη nψ -π

Conjugate Cams

To reduce the pressure angle, several cams can be assembled in the same cam-shaft. We note m the number of cams and β the angle of rotation between two adjacent cams, i.e.,

β = 2π nm

On the Slide-o-Cam mechanism first designed in [START_REF] González-Palacios | The design of a novel pure-rolling transmission to convert rotational into translational motion[END_REF], two conjugate cams with one lobe each and β = π were used. Figure 9 shows two cam profiles with one and two lobes.

Convexity of the Cam Profile

The convexity of the cam influences the accuracy of the machining; for this reason it is preferable to have a convex cam profile for each part of the cam where it drives the roller. In this section, we establish conditions on the design parameters η and a 4 in order to have such a property. We will thus study the sign of the curvature of the cam profile via that of the pitch curve. The curvature of any planar parametric curve, in terms of the Cartesian coordinates u and v, and parameterized with any parameter ψ, is given in [START_REF] López-Cajún | Optimization of Cam Mechanisms[END_REF],

κ = v ′ (ψ)u ′′ (ψ) -u ′ (ψ)v ′′ (ψ) [u ′ (ψ) 2 + v ′ (ψ) 2 ] 3/2 ( 13
)
The sign of κ in Eq.( 13) tells whether the curve is convex or concave at a point: a positive κ implies a convexity, while a negative κ implies a concavity at that point. The computation of the first and second derivatives of the Cartesian coordinates of the pitch curve, defined in Eqs. [START_REF] Carra | Synthesis of cams with negative radius follower and evaluation of the pressure angles[END_REF], with respect to the angle of rotation of the cam, ψ, are

u ′ p (ψ) = ( s ′ (ψ) -e) sin(ψ) + s(ψ) cos(ψ) (14a) v ′ p (ψ) = ( s ′ (ψ) -e) cos(ψ) -s(ψ) sin(ψ) (14b) u ′′ p (ψ) = (2s ′ (ψ) -e) cos(ψ) -s(ψ) sin(ψ) (14c) v ′′ p (ψ) = -(2s ′ (ψ) -e) sin(ψ) -s(ψ) cos(ψ) (14d)
By substituting η = e/p, with Eqs.(14a-d) into Eq.( 13), the curvature κ p of the pitch curve can be obtained as

κ p = - 2n 2 π p [(nψ -π) 2 + 2n 2 (2πη -1)(πη -1)] [(nψ -π) 2 + n 2 (2πη -1) 2 ] 3/2 (15)
provided that the denominator never vanishes for any value of ψ, i.e., if we observe the condition:

η = 1/(2π)
Considering the expression for κ p in Eq.( 15), we have, for every value of ψ,

κ p ≥ 0 if (2πη -1)(πη -1) ≥ 0 and η = 1 2π
whence the condition on η:

κ p ≥ 0 if η ∈ [0, 1 2π [∪[ 1 π , +∞[ (16)
The condition on η given in Eq.( 16) must be combined with the condition established in Eq.( 12), η > 1/(2π). Hence, the final convexity condition of the pitch curve is: We are only interested in the interval of ψ where the cam drives the roller to go to the right; we call this the active interval. An other interval of ψ exists, with the same range, when the cam drives the roller to go to the left as is depicted in Fig. 10.

η ≥ 1 π (17
• A single-cam mechanism: The active interval is:

π n ≤ ψ ≤ 2π n -∆
Indeed, if we start the motion in the home configuration depicted in Fig. 4, with the cam rotating in the ccw direction, the cam begins to drive the roller only when ψ = π/n; the cam can drive the follower until contact is lost, i.e., when ψ = 2π/n -∆, as shown in Fig. 5c. With only one cam, the cam cannot drive the roller throughout one complete turn, as depicted in Fig. 10. This result holds for all values of n. Thus, a cam-follower assembly must have a least two conjugate cams. When -π/n + δ ≤ ψ ≤ 2π/n -δ, contact between two followers and the cam occurs. • A two-conjugate-cam mechanism: The active interval is:

π n -∆ ≤ ψ ≤ 2π n -∆
Indeed, the conjugate cam can also drive the follower when 0 ≤ ψ ≤ π/n -∆; there is, therefore, a common interval, for π/n ≤ ψ ≤ π/n -∆, during which two cams can drive the follower. In this interval, the conjugate cam can drive a roller with lower absolute values of the pressure angle. We assume that, when the two cams drive the rollers, the cam with the lower absolute value of pressure angle effectively drives the follower. Consequently, we are only interested in the value of the pressure angle in the interval π/n -∆ ≤ ψ ≤ 2π/n -∆, as depicted in Fig. 11.

• A three-conjugate-cam mechanism: For the same reason, the active interval is:

4π 3n -∆ ≤ ψ ≤ 2π n -∆
This interval is 33% smaller than a two-conjuage-cam-mechanism, the part of the interval removed having the highest pressure angle, as depicted in Fig. 12.

