E Dortel

L Pecquerie

E Chassot

A Dynamic Energy Budget simulation approach to investigate the eco-physiological factors behind the two-stanza growth of yellowfin tuna (Thunnus albacares)

Keywords: DEB theory, Body-size scaling, Behavioral changes, Ontogeny

The growth of yellowfin tuna has been the subject of considerable research efforts since the early 1960s. Most studies support a complex two-stanza growth pattern with a sharp acceleration departing from the von Bertalanffy growth curve used for most fish populations. This growth pattern has been assumed to result from a combination of physiological, ecological and behavioral factors but the role and contribution of each of them have not been addressed yet. We developed a bioenergetic model for yellowfin tuna in the context of Dynamic Energy Budget theory to mechanistically represent the processes governing yellowfin tuna growth. Most parameters of the model were inferred from Pacific bluefin tuna using body-size scaling relationships while some essential parameters were estimated from biological data sets collected in the Indian Ocean. The model proved particularly suitable for reproducing the data collected during the Pacific yellowfin tuna farming experience conducted by the Inter-American Tropical Tuna Commission at the Achotines Laboratory in Panama.

In addition, model predictions appeared in agreement with knowledge of the biology and ecology

of wild yellowfin tuna. We used our model to explore through simulations two major assumptions that might explain the existence of growth stanzas observed in wild yellowfin tuna: (i) a lower food supply during juvenile stage in relation with high intra-and inter-species competition and (ii) ontogenetic changes in food diet. Our results show that both assumptions are plausible although none of them is self-sufficient to explain the intensity of growth acceleration observed in wild Indian Ocean yellowfin tuna, suggesting that the two factors may act in concert. Our study shows that the yellowfin growth pattern is likely due to behavioral changes triggered by the acquisition of physiological abilities and anatomical traits through ontogeny that result in a major change in intensity of schooling and in a shift in the biotic habitat and trophic ecology of this commercially important tuna species.

Introduction

Yellowfin tuna (Thunnus albacares; YFT) is an epipelagic species widely distributed in the tropical and subtropical waters of the world's major oceans [START_REF] Pecoraro | Putting all the pieces together: integrating current knowledge of the biology, ecol-ogy, fisheries status, stock structure and management of yellowfin tuna (Thunnus albacares)[END_REF]. Owing to its high value as food source, YFT supports important commercial and recreational fisheries. With an annual global catch of about 1.3 million tonnes in the last decade, it ranks among the world's ten most harvested marine species [START_REF]The state of world fisheries and aquaculture[END_REF]. At global scale, YFT has recently been assessed as Near Threatened following the Red list criteria of the International Union for Conservation of Nature [START_REF] Collette | Systematics of the tunas and mackerels (Scombridae). In: Tuna physiology, ecology and evolution[END_REF], while the assessments of the status of its four oceanic stocks conducted within the tuna Regional Fisheries Management Organizations indicate that overfishing currently occurs in the Indian Ocean [START_REF] Iotc | Report of the 21 st Session of the IOTC Scientific Committee. IOTC-2018-SC21-RE Rev1[END_REF].

For assessment and management purpose, growth of wild YFT has been the subject of considerable research efforts since the 1960s through analysis of mark-recapture data, size frequency distribution of commercial fishery catches and ageing from calcified and bony structures [START_REF] Schaefer | Tagging and recovery of tropical tunas[END_REF][START_REF] Diaz | An increment technique for estimating growth parameters of tropical tunas, as applied to yellowfin tuna (Thunnus albacares)[END_REF][START_REF] Wild | The relationship between otolith increments and time for yellowfin and skipjack tuna marked with tetracycline[END_REF]. YFT growth varies between ocean basins and regions and shows in some areas some complex pattern that departs from the traditional von Bertalanffy growth curve used for most fish stocks [START_REF] Pecoraro | Putting all the pieces together: integrating current knowledge of the biology, ecol-ogy, fisheries status, stock structure and management of yellowfin tuna (Thunnus albacares)[END_REF]. In particular, some growth studies conducted in the Atlantic, Indian and Western-Central Pacific Oceans support a two-stanza growth pattern with a sharp acceleration in growth rate at about 60-65 cm fork length [START_REF] Gascuel | Modélisation d'une croissance en deux stances chez l'albacore (Thunnus albacares) de l'Atlantique Est[END_REF][START_REF] Lehodey | Age and growth of yellowfin tuna (Thunnus albacares) from the western and central Pacific Ocean as indicated by daily growth increments and tagging data. 12th Meeting of the SCTB[END_REF][START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF][START_REF] Eveson | Estimating growth of tropical tunas in the Indian Ocean using tag-recapture data and otolith based age estimates[END_REF].

The complex growth pattern observed in some populations of YFT has been assumed to result from a combination of physiological, ecological and behavioral factors but the role and contribution of each of them have not been addressed yet [START_REF] Fonteneau | Croissance de l'albacore (Thunnus albacares) de l'Atlantique Est[END_REF][START_REF] Bard | Croissance de l'albacore (Thunnus albacares) Atlantique d'après les données des marquages[END_REF][START_REF] Gaertner | Observations sur la croissance de l'albacore (Thunnus albacares) dans l'Atlantique Ouest[END_REF][START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF]. A better understanding of these mechanisms is essential to assess the plasticity of YFT growth and eventually strengthen the scientific advice on the stock status and improve the overall quality of current stock assessments.

We investigated the factors governing the growth of YFT with a bioenergetic model that covers the different post-metamorphic life stages of the Indian Ocean YFT within the framework of the Dynamic Energy Budget (DEB) theory [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF]. DEB theory has already proven successful to study growth and reproduction of a large range of marine taxa [START_REF] Van Der Veer | The estimation of DEB parameters for various Northeast Atlantic bivalve species[END_REF][START_REF] Sousa Tânia | From empirical patterns to theory: a formal metabolic theory of life[END_REF][START_REF] Pecquerie | Modeling fish growth and reproduction in the context of the Dynamic Energy Budget theory to predict environmental impact on anchovy spawning duration[END_REF][START_REF] Freitas | Temperature tolerance and energetics: a dynamic energy budget-based comparison of North Atlantic marine species[END_REF] and more recently to study the dynamics of populations and ecosystems [START_REF] Maury | From individuals to populations to communities: A dynamic energy budget model of marine ecosystem size-spectrum including life history diversity[END_REF][START_REF] Pethybridge | Responses of European anchovy vital rates and population growth to environmental fluctuations: An individual-based modeling approach[END_REF][START_REF] Maury | Can schooling regulate marine populations and ecosystems?[END_REF]. Other bioenergetic approaches are available but they are generally specific to a particular life stage and unsuitable for modelling the changes between distinct life stages [START_REF] Hansen | Applications of bioenergetics models to fish ecology and management: Where do we go from here?[END_REF][START_REF] Ney | Bioenergetics modeling today: Growing pains on the cutting edge[END_REF]. In contrast, the DEB theory provides a powerful conceptual framework to describe the lifelong growth in relation to other physiological requirements, such as metabolism and body internal maintenance, development and reproduction, as a function of environmental food conditions and organism state [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF][START_REF] Freitas | Temperature tolerance and energetics: a dynamic energy budget-based comparison of North Atlantic marine species[END_REF].

Materials and methods

Data sources

We collected data sets from the literature for both wild and farmed YFT to compare the outputs of the bioenergetic model with morphometric, ageing and reproduction observations.

