DESIGN OF COMPOSITE FILAMENT WOUND PRESSURE VESSELS: MANAGEMENT OF SINGULARITIES IN MULTI-SEQUENCE DOME LAY-UP SIMULATIONS

Keywords: filament winding, analytical modelling, composite pressure vessels, overlaying of convex areas, overlapping layers

Composite pressure vessels made by filament winding processes are commonly used in aerospace design due to weight saving compared to metal parts. This technique involves complexities in analyzing the geometry especially in the dome section. Tools already exist to predict the geometrical characteristics like winding angle, ply thickness or some singularities. However they are limited to specific sequence dome lay-up. A numerical tool has been developed to simulate windings layer after layer.

The goal is to deal with any kind of sequence dome lay-up calculation. To ensure this there are three main challenges that are seldom cited in scientific literature. The first is when the geometry of the current ply overlaps the inferior ply. The second is the management of convex parts of a ply. Finally, our mathematical tool is able to deal with the accumulation of composite fibers near the polar boss outer radius.

Introduction

The winding process produces composite tanks capable of withstanding high internal pressures. The very high specific properties of these materials lead to structures with remarkable rigidity and strength for a contained mass. The filament winding continuously deposits composite fibers on a mould or a mandrel with a rotational movement. This manufacturing method is particularly suitable for revolution structures. In the space industry this technology is already used for some motor cases or tanks. To best design and optimize these structures it becomes necessary to integrate the simulation of the winding process of composite structures.

The performance of a composite structure depends on the adequacy between the loading directions and the distribution of the fibrous reinforcement. In the case of the filament winding process, a reliable estimation of the mechanical behaviour of the structure is conditioned by the most exact simulation possible of the fiber path, in particular in the domes, considered as the most delicate parts to simulate. Several studies [START_REF] Vasiliev | New generation of filament-wound composite pressure vessels for commercial applications[END_REF][START_REF] Wang | A new method for predicting dome thickness of composite pressure vessels[END_REF][START_REF] Wang | Dome thickness prediction of composite pressure vessels by a cubic spline function and finite element analysis[END_REF][START_REF] Leh | Multi-sequence dome lay-up simulations for hydrogen hyper-bar composite pressure vessels[END_REF] have allowed the development of successive winding models that enable the complex geometry of domes to be managed. However, these models remain limited with regard to the strong disparity of possible windings. The objective of this work is to propose a tool for simulating the filament winding of any winding sequences on generic revolution geometries.

The ultimate aim is to use the model developed in an optimization procedure in which lay-up orientation, stacking sequence and filament winding parameters are significant parameters.

Basic aspects

Winding angle determination

Geodesic winding provides to the best winding stability by finding the shortest fiber path between two points on any surface. However, non-geodesic winding increases the scope of the geodesic winding technique through the use of the slippage tendency of the fibers on their supporting surface. Descriptions of geodesic and non-geodesic winding methods can be found in [START_REF] Carvalho | Optimization of filament wound parts based on non-geodesic winding[END_REF][START_REF] Zu | Design of filament-wound domes based on continuum theory and non-geodesic roving trajectories[END_REF].

For the purposes of future optimization, it is desirable to have a general equation to describe both geodesic and non-geodesic winding. The following equation [START_REF] Zu | Design of filament-wound domes based on continuum theory and non-geodesic roving trajectories[END_REF] is valid:

𝑑𝛼 𝑑𝑧 = 𝜆 [ sin 𝛼 tan 𝛼 𝑟 - 𝑟" 1 + 𝑟 ′ 2 cos 𝛼] - 𝑟 ′ tan 𝛼 𝑟 ( 1
)
where  is the angle between the fiber tow and the meridian direction of the dome,  the slippage coefficient of the fiber tows on the supporting surface and r and z the radial and axial coordinates of the dome, as shown in Fig. 1.

Depending on the sign of , the path will be lengthened or shrunk and thus influence the positioning of the radius at the turn-around of the continuous fiber. Usually this differential equation is solved by a Runge-Kutta method to order 4. The main problem with this method of resolution is the number of integration points required to ensure convergence of results. We will see in the proposed developments that this method of resolution, although used by all the routines allowing the multi-sequence simulation of filament winding, is not sufficiently adapted due to the inherent nature of the differential equation Eq. 1.

Figure 1. Geometry of the liner

To ensure the continuity of the winding between the cylindrical section of the tank and the dome, the following equation [START_REF] Zu | Design of filament-wound domes based on continuum theory and non-geodesic roving trajectories[END_REF] is used:

𝜆 = 𝜆 𝑚𝑎𝑥 cos ( 𝜋 2 𝑟 -𝑟 𝑡𝑢𝑟𝑛 𝑅 -𝑟 𝑡𝑢𝑟𝑛 ) ( 2
)
where R is the external equator radius of the tank (including the thickness of the plies) in the cylindrical section, 𝑟 𝑡𝑢𝑟𝑛 the radius at the turn-around point and 𝜆 𝑚𝑎𝑥 the arbitrary top value of the slippage coefficient. The arbitrary top value 𝜆 𝑚𝑎𝑥 can be an optimization parameter as long as |𝜆 𝑚𝑎𝑥 | ≤ 𝜆 𝑙𝑖𝑚 where 𝜆 𝑙𝑖𝑚 is the slippage limit coefficient which depends on practical winding parameters.

Thickness determination

Once the radius at the turn-around point determined by the resolution of Eq. 1, it is necessary to determine the evolution of the thickness of the composite layer of the cylindrical section up to this radius at the turn-around point. Leh [START_REF] Leh | Multi-sequence dome lay-up simulations for hydrogen hyper-bar composite pressure vessels[END_REF] presented several formulations of thickness. It shows that Wang's formulation [START_REF] Wang | A new method for predicting dome thickness of composite pressure vessels[END_REF] is one of the most recent methods and presents very good results in comparison with the experimental one. It is based on the use of a double formulation to reflect as accurately as possible the thickness evolution:

𝑡(𝑟) = 𝑛 𝑅 𝑚 𝑅 𝑡 𝑝 𝜋 [arccos ( 𝑟 𝑡𝑢𝑟𝑛 𝑟 ) -arccos ( 𝑟 𝑏 𝑟 )] for 𝑟 2𝑏 ≤ 𝑟 ≤ 𝑅 (3)
𝑡(𝑟) = 𝐴 + 𝐵𝑟 + 𝐶𝑟 2 + 𝐷𝑟 3 for 𝑟 𝑡𝑢𝑟𝑛 ≤ 𝑟 ≤ 𝑟 2𝑏 (4)
where 𝑛 𝑅 , 𝑚 𝑅 are the number of pseudo-plies (±𝛼 𝑅 ) and their quantity in the cylindrical section, 𝑡 𝑝 is the thickness of a fiber tow or tape, b is the width of a fiber tape, 𝑡 𝑅 and 𝛼 𝑅 are the thickness of the layer and the initial winding angle on the cylindrical part, respectively. The parameters 𝑟 𝑏 and 𝑟 2𝑏 are the radius at one and two tape-width distances, respectively, from the turn-around point, and A, B, C and D the four coefficients of the method presented in [START_REF] Wang | A new method for predicting dome thickness of composite pressure vessels[END_REF] and defined using boundary conditions (pseudo-ply thickness at the turn-around radius, continuity of the thickness and the curvature in 𝑟 2𝑏 between Eqs. 3 and 4, and constant material volume between 𝑟 𝑡𝑢𝑟𝑛 and 𝑟 2𝑏 ).

Lay-up determination

In the multi-sequence lay-up of a dome, prediction is a tricky process, given that the determination of both winding angle and thickness is based on the current supporting surface topology, which, in a multi-layer lay-up, depends on the previous supporting surface topologies. The classical process presented by Leh [4] is as follows:  recovery and storage in internal variables of geometric data (in the form of elliptical liner definition parameters) and winding sequence parameters,  a definition of the coordinates according to z and r and a discretization of this geometry,  the calculation of first and second derivatives of current shape line by the finite difference method to solve Eq. 1,  the resolution of Eq.1 by a Runge-Kutta method to order 4 in order to obtain the value of the radius at the turn-around of the current layer,  the calculation of the overthickness to be applied to the previous layer thanks to the Wang's formulation,  normal calculation and application of the previously determined overthickness at each integration point,  smoothing of new points calculated by a B-spline,  the looping of all these steps for each layer described by the sequence of the winding angles. This process has some limitations not compatible with configuration extensions. Thus the tool is not able to deal with winding lay-up other than increasing initial winding angle lay-up. It consequently does not make possible to cover a layer n with the layer n + 1. It is also highly unstable when covering a convex shape. As shown in Fig. 2, if the integration step is too large, there is a possibility that the normals at two (or more) integration points intersect. This results in the creation of points for the new layer that will no longer be ordered in an increasing z when the overthickness is added.

Finally, another critical point consists in the absence of possibility to model the covering of the barrel of the boss by several layers of composite. Indeed this routine foresees that only the first layer reaches the barrel of the boss. Then, because only sequences of increasing winding angles are managed, the other layers can only be modelled with increasing radii at the turn-around point. All the identified authors use the Runge-Kutta algorithm at order 4 (RK4) to solve Eq. 1 for determining the radius at the turn-around point of the current layer. The main problem with this resolution method is it is highly dependent on the number of integration points resulting from the geometric discretization of the current layer.

As we can see in Fig. 3, the problem is strongly non-linear: the closer we get to 𝛼 = 90°, the more the influence of the discretization step is preponderant. This is problematic because the calculation error via RK4 is strongly related to discretization and consequently to calculation time. The resolution model should also be validated by optimizing the calculation step to reach a cumulative error below an admissible threshold.

