C Adloff

M Chefdeville
email: chefdevi@lapp.in2p3.fr

A Espargilière

R Gaglione

Environmental study of a Micromegas detector

We report on measurements of the basic performance of a Micromegas detector for a digital hadronic calorimeter. Electron collection efficiency, energy resolution and gas gain were measured in various mixtures of Ar and CO 2 . Also the dependence of the gain on environmental variables (pressure, temperature), gas parameters (flow, mixing ratio) and geometry (amplification gap size) is studied. Eventually, predictions on the impact of these variables on the detection efficiency of thin Micromegas detectors are drawn.

Introduction

Calorimetry at a future electron collider

Several important physics measurements that could be realized at a future electron linear collider (ILC or CLIC) would require a very good jet energy resolution (3 % at 100 GeV) [1]. The Particle Flow Approach (PFA) has been proposed to reach such a resolution. It consists in measuring the energy of the each particle contained in a jet with the tracker or with the calorimeters depending on its charge. While charged particle energy is measured with the tracker (which is more precise than the calorimeters), neutral particle energy is deduced from the energy deposit in the calorimeter from which the contribution of charged particles is subtracted. This calls for finely segmented calorimeters with single shower imaging capability.

A digital hadronic calorimeter (DHCAL) is a candidate at a future linear collider which would balance the cost of an increased number of readout channels (with a cell size of 1 cm 2 and a total active area of 3•10 3 m 2 , 3•10 7 cells should be read out) by a simpler readout circuitry (1 bit information). The energy of a hadron would then be measured by summing up the hits, taking into account the calibration constant of the detector. A DHCAL can be instrumented with scintillating or gaseous layers. In the latter case, GEMs, RPCs and Micromegas are being considered [START_REF] Yu | GEM DHCal Development[END_REF][START_REF] Bilki | [END_REF][START_REF] Adloff | Development of Micromegas for a Digital Hadronic Calorimeter[END_REF]. Some benefits of a Micromegas DHCAL are a potentially high efficiency for MIPs, a low hit multiplicity (close to 1) [START_REF] Adloff | Development of Micromegas for a Digital Hadronic Calorimeter[END_REF], low working voltages (400-500 V which is low w.r.t. GEMs and RPCs), a thin sensitive layer (3 mm of Ar), a very good long-term irradiation behaviour and a high rate capability (no gain saturation up to GHz/mm 2 with thin gas layers) [START_REF] Derré | [END_REF]. A disadvantage of thin gap Micromegas is the small charge that the readout circuits have to accommodate (a few tens of fC for a MIP).

Detection efficiency of a Micromegas DHCAL

The performance of a Micromegas DHCAL (single particle energy resolution and shower separation capability) should depend on the detection efficiency and the hit multiplicity. We derive here a simple formula for the detection efficiency as a function of signal and threshold. This will motivate the study of the gain sensitivity to various parameters.

For a given electronic threshold t and total charge distribution on a pad p(Q t ), the efficiency η can be simply expressed as:

η = ∞ t p(Q t )dQ t (1)
where the distribution p(Q t ) is normalized. It is the convolution of the primary charge distribution p(Q p ) and the gas gain distribution p(G). For a charged particle of given energy, a parametrization of p(Q p ) is the Landau distribution [START_REF] Sauli | Principles of operation of Multiwire Proportional and Drift Chambers CERN Yellow Reports[END_REF]. The gas gain distribution is generally assumed to be a decreasing exponential distribution [START_REF] Blum | Particle detection with drift chambers[END_REF]. Eventually, p(Q t ) can also be described by a Landau distribution.

The solution of Equation 1 is not known but can be calculated by numerical integration. For this purpose, the two parameters of the Landau distribution p(Q t ) (most probable value MPV and a scale parameter) are deduced from the analysis of data recently collected in a beam test at CERN (200 GeV muons traversing mostly at normal incidence 3 mm of Ar/iC 4 H 10 95/5). The Landau distribution as adjusted to the measurements is shown in Figure 1 Using these results, the efficiency of HARDROC [START_REF] Boudry | HARDROC1, Readout chip of the Digital Hadronic Calorimeter of ILC[END_REF] or DIRAC-based [START_REF] Gaglione | DIRAC: a Digital Readout Asic for hAdronic Calorimeter[END_REF] Micromegas chambers can be predicted. Assuming an electronic threshold of 20-30 fC, the efficiency would lie between 0.6-0.8. Still, this figure applies for 200 GeV/c muons which are 50 % more ionising than true MIPs (300 MeV/c muons): the efficiency for MIPs would be worse. For a DHCAL, the efficiency should be as high as possible. Because the MPV is proportional to the gas gain, the gain sensitivity to changes in, for instance, pressure should be known. In particular the drop of gain due to a given increase of pressure or decrease of temperature should be quantified.