Wild Indian Ocean yellowfin tuna

For wild YFT, we considered a length-at-age data set collected throughout the Regional Tuna Tagging Project (RTTP), a large scale mark-recapture program conducted in the Western Indian Ocean between 2005-2007, and the West Sumatra Tuna Tagging Project (WSTTP), a tagging program conducted off the western Indonesian coast in August 2007 [START_REF]West Sumatera Tuna Tagging Project 2006-2007[END_REF][START_REF] Murua | The Indian Ocean Tuna Tagging Programme: Building better science for more sustainability[END_REF]. All lengths were measured with a measuring board to the nearest 0.1 cm in fork length (L F ), i.e. the projected straight distance measured from the tip of the upper jaw to the shortest caudal ray. Fish ages were estimated from readings of saggital otoliths collected from (i) 128 YFT caught through the RTTP and measuring between 43-72 cm L F at tagging and 47.9-135.4 cm L F at recapture, (ii) 18 fish caught through the WSTTP measuring between 19-29 cm L F , (iii) 35 fish, including 7 females, collected from 2008-2009 at the Indian Ocean Tuna Ltd. cannery and measuring between 31-147.5 cm L F [START_REF] Morize | Preliminary growth studies of yellowfin and bigeye tuna (Thunnus albacares and T. obesus) in the Indian Ocean by otolith analysis[END_REF][START_REF] Sardenne | Determining the age of tropical tunas in the Indian Ocean from otolith microstructures[END_REF].

Otoliths were prepared for age analysis and read at the "Laboratoire de Sclérochronologie des Animaux Aquatiques" (Brest, France) following the method described by Stéquert [START_REF] Stéquert | Détermination de l'âge des thons tropicaux à partir de leurs otolithes: exemple du Yellowfin (Thunnus albacares)[END_REF]. The number of otolith growth increments was counted repeatedly for different otolith sections [START_REF] Morize | Preliminary growth studies of yellowfin and bigeye tuna (Thunnus albacares and T. obesus) in the Indian Ocean by otolith analysis[END_REF][START_REF] Sardenne | Determining the age of tropical tunas in the Indian Ocean from otolith microstructures[END_REF]. The ages were estimated from these counts using an ageing error model that explicitly accounts for process and interpretation errors in otolith readings [START_REF] Dortel | Accounting for age uncertainty in growth modeling, the case study of yellowfin tuna Thunnus albacares of the Indian Ocean[END_REF].

We also considered length-weight measurements from defrosted YFT caught in the whole Indian Ocean basin. Fork length was measured with a caliper to the nearest 0.1 cm and total weight was measured with a scale to the nearest 0.1 kg [START_REF] Bodin | Ecological data for western Indian Ocean tuna[END_REF] Finally, we used a data set of batch fecundity, i.e. number of eggs released per batch, derived from gravimetric method applied to 40 YFT females in the size range 79-146.9 cm L F that were collected on board commercial purse seiners in the western Indian Ocean during 2009-2010 [START_REF] Zudaire | Reproductive potential of yellowfin tuna (Thunnus albacares) in the western Indian Ocean[END_REF]. The batch fecundity varied from 0.32 million to 6.91 million eggs with a mean of 3.07 million eggs.

Farmed Pacific yellowfin

We used data from YFT farming experiment conducted by the Inter-American Tropical Tuna Commission (IATTC) in Achotines Laboratory in the Republic of Panama [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. This experiment lasted over 3 years and involved 55 YFT (53% females, 46% males) caught in the coastal waters of the northwest Panama Bight, in 1996 and 1998, and then transferred in a broodstock YFT tank. The tank was filled with seawater and semi-open to ocean, thus the temperature and salinity in tank were influenced by the seasonal fluctuations in ocean upwelling and rainfall. During the experiment, daily temperatures in tank ranged from 20.1 to 29.7 • C with an average of 27.5 • C. YFT were fed once per day with one of five prey species whose caloric values were calculated based on metabolizable energy values of protein (17.7 kJ.g -1 ) and lipid (33.5 kJ.g -1 ). The length and wet weight of fish were measured at the time of the transfer in the tank and at their death. The food daily rations (F DR, % body wet weight day) are also available from the literature [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. They were calculated as the ratio between the wet weight of ingested food and the wet weight of an individual at each time step t of the simulation, i.e. 1 day. A food conversion ratio (F CR) was estimated as the ratio between the wet weight of ingested food on increasing biomass over given times intervals of 12-103 days for a constant number of YFT.

Description of the yellowfin tuna DEB model

State variables and dynamics

The yellowfin life cycle includes five different stages: embryo, larva, early juvenile, juvenile and adult. In this study, we implemented a bioenergetic model based on DEB theory [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF] that only describes three life stages of YFT: early juvenile, juvenile and adult. This model was focused on an individual fish defined by four state variables: reserve (E in J), structural length (L V in cm), maturation (E H in J) and reproduction (E R in J). Instead of the dynamics of the reserve E, we followed the dynamics of the scaled reserve density e which is driven by the dynamics of E. The scaled reserve density e = [E]/[E m ] is a dimensionless quantity where [E] = E/L 3 V is the reserve density and [E m ] represents the maximum reserve density.

The four state variables and the parameters used are described in the Table 2. The dynamics of the state variables (Figure 1) are controlled by various energy fluxes varying with size and body temperature (Table 3). The energy derived from ingested food is directly assimilated into the reserve ( ṗA ). From there, the energy is mobilized according to the κ-rule. A constant fraction (κ) is allocated to somatic maintenance ( ṗS ) and growth ( ṗG ), with priority for somatic maintenance.

The rest is allocated to maturity maintenance ( ṗJ ) and maturation or reproduction in adults ( ṗR ), with priority for maturity maintenance. Maturation explicitly refers to the fish development and the transitions from one stage to another occur for defined maturity thresholds, E e H from early juvenile to juvenile and E p H from juvenile to fully developed adult.

The beginning of the juvenile stage is marked by the acquisition of thermoconservation ability, a notable property of most tuna species compared to other fish species [START_REF] Brill | A review of temperature and oxygen tolerance studies of tunas pertinent to fisheries oceanography, movement models and stock assessments[END_REF]. From about 20 cm L F , YFT is able to raise the temperature in some parts of its body above the seawater temperature by the storage of the metabolic heat through vascular heat exchanger systems [START_REF] Wexler | Temperature and dissolved oxygen requirements for survival of yellowfin tuna, Thunnus albacares, larvae[END_REF][START_REF] Shadwick | Physiology of swimming and migration in tunas[END_REF]. We assumed that the thermoconservation ability of YFT involved energetic costs proportional to the square of structural length. The initiation of the thermoconservation ability occurs as a point in time once E H reaches E e H (Eq. D2 in Table 2). This assumption slightly simplifies the model compared to the DEB-based model developed for Pacific Bluefin tuna (Thunnus orientalis) which considered a gradual increase of the energetic costs associated with regional endothermy [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF].

Once at the adult stage, the maturity level does not increase any further and the fish stores energy in a reproduction buffer throughout the year. During a spawning season, the energy stored in the reproduction buffer is converted to eggs according to species-specific handling rules. Two spawning seasons are observed, a main season from November to March and a second minor period in June-July [START_REF] Stéquert | Gonadosomatic index and seasonal variations of plasma sex steroids in skipjack tuna (Katsuwonus pelamis) and yellowfin tuna (Thunnus albacares) from the Western Indian Ocean[END_REF][START_REF] Zudaire | Reproductive potential of yellowfin tuna (Thunnus albacares) in the western Indian Ocean[END_REF]. As observed for Pacific YFT, we considered that spawning is triggered when water temperature reaches 24 • C and ends above 28 • C [START_REF] Itano | The reproductively biology of yellowfin tuna (Thunnus albacares) in Hawaiian waters and the Western Tropical Pacific Ocean: Project Summary[END_REF][START_REF] Margulies | Spawning and early development of captive yellowfin tuna (Thunnus albacares)[END_REF]. Spawning was considered to occur almost daily from December-March and June-July and for seawater temperatures between 24 • C and 28 • C when the reproduction buffer contains sufficient energy to produce a batch.