Eq. 1 is an equation whose sensitivity to parameters makes it difficult to solve by explicit numerical methods. The closer we get to 𝛼 = 90° the more sensitive it is and the less relevant the RK4 algorithm is to solve it. Works using this algorithm do not quantify the error associated with the resolution of this problem with this method. The solution to enable the algorithm to converge is then to define an extremely fine step without obtaining information on the cumulative error. There are many powerful resolution algorithms, originally optimized to run on low-performance machines, alternative to explicit methods. This is the case with the LSODA [START_REF] Brown | Reduced storage matrix methods in stiff ODE systems[END_REF] algorithm, which offers many advantages. The first is that it is able to manage different resolution methods in the same calculation. Thus, it can switch from an Adams method to a BDF method automatically depending on the degree of stiffness of the differential equation. The algorithm is also independent of the calculation step imposed. The code automatically manages the step needed to solve the problem based on the calculated error to ensure convergence. Once solved on the number of steps strictly necessary, the function then re-interpolates at the integration points specified initially.

Instabilities due to convex zones

Due to the reduction in the number of integration points compared to a purely explicit routine, the code is then sensitive to the problem of overlapping instability of convex areas (Fig. 2) : physically meaningless loops may appear (Fig. 4a, Fig. 4b). It is therefore necessary to introduce a specific function to erase these loops (Fig. 4b). This function detects each segment consisting of two successive integration points intersecting another segment of the original curve. In order to limit the calculation time, the function must only take into account the convex parts of the curve to be checked. The outputs of this function are all the intersection points of the curve as well as two series of z and r coordinates of the erased instability loops. In addition, a smoothing function is required to ensure the stability of the next layer simulation (Fig. 5a). The degree of the smoothing function will depend on the geometric dimensions of the wound body and the thickness of the composite layers. This smoothing function has to manage the different possible winding angle sequences:  sequence with equal initial angles (Fig. 5b),  sequence with decreasing initial angles (Fig. 5c),  sequence with increasing initial angles (Fig. 5d),  sequence with alternation of the three previous types.

Layers overlapping the barrel of the boss

Modelling the helical layers overlapping the barrel of the boss is a development which is less dealt with in the literature, although [START_REF] Moussaïd | Analyse et intégration des spécificités liées au procédé de fabrication dans les modèles de calcul des structures composites[END_REF] addresses this issue. In order to take into account these overlappings which may occur near the boss (Fig. 6), a method is proposed to transfer the overthickness. [START_REF] Moussaïd | Analyse et intégration des spécificités liées au procédé de fabrication dans les modèles de calcul des structures composites[END_REF] considers a linear extension tangent to the curve at the radius of the boss 𝑟 𝑏𝑜𝑠𝑠 to be the support for the calculation of the current layer. It then proposes that the volume defined by the winding for 𝑟 < 𝑟 𝑏𝑜𝑠𝑠 be reported with respect to the location of the curve presenting the maximum thickness (𝑟 𝑡𝑚𝑎𝑥 ) and satisfying several conditions:  the thickness is redefined by:

𝑡(𝑟) = 𝐴 + 𝐵𝑟 + 𝐶𝑟 2 for 𝑟 𝑏𝑜𝑠𝑠 ≤ 𝑟 ≤ 𝑟 𝑡𝑚𝑎𝑥 (5)
 the volume for a radius smaller than 𝑟 𝑏𝑜𝑠𝑠 is distributed between 𝑟 𝑏𝑜𝑠𝑠 and 𝑟 𝑡𝑚𝑎𝑥 ,  continuity of thickness at 𝑟 𝑡𝑚𝑎𝑥 ,  continuity of slope at 𝑟 𝑡𝑚𝑎𝑥 .

The problem with this approach is twofold. First of all, the expression of the thickness in 𝑟 2 does not allow to take into account geometries with an inflection point, which is often the case. On the other hand, as can be seen in Fig. 7, this approach does not work if the radius at maximum thickness (𝑟 𝑡𝑚𝑎𝑥 ) is also less than 𝑟 𝑏𝑜𝑠𝑠 . A satisfactory routine should be able to handle overlapping layers on the boss. The most common function whose curve has an inflection point is a polynomial function of degree 3. Rather than reasoning about the redefinition of the overthickness to be applied, the geometry of the current layer is directly modified once it has been calculated. It is therefore first necessary to extend the spatial domain of the lower layer which will serve as a support for the calculation of the current layer for the values of 𝑟 < 𝑟 𝑏𝑜𝑠𝑠 . Note that a second order spline has been finally chosen (Fig. 8a).

As one of the conditions on the polynomial of degree 3 used to smooth this kind of layer regards a volume equivalence, it is necessary to be able to perform a numerical integration. To do this, a polynomial definition of z is chosen as a function of r. A new geometry is then defined as:

𝑧 = 𝐴 + 𝐵𝑟 + 𝐶𝑟 2 + 𝐷𝑟 3 (6)
This new geometry is defined over a transfer length 𝐿 𝑒𝑥 . To determine the parameters A, B, C and D it is necessary to introduce 4 conditions, namely:  continuity at fixed 𝐿 𝑒𝑥 ,  continuity of the slope at 𝐿 𝑒𝑥 ,  slope equal to the one of previous layer at 𝑟 = 𝑟 𝑏𝑜𝑠𝑠 ,  conservation of the area between 𝑟 𝑏𝑜𝑠𝑠 and 𝑟 𝑏𝑜𝑠𝑠 + 𝐿 𝑒𝑥 for initial formulation initiale and polynomial formulation.

Once the equation system is solved, it is then possible to switch back to a definition of r as a function of z to establish the new geometry. The routine is then able to manage the layers overlapping the barrel of the boss (Fig. 8a) and to propose a redefinition of the geometry in a polynomial of degree 3 (Fig. 8b). From a physical point of view, the only value that the winding angle can take in the vicinity of the boss end is 𝛼 = 90°. This corresponds to the tilting of the fiber during winding at the radius at the turn-around point. Indeed, a value lower would mean that the winding head hits the boss barrel.

If the initial winding angle at the cylinder/dome junction is fixed by the process, there is only one parameter left to ensure these conditions, i.e. . This parameter is then optimized to ensure that the winding is feasible. To do this, the routine solves Eq. 1 backwards from the radius at the turn-around point 𝑟 𝑡𝑢𝑟𝑛 = 𝑟 𝑏𝑜𝑠𝑠 with a definition of  set by a value of 𝜆 𝑚𝑎𝑥 . The idea is then to compare the value of the calculated initial winding angle with the value of the imposed initial winding angle and this by modifying the value of the parameter 𝜆 𝑚𝑎𝑥 during optimisation process. For this one must defined a cost function: Δ = 𝛼 0 (𝜆 𝑚𝑎𝑥 ) -𝛼 0 [START_REF] Brown | Reduced storage matrix methods in stiff ODE systems[END_REF] Thanks to an adapted algorithm, it's possible to optimize this calculation by minimizing this cost function, i.e. by searching for the root of: Δ = 0, -𝜇 𝑠 < 𝜆 𝑚𝑎𝑥 < 𝜇 𝑠 [START_REF] Moussaïd | Analyse et intégration des spécificités liées au procédé de fabrication dans les modèles de calcul des structures composites[END_REF] where 𝜇 𝑠 is the static friction coefficient, the maximum value of the sliding coefficient beyond which it becomes impossible to wind. Thus it is possible to identify the 𝜆 𝑚𝑎𝑥 value that satisfies both 𝛼 = 𝜋 2 at the barrel of the boss and α = 𝛼 0 for 𝑟 = 𝑅.

Conclusions

The work presented here allows the development of a generic filament winding routine to simulate any sequence. It is not limited to specific geometries (which must be axisymmetric) and is suitable to any winding sequence. This is made possible by a number of functions developed such as smoothing or winding stabilization functions on convex geometries. Particular attention is paid to a physical interpretation of certain parameters such as the definition of the slip coefficient or the management of the winding angle in the vicinity of the boss.

The interest of this approach is to propose the integration of a simulation module of the winding process, flexible enough to be adaptable to the geometrical peculiarities of the motor cases or tanks, in a design and optimization tool for wound composite structures sizing.

  and ECCM18 -18 th European Conference on Composite Materials Athens, Greece, 24-28 th June 2018 4 Bizet L., Saffré P., Mathis K., Halm D., Gueguen M. and Francescato P.

Figure 2 .Figure 3 . 1 .

 231 Figure 2. Overlap instability for convex areas due to the interweaving of normals close to integration points

Figure 4 .

 4 Figure 4. Resolution of instabilities related to convex shapes of geometries forthe current layer

6 Figure 5 .

 65 Figure 5. Smoothing routine of layer n with layer n -1 (a), in sequence of equal (b), decreasing (c) and increasing (d) initial winding angles

Figure 6 .

 6 Figure 6. Micrograph of storage tanksstudied in[START_REF] Moussaïd | Analyse et intégration des spécificités liées au procédé de fabrication dans les modèles de calcul des structures composites[END_REF]

Figure 7 .

 7 Figure 7. Volume transfer defined by a winding for r turn < r boss

Figure 8 .

 8 Figure 8. Layer overlapping the barrel of the boss (a) and geometrical definition in the form of polynomial of degree 3 (b)

  th European Conference on Composite Materials Athens, Greece, 24-28 th June 2018 8Bizet L., Saffré P., Mathis K., Halm D., Gueguen M. and Francescato P.

ECCM18 -18 th European Conference on Composite Materials Athens, Greece, 24-28 th June 2018

Acknowledgments

This research is supported by the French Space Agency (CNES).