In the next section we discuss some variables that impact on the gain. The purpose of our study is to measure (or predict) how sensitive the gain is to variations of each of them. Gain variations can be measured relatively simply by monitoring the signal from 55 Fe quanta conversion in the gas. Indeed, variations of Q t are a direct indication of gain variations because the energy absorbed in the gas is fixed in this case (in opposition with traversing charged particles). The experimental setup is described in section 3 and the measurements are reported in sections 4 and 5. As a conclusion, an attempt is made to predict efficiency variations from gain variations.

Gas gain variations

Gas and environmental variables

The following variables impact on the gas gain:

• atmospheric pressure. Our detector is constantly flushed with a mixture of gas, the gas pressure variations hence follow atmospheric pressure variations. When the pressure decreases, so is the gas molecular number density: avalanche electrons get more energy between collisions with the gas molecules and the fraction of ionizing collision increases. At the same time, the mean distance between ionisation increases (because the ionisation cross section and the gas number density decrease). As a result, there is a maximum of gain as a function pressure. The pressure at which the gain is maximum depends on the temperature and the amplification gap thickness [START_REF] Giomataris | [END_REF];

• gas temperature. Similarly to the pressure, the temperature impacts on the molecular number density and thus on the gain. According to the ideal gas law, the number density is n = P /(k B T );

• amplification gap size. Gap variations affect the gain uniformity over the mesh area and depend on the fabrication process of the Bulk Micromegas. At given grid voltage, pressure, temperature and in a given gas mixture, the number of ionising collisions per unit path (socalled Townsend coefficient α) depends on the field strength. Hence, the gain depends on the field and the gap size g: G = exp(αg). Similarly to the pressure dependence, the gain exhibits a maximum as a function of gap: in Ar-based mixtures this gap was measured to be close to 55 µm [START_REF] Chefdeville | Development of Micromegas like gaseous detectors using a pixel readout chip as collectinganode[END_REF]. Above this value, the gain is a decreasing function of gap as the lower field does not compensate for the larger distance available for the avalanche development;

• gas mixing ratio. Proportional gas detectors are operated in a mixture of a noble and a molecular gas (so-called quencher). When the mixture is made with mass flow controllers, the relative concentrations of the two species may change. Quencher gases efficiently absorb the electron energy because of vibration and rotation modes available at low energy (a few tenth to a few eV) [START_REF] Sharma | Properties of some gas mixtures used in tracking detectors[END_REF]. As a result, for quencher concentration above a few percents the gain decreases with the quencher concentration ( [START_REF] Bronić | [END_REF]);

• gas flow. At low flow, the concentration of electronegative impurities (e.g. O 2 , H 2 O) may be responsible for signal loss by electron attachment. The attachment cross-section of a given electronegative species depends on the electron energy and thus the field strength. It may thus happen in the drift or amplification region.

As will be shown in the next section, the functional dependence of the gain on the grid voltage, gap size and P, T variables is similar.

Gas gain model

In this section a parametrization of the gain in terms of the variables of interest is proposed. The parameters will be deduced from two independent measurements: gain as a function of grid voltage (section 4) and gain, pressure and temperature as a function time (section 5).

The gain is equal to the exponential of the number of ionisation produced by the avalanche initiating electron:

G = exp(αg) ( 2
)
where α is the number of ionisation per unit distance (so-called Townsend coefficient) and g is the amplification gap size. A common parametrization of α was proposed by Rose and Korff [START_REF] Bateman | Gain stabilisation in proportional counters[END_REF]:

α/n = A 0 exp(-B 0 n/E) (3)
with n the gas number density and A 0 , B 0 two constants that depend on the gas. Writing N A the Avogadro's number and R the ideal gas constant, the gas number density can be expressed as:

n = N A P RT (4)
Combining Equations 2-4:

G = exp AP g T exp(- BP g T V ) ( 5
)
where A and B depend on the gas mixture and V is the mesh voltage (V = Eg). According to Equation 5 the gain goes through a maximum for:

P g T * = V B (6)
which is the well-known property of Micromegas (and more generally of any parallel plate detector working in proportional mode) that at a given voltage and in a given gas mixture, the gain goes through a maximum for a particular set of gap, pressure and temperature [START_REF] Giomataris | [END_REF]. At NTP in Ar/CH 4 90/10 and Ar/iC 4 H 10 80/20, this maximum was measured for a gap of about 55 µm [START_REF] Chefdeville | Development of Micromegas like gaseous detectors using a pixel readout chip as collectinganode[END_REF]16], the mesh voltage was about 400 V. Measurements of gain as a function of pressure in Ar/C 6 H 12 93/7 at V = 270 V shown that the gain goes through a maximum at a pressure of 500-550 mbar [START_REF] Giomataris | [END_REF].

The gain sensitivity to changes in ambient variables or gap size can be calculated from Equation 5:

∆G G = C P ∆P + C T ∆T + C g ∆g (7)
where

C P = 1 G • ∂G ∂P = exp(- BP g T V ) • ( Ag T - ABP g 2 T 2 V ) ( 8
)
C T = 1 G • ∂G ∂T = exp(- BP g T V ) • ( ABP 2 g 2 T 3 V - AP g T 2 ) ( 9
)
C g = 1 G • ∂G ∂g = exp(- BP g T V ) • ( AP T - ABgP 2 T 2 V ) ( 10
)
The parameters A and B will be deduced from gas gain measurements in section 4 and used to calculate (Pg/T ) * , C P , C T and C g . In section 5, C P and C T are measured and compared with their calculated values.

3 Experimental setup

Chamber

The test chamber is formed by a 6×16 cm 2 anode plane segmented into 96 pads, a Micromegas mesh maintained 128 µm above the anode PCB by insulating pillars (so-called Bulk Micromegas [17]), a 3 mm thick plastic frame which defines the drift region and a grounded steel cover. The drift electrode is a copper foil glued on a kapton, itself glued on the cover inside surface. The cover is perforated in some spots for X-ray characterization. The gas is flushed in the chamber through two holes in the plastic frame. This chamber is part of a stack of four chambers (Figure 2) built for efficiency and multiplicity studies in beam tests [START_REF] Adloff | Development of Micromegas for a Digital Hadronic Calorimeter[END_REF]. Among the four chambers, only one is used for our measurements.

Gas system

The gas system consists of two bottles of Ar and CO 2 , two mass flow controllers, a mixing barrel, four rotameters and bumbles (with the four chambers in between) and one exhaust. The mass flow controllers are calibrated for the two gas species. The Ar and CO 2 controllers deliver a maximum flow of 3 and 0.96 l/h respectively. The flow can be controlled to a 1 % accuracy. A third controller calibrated for iC 4 H 10 is available but not used. The rotameter valves are opened completely and can be used to measure the gas flow and check that it complies with the controller settings. A good agreement was found up to gas flows of 1.5 l/h above which the rotameter measurement saturates. This may be an indication of a leak between the controllers and the rotameters. In typical working conditions the flow rate is about 1 l/h so we are confident that the mixing ratio in the chamber is precisely known.

Signal readout and DAQ

Each primary electron from 55 Fe conversions reaching the amplification region initiates an avalanche. The motion of avalanche electrons and ions in the gap induces a signal of opposite polarity on the pad underneath and the mesh. The signals from individual avalanches are picked up on the mesh by an ORTEC preamplifier, followed by an ORTEC amplifier/shaper. The output pulse is digitized by a 12 bit VME ADC housed in a crate. A fast output of the ORTEC preamplifier is used to trigger the digitization when the pulse amplitude is maximum. The ADC is read by the CENTAURE program which takes care of spectrum display and saving. The calibration constant of the readout chain was precisely measured to be 2.199 ± 0.026 ADU/fC (where ADU stands for ADC Units) [START_REF] Chefdeville | Technical note on LAPP LC Detector cosmic bench calibration[END_REF].

The CENTAURE acquisition settings, source activity and collimation are such that the event rate is about 100 Hz and that 1 MBytes files are written every 10-30 minutes depending on the gas pressure. Each file contains about 50000 events. In parallel to that, a slow control of the atmospheric pressure, chamber over-pressure and gas temperature is done. These variables are written to file every 100 seconds.