Consistent with the multiple batch spawning behaviour of YFT, the energy stored in reproduction buffer is released through several batches with an increasing energy requirement per batch as fish grows. The energy required to produce one batch (E B in J) includes the energy content of eggs and energy losses to convert the energy of reproduction buffer into yolk material:

E B = [E B ]L 3 V κ R (1)
where [E B ] is the energy density of one batch (J.cm -3 ). The number of eggs released per batch (N B ) can be estimated from the energy requirement per batch and the energy content of one egg (E 0 ) following:

N B = E B E 0 (2)
where E 0 was estimated to have a mean value of 1.295 J based on the mean dry weight of an egg of captive yellowfin (about 0.043 mg; [START_REF] Margulies | Spawning and early development of captive yellowfin tuna (Thunnus albacares)[END_REF]) and the lipid contents of the eggs of wild Atlantic bluefin tuna (Thunnus thynnus) and bonito (Sarda sarda) estimated to be 7.953 J.mg -1 , i.e. 26.4% of the total caloric value of a freshly spawned egg or approximately 20% of its dry mass [START_REF] Ortega | Comparison of the lipid profiles from wild caught eggs and unfed larvae of two scombroid fish: northern bluefin tuna (Thunnus thynnus L., 1758) and Atlantic bonito (Sarda sarda Bloch, 1793)[END_REF].

Forcing variables

Consistent with DEB theory, two forcing variables were considered, the body temperature T (K)

and the food density X (J.m -3 ). Body temperature affects all the metabolic fluxes including the maximum assimilation { ṗAm }, the somatic maintenance rates [ ṗM ] and { ṗT } as well as v and kJ .

The values of those rates fluctuate within a species-specific tolerance range as described by the Arrhenius relationship. Let k(T ), the value of a metabolic rate k at any temperature T (K), T A the species-specific Arrhenius temperature (K) and T 1 a chosen reference temperature (K):

k(T ) = k(T 1 ) exp T A T 1 - T A T (3)
The food density is converted into ingestion according to a Holling type II scaled functional response, a dimensionless quantity defined as:

f = X X + X k (4)
X k , the half-saturation constant, is the food density for which the ingestion rate is half its maximum value [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF].

Link between state variables and fishery observations

The DEB state variables cannot be measured directly, only indirectly with additional assumptions to relate these state variables with the observations such as length, weight and fecundity. The structural length L V is related to the physical length expressed in fork length L F (cm) using the species-specific shape coefficient δ j :

L F = L V δ j (5)
The wet weight W W (g) has contributions from structure W V , reserve W E and also from reproduction buffer for adults W E R and it expresses as follows:

W W = W V + W E + W E R = d V L 3 V + E + E R ρ E (6)
where d V is the density of the structural volume (g.cm -3 ) and ρ E is the energy content per gram of reserve or reproduction buffer.

We also define the reproductive investment I R as the ratio between the weight of reproduction buffer and the wet weight of fish. This latter provides information on the weight losses due to the spawning activity.

I R = 100 × W E R W W = 100 × E R d V ρ E L 3 V + E + E R (7)

Model calibration

Body-size scaling relationships provide rules for how primary parameters vary among phylogenetically related species [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF]. The intensive parameters are related to biochemical processes and they are assumed roughly invariant among related species, while the extensive parameters ({ ṗAm } and maturity thresholds) depend on maximum size [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF][START_REF] Nisbet | Integrating dynamic energy budget (DEB) theory with traditional bioenergetic models[END_REF]. Thus, the extensive parameters can be directly adjusted using a zoom factor z defined as the ratio of the maximum structural length of species of interest to that of the reference species.

Our model is very close to the model developed for Pacific Bluefin Tuna (PBT) [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF], with a slight difference in the energetic process related to the acquisition of thermoconservation ability. In both models, this acquisition resulted in an additional energy cost which gradually increases from metamorphosis until the end of the early juvenile stage for Jusup et al. [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF] while this additional cost appears at the end of the early juvenile stage in our model. Thus, the parameters of adult PBT were used as a first approximation of primary parameters. First, the body-size scaling relationships were used to infer the values of the surface-area specific maximum assimilation rate { ṗAm } and the maturity thresholds E j H , E e H and E p H from those of PBT. Under constant food and temperature conditions, the maximum structural length can be related to the physical asymptotic length L F∞ . Thus, the maximum structural length of YFT was estimated from an asymptotic length fixed to 200 cm L F as follows:

L m = δ j L F∞ f (8)
Then the shape parameter δ j , the structural volume density d V and the weight-energy coupler ρ E , can be estimated from the length-weight relationship of juveniles. As juveniles do not store energy for reproduction, the wet weight (W W ) can be expressed from the physical length and the reserve density [E] = E/L 3 V (J.cm -3 ) as follows:

W W = δ 3 j L 3 F d V + [E] ρ E (9)
This length-weight relationship was fitted to data of wild immature YFT including 87 fish measuring 29-61 cm and weighing 0.39-4.2 kg in a Bayesian framework. Without prior knowledge, an uninformative inverse gamma distributions was assigned to [E]. d V was assigned a normal distribution centered on 1. δ j was assumed to vary according to a beta prior distribution between 0 and 1. ρ E was fixed such that the ratio of ρ E to δ j was the same as in Jusup et al. [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF]. The parameters estimate was based on sampling approach of Markov Chain Monte Carlo simulations as implemented in OpenBugs version 3.2.1 [START_REF] Spiegelhalter | OpenBUGS version 3.2.1 user manual[END_REF]. Three MCMC chains of 100,000 samples, thinned to one draw every 1000 th sample, were used to estimate the parameters. The convergence to the stationary posterior distribution was assessed by the Gelman-Rubin diagnostic from the second half of MCMC simulation sample [START_REF] Gelman | Inference from iterative simulation using multiple sequences[END_REF].

The Arrhenius temperature T A was deduced from the value of the van't Hoff coefficient Q 10 estimated for juveniles Pacific YFT, i.e. 1.65 at 18-24 • C and 1.67 at 24-30 • C [START_REF] Dewar | Studies of tropical tuna swimming performance in a large water tunnel -Energetics[END_REF]. The Q 10 is a correcting factor applied to metabolic rates for every 10 • C increase in temperature. For any body temperature T , Q 10 and T A are related according to the following relationship:

Q 10 = exp 10T A T (T + 10) (10)
The maturity thresholds, E j H , E e H and E p H were adjusted such that the corresponding physical lengths are consistent with available knowledge from literature. The metamorphosis of YFT occurs at about 13 mm standard length (S L ) [START_REF] Kaji | Growth and morphological development of laboratory-reared yellowfin tuna Thunnus albacares larvae and early juveniles, with special emphasis on the digestive system[END_REF], the end of early juvenile stage corresponds to a size of 20 cm L F [START_REF] Wexler | Temperature and dissolved oxygen requirements for survival of yellowfin tuna, Thunnus albacares, larvae[END_REF] and the puberty to a size close to 75 cm L F [START_REF] Zudaire | Reproductive potential of yellowfin tuna (Thunnus albacares) in the western Indian Ocean[END_REF].

Finally, the energy density for a single batch [E B ] was estimated from the relationship between the fork length and the batch fecundity observed for wild YFT.