µ (degree) µ (degree) ψ(rad) ψ(rad)

Pressure Angle and Design Parameters

We study here the influence of design parameters η, a 4 and n on the values of the pressure angle, whith the cam driving the roller, as reported in [START_REF] Zhang | A parametric study of planar cam-roller speed reducers[END_REF]. We also study the influence of the number of conjugate cams on the value of pressure angle.

• Influence of parameter η

Figure 13 shows the influence of the parameter η on the pressure angle, with a 4 and p being fixed for several numbers of cams and lobes. From these plots we have the result:

The lower η, the lower the absolute value of the pressure angle, with η ≥ 1/π, as defined in Eq.17.

This result is the same if we consider a single cam with several lobes or a two-or three-conjugate-cam mechanism.

• Influence of the radius of the roller a 4 Parameter a 4 does not appear in the expression for the pressure angle, but it influences the value of the extended angle ∆, and hence the plot boundaries of the pressure angle, as shown in Fig. 14 for two value of a 4 .

By computing the value of the extended angle ∆ for several values of a 4 , we notice that the higher a 4 , the lower the absolute value of ∆. Consequently, since the boundaries to plot the pressure angle for a system with two conjugate cams are π/n -∆ and 2π/n -∆ or 4π/3n -∆ and 2π/n -∆ for three conjugate cams, we notice that when we increase a 4 , -∆ decreases or, equivalently, ∆ increases and the boundaries are translated toward the left, i.e., toward higher absolute values of the pressure angle. We thus have the result:

The lower a 4 , the lower the absolute value of the pressure angle. This result is valid for single or conjugate-cam mechanisms and is independent of the number of lobes.

• Influence of the number of lobes n By computing the value of the pressure angle for several values of n, as depicted in Fig. 13, we have the result:

The lower n, the lower the absolute value of the pressure angle.

For at least two lobes, contact is lost when the pressure angle is greater than 20 • .

• Influence of the Number of Conjugate Cams

Figures 11(a) and 12 show that the higher the number of cams, the lower the active interval. Especially, the number of cams reduces the maximum absolute value of the pressure angle. Therefore, The higher the number of conjugate cams, the lower the absolute value of the pressure angle. The optimum solution is found whenever (i) the number of lobes is one, (ii) for the higher number of conjugate cams, (iii) for the smallest value of the radius of the roller a 4 and (iv) for the smallest value of η. However, to minimize the deformation of the roller pin, a 4 have to be computed to resist to efforts during the motion as is made in [START_REF] Renotte | The Design of a Novel Prismatic Drive for a Three-DOF Parallel-Kinematics Machine[END_REF]. Further research is currently underway to evaluate the influence of variations in the design parameters on the pressure angle and the active interval and the design of cam with non-convex shape.

Figure 3 :Figure 4 :

 34 Figure 3: Parameterization of Slide-O-Cam

Figure 5 :

 5 Figure 5: Extended angle ∆ for different values of ψ, for a cam with one lobe: (a) ψ = π; (b) ψ = 2π; (c) ψ = 2π -∆; and (d) ψ = ∆ Consequently, the cam profile closes within ∆ ≤ ψ ≤ 2π/n -∆, with a corresponding relation for the remaining n -1 lobes.

Figure 6 : 4 Figure 7 :

 647 Figure 6: Cam profile (in red) and pitch curve (in blue), within ∆ ≤ ψ ≤ 2π/n -∆, with a 4 = 4, p = 50 and e = 9, for: (a) n = 2; (b) n = 3; (c) n = 4; and (d) n = 5.

Figure 8 :

 8 Figure 8: Design parameter space for P = 50mm and b = 4.25mm

Figure 9 :

 9 Figure 9: Assembly of two cams with p = 50, a 4 = 10 and e = 9: (a) one lobe; (b) two lobes

) 3

 3 Influence of the Number of Conjugate Cams and the Number of Lobes on the Pressure Angle 3.1 Active Angular Interval of the Pressure Angle

Figure 10 :

 10 Figure 10: Pressure-angle distribution for one cam with one lobe, with p = 50, a 4 = 10 and e = 9

Figure 11 : 9 µFigure 12 :

 11912 Figure 11: Pressure-angle distribution for two conjugate-cams with (a) one lobe and (b) two lobes, with p = 50, a 4 = 10 and e = 9

Figure 13 :Figure 14 :

 1314 Figure 13: Influence of parameter η on the pressure angle µ (in degree), with p = 50 mm and a 4 = 10 mm, for one or two lobes and one or two cams, for (a) one cam and one lobe; (b) two cams and one lobe; (c) one cam and two lobes; and (d) two cams and two lobes

It is possible to have a 4 = ηpb if the cams and the shaft are machined from the same block. Thus, there is a common point, referred to as B in Fig.7b, between the shaft and the cam profile. This will not be the solution chosen for our design, because it is too complicated to machine the cams and the shaft on the same block.

If in the initial configuration the roller were on the upper side of the x-axis, the input-output function would be s(ψ) = (p/2π)ψ + p/2, and we would have the constraint vc(0) ≥ 0, which would lead to the same constraint η > 1/(2π).