Anastazia Banaszak

David O Obura
email: dobura@cordioea.net

Lionel Bigot

Francesca Benzoni

Coral responses to a repeat bleaching event in Mayotte in 2010

Keywords: Coral bleaching, Western Indian Ocean, Acropora, Recovery, Northern Mozambique Channel, Coral reef, Eastern Africa, Climate change, Resilience

Background. High sea surface temperatures resulted in widespread coral bleaching and mortality in Mayotte Island (northern Mozambique channel, Indian Ocean: 12.1 • S, 45.1 • E) in April-June 2010. Methods. Twenty three representative coral genera were sampled quantitatively for size class distributions during the peak of the bleaching event to measure its impact. Results. Fifty two percent of coral area was impacted, comprising 19.3% pale, 10.7% bleached, 4.8% partially dead and 17.5% recently dead. Acropora, the dominant genus, was the second most susceptible to bleaching (22%, pale and bleached) and mortality (32%, partially dead and dead), only exceeded by Pocillopora (32% and 47%, respectively). The majority of genera showed intermediate responses, and the least response was shown by Acanthastrea and Leptastrea (6% pale and bleached). A linear increase in bleaching susceptibility was found from small colonies (<2.5 cm, 83% unaffected) to large ones (>80 cm, 33% unaffected), across all genera surveyed. Maximum mortality in 2010 was estimated at 32% of coral area or biomass, compared to half that (16%), by colony abundance. Discussion. Mayotte reefs have displayed a high level of resilience to bleaching events in 1983, 1998 and the 2010 event reported here, and experienced a further bleaching event in 2016. However, prospects for continued resilience are uncertain as multiple threats are increasing: the rate of warming experienced (0.1 • C per decade) is some two to three times less than projected warming in coming decades, the interval between severe bleaching events has declined from 16 to 6 years, and evidence of chronic mortality from local human impacts is increasing. The study produced four recommendations for reducing bias when monitoring and assessing coral bleaching: coral colony size should be measured, unaffected colonies should be included in counts, quadrats or belt transects should be used and weighting coefficients in the calculation of indices should be used with caution.

INTRODUCTION

Coral bleaching is an increasingly common phenomenon on tropical coral reefs as background warming occurs and inter-annual modes of climate variability intensify [START_REF] Mcphaden | ENSO as an integrating concept in earth science[END_REF][START_REF] Hoegh-Guldberg | Coral reefs under rapid climate change and ocean acidification[END_REF]. Repeated major coral bleaching events are now standard occurrence in most world regions [START_REF] Donner | A new, high-resolution global mass coral bleaching database[END_REF][START_REF] Hughes | Spatial and temporal patterns of mass bleaching of corals in the Anthropocene[END_REF], including the Caribbean [START_REF] Eakin | Caribbean corals in crisis: record thermal stress, bleaching, and mortality in 2005[END_REF][START_REF] Jackson | Part I: overview and synthesis for the wider Caribbean[END_REF], the Pacific [START_REF] Chin | Status of coral reefs of the Pacific and outlook[END_REF], and the western Indian Ocean [START_REF] Mcclanahan | Biogeography and change among regional coral communities across the Western Indian Ocean[END_REF]Obura et al., 2017), and recently in concurrent years on the Great Barrier Reef (Hughes et al., 2017b). The Western Indian Ocean (WIO) has suffered repeated bleaching events, with the most extreme being in 1998 [START_REF] Wilkinson | Ecological and socioeconomic impacts of 1998 coral mortality in the Indian Ocean: an ENSO impact and a warning of future change?[END_REF][START_REF] Goreau | Conservation of coral reefs after the 1998 global bleaching event[END_REF], but with smaller events before, most notably in 1983 [START_REF] Faure | Sur un phénomène remarquable de blanchiment et de mortalité massive des madréporaires dans le complexe récifal de l'île Mayotte (SW Ocean Indien)[END_REF]) and since, in 2005[START_REF] Mcclanahan | Effects of geography, taxa, water flow, and temperature variation on coral bleaching intensity in Mauritius[END_REF]), 2007(D Obura, 2007, unpublished data), 2010 [START_REF] Eriksson | Coral bleaching and associated mortality in Mayotte, Western Indian Ocean[END_REF]) and 2016 [START_REF] Nicet | Intensité et impact du blanchissement corallien massif de 2016 sur les récifs coralliens français de l'océan Indien. Projet BECOMING 2016 Rapport MAREX[END_REF]Obura et al., 2017). The increasing frequency of major bleaching events [START_REF] Hughes | Spatial and temporal patterns of mass bleaching of corals in the Anthropocene[END_REF] is calling into question the long term survival of coral reef ecosystems, with most world regions predicted to experience severe bleaching conditions on an annual basis within the next 40-80 years [START_REF] Van Hooidonk | Local-scale projections of coral reef futures and implications of the Paris agreement[END_REF].

How well coral reefs will cope with these conditions is a major question in current research (Hughes et al., 2017a). The WIO was among the worst affected regions during the 1998 bleaching event. In that year three phenomena coincided-1998 was an unusually hot year globally, and strong positive phases of the El Niño Southern Oscillation (ENSO) and Indian Ocean Dipole (IOC) occurred in phase [START_REF] Saji | A dipole mode in the tropical Indian Ocean[END_REF][START_REF] Mcphaden | ENSO as an integrating concept in earth science[END_REF]. As a consequence, coral mortality at different reef sites varied between 50-80%, and accounted for a loss of 16% of healthy reefs (reviewed in [START_REF] Wilkinson | Status of coral reefs of the world[END_REF]. The Northern Mozambique Channel, one of the least known regions of the WIO, had variable levels of coral bleaching reported in the 1998 event [START_REF] Souter | Coral reef degradation in the India Ocean. Status reports[END_REF][START_REF] Wilkinson | Status of coral reefs of the world[END_REF]Obura et al., 2018). While it is a center of diversity and accumulation for coral species (Obura, 2012;Obura, 2016), it may show some characteristics of being a climate refuge because of its low rate of temperature rise (McClanahan et al., 2007a;McClanahan et al., 2007b;[START_REF] Mcclanahan | Biogeography and change among regional coral communities across the Western Indian Ocean[END_REF]. However, larger scale studies present it as among the earliest parts of the WIO to face severe thermal stress conditions [START_REF] Sheppard | Predicted recurrences of mass coral mortality in the Indian Ocean[END_REF][START_REF] Van Hooidonk | Local-scale projections of coral reef futures and implications of the Paris agreement[END_REF].

Mayotte (12.1 • S, 45.1 • E) is the oldest island in the Comoro archipelago, located in the center of the Northern Mozambique Channel. It has somewhat lower coral diversity than surrounding mainland and Madagascar coasts on account of the island area effect (Obura, 2012). However, it has among the highest geomorphological diversity of reef habitats in the western Indian Ocean. It is highly eroded with high sedimentation and turbidity in the lagoon [START_REF] Thomassin | Mayotte (SW Indian Ocean): recent demographical and economical rise induced by anthropisation and/or climatic variations on the littoral and on lagoonal ecosystems[END_REF]. Sea surface temperatures (SSTs) around Mayotte are warm and bimodal [START_REF] Ateweberhan | Relationship between historical sea-surface temperature variability and climate change-induced coral mortality in the western Indian Ocean[END_REF] determined by the monsoons and interactions of the South Equatorial Current with Madagascar, varying between 25.5 and 29 • C. Coral bleaching on Mayotte has been well documented in 1983 and 1998 [START_REF] Faure | Sur un phénomène remarquable de blanchiment et de mortalité massive des madréporaires dans le complexe récifal de l'île Mayotte (SW Ocean Indien)[END_REF]Quod et al., 2002), as well as in 2010 [START_REF] Eriksson | Coral bleaching and associated mortality in Mayotte, Western Indian Ocean[END_REF] [START_REF] Nicet | Intensité et impact du blanchissement corallien massif de 2016 sur les récifs coralliens français de l'océan Indien. Projet BECOMING 2016 Rapport MAREX[END_REF], L Bigot & D Obura, 2010, unpublished data).

This study is based on surveys in early June 2010 following the peak months of high sea furface temperatures from February to April (Fig. 1). Bleaching and mortality of corals were observed throughout the island's reefs. This paper focuses on comparisons among coral genera and coral colony size classes in bleaching and mortality patterns, and estimation

METHODS

Coral reef structures on Mayotte Island are diverse, including outer reef banks and slopes, inner slopes of the barrier, a second inner barrier reef at the southwest of the island and fringing reefs around the main island and smaller islets (Fig. 2). The island is densely populated, with high pressure on the lagoon from fishing, and sedimentation from runoff and land-use change [START_REF] Thomassin | Mayotte (SW Indian Ocean): recent demographical and economical rise induced by anthropisation and/or climatic variations on the littoral and on lagoonal ecosystems[END_REF][START_REF] Bigot | The French territories in the South Western Indian Ocean[END_REF]. Sampling for coral bleaching was conducted on the Tara Oceans Expedition from 30 May-17 June 2010, following peak bleaching months. Twenty seven of the 34 sites (Table 1, Fig. 2) were sampled between a depth of 8 and 12 m, corresponding to the zone of maximum reef development, though on two fringing and one inner barrier sites the reef profile forced sampling to be done at 4-7 m, and on two outer barrier and the two bank reefs, sampling was done at 16-20 m. This paper reports on two survey methods collected as part of a more comprehensive dataset (Obura & Grimsdith, 2009). Coral size class structure was sampled with 1 m wide belt transects, including bleaching and mortality observations by colony, and visual estimation of abundance of coral genera on a 1-5 scale, and of percent coral cover.

Belt transects 1 m wide were used for sampling coral colony sizes. Coral colonies whose center fell within the belt and quadrats) were counted. The largest colony diameter was recorded in the size class bins: 11-20, 21-40, 41-80, 81-160, 161-320 and >320 cm. For corals smaller than 10 cm, subsampling was done using 1 m 2 quadrats at the 0, 5, 10, 15, 20, and 25 m transect marks, in size class bins 0-2.5, 3-5 and 6-10 cm. Colony condition was recorded as unaffected (no visible effect of thermal stress), pale, bleached, partially dead or fully dead, as exclusive categories by colony. Colonies were assigned to the most severe condition observed if that condition occupied more than 1/5 of the colony surface; i.e., a colony with bleaching and partial mortality was classed as partially dead if more than about 1/5 was dead. A 1 m stick was used to help guide estimation of transect width and mark the 1 m 2 quadrats, and the stick was marked at 10, 20, 40 and 80 cm to guide size estimation of coral colonies. A standard transect length of 25 m was targeted, but length was often limited by a high density of corals and time available per dive. Seventeen of the 34 sites were sampled with 25-m transects (Table 1). The smallest sampling was of two 1 m 2 quadrats and a 7-m belt transect, recording 18 small and 94 large corals, respectively. The minimum counts of 11 small and 86 large corals were recorded within a 25-m transect.