Model validation using simulation

Different simulations were performed to assess the model suitability. Firstly, the model was used to simulate the growth in length, the length-weight relationship, the food daily ration (F DR) and the food conversion ratio (F CR). The simulation outputs were then compared to the observations described in the study of Wexler et al. [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. The F DR (% body wet weight per day) was calculated as the ratio between wet weight of ingested food W X and the wet weight of an individual W W at each time step of the simulation, i.e. 1 day. At any time t:

F DR = 100 × W Xt W Wt (11)
The F CR was calculated as the ratio between the wet weight of ingested food and increased wet weight of an individual over time interval of 1 day ignoring the weight of the reproduction buffer:

F CR = W Xt-1 (W vt + W Et ) -(W vt-1 + W Et-1 ) (12)
The wet weight of ingested food at time t was calculated using the following equation:

W Xt = f { ṗAm }L 2 vt κ X ρ X ( 13
)
ρ X is the energy content of 1 g of food estimated from the caloric value of prey given in Wexler et al. [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. As the prey provided to farmed YFT for a given day was not specified, three values illustrating the range of possible levels in terms of energy were used for ρ X . These values corresponded to the most energetic species (6471 J.g -1 ), the less energetic species (2980 J.g -1 ) and the mean (4910 J.g -1 ).

This first simulation was performed for a constant food density expressed through a scaled functional response f = 0.925 and for body temperatures considered equivalent to the daily temperatures in the experimental tank [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. The age and length at the experiment start were considered as initial conditions, i.e. an age of 395 days and a length of 53 cm L F from which we deduced a structural length L V = 13.53 cm and a maturity level E H = 2, 460, 526 J.

The seasonal reproductive investment (I R , Eq. 7) was explored with another simulation. This latter simulation was run under constant body temperature of 27 • C and constant food density (f = 0.925) from metamorphosis to fish death. The initial state of the system agreed with the fish state at metamorphosis, i.e. a maturity level E j H , age of 30 days and fork length of 1.3 cm from which we deduced a structural length of 0.33 cm.

Both simulations were performed with a 1-day time step using the dde function, a numerical solver for delay differential equations implemented in the PBSddesolve package [START_REF] Couture-Beil | PBSddesolve: solver for delay differential equations[END_REF] of the R software [START_REF]R: A Language and Environment for Statistical Computing[END_REF].

Investigating eco-physiological factors behind the growth stanzas

In this section, we investigated through simulations two main assumptions that could explain the growth stanzas observed in wild YFT: (i) a decrease of competition for food resource due to change in schooling behavior and (ii) a change in food diet due to habitat change in relation with the improvement of some physiological traits. For both assumptions, the growth in length was simulated with a 1-day time step, under constant temperature of 27 • C from metamorphosis to fish death. The average annual growth rates by 10 cm size-class, from 20 to 90 cm, were computed and compared to those of wild Indian Ocean YFT as estimated by Dortel et al. [START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF] with otolith data and otolith and length-frequency data. The relative root mean square error (RMSE), a normalized indicator that measures the discrepancy between the average annual growth rates simulated (r * ) and estimated for wild Indian Ocean YFT (r) was used:

RM SE = (r * -r) 2 r * (14)

Competition for food resource

Juveniles of YFT form mixed schools with juveniles of bigeye and skipjack under floating objects such as the fish aggregating devices (FADs) deployed in purse seine fisheries to increase fishing success [START_REF] Fonteneau | Global spatio-temporal patterns in tropical tuna purse seine fisheries on drifting fish aggregating devices (DFADs): Taking a historical perspective to inform current challenges[END_REF]. Within the schools, the inter-and intra-species competition for food resource is assumed to be strong which could result in lower food intake by young YFT as compared to adults [START_REF] Potier | Preliminary results of tuna diet studies in the west equatorial Indian Ocean[END_REF][START_REF] Hallier | Drifting fish aggregation devices could act as an ecological trap for tropical tuna species[END_REF][START_REF] Sardenne | Trophic niches of sympatric tropical tuna in the Western Indian Ocean inferred by stable isotopes and neutral fatty acids[END_REF]. In our simulations, this was expressed by a lower value of food density modeled through a lower scaled functional response below 70 cm L F which corresponds to the size at which YFT leave the schools associated with floating objects [START_REF] Fonteneau | Global spatio-temporal patterns in tropical tuna purse seine fisheries on drifting fish aggregating devices (DFADs): Taking a historical perspective to inform current challenges[END_REF]. Two functional responses, f j and f p for juvenile and adult stages, respectively, were defined such that f j < f p and the effect of an abrupt transition from f j to f p was investigated. Four values of f j were tested, 0.8, 0.825, 0.85 and 0.875, for a value of f p fixed to 0.925.

Food diet change

In the Indian Ocean, YFT feed on a wide variety of epipelagic and mesopelagic fish, crustaceans and cephalopods and their diet varies according to size and foraging depth [START_REF] Maldeniya | Food consumption of yellowfin tuna, Thunnus albacares, in Sri Lankan waters[END_REF][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][START_REF] Zudaire | Variations in the diet and stable isotope ratios during the ovarian development of female yellowfin tuna (Thunnus albacares) in the Western Indian Ocean[END_REF]. Adult YFT are dispersed over a wider geographical area covering deeper and colder waters and, contrary to juveniles, they can dive below the thermocline to forage [START_REF] Schaefer | Movements, behavior, and habitat utilization of yellowfin tuna (Thunnus albacares) in the Pacific Ocean off Baja California, Mexico, determined from archival tag data analyses, including unscented Kalman filtering[END_REF]. Thus, juveniles might have access to a range of prey which might be not as nutritious as for adults. This was expressed in our simulations through a lower surface-area specific maximum assimilation rate for YFT smaller than 60 cm L F , the minimum size of fish exhibiting deep dives [START_REF] Schaefer | Movements, behavior, and habitat utilization of yellowfin tuna (Thunnus albacares) in the Pacific Ocean off Baja California, Mexico, determined from archival tag data analyses, including unscented Kalman filtering[END_REF]. Two assimilation rates, { ṗAm } j and { ṗAm } p , were defined for juveniles ans adults respectively such that { ṗAm } j < { ṗAm } p and the effect on growth of an abrupt transition from { ṗAm } j to { ṗAm } p was investigated. We considered two values of { ṗAm } j , 3900 and 4000 J.day -1 .cm -2 , for a value of { ṗAm } p fixed to 4370 J.day -1 .cm -2 .

Results

Validation of a reference DEB model for yellowfin tuna

The reference DEB model successfully reproduced the growth in length and the length-weight relationship observed for farmed Pacific YFT as described in the study of Wexler et al. [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]. Furthermore, the observed values of F DR and F CR were well within the simulated range (Figure 2).

Thus, the set of parameters, some of which were derived from PBT [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF], appeared suitable to assess the effects of variations in the quantity and quality of food on YFT growth (Table 1).

From the length-weight relationship for immature YFT, the value of d V was estimated to be significantly higher than 1 (Table 4). This is in agreement with the observation that tuna have a higher density than seawater [START_REF] Magnuson | Comparative study of adaptations for continuous swimming and hydrostatic equilibrum of Scombroid and Xiphoid fishes[END_REF]. The value of ρ E was found to be lower than for PBT, reflecting a lower energetic value for YFT due to a lower mass of red muscle.

From the Q 10 values observed for juvenile YFT in Pacific Ocean, the Arrhenius temperature was estimated at 4622 ± 207 K. This value was lower than that of PBT estimated at about 5298 K, which could indicate that YFT are less sensitive to temperature variations within their temperature tolerance range, although more temperature data would be required to validate this assumption.

Yellowfin growth according to the reference DEB model

For a constant temperature of 27 • C and a constant food density f=0.925, the reference DEB model forecasted particularly fast growth until the end of the early juvenile stage. Then, it predicted a Von Bertalanffy type growth. Regarding wild Indian YFT, the model appeared suitable for adult growth but overestimated juvenile growth (Figure 3.A). The puberty was reached at a length of abour 75 cm corresponding to an age of 1.4 years. The evolution of weight relative to length was higher than expected, which may indicate a too high level of food density (Figure 3.B).