Data analysis

The analysis consists in reconstructing 55 Fe spectra of 10 4 events and determining the position and width of the photopeak. This is realized by fitting the parameters of a dedicated function to the spectra. The function is the sum of three gaussians: one accounting for the escape peak, two for the K α and K β lines of the photopeak (an 55 Fe source emits 5.9 and 6.5 keV photons in the ratio 8.5/1 [11]). The parameters of the K β peak are fully constrained by the K α ones:

µ β = µ α 6.5 5.9 (11)
σ β = σ α 6.5 5.9 (12)

h β ∼ h α 7 ( 13
)
where µ, σ and h are the mean, standard deviation and maximum of a gaussian. Finally the number of free parameters reduces to 6: half for the escape peak, half for the photopeak. A measured spectrum with the fitted function is shown in Figure 3.

In the next two sections we present measurements of some basic properties of the Micromegas used in Ar/CO 2 and results of the environmental study in Ar/CO 2 80/20.

4 Micromegas basic properties in Ar/CO 2 mixtures

Electron collection efficiency

The electron collection efficiency is the probability that a primary electron enters the amplification region. It depends mainly on the field configuration and to a lower extend on the electron transverse diffusion. In Micromegas the electron collection improves with the compression of the field lines from the drift to the amplification region. The compression factor (more precisely the ratio of the funnel cross section areas at the drift electrode and at the anode S D /S A ) is equal to the ratio of the fields (E A /E D ). Accordingly, at a given grid voltage (E A constant) the collection efficiency increases when lowering the drift field. Above a certain field ratio, the collection efficiency may be close to 1 as all electrons should enter the amplification region. The 55 Fe photopeak position was measured as a function of drift field in Ar/CO 2 mixtures. In each mixture, the mesh voltage is adjusted so as to have signals of similar magnitudes: V mesh = 485 V, 520 V and 570 V in Ar/CO 2 5/95, 90/10 and 80/20 respectively. The measurements are shown in Figure 4. It is observed that the peak position first rises with the field ratio (FR) as the collection efficiency improves. Instead of the expected plateau the position then decreases above FR = 200, reaches a local minimum at FR = 500 and rises again. This trend is observed for the three gas mixtures and is more pronounced at larger CO 2 fractions. It can be explained by attachment of some primary electrons in the drift region by electronegative impurities. This explanation is supported by the fact that the local minimum is reached at similar values of the drift field (E D = 75-90 V/cm). Attachment in mixtures containing a significant fraction of CO 2 is a three-body process that involves oxygen molecules [START_REF] Blum | Particle detection with drift chambers[END_REF]: the attachment probability depends on the CO 2 and O 2 concentration. Hence, if our explanation is correct the dip may disappear if the gas tightness of the chamber is improved. The maximum of the signal is obtained at a field ratio of 200 which corresponds to drift fields of 190, 203 and 223 V/cm in Ar/CO 2 95/5, 90/10 and 80/20. At such field ratios, the collection efficiency may be maximum.

ADC counts

Gas gain

The gas gain is calculated from the measured peak position, the known calibration constant of the readout chain (cf. section 3.3) and the expected number of primary electrons in the gas mixture. Using W from literature (W (Ar) = 25.6 eV and W (CO 2 ) = 36.0 eV [START_REF] Christophorou | Atomic and Molecular Radiation Physics[END_REF]), a simple weighted average (which neglects the different values of the ionisation cross section of Ar and CO 2 ) yields 221, 219 and 212 electrons in Ar/CO 2 95/5, 90/10 and 80/20. At a drift field of 30 V/cm, the peak position is measured as a function of grid voltage until the spark rate reaches about 0.1 Hz. The pressure variation throughout the measurement period is 2 mbar. The measurements corrected for this variation are shown in Figure 5. As expected the gain decreases with the CO 2 fraction. The maximum gain is about 10 4 in Ar/CO 2 80/20 and 90/10 and slightly lower in the 95/5 mixture probably due to a poorer quenching.

The parameters of Equation 5 are adjusted to the measured G(V ) trend for each mixture and are listed in Table 1. The other variables that enter Equation 5 were fixed to P = 963 mbar, T = 298 K and g = 128 µm (P and T were measured, g is assumed equal to the pillar thickness). The calculated (Pg/T ) * , C P , C T and C g (Equations 6-10) appear in Table 2. In Ar/CO 2 80/20, they are equal to -0.5, 1.5 and -3.5 % respectively. These figures decrease with the quencher fraction (or increase with the steepness of the gain curve).