Sampling focused on coral genera that were already known to cover a range of bleaching susceptibility from high to low and that are generally common on East African reefs (Obura & Grimsdith, 2009): 1-low resistance to bleaching: Acropora (including Isopora), Montipora, Pocillopora, Seriatopora, Stylophora; 2-intermediate resistance to bleaching: Echinopora, Dipsastraea, Favites, Goniastrea, Leptastrea, Platygyra, Acanthastrea, Coscinaraea, Fungia, Galaxea, Hydnophora, Lobophyllia, Oxypora, Pavona, Plerogyra ; 3high resistance to bleaching: Porites (massive and branching morphologies recorded separately) and Turbinaria.

The relative abundance of all coral genera at a site was recorded by visual estimate, on a five-point scale (rare, uncommon, common, abundant, dominant) following [START_REF] Devantier | Species richness and relative abundance of reef-building corals in the Indo-West Pacific[END_REF]. An index of relative abundance for each genus was calculated from these estimates as the average of: the proportion of sites at which a genus was present, its average abundance across all sites, and maximum abundance at any site, all converted to 0-5 scale (Obura & Grimsdith, 2009).

For analysis of the coral size class data, all densities per genus were transformed to a standard area of 100 m 2 . The number and area of colonies per 100 m 2 was used. Colony area was calculated for each size class using its median diameter and assuming the area of a coral colony is approximated by an ellipse with the second diameter half of the maximum (area = 1/2*pi*r 2 ). Proportions of each of the colony condition classes were calculated based on abundance and area of each in each size class. For some analyses the number of classes was reduced to three, by aggregating pale and bleached (termed bleached) and partially and full dead (termed dead).

To investigate the impact of different approaches to monitoring and reporting bleaching, two variations of the bleaching and mortality data were analyzed. First, we tested the effect of excluding the proportion of normal colonies, because many programmes count or report only colonies showing some level of bleaching and/or mortality, and not count the number of normal corals. Second, we tested the effect of weighting conditions of increasing severity, i.e., from pale, to bleached, to partially dead to dead (see McClanahan et al., 2007b). We applied the following weights: 1*pale, 2*bleaching, 3*partial mortality and 4*mortality.

Cluster analysis using the Bray-Curtis similarity index was found to give results easy to interpret for the pros and cons of the different approaches. Analysis was conducted in PRIMER v6.0 with significant groupings identified using SIMPROF [START_REF] Clarke | Change in marine communities: an approach to statistical analysis and interpretation[END_REF].

Estimates of the potential final mortality of corals were obtained from the size class data based on the assumption that minimum mortality from the event would be equivalent to current levels of partial and full mortality. This was obtained by subtracting the sum of partial and full mortality from the total counts for all size classes for each genus. By contrast, the assumption that all currently bleached corals would die gives an estimate of maximum mortality from the event, thus subtracting the counts of bleached corals plus partial and fully dead corals. We assumed that pale corals survived.

RESULTS

Thirty-four sites were surveyed overall, in seven reef zones (Table 1, Fig. 2). In total, 8,439 colonies were sampled. Coral cover varied from a maximum of almost 80% on the offshore Banc d'Iris, between 35-45% for outer barrier, inner barrier and small-island reefs within the lagoon, slightly over 30% for fringing reefs on the main island, and <30% in the Passe en 'S', the only channel site surveyed (Fig. 3A). Nevertheless, coral cover was not significantly different by reef zone (One Way ANOVA, (F = 1.338, P = 0.275). No patterns were observed, nor have been reported in the literature, of differential coral genus distributions around the island unrelated to reef zone, so all sites were lumped together in subsequent analyses, to focus on patterns in bleaching by coral traits (taxonomy, colony size). Across all sites, average coral colony abundance was 2327 (±1,253) and area was 41 m 2 (±21.9) per 100 m 2 of reef. By abundance, 35% of coral colonies were affected by the bleaching event, with 18.4% being pale, 9.3% bleached, 2% partially dead and 5.3% dead (Fig. 3B). By area, 52% of coral area or biomass was affected by bleaching, with 19.3% being pale, 10.7% bleached, 4.8% partially dead and 17.5% dead. A total of 60 coral genera were recorded (Fig. 4A), with clear dominance by Acropora, followed by Porites. All of the most abundant nine genera, and 17 of the top 25, were among those targetted for size class sampling (Fig. 4B). Genera omitted from size class sampling, but that were present at moderate levels of abundance included Leptoseris (rank = 10), Echinophyllia (11), Halomitra (15), Mycedium (17), Pachyseris (18) and Podabacia (20). Acropora contributed 58% of the total area of sampled corals (Fig. 4B), about five times greater than Porites, the second genus. Sixteen genera were recorded at abundances between 1-10 m 2 per 100 m 2 of reef, with four at low abundance between 0.1-1 m 2 , and two at very low abundance (Stylophora and Turbinaria). For the genera sampled using both methods, the logarithm of the area sampled and their visually assessed relative abundance index were strongly correlated (r 2 = 0.894), though some genera switched ranks between the two methods (Fig. 4).

Bleaching and mortality varied widely across genera, from 100% bleached in Turbinaria to 6% pale in Acanthastrea. Genera with fewer than 20 colonies sampled (i.e., Turbinaria, Oxypora, and Stylophora) are excluded from further analysis, so the following results are for the 20 remaining genera. The most susceptible genera to combined bleaching and mortality (Fig. 5A) were Pocillopora and Montipora (>70% total impact) followed by Lobophyllia, Porites (massive species), Dipsastrea, Goniastrea, Acropora and a range of others (40-65%). Leptastrea and Acanthastrea showed lowest levels of combined pale and bleached colonies (≈6%), with no mortality. Pocillopora and Acropora were the only genera with high levels of mortality (47% and 32%, respectively, combined partially dead and dead), though mortality was also observed in Porites (massive and branching species), Echinopora, Favites, Seriatopora, Platygyra, and Hydnophora. Pocillopora showed the most extensive and complex response (18% pale, 14% bleached, 21% partially dead and 26% dead), while Acanthastrea was the least impacted (6% pale). Most genera showed a higher degree of paling (up to 50% in Dipsastraea) than full bleaching (up to 31% in Montipora). Simplified into three categories (unaffected, bleached and dead) and illustrated in a ternary plot, Acropora and Pocillopora showed intermediate levels of all three (Fig. 5B). Other genera occupy a zone near the axis marking low mortality, at varied levels of unaffected and bleached.

Cluster analyses showed the impact of excluding unaffected colonies, and the use of weighting coefficients (Fig. 5C). Without weights, including 'unaffected' as a category distinguished three significant clusters (case 1): (a) Acropora and Pocillopora due to their high combined bleaching and mortality, (b) a large group of intermediate genera showing some bleaching and limited mortality, and (c) a smaller group of five genera mostly affected by bleaching, but at lower abundance. Excluding 'unaffected' (case 2) shifted the boundary between (b) and (c) such that (c) was reduced to two genera (Leptastrea and Acanthastrea), the other three genera shifting into group (b). Using weighting coefficients and including 'unaffected' (case 3) altered case 1 by splitting the five low-response genera into two groups-the three that were pushed into group (b) in one group, and Acanthastrea and Leptastrea in a separate group. Finally, using weighting coefficients but excluding 'unaffected' (case 4) put all of the genera except Acanthastrea and Leptastrea into a single group, and these two in their own small group.

Two conclusions can be drawn from the above. First, that including normal or 'unaffected' colonies in field counts and analysis is important, as cases 2 and 4 both showed less discrimination of bleaching responses (less discrimination of the first order spatial patterns in Fig. 5B) than cases 1 and 3, respectively. Second, that including weighting coefficients alters the results, but varying with the set of variables included. Comparing cases 2 and 4, adding weighting coefficients worsened the result, removing the distinction between Acropora and Pocillopora (bleaching with mortality) from the main group of genera (paling and bleaching, no mortality). Comparing cases 1 and 3, adding weighting coefficients added resolution in the low-response low-abundance group, distinguishing Leptastrea and Acanthastrea (with only minor paling and bleaching, <6%, Fig. 5A) from Porites (branching), Favites and Seriatopora (from 20-33% impacted, and varied amounts of paling, bleaching and mortality). While the latter result adds value to interpretation, weighting should be used with caution and tested on each dataset to which it is applied, to fully understand how it influences the results. By area, the coral community was dominated by mature colonies in the 81-160 cm size class (Fig. 6A) followed by younger colonies from 11-80 cm and then the largest colonies above 1.6 m. Numerically, 6-10 and 11-20 cm corals were most abundant. Bleaching and mortality varied by colony size (Fig. 6B). Both bleaching and mortality showed a strong linear increase with coral colony size, up to 1.6 m. Of the smallest colonies only 17% were affected by bleaching, with 3% suffering mortality. These proportions increased progressively to the most impacted size class, 81-160 cm, for which 67% of colonies were affected with 35% mortality. The sample size for corals >160 cm was low, with 20 and 1 in the 160-320 and >320 cm size classes, respectively, compared to from 131 to 4257 colonies in the smaller size classes. For corals <10 cm, the ratio of mortality to bleaching was <0.2, which increased to 0.3-0.5 for intermediate colonies from 10-80 cm. For larger corals, 81-160 and 161-320 cm, mortality exceeded bleaching, with ratios of 1.6 and 1.1 respectively.

DISCUSSION

Bleaching and mortality averaged 30 and 22%, respectively, across all sites (Fig. 3B) though with high levels of variability from near zero at sites in the south to maximum mortality levels at sites in the north and east of Mayotte. These results agree largely with [START_REF] Eriksson | Coral bleaching and associated mortality in Mayotte, Western Indian Ocean[END_REF] who found 10% bleached and 40% dead corals on the more highly impacted northern and eastern reefs during May 2010. Percent coral cover varied over a wide range across reef zones in Mayotte (Fig. 3A) though without statistical significance. Accordingly, we aggregate the coral community of Mayotte to analyze variance in bleaching response among genera.