The energy stored in the reproduction buffer was also converted in weight to determine the seasonal reproductive investment. This latter increased with fish size up to 27.8% of total body wet weight for fish of 89 kg (150 cm L F ) and then slightly decreased to 25.9% of body wet weight (Figure 4).

Effect of food density

A decrease in food supply during the juvenile stage resulted in lower growth rates, from 20% to 60% depending on the f values. For a value between 0.825 and especially 0.85, the model predicted a two-stanza growth which better captured the juvenile growth, leading to 52-71% and 81-87%

decrease in RMSE values between 30-70 cm compared to the reference model (Table 5). But as shown in Figure 5.A, the model underestimated the magnitude of growth acceleration. The age at puberty increased at 1.83, 2.11, 2.5 and 3.14 years for a decrease in f from 0.875 to 0.8, while the fork length at puberty remained almost constant and varied around 74.9 cm.

Effect of prey quality

A reduced energy intake in young YFT led to an average decrease in the growth rate of 37% and 29% for { ṗAm } j values of 3900 and 4000 J.day -1 .cm -2 respectively. For both values, the model was able to capture the juvenile growth, as shown by the 66-72% and 86-87% decrease in RMSE values between 30-60 cm (Table 5), but it underestimated the magnitude of growth acceleration (Figure 5.B). The predicted age and fork length at puberty were 2.2 years for 74.79 cm and 1.96 years for 74.77 cm for { ṗAm } j values of 3900 and 4000 J.day -1 .cm -2 respectively.

Discussion

We formulated a bioenergetic model covering the juvenile and adult stages of YFT in the context of Dynamic Energy Budget theory. Some essential parameters of the model were estimated with Bayesian analysis from morphometric, age, and reproduction data sets collected from YFT sampled in the Indian Ocean while other parameters were derived from the DEB model developed for PBT [START_REF] Jusup | A full lifecycle bioenergetic model for bluefin tuna[END_REF]. Our results show that both lower food availability in the juvenile stage and ontogenetic changes in food diet characterized by the consumption of more energetic prey in older fish can explain the complex two-stanza growth observed in wild but not captive YFT. Both assumptions could explain the growth acceleration observed in wild YFT but none of them was self-sufficient to reflect the level of change observed. We thus assumed that the observed acceleration resulted from a combination of both processes.

Yellowfin DEB model

DEB theory allows to describe continuously growth and reproduction over the organism lifespan [START_REF] Nisbet | Integrating dynamic energy budget (DEB) theory with traditional bioenergetic models[END_REF]. Hence, DEB theory appeared as a particularly suitable framework for the study of YFT growth throughout different life stages. However, the experimental data currently available on YFT physiology are insufficient for estimating a comprehensive set of parameters. It was however possible to benefit from the work on PBT through the use of the body-size scaling relationships [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF][START_REF] Pecquerie | Analyzing variations in lifehistory traits of Pacific salmon in the context of Dynamic Energy Budget (DEB) theory[END_REF]. Under these relationships, YFT and PBT have similar metabolic and physiological processes and the differences between the two species would mainly result to lower energy assimilation for YFT. This assumption may not be quite accurate. Indeed, bluefin tuna have greater abilities for conservation of metabolic heat due to more developed vascular heat exchanger systems and a higher proportion of red muscle resulting in a more extensive thermal-niche than YFT [START_REF] Blank | Influence of swimming speed on metabolic rates of juvenile pacific bluefin tuna and yellowfin tuna[END_REF][START_REF] Shadwick | Physiology of swimming and migration in tunas[END_REF]. As the internal heat production is scaled with the surface, the surface area-specific maintenance costs, { ṗT }, are probably lower for YFT. Despite that, the set of parameters approximated from primary parameters of PBT has proved particularly suitable for reproducing the experiments on captive YFT and the model predictions appeared in agreement with the knowledge of the species biology.

In constant conditions of temperature and food, the reference model forecasted particularly fast initial growth until the end of the early juvenile stage. Then, as was expected for farmed fish, it complied with a Von Bertalanffy growth, with relatively high growth rates for juveniles and young adults which gradually slowed as asymptotic length approaches. The warming ability of tissues from retention of metabolic heat appears toward the end of the early juvenile stage [START_REF] Shadwick | Physiology of swimming and migration in tunas[END_REF]. According to our assumption, this ability is associated with an additional somatic maintenance cost and it led to a change in the energy balance resulting in a growth slowdown.

Regarding wild YFT, the growth pattern predicted by the reference model was consistent with the adult growth over 90 cm L F but it overestimated growth in juvenile and pre-adult stages.

Therefore, the puberty was reached at a length of about 75 cm L F , consistent with observations of length at maturity in the Indian Ocean [75 -90 cm 62, 79], but at 1.4 years, an earlier age than expected by the growth model of Indian Ocean YFT [2.2-2.5 years 11]. By contrast, the predicted length-weight relationship appeared suitable for juveniles, but the model overestimated the weight gain over 100 cm L F , suggesting an excessive weight contribution of the reproduction buffer.

Growth and population dynamics

A comparison of the biological parameters used in population dynamics models shows major differences in growth pattern and rates between the four world stocks of YFT [START_REF] Pecoraro | Putting all the pieces together: integrating current knowledge of the biology, ecol-ogy, fisheries status, stock structure and management of yellowfin tuna (Thunnus albacares)[END_REF]. The plasticity in tuna growth has generally been attributed to differences in oceanographic parameters (e.g. sea temperature) and prey availability [e.g. 12] and the environment and oceanic food webs that support YFT vary substantially across oceans and regions [START_REF] Pethybridge | A global meta-analysis of marine predator nitrogen stable isotopes: Relationships between trophic structure and environmental conditions[END_REF]. The availability of data sources and choice of estimation method may, however, also modify the growth models and affect our perception of tuna growth [START_REF] Kolody | Modelling growth in tuna RFMO stock assessments: Current approaches and challenges[END_REF].

In the Indian Ocean, historical and recent studies based on multiple data sources support a marked two-stanza growth pattern with a sharp acceleration in growth rate at about 60-65 cm fork length [START_REF] Marsac | Preliminary study of the growth of yellowfin estimated from purse seine data in the Western Indian Ocean[END_REF][START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF][START_REF] Eveson | Estimating growth of tropical tunas in the Indian Ocean using tag-recapture data and otolith based age estimates[END_REF]. In the Atlantic Ocean, the two-stanza growth model derived in the early 1990s from length-frequency data and used for several assessments of the stock status of YFT [START_REF] Gascuel | Modélisation d'une croissance en deux stances chez l'albacore (Thunnus albacares) de l'Atlantique Est[END_REF] has been challenged by direct age estimates from counts of daily growth increments in sagittal otoliths that support a von Bertalanffy growth curve [START_REF] Shuford | Age and growth of yellowfin tuna in the Atlantic Ocean[END_REF]. Results from otolith reading however appear inconsistent with the maximum length of Atlantic YFT and growth rates derived from past tagging programs conducted by the International Commission for the Conservation of Atlantic Tunas, calling for further studies integrating information from multiple data sources in this ocean [START_REF] Fonteneau | An overview of yellowfin tuna growth in the Atlantic ocean: Von Bertalanffy or multistanza growth?[END_REF]. Recent mark-recapture data collected in the Atlantic Ocean indicate that tagged fish released with lengths of less than 65 cm tend to grow slower than expected from a von Bertalanffy model while fish above 65 cm tend to grow faster, a pattern modelled with a Richards growth curve in the most recent assessment of Atlantic YFT [START_REF]Report of the 2019 ICCAT yellowfin tuna stock assessment meeting. IC-CAT, Grand-Bassam, Cote d'Ivoire[END_REF]. As part of the Atlantic Ocean Tropical Tuna Tagging Program, some work is ongoing to develop an integrated growth model accounting for the new data set collected on YFT growth.