Mixing ratio

A ± ∆A (K/mbar/µm) B ± ∆B (V•K/mbar/

Energy resolution

The energy resolution of our proportional chamber to 55 Fe signals is governed by primary charge fluctuations, electron collection efficiency, gain fluctuations and electronic noise. Assuming full collection, the resolution R can be expressed as:

R = W (F + b) E 0 + R n (14)
where E 0 is the energy absorbed in the gas, F the Fano factor, b the relative variance of the gain distribution and R n the contribution from the noise. Using F = 0.2 [START_REF] Chefdeville | Development of Micromegas like gaseous detectors using a pixel readout chip as collectinganode[END_REF], b = 0.5 [START_REF] Alkhazov | [END_REF], W = 27 eV, the theoretical limit of the resolution at 5.9 keV in Ar/CO 2 mixtures is about 5.6 % r.m.s.. For every measured 55 Fe spectrum, the resolution is calculated as the relative width of the photopeak. The trend of the resolution as a function of gas gain is shown in Figure 6. The measured resolution is equal to 11-15 % which is twice larger than the theoretical limit which can be explained by the important noise at the preamplifier input. The latter results from the detector capacitance which due to the large mesh area is high (C detector (96 cm 2 ) ∼ 0.5-1 nF, neglecting the pillar dielectric permittivity). 5 Environmental study

Signal and gas flow

The study of the signal dependence on gas flow was conducted prior to the installation of the mass flow controllers. The 55 Fe photopeak position was measured in a mixture of Ar/iC 4 H 10 95/5 available from a pre-mixed bottle. In that configuration, the mixture was flushed through the rotameters and the chambers. The rotameters were calibrated for that mixture and were used to change the flow between 0.05 and 0.3 l/h. The measurements are shown in Figure 7. The peak position increases with the flow probably because of a decrease of the concentration of electronegative impurities inside the chamber. It is not clear if it is the gain or the collection efficiency or both that increase with the flow. Still, we observe that the peak reaches a stable value for flows larger than 0.25-0.3 l/h. As the gas is flushed through three chambers of 0.029 l and one of 0.115 l the effective flow through the chamber of interest is one seventh of the flow indicated in Figure 7 (assuming the same gas tightness for all chambers). Accordingly, the saturation is reached for a flow of 1.2-1.5 chamber volume per hour.

Two week study

During two weeks the amplitude of some 10 8 pulses from 55 Fe quanta conversions in Ar/CO 2 80/20 were recorded, enabling a precise monitoring of the detector gain as a function of time. In parallel, gas pressure and temperature were also recorded. The grid voltage is set to 570 V at which a gain of about 5•10 3 is expected. The drift field is kept at 100 V/cm which corresponds to a field ratio of 445. The Ar and CO 2 gas flows are equal to 0.97 and 0.24 l/h, yielding a total flow of 1.21 l/h. The trends of the peak position, gas pressure and temperature as a function of time are shown in Figure 8 (a)-(c). The three peaks in the temperature trend at 1.6 and 4.8 and 5.0 days result from a voluntary change of the room temperature by means of an air conditioner. After 5.2 days, the room windows were masked to prevent sun light from reaching our setup. As a result, the temperature variations are significantly reduced. The data analysis is based on the assumption that the gain dependence on the pressure P and the temperature T are independent. Hence, we first measure G(P ) when the temperature varies little and correct G(t) for pressure variations. The corrected variations of G(t) should then be caused by temperature variations only and the relation G(T ) can be determined.

Gain and gas pressure

Ideally the gain dependence on pressure should be determined at constant temperature. In our case, the temperature changes with time and therefore the gain and pressure variations should be studied over small temperature ranges (e.g. ∆T = 0.1 • C). On the other hand, the relation G(T ) will be more accurately studied if one allows for a larger ∆T because the pressure variations ∆P will be larger. Accordingly we chose a temperature range of 1 • C which is the largest variation measured between 5.2 and 16 days. A scatter plot of peak position (proportional to gain) and pressure measured during this time is shown in Figure 9 (a) (T = 24.5-25.5 • C). According to Equation 5, we use the following parametrization:

G(P ) = exp(A 1 P exp(-B 1 P )) (15)
with

A 1 = Ag T (16)
and

B 1 = Bg T V (17) time (days)

4).