Do genus and size affect coral bleaching?

Pocillopora and Acropora were the only coral genera to show significant mortality as a result of the 2010 event, setting them apart from the others as the most susceptible genera to bleaching, in accordance with findings throughout the Indo-Pacific [START_REF] Marshall | Bleaching of corals on the Great Barrier Reef: differential susceptibilities among taxa[END_REF]Obura, 2001;[START_REF] Loya | Coral bleaching: the winners and the losers[END_REF][START_REF] Mcclanahan | Comparing bleaching and mortality responses of hard corals between southern Kenya and the Great Barrier Reef, Australia[END_REF][START_REF] Van Woesik | Revisiting the winners and the losers a decade after coral bleaching[END_REF]. Cluster analysis of the bleaching responses identified two additional groups of species-a large group with intermediate responses characterized by variable but low mortality and variable levels of pale and bleached colonies (group 'b', case 1, Fig. 5C), and a smaller group with low levels of bleaching and variable but low mortality (group 'c'). The middle group included a wide range of genera including Porites (massive species), various merulinids, agariciids, siderastreids, and fungiids. The last group included Porites (branching species), Favites, Seriatopora, Acanthastrea, and Leptastrea. These groups are in broad agreement with previous reports from East Africa (e.g., massive Porites is usually ranked among the least susceptible to bleaching and mortality, branching Porites among the most susceptible -Obura, 2001;[START_REF] Mcclanahan | The relationship between bleaching and mortality of common corals[END_REF]. We have not analyzed the symbiont characteristics of genera sampled here, which has significant effects on bleaching susceptibility and mortality, particularly when different host-symbiont combinations may be possible (e.g., see Stat et al., 2009;[START_REF] Baker | Multiple symbiotic partnerships are common in scleractinian corals, but not in octocorals: comment on Goulet (2006)[END_REF]Oliver & Palumbi, 2009). Overall, though this appeared to be a severe bleaching event for reefs in Mayotte due to the dominance of Acropora and consequent loss of total coral cover, and visual dominance of bleached and dead Acropora and Pocillopora, the thermal stress to other genera was not sufficient to result in significant mortality.

A clear size differential in bleaching and mortality susceptibility was found, with susceptibility being greatest in large corals. For corals in the three smallest size classes (Fig. 6B) the proportion of bleached colonies increased (15 to 25%) with size, with little increase in the proportion of mortality (<5%). For adult corals from 10 cm and above, the proportion of bleached corals remained relatively stable (25%), but the proportion of mortality increased (10 to 40%), and exceeded the proportion bleached. This response was strongly determined by the size-dependent response of Acropora, but it did occur in other genera (Fig. 7H). Lower bleaching levels in small colonies, and particularly recruits, has been noted in other locations and coral taxa, particularly Oculina patagonica in the Mediterranean [START_REF] Shenkar | Size matters: bleaching dynamics of the coral Oculina patagonica[END_REF], and three dominant species in the Florida Keys (Colpophyllia natans, Montastrea faveolata, and Siderastrea siderea, [START_REF] Brandt | The effect of species and colony size on the bleaching response of reefbuilding corals in the Florida Keys during the 2005 mass bleaching event[END_REF]. However, in a mild bleaching event Ortiz, Del C Gomez-Cabrera & Hoegh-Guldberg (2009) found no relation of size to bleaching extent, and [START_REF] Bak | Population structure as a response of coral communities to global change[END_REF] predicted that bleaching impacts would be selectively higher on small rather than large corals.

From a methodological point of view, these results suggest several important considerations for measuring the impact of bleaching events at colony level, and extrapolating to community level indices. First, colony size is important in the bleaching susceptibility of corals for two reasons-smaller corals are less susceptible to bleaching, and larger corals have an exponentially greater contribution to biomass and area. Second, it is important to include unaffected colonies in the counts, as this improves identification of differential bleaching responses (Fig. 5C), and without total numbers of colonies the prevalence of bleaching and mortality in relation to the total population cannot be determined. Third, a corollary of the first two, is the importance of an unbiased sample of all corals, unaffected and bleached, and across all size classes. Without a strict fixed-area sampling method, such as with a physical transect or quadrat as a guide, unconstrained or haphazard counts are likely to both (a) oversample colonies that show a response over unaffected ones, as observers will tend to count what they are looking for (bleached and dead corals) and many normal colonies are brown and inconspicuous, and (b) small colonies will always be under-sampled compared to larger colonies, particularly when including sizes under 10 cm. Fourth, differential weighting of bleaching and mortality categories can have a strong influence on results, so should be done with caution in each case. In our results it improved discrimination of low-response groups when all colonies (impacted and unaffected) were included, but when unaffected corals were excluded it worsened the result. Thus where sampling is not restricted by physical quadrats or transects, and the size of corals is not measured (one and three above), the impact of weighting on results may not be possible to ascertain.

Estimating the impact of a bleaching event

The surveys took place in early June, over 1 month after the end of peak temperatures at the end of April 2010 (Fig. 1), so this dataset likely presents peak levels of combined bleaching and mortality. This is a common challenge in interpreting the eventual impact of a bleaching event, as surveys are often targeted for peak bleaching conditions, but unless follow up surveys are done, it cannot be known if bleached corals recovered or died. However, the envelope of possible outcomes of a bleaching event can be estimated from peak bleaching levels assuming further mortality from bleaching is either zero or maximal. That is, currently bleached corals all recover, or die, respectively.

The present dataset allows this to be done across coral size classes and genera, to estimate minimum and maximum potential impact of the bleaching event (Fig. 7). By number of colonies the minimum and maximum loss of corals appears minor (Fig. 7A) but is significant by area (Fig. 7B), particularly for 81-160 cm corals. Maximum mortality by colony abundance is estimated at 10-15% for Acropora, Porites (massive) and other genera (Fig. 7C), but at 50% for Pocillopora. By contrast, maximum mortality by colony area is estimated at 40% for Acropora, 25% for Porites (massive), <20% for other genera (Fig. 7C), but at >60% for Pocillopora. By colony abundance, maximum mortality overall is estimated at 16%, but by colony area, 32%, a major difference in result affected by consideration of colony area. This is because the bleaching event preferentially impacted larger colonies, paticularly Acropora, eliminating the dominance of 81-160 m colonies (Fig. 7D). Reefs in Mayotte are strongly dominated by staghorn and tabular Acropora species, and dominance of the 81-160 cm size class in April suggests the community was approaching maturity where the community would be dominated by even larger stands of tabular and staghorn Acropora. The bleaching event strongly flattened the size class distribution with similar area in 41-320 cm classes after the event.

Pocillopora populations were strongly dominated by 21-40 cm and 11-20 cm colonies pre-bleaching (Fig. 7E), providing the corresponding peak in the overall population curve (Figs. 5A, 6B); their area was reduced by about 60% by the bleaching event (Fig. 7C), though they still remained dominant over other size classes of Pocillopora. The population of massive Porites was strongly dominated by 81-160 cm corals, which remained dominant after bleaching. The remaining genera were most strongly represented by 11-40 cm colonies, and mortality was relatively minor and evenly spread across them, maintaining the same size class distribution following bleaching. Finally, estimated maximum mortality increased with size for all corals and for the three key genera Acropora, Pocillopora, and Porites (Fig. 7H). Interestingly, these three genera all show a decline in mortality for their largest size class.

Recurring bleaching in Mayotte

The coral reefs of Mayotte have been impacted by multiple significant bleaching events. In May-June 1983 bleaching was documented at 0-18% for fringing reefs, 30-45% for lagoon reefs and 30-75% for outer barrier reefs, though an indication of the final mortality was not noted [START_REF] Faure | Sur un phénomène remarquable de blanchiment et de mortalité massive des madréporaires dans le complexe récifal de l'île Mayotte (SW Ocean Indien)[END_REF]. In 1998, 36% of the reefs had not recovered from the 1983 bleaching event, and mortality of >80% of Acropora tables on outer reef slopes was reported from April to August (Quod et al., 2002), which judging by patterns in this study may have reflected approximately 50% mortality including inner reefs. The 2010 event recorded here resulted in 32% mortality of all corals, and just under 40 % for Acropora alone (Fig. 7C).

Observations in 2016 suggest that mortality of corals was between 25 and 50% using the same methods used here (by area, D Obura, pers. obs., 2016), and was recorded at 10-30% by colony number using other methods [START_REF] Nicet | Intensité et impact du blanchissement corallien massif de 2016 sur les récifs coralliens français de l'océan Indien. Projet BECOMING 2016 Rapport MAREX[END_REF], so roughly comparable to the 2010 event.

In all four events, higher impact occurred on outer barrier reefs and lower levels in the lagoon and on fringing reefs, and the most dramatic bleaching and mortality was of tabular and staghorn Acropora colonies on outer reefs. Because of its dominance of the coral community, Acropora's response to thermal stress dominated the overall community response. The intervals between these major bleaching events has progressively declinedfrom 16 years to ten and six years. Across all of these events the reefs have apparently achieved considerable recovery and maintained the same dominance by Acropora. This suggests a high degree of community resilience, likely partly a result of high levels of connectivity [START_REF] Crochelet | A modelbased assessment of reef larvae dispersal in the Western Indian Ocean reveals regional connectivity patterns-potential implications for conservation policies[END_REF] due to the complex eddies that maintain high selfseeding of reefs within the northern Mozambique channel (Obura et al., 2018;[START_REF] Bigot | The French territories in the South Western Indian Ocean[END_REF].

However, anthropogenic stresses in Mayotte from both fisheries and water quality degradation [START_REF] Wickel | Les récifs coralliens frangeants de l'île de Mayotte (Grande Terre): bilan de l'état de santé en 2004 et évolution depuis[END_REF] are increasingly evident in the greater prevalence of coral disease and chronic mortality from unknown sources (D Obura, pers. obs., 2016). This will likely undermine the natural resilience of the reefs (Obura, 2005;[START_REF] Hughes | Rising to the challenge of sustaining coral reef resilience[END_REF] and may reduce their ability to recover from the 2016 bleaching event.