Tropical tuna growth is the major focus of current research in the Pacific Ocean [START_REF] Farley | Progress on yellowfin tuna age and growth in the WCPO[END_REF]. In the Western-Central Pacific Ocean, the growth model used for assessing the stock of YFT predicts a growth rate slowing between about 40 and 70 cm, similarly to that observed in the Indian Ocean [START_REF] Lehodey | Age and growth of yellowfin tuna (Thunnus albacares) from the western and central Pacific Ocean as indicated by daily growth increments and tagging data. 12th Meeting of the SCTB[END_REF][START_REF] Tremblay-Boyer | Stock assessment of yellowfin tuna in the western and central Pacific Ocean[END_REF].

Changes in configuration of the YFT population model also suggest some regional differences in estimated growth rates consistently with direct age estimates and mark-recapture experiments [START_REF] Tremblay-Boyer | Stock assessment of yellowfin tuna in the western and central Pacific Ocean[END_REF].

Preliminary results from annual increment counts in otoliths suggest that the degree of two-stanza might however be lower than previously considered [START_REF] Farley | Progress on yellowfin tuna age and growth in the WCPO (Project 82)[END_REF]. In the Eastern Pacific Ocean, the stock of YFT shows a more classical growth pattern than that observed in the other ocean basins [START_REF] Minte-Vera | Yellowfin tuna in the eastern Pacific Ocean, 2019: Benchmark assessment[END_REF].

The ageing technique based on daily increment counts for this stock has however recently been challenged [START_REF] Farley | Progress on yellowfin tuna age and growth in the WCPO (Project 82)[END_REF]. Ongoing research focusing on annual increments in otoliths is expected to shed the light on the growth of the two stocks of Pacific YFT.

Behavioral changes behind the two-stanza growth

According to the experiment of Wexler et al. [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF], farmed YFT did not exhibit the growth acceleration occurring around 65 cm L F for wild fish. Similarly to the experiment of Achotines, the growth pattern of YFT held in captivity for several years at the Tuna Research and Conservation Center (TRCC) in Pacific Grove, California is well described by a von Bertalanffy model [START_REF] Estess | Bioenergetics of captive yellowfin tuna (Thunnus albacares)[END_REF]. In this experiment, YFT were however found to grow significantly slower than those of the IATTC experiment. These observations supported that the origin of the stanza growth would rather be behavioral or ecological than physiological. In captivity, the food intake is adjusted to comply with the nutritional requirements of tunas which do not suffer from competition for food. Thus, the growth stanza would be caused by a behavioral change related to feeding. Indeed, both assumptions, i.e. a low food availability and the consumption of less energetic prey during the juvenile stage, led to a two-stanza growth with growth acceleration in relation with behavioral change. For a decrease in food density X from 54% to 62% or a reduction in the assimilation rates from 8.5% to 10.75%, the model captured the juvenile growth but the intensity of growth acceleration and overall adult growth were underestimated. In addition, YFT reached puberty at similar sizes as those predicted by the reference model but at older ages, leading to age values in better agreement with those expected, i.e. between 2.2 and 2.5 years according to the growth curve of Indian Ocean YFT [START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF].

Several lines of evidence support low feeding rates for juveniles of YFT when they occur in mixed schools associated with drifting objects. In particular, stomach contents analysis point out to a much higher proportion of empty stomachs for fish caught from associated schools which are significantly slimmer than YFT caught in free swimming schools [START_REF] Potier | Preliminary results of tuna diet studies in the west equatorial Indian Ocean[END_REF][START_REF] Hallier | Drifting fish aggregation devices could act as an ecological trap for tropical tuna species[END_REF][START_REF] Jaquemet | Do drifting and anchored Fish Aggregating Devices (FADs) similarly influence tuna feeding habits? A case study from the western Indian Ocean[END_REF]. This might be explained by the high level of intra-species competition and decreased foraging success in YFT when they occur in large schools [START_REF] Maury | Can schooling regulate marine populations and ecosystems?[END_REF].

YFT is an opportunistic species that feeds on a large number of prey categories and trophic studies underline a change in diet with size as well as the ability of larger YFT to forage in deeper waters [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][START_REF] Potier | Forage fauna in the diet of three large pelagic fishes (lancetfish, swordfish and yellowfin tuna) in the western equatorial Indian Ocean[END_REF]. Firstly, YFT feed mainly on planktonic crustaceans and then, from 50 cm L F , they become essentially piscivorous, with a decline in crustaceans consumption and a strong consumption of cephalopods between 80-100 cm [START_REF] Maldeniya | Food consumption of yellowfin tuna, Thunnus albacares, in Sri Lankan waters[END_REF][START_REF] Potier | Preliminary results of tuna diet studies in the west equatorial Indian Ocean[END_REF]. Recent analysis of fatty acids and stable isotopes of carbon and nitrogen showed some major ontogenetic changes in the diet of Indian Ocean YFT [START_REF] Sardenne | Trophic niches of sympatric tropical tuna in the Western Indian Ocean inferred by stable isotopes and neutral fatty acids[END_REF]. In particular, the shift in concentration of neutral fatty acids observed in the liver of YFT at around 70-80 cm might reflect the change of habitat from surface to deeper layers where rich-MUFA prey of YFT such as myctophids and their predators are more abundant [START_REF] Sardenne | Trophic niches of sympatric tropical tuna in the Western Indian Ocean inferred by stable isotopes and neutral fatty acids[END_REF]. Investigating the caloric value of prey found in YFT stomach contents at various size classes would be useful to quantitatively assess the energetic contribution of each prey group to the ontogenetic changes in diet.

Variability in YFT growth

Most tuna stock assessments assume a constant individual growth described with some individual variability around the mean [START_REF] Kolody | Modelling growth in tuna RFMO stock assessments: Current approaches and challenges[END_REF]. Changes in population abundance and environmental conditions may however affect growth rates over space and time as evidenced for southern bluefin tuna [START_REF] Lorenzen | Toward a new paradigm for growth modeling in fisheries stock assessments: Embracing plasticity and its consequences[END_REF][START_REF] Polacheck | Increase in growth rates of southern bluefin tuna (\textitThunnus maccoyii) over four decades: 1960 to 2000[END_REF].

Growth parameters used for the assessment of Indian Ocean YFT are considered constant over almost seven decades when the stock has experienced a strong decline in abundance [START_REF] Iotc | Report of the 21 st Session of the IOTC Scientific Committee. IOTC-2018-SC21-RE Rev1[END_REF] and YFT fishing grounds have been characterized by major temporal changes in temperature and primary productivity [START_REF] Roxy | A reduction in marine primary productivity driven by rapid warming over the tropical Indian Ocean[END_REF]. Furthermore, the increasing use of FADs to aggregate juveniles of YFT and skipjack tuna since the early 1990s might have increased intra and inter-species competition and led to changes in growth rates [START_REF] Marsac | Drifting FADs used in tuna fisheries : an ecological trap ?[END_REF][START_REF] Hallier | Drifting fish aggregation devices could act as an ecological trap for tropical tuna species[END_REF]. Assessing the influence of FAD density on YFT growth is very difficult due to the lack of routine monitoring for ageing tunas linked to the high costs and logistics of sampling and otolith reading. Developing standard ageing procedures across tuna RFMOs with otolith reference collections constitutes a first step to monitor changes in age-length relationships and promote the comparison of tuna growth between oceans [START_REF] Sardenne | Determining the age of tropical tunas in the Indian Ocean from otolith microstructures[END_REF][START_REF] Farley | Progress on yellowfin tuna age and growth in the WCPO (Project 82)[END_REF]. In addition, we recommend the systematic implementation of comprehensive ageing projects in conjunction with large-scale tagging programs every five or ten years to track changes in growth rates and better account for growth variability in the demographic analysis of these socio-economically important species.

Conclusion & perspectives

Building on the Dynamic Energy Budget modelling framework that mechanistically links the biology of individual organisms to abiotic drivers [START_REF] Kooijman | Dynamic Energy Budget Theory for Metabolic Organisation[END_REF], our model suggests that the unique growth pattern observed in wild YFT could stem from a major change in intensity of schooling and associative behavior with size, which would substantially modify both the quantity and quality of food available for maintenance, reproduction and growth. This change would be triggered by the acquisition of physiological abilities and anatomical traits with age and result in an ontogenetic shift in the biotic habitat and trophic ecology of YFT. In addition, the intensity of schooling might depend on population abundance, with higher density resulting in the formation of larger schools described by a higher degree of competition [START_REF] Maury | From individuals to populations to communities: A dynamic energy budget model of marine ecosystem size-spectrum including life history diversity[END_REF][START_REF] Maury | Can schooling regulate marine populations and ecosystems?[END_REF]. 3 and1, respectively. [START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]: growth in length (A), mean length-weight relationship (B), food daily ration (F DR, C) and food conversion ratio (F CR, D). To determine the F DR and F CR, different energetic levels of prey were used: minimal (dashed line), medium (solid line) and maximal (dotted line). Simulation was performed with a constant functional response of 0.925 and with daily body temperatures equal to the seawater temperature variations in tank (see text for details). Tables Table 1: List of state variables and parameters used in DEB model. The parameters values for adult Pacific bluefin tuna [PBT; 31] and yellowfin tuna (YFT) at given for a reference temperature of 25 ) of juvenile yellowfin tuna (10-90 cm) consecutive to a decrease in food supply and reduced energy intake simulated through changes in the scaled functional response (f ) and the surfacearea specific maximum assimilation rate ({ ṗAm }) respectively. The RMSE values between the average growth rates simulated and estimated for the wild Indian Ocean YFT by Dortel et al. [START_REF] Dortel | An integrated Bayesian modeling approach for the growth of Indian Ocean yellowfin tuna[END_REF] are indicated in parentheses.

dE R dt =    (1 -κ) × { ṗAm } e+g × e g + L V +L T Lm × L 2 V -kJ × E p H -E B if a batch is released (1 -κ) × { ṗAm } e+g × e g + L V +L T Lm × L 2 V -kJ × E p H else D4

  . The data set included: (i) 5,089 fish from the Indian Ocean Tuna Ltd. cannery caught during 2005-2012 (66-165 cm L F , 5.8-87.5 kg) and (ii) 100 additional fish collected in 2010-2013, of which 35 females (54.7-148.2 cm L F , 3.4-61.5 kg) and 65 individuals whose sex was indeterminate (29-61 cm L F , 0.4-4.1 kg).

Figures

  Figures

Figure 1 :

 1 Figure 1: Conceptual representation of a Dynamic Energy Budget model for yellowfin tuna. Energy fluxes and parameters are defined in Tables3 and 1, respectively.

Figure 2 :

 2 Figure 2: Comparison of simulation outputs from the DEB model with farmed yellowfin tuna data [grey triangles;[START_REF] Wexler | Tank culture of yellowfin tuna, Thunnus albacares: developing a spawning population for research purposes[END_REF]: growth in length (A), mean length-weight relationship (B), food daily ration (F DR, C) and food conversion ratio (F CR, D). To determine the F DR and F CR, different energetic levels of prey were used: minimal (dashed line), medium (solid line) and maximal (dotted line). Simulation was performed with a constant functional response of 0.925 and with daily body temperatures equal to the seawater temperature variations in tank (see text for details).

Figure 3 :Figure 4 :

 34 Figure 3: Comparison of yellowfin growth in length (A) and length-weight relationship (B) as simulated by the reference model in constant condition of temperature (27 • C) and food density (f = 0.925) with data from wild yellowfin tuna of the Indian Ocean. Black circles correspond to females and grey circles to fish of indeterminate sex. In B, the solid line represents the mean length-weight relationship and the dotted lines represent the variations around this mean due to status of reproduction buffer (see text for details).

Table 2 :

 2 • C. Dynamics of state variables in the DEB model of yellowfin tuna.

	Definition	Notation Unit	PBT	YFT
	State variables
	Scaled reserve density	e	-
	Structural length	L V	cm
	Maturity level	E H	J
	Reproduction buffer	E R	J
	Primary parameters
	Fraction of mobilised reserve allocated to soma	κ	-	0.7807	0.7807
	Surface-area specific maximum assimilation rate	{ ṗAm }	J.d -1 .cm -2 4783.707	4370.394
	Energy conductance	v	cm.d -1	7.056	7.056
	Volume-specific somatic maintenance rate	[ ṗM ]	J.d -1 .cm -3 17.395	17.395
	Surface area-specific somatic maintenance rate	{ ṗT }	J.d -1 .cm -2 2215.415	2215.415
	Volume-specific costs of structure	[E G ]	J.cm -3	8563.387	8563.387
	Maturity maintenance rate coefficient	kJ	d -1	0.0612	0.0612
	Maturation threshold for metamorphosis Maturation threshold for end of early juvenile stage E e E j H H Maturation threshold for puberty E p H	J J J	6902.209 969476 25484395.636 5053598 1368.719 192248.7
	Fraction of food energy fixed in reserve	κ X	-	-	0.8
	Link parameters
	Shape parameter for juvenile and adult stages	δ j	-	0.2704	0.2559
	Structural volume density	d V	g.cm -3	1	1.0821
	Weight-energy coupler	ρ E	J.g -1	7763.975	7346.034
	Arrhenius temperature	T A	K	5298.838	4622.495
	Half saturation constant	X k	J.cm -3	0.0004463	-
	Reproduction module parameters
	Energy costs of one egg	E 0	J	3.75	1.295
	Fraction of reproduction energy fixed in eggs	κ R	-	0.95	0.95
	Energy density of one batch	[E B ]	J.cm -3	-	1.794
	Compounds parameters
	Energy investment ration	g	-	g = v[E G ] κ{ ṗAm }
	Structural heating length	L T	cm	L T = { ṗT } [ ṗM ]
	Maximum structural length	L m	cm	L m = κ{ ṗAm } [ ṗM ]
	Maximum reserve density	[E m ]	J.cm -3	[E m ] = { ṗAm } v

Table 4 :

 4 Mean, standard deviation (Std.dev) and credibility interval of link parameters as estimated by Bayesian inference.

	Parameter Mean	Std.dev	Posterior quantiles 2.5% 97.5%
	δ j	0.2559	0.0048	0.2455	0.2628
	d V	1.0821	0.0643	1.0013	1.2337
	ρ E	7346.034 136.6962 7046.95 7545.025

Table 5 :

 5 Average growth rates (cm.month-1

Next steps of the work will include the development of a comprehensive calibration and sensitivity analysis to identify the most influential parameters of the model and estimate the ranges of values driving different shapes of growth curves and the associated age-length keys that are used for assessing the status of YFT stocks. To this end, the data collected at TRCC would be useful to estimate a comprehensive set of parameters as well as to disentangle the relative contributions of water temperature and low energy diet on the low growth rates of YFT observed at TRCC. More generally, tuna morphometric and bioenergetic data collected for scientific purpose under controlled environmental conditions are extremely difficult and costly to obtain [START_REF] Kaji | Growth and morphological development of laboratory-reared yellowfin tuna Thunnus albacares larvae and early juveniles, with special emphasis on the digestive system[END_REF][START_REF] Klinger | The effect of temperature on postprandial metabolism of yellowfin tuna (Thunnus albacares)[END_REF][START_REF] Estess | Bioenergetics of captive yellowfin tuna (Thunnus albacares)[END_REF][START_REF] Margulies | Chapter 5 -Research on the reproductive biology and early life history of yellowfin tuna Thunnus albacares in Panama[END_REF]]. Making such valuable information available and accessible (e.g. with standard metadata) would foster their use as inputs for models and make the most of the data.