Between latitude S 12 -13.5 and longitudes E 44.5-45.5 (a box around Mayotte) SST has warmed by 0.096 • C per decade for the thirty-year period from 1981-2010(Reynolds et al., 2002)), slightly less than the global level of 0.147 • C per decade (Rayner et al., 2003). Mayotte is in the region with the lowest SST rise in Eastern Africa (McClanahan et al., 2007a). [START_REF] Donner | Coping with commitment: projected thermal stress on coral reefs under different future scenarios[END_REF] estimated a rate of SST rise to which corals must adapt to avoid catastrophic coral decline, of 1.5 • C in 50-80 years. This is 0.2-0.3 • C per decade, some two to three times higher than the rise in temperatures that coral reefs in Mayotte have experienced over the course of four bleaching events. While Mayotte's reefs have shown remarkable resilience so far, it is not clear that they are acclimating or adapting sufficiently to the rise experienced of 0.1 • C per decade. Further, the shorter intervals for recovery between events, matching the global pattern now at sub-decadal levels [START_REF] Hughes | Spatial and temporal patterns of mass bleaching of corals in the Anthropocene[END_REF], is approaching limits for recovery consistently used in framing the onset of 'catastrophic' bleaching [START_REF] Sheppard | Predicted recurrences of mass coral mortality in the Indian Ocean[END_REF][START_REF] Van Hooidonk | Local-scale projections of coral reef futures and implications of the Paris agreement[END_REF].

CONCLUSION

The most recent analysis identifies 2030 as approximately the year in which Mayotte will experience Annual Severe Bleaching under RCP 8.5 (business as usual scenario, equivalent to today's CO2 emission rates; [START_REF] Van Hooidonk | Local-scale projections of coral reef futures and implications of the Paris agreement[END_REF]. That the reefs are already experiencing decadal severe bleaching only 15-20 years earlier is strong indication that the trajectory for coral reefs in the region towards decline may be inexorable on the time scales at hand, and that the reefs cannot withstand annual occurrence of the scale of bleaching documented in 1983, 1998and 2010, and repeated in 2016[START_REF] Nicet | Intensité et impact du blanchissement corallien massif de 2016 sur les récifs coralliens français de l'océan Indien. Projet BECOMING 2016 Rapport MAREX[END_REF]. Although the Northern Mozambique Channel may be a center of diversity and of key significance to the Western Indian Ocean at large (Obura, 2012;Obura et al., 2018), it may not be a refuge from warming for coral reefs [START_REF] Mcclanahan | Biogeography and change among regional coral communities across the Western Indian Ocean[END_REF] and urgent and emergency planning is needed to identify what can be done to secure the best possible future not just for the reefs of Mayotte or the WIO, but also more broadly on a global scale [START_REF] Beyer | Long-term risk-sensitive planning for conserving coral reefs under rapid climate change[END_REF].

(http://oceans.taraexpeditions.org). This article is contribution number 77 of the Tara Oceans Expedition 2009-2012.

Figure 1

 1 Figure 1 Sea Surface Temperature (SST) in the Western Indian Ocean, by month, from January to July 2010. Mean monthly level 3 data at 4 km resolution, source: Moderate Resolution Imaging Spectroradiometer (MODIS), https://oceancolor.gsfc.nasa.gov/cgi/l3. Full-size DOI: 10.7717/peerj.5305/fig-1

Figure 2

 2 Figure 2 Map of Mayotte, showing the reef and lagoon structure, and sampling sites coded by reef zone (UNEP-WCMC, 2010). Full-size DOI: 10.7717/peerj.5305/fig-2

Figure 3

 3 Figure 3 Coral cover and condition in Mayotte, June 2010. (A) Coral cover by reef zone (mean ± se). The number of sites sampled in each zone is shown in parentheses in the x axis labels. (B) Proportional bleaching and mortality of corals in Mayotte, in June 2010, by number and area of colonies sampled. Full-size DOI: 10.7717/peerj.5305/fig-3

Figure 4

 4 Figure 4 Coral genera abundance, Mayotte, 2010. (A) Relative abundance of all coral genera identified, aggregated across sites. The index of relative abundance ranges from a maximum of 5, with zero representing absence. Site level scores were: rare (1), uncommon (2), common (3), abundant (4) and dominant (5). (B) Area (biomass) of each genus measured in size class transects (in m 2 per 100 m 2 of reef area), ranked from highest to lowest. Genera sampled in both methods (A and B) shaded red, genera sampled only in visual estimates (in A) shaded blue. Full-size DOI: 10.7717/peerj.5305/fig-4

Figure 5 Obura

 5 Figure 5 Bleaching and mortality of coral genera. (A) Proportion of bleaching and mortality by genus, excluding the two least abundant genera. (B) Ternary plot of unaffected, bleached (pale plus bleached) and dead (partial plus full mortality) for genera sampled in the study. Key genera are indicated using three-letter codes that correspond to the first three letters of genus names in (A) and (C), except for 'fat' (Favites), 'mtp' (Montipora), 'porb' (branching Porites) and 'porm' (massive Porites). (C) Cluster analysis results with SIMPROF test to show significant clusters of genera at p = 5% level for four cases: unweighted proportions of pale, bleached, partial and full mortality analyzed with (case 1) and without (case 2) the proportion of unaffected colonies; and with weights applied to bleached (x2), partial mortality (x3) and full mortality (x4) (see methods), also with (case 3) and without (case 4) the proportion of unaffected colonies. Letters show significant groups within each test. Full-size DOI: 10.7717/peerj.5305/fig-5

Figure 6 OburaFigure 7

 67 Figure 6 Bleaching and mortality of corals, by colony size. (A) Size class distributions of corals from all sites sampled, by number of colonies (left axis, closed circles) and area (in m 2 ) of colonies (right axis, open triangles) per 100 m 2 of reef area. (B) Overall bleaching and mortality proportions by colony area in each size class . Data is combined across all coral genera sampled in Mayotte, in June 2010. Full-size DOI: 10.7717/peerj.5305/fig-6

Table

1 Sampling details of Tara Oceans Expedition to Mayotte, 2010.

  The table shows the depth characteristics of each site, the area of small coral quadrats and large coral transects samples, actual number of coral colonies counted, and standardized number and area of colonies (to 100 m 2 ).

	SITE			Area sampled (m 2 )	# colonies	Standardized to 100 m 2
		Zone	Depth	<10 cm	>10 cm	counted	# colonies	area (m 2 )
	MA05	bank	20	3	12	132	1,625	68.5
	MA24	bank	20	6	25	97	527	89.0
	MA03	outer barrier	10	3	13	299	5,691	59.8
	MA14	outer barrier	10	3	10	242	3,983	48.4
	MA15	outer barrier	9	5	20	347	2,980	30.8
	MA16	outer barrier	10	5	22	303	2,366	75.2
	MA18	outer barrier	10	3	12	338	5,467	70.5
	MA22	outer barrier	11	6	25	208	1,529	42.8
	MA27	outer barrier	16	6	25	217	1,349	30.7
	MA28	outer barrier	16	6	25	328	2,832	19.6
	MA29	outer barrier	12	6	25	315	2,666	20.2
	MA30	outer barrier	12	6	25	357	2,644	24.4
	MA19	channel	10	5	20	256	2,165	27.0
	MA08	inner barrier	10	4	20	341	3,045	40.2
	MA10	inner barrier	10	3	11	248	4,824	36.7
	MA32	inner barrier	6	6	25	186	1,175	90.3
	MA12	double barrier	8	4	15	173	1,923	27.6
	MA13	double barrier	10	3	10	190	2,880	47.6
	MA04	inner	10	4	20	246	2,430	43.8
	MA06	inner	10	5	20	305	2,665	72.6
	MA07	inner	10	5	20	361	2,945	41.0
	MA17	inner	9	6	25	208	1,351	17.9
	MA20	inner	8	6	25	218	1,404	48.2
	MA23	inner	11	6	25	170	1,643	22.4
	MA25	inner	10	6	25	513	3,673	36.4
	MA26	inner	11	6	25	236	1,932	10.5
	MA34	inner	9	12	50	308	1,028	31.3
	MA01	fringe	10	2	7	112	2,243	41.5
	MA02	fringe	10	6	25	144	1,083	14.2
	MA09	fringe	10	6	25	209	1,710	13.0
	MA11	fringe	8	5	23	144	1,049	32.2
	MA21	fringe	8	6	25	231	1,076	25.8
	MA31	fringe	7	6	25	173	1,135	28.8
	MA33	fringe	4	6	25	284	2,073	74.3
				Totals			Averages:
				176	730	8,439	2,327	41.3

Obura et al. (2018), PeerJ, DOI 10.7717/peerj.5305 5/22

Obura et al. (2018), PeerJ, DOI 10.7717/peerj.5305

Obura et al. (2018), PeerJ, DOI 10.7717/peerj.5305 20/22

ACKNOWLEDGEMENTS

We are grateful to Etienne Bourgois, the Tara schooner and its captain Hervé Bourmand, Mathieu Oriot (Diving Officer), the Tara crew, the Oceans Consortium, and James Mbugua. Tara Oceans would not exist without continuous support from 23 institutes

Data Availability

The following information was supplied regarding data availability:

The raw data are provided in a Supplemental File.

ADDITIONAL INFORMATION AND DECLARATIONS Funding

Funding for David Obura was provided by the Western Indian Ocean Marine Science Association, Grant No: MASMA/OR/2008/05. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Grant Disclosures

The following grant information was disclosed by the authors: Western Indian Ocean Marine Science Association: MASMA/OR/2008/05.

Competing Interests

The authors declare there are no competing interests.

Author Contributions

• David O. Obura conceived and designed the experiments, performed the experiments, analyzed the data, contributed reagents/materials/analysis tools, prepared figures and/or tables, authored or reviewed drafts of the paper, approved the final draft.

• Lionel Bigot performed the experiments, contributed reagents/materials/analysis tools, authored or reviewed drafts of the paper, approved the final draft.

• Francesca Benzoni conceived and designed the experiments, approved the final draft, expedition leader and funding.

Supplemental Information

Supplemental information for this article can be found online at http://dx.doi.org/10.7717/ peerj.5305#supplemental-information.


Eitan Altman

Alexandre Reiffers

Daniel S Menasch

Mandar Datar

Swapnil Dhamal

Corinne Touati

Mining competition in a multi-cryptocurrency ecosystem at the network edge: a congestion game approach

We model the competition over several blockchains characterizing multiple cryptocurrencies as a non-cooperative game. Then, we specialize our results to two instances of the general game, showing properties of the Nash equilibrium. In particular, leveraging results about congestion games, we establish the existence of pure Nash equilibria and provide efficient algorithms for finding such equilibria.

INTRODUCTION

The blockchain is a distributed synchronized secure database containing validated blocks of transactions. A block is validated by special nodes called miners and the validation of each new block is done via the solution of a computationally difficult problem, which is called the proof-of-work puzzle. The miners compete against each other and the first to solve the problem announces it, the block is then verified by the majority of miners in this network, trying to reach consensus. After the propagated block reaches the consensus, it is successfully added to the distributed database. The miner who found the solution receives a reward either in the form of cryptocurrencies or in the form of a transaction reward.

Because of the huge energy requirement necessary to be the first to solve the puzzle, blockchain mining is typically executed in specialized hardware. In [START_REF] Zhang | Offloading in software defined network at edge with information asymmetry[END_REF] an Edge computing Service Provider (ESP) is introduced to support proof-ofwork puzzle offloading by using its edge computing nodes. In [START_REF] Xiong | Edge computing resource management and pricing for mobile blockchain[END_REF] a game is formulated between the miners in the presence of a single ESP and then a Stackelberg game is used to compute the pricing that maximizes the revenue of the ESP.

Our work addresses the following two questions: 1) given a single blockchain, how should rational users contribute to the mining process, possibly counting on thirdparty ESPs or mining pools to offload infrastructure costs?

2) given multiple blockchains, e.g., in a multi-cryptocurrency ecosystem, how should rational users distribute their monetary and/or computational budget towards mining?

In this note, we focus on the competition between miners while addressing the two questions above. We model the competition over several ESPs and over several blockchains characterizing multiple cryptocurrencies as a non-cooperative game. Then, we specialize our results to two instances of the general game, showing properties of the Nash equilibrium.

In the first game, there is a single blockchain (e.g., cryptocurrency) and any of the M ESPs (or mining pools) can be used by the miners to solve the puzzle. In the second game, we consider K opportunities, each of which corresponding to another blockchain. At each time slot of duration T (which corresponds to a new puzzle to be solved) each of the miners decides which of K puzzles to solve. We formulate both games and establish conditions for the existence of a pure Nash equilibrium for the association problem between miners and ESPs, providing an efficient algorithm for solving it. We summarize our contributions as follows:

Congestion game for mining competition: we model the competition among users searching for a solution to the mining puzzle as a game (Section 2). In essence, as the number of users willing to mine increases, the chances that a given user is the first to succeed in solving the mining puzzle and wins a reward decreases (i.e., the system becomes congested ). In particular, we assume that users can count on third-parties to offload infrastructure costs, and can mine multiple cryptocurrencies. Under the assumption that such third-parties are roughly indistinguishable, we further show that when there is one single cryptocurrency of interest the congestion game admits a simple equilibrium accounting for users that must decide whether to mine or otherwise not join the system (Section 3).

Analysis of multi-cryptocurrency ecosystem: we analyze the congestion game involving multiple cryptocurrencies. In that case, miners compete against those that decide to mine the same cryptocurrency (Section 4). We show that the proposed game admits a potential, and discuss a number of extensions, such as accounting for dynamic puzzle complexity and mining pools (Sections 5 and 6).

BLOCKCHAIN COMPETITION GAME

We consider a population of M ESPs and a set of K blockchains. We assume that in ESP m, the amount of service R k,m,i requested by miner i to solve puzzle k is exponentially distributed with expectation 1/µ k,m . R k,m,i are independent RVs. Thus, if there are k,m miners associated to ESP m mining currency k, the time it takes for the fastest of them to solve the puzzle corresponding to currency k is exponentially distributed with expectation 1/( m µ k,m k,m ).

Miners, mining servers and puzzles. We denote by N := {1, 2, . . . , N } the set of miners. There is a finite population of miners, and if a miner changes his strategy this will cause a change in the utilities of other miners. Let K := {1, 2, . . . , K} be the set of puzzles, each of which associated with a different cryptocurrency that the miners are trying to solve. We assume that each cryptocurrency corresponds to exactly one puzzle. Let M := {1, 2, . . . , M } denote the ESPs, also referred to as mining servers, that miners can rely on. Notation is summarized in Table 1.

Strategies. Set Si ⊂ K × M denotes the set of ordered pairs (puzzle, ESP), corresponding to ESPs that miner i can rely on to solve puzzles of a given type. The set Si can differ across miners due to political or economic restrictions. For instance, certain countries do not allow investment in certain cryptocurrencies. Alternatively, the set of available ISPs for two different miners may not be the same. A strategy for miner i is denoted by si ∈ Si, corresponding to the puzzle (cryptocurrency) which the miner intends to solve using a given infrastructure. A strategy vector s := (si)i∈N produces a load vector := ( k,m ) k,m .

Rewards, costs and utilities. Let η k be the load of miners across all ESPs towards cryptocurrency k. Then,

η k := m ∈M k,m µ k,m . (1)
Recall that for a given load vector , the time to solve the puzzle of the k th cryptocurrency is exponentially distributed with expectation 1/η k . Let q k be the probability that puzzle k is solved by time T ,

q k = 1 -exp(-T η k ). (2)
Let pk,m denote the probability that a miner using ESP m is the first to solve puzzle k. Then, pk,m = q k k,m µ k,m /η k . Throughout this paper, 0/0 = 0. In the expression of pk,m , for instance, if η k = 0 and k,m = 0, then pk,m = 0/0 = 0. The probability that a given miner using ESP m is the first to solve puzzle k is

p k,m = 1 k,m >0 q k µ k,m /η k , (3)
where 1c equals 1 if condition c holds and 0 otherwise. Let U k,m ( ) denote the utility to a miner who tries to find the solution of the current puzzle associated to cryptocurrency k, using ESP m. The utility is the weighted sum between the probability to solve first the puzzle and the cost associated with it. We denote by γ k,m the cost of mining blockchain k at ESP m. Thus,

U k,m ( ) = p k,m -γ k,m if p k,m > γ k,m , 0 otherwise. ( 4
)
User i utility is Ũi(si, s-i) = (k,m)∈S i 1 s i =(k,m) U k,m (
), where s-i := (s1, s2, . . . , si-1, si+1, . . . , sN ) is the vector of strategies of all miners except miner i. Given the ingredients above, the blockchain competition game is characterized by

C = N , K × M, (Si)i∈N , (U k,m ) (k,m)∈K×M .
In Sections 3 and 4 we analyze two special instances of this game.

Congestion games and potentials

Next, we briefly introduce some basic background on congestion games, crowding games and potentials. Such background is instrumental in the analysis of the blockchain competition game that follows.

Congestion games [START_REF] Monderer | Potential games[END_REF] are equivalent to routing over an arbitrary graph, when all routed objects have the same size, and are non splittable. The cost of using an edge is the same for all players. Crowding games [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF] are congestion games In our setup, the routed object is the mining power. In the simplest setup, wherein we have one single cryptocurrency, the vertices of the graph are users and ESPs (with a virtual ESP corresponding to the option of not mining). The cost incurred by a user who decides to mine through a given ESP is the cost of an edge between the user and the ESP.

A congestion game without player specific payoff functions is guaranteed to admit a standard potential and a pure equilibrium [START_REF] Monderer | Potential games[END_REF]. A game that does not admit a standard potential may still admit an ordinal potential. A game with an ordinal potential can have any finite subset of actions available to a player, still admitting a pure equilibrium.

Milchtaich [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF] proves the existence of a pure Nash equilibrium given user dependent costs in crowding games. In this paper, we are interested in user dependent strategy sets. Nonetheless, one can show an equivalence between user dependent costs and user dependent strategy sets, and henceforth we use interchangeably the two notions.

ESP ASSOCIATION GAME

In this section, we introduce the ESP association game and analyze some properties of its equilibria.

Next, we consider the special case where we have only one cryptocurrency, which we denote by . Given vector , where ,m denotes the number of miners associated to ESP m, the time T (1) when the fastest solves the puzzle is exponentially distributed with expectation of E(T (1) ) = 1/η , where η := m ∈M ,m µ ,m . Note that if ,m = 0 for all m then η = 0 and E(T (1) ) = ∞. In Section 5 we discuss heuristic strategies adopted by existing cryptocurrencies to prevent the time to solve a puzzle to grow unboundedly.

The probability that a given miner associated with ESP m wins is given by (3), p ,m = 1 ,m >0 q µ ,m/η where q is given by [START_REF] Eyal | The miner's dilemma[END_REF].

In what follows, we analyze the ESP game proposed above. We assume that a miner has always the option of not associating to any ESP and in that case its utility is zero.

Theorem 1 (existence).

A pure Nash equilibrium exists. We denote it by s * := (s * i )i∈N .

Proof. First, consider the case where for all i and j, Si = Sj and si does not depend on i. Then, the ESP association game is a congestion game, in the sense of [START_REF] Rosenthal | A class of games possessing pure-strategy Nash equilibria[END_REF], and the theorem follows. Otherwise, the game is a crowding game, in the sense of [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF]. As shown in [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF], as the game has playerspecific payoffs it may not admit a standard potential [START_REF] Monderer | Potential games[END_REF], but it still admits pure Nash equilibria.