Acknowledgments: This study was funded by the European Union (Reg.199/2008, 665/2008 and SI2.604453), the French Ministry of Agriculture and Fisheries (301629/00), ORTHONGEL, the Institut de Recherche pour le Développement, and the French ANR funded project EMOTION (ANR 11JSV7 007 01). We wish to acknowledge the contributions of all the people who have been involved in the Regional Tuna Tagging Project of the Indian Ocean, funded under the 9 th European Development Fund (9.ACP.RSA.005/006) of the European Union and the West Sumatra Tuna Tagging Project funded by the Government of Japan. We are grateful to F Sardenne, G Le Croizier, E Morize, JM Munaron, C Geffroy and E Dabas for their contribution to otolith analysis, M Herrera, C Assan, and A Delgado de Molina for data on YFT size measurements, and P Dewals and his team for their work on recoveries and sex identification. We particularly thank M Jusup for his help with the code of the PBT model and I Zudaire for advice on YFT spawning behavior. The manuscript benefited from fruitful discussions with O Maury, D Margulies, N Bodin, F Sardenne, D Gaertner, A Fonteneau, O Gimenez, N Bousquet and M Huret. We finally thank the reviewers for their useful comments that greatly improved the manuscript.

Metabolic process Energy flux

Maturity maintenance ṗJ = kJ E H Maturation or reproduction ṗR = (1 -κ) ṗC -ṗJ


Erdogan Nohuz
email: erdogan.nohuz@chu-lyon.fr

Karine Lebail-Carval

Philippe Chabert

Géry Lamblin

Thierry Michy

Gautier Chêne

Erdogan Nohuz 1©

Mastectomy

Technical trick to limit excess skin ("dog-ears") at the ends of a mastectomy scar

Keywords: Mastectomy, scar, breast cancer, surgical technique, plastic surgery, reconstructive surgery

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

Introduction

Breast cancer remains the most common malignant tumor in terms of incidence and mortality in women and represents a major public health issue [START_REF] Inca | Les chiffres du cancer en France[END_REF]. Surgery is the first step of the treatment is most cases. Despite a tendency to surgical de-escalation seen in recent decades, there are still many indications of breast removal. Historically, the delabrating radical operation of Halsted involving the removal of the pectoral muscles has been replaced by that of Patey (resection of the pectoralis minor muscle) itself supplanted by the modified radical mastectomy which consists of resection of the mammary gland preserving the pectoralis minor muscle. This is currently the standard radical treatment [START_REF] Patey | The prognosis of carcinoma of the breast in relation to the type of operation performed[END_REF]. Thus, when performing a mastectomy, the unsightly excess skin located at the two ends of the scar ("dog-ears"), resulting from oval or circular incisions beyond aesthetic discomfort that they generate, can be responsible for functional hindrance, especially in the axillary region [START_REF] Lehmann | Problèmes psychologiques posés par les exérèses ovariennes et mammaires et par les reconstructions mammaires[END_REF]. These excesses, although attenuated when using breast expansion implant, are increased by too economical incisions and require then additional resections by an incision at their base, at the cost of an extension of the original incision, sometimes in a second time (Fig. 1). We use a simple trick that can help to reduce this scar ransom and to obtain more aesthetically pleasant results.

Surgical technique

The patient is positioned supine, the arm homolateral to the mastectomy being in abduction at 90 degrees on a tablet, to preserve the shoulder, the brachial plexus and the points of support. Once the antiseptic brushing has been performed, a sterile draping is carried out. An elliptical skin incision, slightly oblique at the bottom and inside, is made. This incision encompasses the nipple-areolar complex as well as the skin facing the lesion and carrying away any anterior scars consecutive to a lumpectomy or biopsies. We use a trick to perform a pre-location of the incision line. The principle is to use a strand of braided 0 or 1 suture previously dipped in a cup filled with iodized polyvidone (in the absence of allergy) (Fig. 2). The 2 ends of the wire are then firmly applied to the top and below the areolar area, for 3 to 4 seconds, according to the size of the breast and the location of the lesion (Fig. 3).

The impregnated wire with iodized polyvidone thus draws a spindle of resection in the image of the contours "of an eye" which meet themselves at the external (facing the axillary hollow) and internal (facing the sternum) extremities. The direction of the pre-thoracic incision is drawn in the vertical way as much as possible so that the scar can be hidden in the bra. The marking by the wire is therefore highlighted by the iodized solution and persists for a longer period than with the wire pressure alone. The skin marking is thereafter tinted and preserved using a sterile dermographic pencil and the incision made (Fig. 4 and5). The mastectomy is performed in a conventional manner (Fig. 6).

The skin closure time is left to the discretion of the surgeon. For our part, we perform it on one or two aspirative drains (depending on whether or not a concomitant axillary lymph node dissection has been performed), in two planes: a subcutaneous plan of skin approximation with a 2/0 or 3/0 absorbable suture and a skin plan with a 3/0 absorbable single-strand continuous suture.

The two cutaneous walls then come face to face without tension or unsightly excess skin at the ends of the incision (Fig. 6F). Note that in the era of enhanced recovery after surgery, aspirative drainage can be avoided by using suture closure of the dead spaces (subcutaneous padding) in order to shorten hospital stay [START_REF] Classe | Randomized Clinical Trial Comparing Axillary Padding With Closed Suction Drainage for the Axillary Wound After Lymphadenectomy for Breast Cancer[END_REF][START_REF] Hejl | Quality of Life, Anxiety, and Postoperative Complications of Patients Undergoing Breast Cancer Surgery as Ambulatory Surgery Compared to Non-Ambulatory Surgery: A Prospective Non-Randomized Study[END_REF]. This technique can also be used when a scar resection is indicated in case of invasion of the cutaneous walls objectified during the final anatomopathological examination, although the tracing of an arc of circle is easier in this situation. The wire impregnated with iodized polyvidone is then applied above and below the mastectomy scar, according to the same principles. Post-operative adhesions can however induce discomfort up or down as the skin to be resected. Once the scar has been excised, the two cutaneous edges meet without excess skin at the internal and external end of the incision (Fig. 7).

Conclusion

A radical gesture such as a mastectomy is experienced as a trauma, both physical and psychological, regardless of the age of the patients. The attention paid to the incision contributes favorably to their care plan [6]. Certain patients wishing to see before the intervention the quantity of skin which will be removed, this process can also be carried out preoperatively in their room. This alternative thus rallies two other objectives: that of being able to shorten the operating time on the one hand and to mark the breast to be operated on the other. However, this is only conceivable in women who accept the possibility of it so as not to inflict additional preoperative mental suffering.

Not intended to solve all the problems encountered when closing mastectomy incisions, this technique of pre-incisional identification remains a simple trick that can improve cosmetic results by providing aesthetic scars and not subject to excessive tension.

Funding sources: None

Figures legendsFigure 1 :

 1 Figures legends

Figure 2 :

 2 Figure 2: Wire impregnation

Figure 3 :

 3 Figure 3: Skin marking

Figure 4 :Figure 5 :

 45 Figure 4: Highlighting of iodized marking using a dermographic marker

Figure 6 :

 6 Figure 6: Mastectomy and skin closure (left side)

Figure 7 :

 7 Figure 7: Technique used during a mastectomy scar resection (right side)

6. Mathelin C, Nisand I. Trop vieilles pour ça ? Seuls les autres le croient. Gynecol Obstet Fertil Senol. 2019;47:547-548.

Mastectomy