Let

be the number of miners that decide to associate to an ESP, = M m=1 i∈N

1 s * i =( ,m) . (5)
Then, N -is the number of users that decide not to mine. When all µ ,m are equal we denote them by µ . Then, equation (3) reduces to

p ( ) = 1 >0 (1 -exp(-T µ ))/ . ( 6
)
where p is the probability that a user that decides to connect to an ESP is the first to solve the puzzle. Let γ ,m be a fixed cost for associating with ESP m.

In the following theorem, we assume that µ ,m and γ ,m are the same for all m (therefore µ ,m = µ and γ ,m = γ for all m). Let the utility for a miner associating to ESP m be given by (4).

Theorem 2 (no player-specific strategies). If for all i and j, Si = Sj and si does not depend on i, the Nash equilibrium is given by the solution of the following optimization problem,

argmin s Φ(s) := l=1 p (l) -γ * (7)
subject to:

≤ N, ≥ 0. ( 8
)
Equation ( 11) is the game potential function. The optimization problem (11)-( 8) is equivalent to a bin-packing problem with concave costs.

Proof. This is a congestion game in the sense of Rosenthal [START_REF] Rosenthal | A class of games possessing pure-strategy Nash equilibria[END_REF] and therefore has a potential. Indeed, in this game each player can decide to associate or not with an ESP. Thus all associations to the M ESPs can be aggregated to a single route that represents the choice of mining and the option of not associating represents the second route.

Theorem 3 (player-specific strategies). If si depends on the identity of user i, the game may not admit a standard potential, but still admits pure Nash equilibria.

Proof. The game is a crowding game, and the result follows from [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF][START_REF] Milchtaich | Crowding games are sequentially solvable[END_REF].

BLOCKCHAIN ASSOCIATION GAME

In this section, we introduce the multiple cryptocurrencies game and derive structural properties of the associated set of equilibria.

Here we assume that there are K cryptocurrencies. To simplify presentation, we consider a single ESP, and drop subscript m from all variables.

For a given load vector , the time it takes till the fastest puzzle to be solved is exponentially distributed with expectation 1/(µ k k ). Thus, the probability that a miner is the first to solve the puzzle is

p k ( k ) = (1 -exp(-T µ k k ))/ k (9)
Note that p k = 0 if k = 0 (recall that we assume 0/0 = 0 throughout this paper). The utility of a tagged miner to mine a cryptocurrency k when there are k miners associated with the same cryptocurrency (including the tagged miner) is given by ( 4), where

U k ( k ) = p k -γ k , if p k -γ k ≥ 0. ( 10
)
We add to it the constraint that a miner does not participate in solving the puzzle if its utility is negative. In that case the equilibrium is characterized by the condition k * k ≤ N, * k ≥ 0, for k = 1, . . . , K. This game is referred to as an elastic game. If the equilibrium vector * saturates the constraint

( k * k = N, * k ≥ 0, k = 1, ..., K) then for each k for which * k > 0, and each k , U k ( * k -1) ≥ U k ( * k + 1
). Similar theorems as those presented in the previous section establishing the existence of pure Nash equilibria and characterizing the equilibria still hold under the blockchain association game. The statements of the theorems and the proofs are similar to those in the previous section, and are omitted for conciseness. Recall that in Theorem 2 for all i and j, Si = Sj and si does not depend on i. Then, in this case the number of miners associated to each cryptocurrency * k = i∈N 1 s * i =k is now the solution of the following optimization problem,

argmin s Φ(s) := k∈K k l=1 p k (l) -γ k (11)
subject to:

k∈K k ≤ N, k ≥ 0. ( 12
)
Theorem 2 holds replacing ( 7)-( 8) by the equations above.

DISCUSSION

Positive and negative externalities. In the models proposed in this paper, we assumed that users who contribute to the system by mining cryptocurrencies generate negative externalities towards their mining peers. Indeed, the competition among miners is a very fundamental aspect of the mining process [START_REF] Dimitri | Bitcoin mining as a contest[END_REF]. Nonetheless, by incorporating more miners, the blockchain becomes more robust [START_REF] Garay | The bitcoin backbone protocol[END_REF]. Such robustness, in turn, may translate into an increase in the real value of the cryptocurrency under consideration [START_REF] Shah | Bayesian regression and Bitcoin[END_REF][START_REF] Raval | Ethereum Price Prediction[END_REF]. Therefore, by increasing the pool of miners, each miner is also contributing with positive externalities towards the system, and we leave such aspect as subject for future work.

Mining pools. Mining pools play a key role in todays' public blockchain systems [START_REF] Eyal | The miner's dilemma[END_REF]. 1 The competition analyzed in this paper applies to mining pools under two scenarios. First, from the perspective of the mining pool, it can use cloud resources for mining purposes. Therefore, the mining pools assume the role of players as considered in this work. Alternatively, the players are the end users, who contract mining pool services. Then, mining pools assume the role of ESPs. In the first case, we consider competition among mining pools, at the macro level, and in the latter case, we consider the micro-competition among end-users.

Multi-cryptocurrency ecosystem. In the cryptocurrency ecosystem, large mining pools typically decide, dynamically, which blockchain to mine. Such decisions are made based on different thresholds related to the value of the cryptocurrencies and the costs for mining (mining complexity). The churn of computational power across blockchains is a well-known source of price volatility, and different mechanisms have been developed to counteract migrations of miners across platforms [START_REF] Ulrich | Attacking Bitcoin[END_REF]. One of those mechanisms is referred to as emergency difficulty adjustment (EDA), which reduces the difficulty of the puzzle when there are not many miners in the system, preventing the blockchain from dying.

Puzzle complexity. In this work, we assume that the time to mine the next block is monotonically decreasing with respect to the number of miners. In Bitcoin, puzzle difficulty (complexity) is dynamically adjusted so that the time to mine a block varies between certain pre-established time bounds. Bitcoin target block generation rate is of 10 minutes.

In theory, due to the dynamic adjustment of puzzle complexity, Bitcoin throughput (number of blocks generated per time unit) does not depend on the number of miners. Nonetheless, in practice the throughput does vary over time and an increase in the number of miners can decrease the time between generation of blocks [START_REF] Meshkov | Revisiting difficulty control for blockchain systems[END_REF][START_REF] Wisdom | Bitcoin Difficulty[END_REF]. In [START_REF] Huberman | Monopoly without a monopolist[END_REF], the authors argue in favor of adjusting the frequency at which blocks are generated as a function of the congestion in the network. Our models can be extended to account for fixed and dynamic block generation rates, and we leave a further investigation of that subject for future work.

RELATED WORK

There is a vast literature investigating game theoretical aspects of blockchain systems [START_REF] Kiayias | Blockchain mining games[END_REF][START_REF] Spiegelman | Game of coins[END_REF][START_REF] Huberman | Monopoly without a monopolist[END_REF]. Nonetheless, the literature on congestion games applied to such systems is scarce. In particular, to the best of our knowledge, there is no prior work investigating the competition at the network edge among miners as a congestion game, and its connection to multi-cryptocurrency markets.

Congestion games have been applied in the field of networking to account for security aspects [START_REF] Maillé | Interplay between security providers, consumers, and attackers: a weighted congestion game approach[END_REF], link congestion [START_REF] Johari | Network resource allocation and a congestion game[END_REF] and pricing of infrastructures and users [START_REF] Hassin | Equilibrium threshold strategies: The case of queues with priorities[END_REF]. In [START_REF] Huberman | Monopoly without a monopolist[END_REF], the authors study Bitcoin as a congestion game, where the congestion occurs due to an increase in the number of transaction requests from users. In particular, the authors abstract away from several aspects of the competition between miners. In this paper, in contrast, we focus on the competition between miners.

Spiegelman et al. [START_REF] Spiegelman | Game of coins[END_REF] adopted the framework of congestion games to model competition between miners of multiples cryptocurrencies who try to maximize utilities by choosing which puzzle (cryptocurrency) to mine [START_REF] Spiegelman | Game of coins[END_REF]. The authors prove that there is no standard potential function for the game they propose, but that an ordinal potential always exists, implying that best response converges to a pure Nash equilibrium. Our work captures different aspects of the problem, and is complementary to [START_REF] Spiegelman | Game of coins[END_REF]. An important similarity between the two works consists of establishing conditions under which pure Nash equilibria exist even when the game does not admit a standard potential function. The major differences between our work and [START_REF] Spiegelman | Game of coins[END_REF] are: 1) in the modeling of the probability to succeed in solving a puzzle (see Section 5); 2) in the ESP decision, which is out of the scope of [START_REF] Spiegelman | Game of coins[END_REF]; 3) in the action space (mining power), which is continuous in [START_REF] Spiegelman | Game of coins[END_REF], precluding the use of crowding game results, and discrete in this paper, allowing us to rely on [START_REF] Milchtaich | Congestion games with player-specific payoff functions[END_REF] to prove existence of pure Nash equilibria. We refer the reader to [START_REF] Spiegelman | Game of coins[END_REF][START_REF] Ulrich | Attacking Bitcoin[END_REF] for additional references on the multi-cryptocurrency ecosystem and its security challenges.

CONCLUSION

We modeled the competition over several ESPs and over several blockchains characterizing multiple cryptocurrencies as a non-cooperative game. Then, we specialized our game to two cases: the ESP connection game and the cryptocurrency selection game. For each game, we showed properties of the Nash equilibrium. In particular, leveraging results about congestion games, we establish the existence of pure Nash equilibria and characterize such equilibria through problems that admit efficient algorithmic solutions.

Table 1 :

 1 Table of notation variable description k,m number of users mining blockchain k at ESP m p k,m ( ) probability that user is first to mine a block U k,m ( ) utility of user mining blockchain k at ESP m γ k,m mining cost associated to blockchain k at ESP m with more restricted topology (parallel links) but more general costs (user dependent).

For instance, https://miningpoolhub.com/.

Acknowledgment Research conducted within the context of the THANES Associate Team, jointly supported by Inria (France) and FAPERJ (Brazil).