Adjusting A 1 and B 1 to the data points, we find A = 0.11 K/µm/mbar and B = 2.34 KV/µm/mbar which are compatible with the values determined from the gain curve 1 . Accordingly, the relative gain variation for a pressure change of 1 mbar is C P = -0.63 %/mbar (cf. Equation 8). This is in good agreement with the gain curve calculated value of -0.46 %/mbar.

Gain and temperature

The peak position p (or gain) at a time t is corrected for pressure variations using Equation 7:

p cor (t) = p(t 0 ) + ∆p(t) ∼ 13.4 • (G(t 0 ) + ∆G) ∼ 13.4 • G(t 0 )(1 + C P ∆P ) (18)
1 The parameters A 1 and B 1 relates to the parameters p 0 and p 1 of Figure 9 The obtained trend of p(t) is shown in Figure 8 (d) where the peak variations should be caused by changes of temperature only. A scatter plot of the pressure corrected peak position and temperature is shown in Figure 10. This graph includes data collected between 4.0 and 5.1 days when the temperature variations were the largest. Using Equation 5:

G(T ) = exp(A 2 /T exp(-B 2 /T )) (19)
and fitting A 2 and B 2 to the data, we obtain A = 0.11 K/µm/mbar and B = 2.33 KV/µm/mbar. The gain relative sensitivity to temperature variations is thus C T = 2.01 %/K (Equation 9) which agrees relatively well with the value of 1.50 %/K deduced from the gain curve.

Gain and amplification gap size

The measurement of the gain sensitivity to gap variations is difficult as it requires Micromegas of various gap sizes. Such a measurement could be performed with three integrated Micromegas (so-called InGrid detectors) of gap sizes ranging from 45 to 70 µm [START_REF] Chefdeville | Development of Micromegas like gaseous detectors using a pixel readout chip as collectinganode[END_REF]16], showing that in Ar-based mixtures at NTP and at a voltage of 390-440 V the gain is maximum for a gap size of about 55 µm. This study can not be performed here, nevertheless the gain sensitivity to gap variations can be predicted using the values of A and B deduced from the gain curve or from the environmental study. At a pressure of 963 mbar, a temperature of 298 K, a grid voltage of 570 V and for a gap size of 128 µm, the gain sensitivity to gap size variation C g lies between 3.5-4.7 %/µm. The gap size uniformity of our Bulk Micromegas is not precisely known but is probably in the order of one to a few microns. Therefore the gain should vary at least by a few percents across the mesh area. This prediction can be put to the test if several measurements of the gain across the mesh area are performed. These are not available yet, however, they should be done with the 32×48 cm 2 ASU test box (perforated aluminium cover).

Another prediction concerns the product (Pg/T ) * for which the gain (at a given mesh voltage and in a given gas mixture) is maximum. Using the measured value of B = 2.12 KV/µm/mbar we find at 570 V: (Pg/T ) * = 268 K/µm/mbar which yields an optimum gap of 83 µm at NTP. To reach this optimum with our current gap size of 128 µm and at room temperature, a pressure of 635 mbar or a working voltage of 880 V would be needed. The optimum working point can not therefore be obtained with our detector2 .

Gain and gas mixture

When two gases from different bottles are mixed, variations of their concentrations and hence of the gain can occur. The gain sensitivity to quencher fraction variations was determined by changing the CO 2 fraction from 20 % down to 12 % by small steps of 2 % and measuring a gain curve. The measurements were carried out over two days and are shown in Figure 11. For each mixture the gain is corrected for pressure and temperature, taking 958 mbar and 298 K as reference conditions. The gain at a mesh voltage of 550 V is plotted as a function of the CO 2 fraction in Figure 12.

The trend is well described by a decreasing exponential function and we use the following parametrization to calculate the sensitivity:

G(f ) = exp(A 3 + B 3 f ) ( 20
)
where f stands for the CO 2 fraction. Adjusting A 3 and B 3 to the data points we obtain (B 3 ± ∆B 3 ) = (-0.177 ± 0.004) /% from which the gain sensitivity is: weeks of the environmental study the Ar and CO 2 flows slightly changed. The largest measured deviation from the initial 80/20 mixture was 79.94/20.06 (the mixing ratio was measured every day). This should have changed the gain by 1 %. As a conclusion, our gas system delivers a steady mixture. Also, the measurements presented in Figure 8 should be affected very little by such small variations.

∆G G = -0.177∆f (
6 Conclusion and extrapolation to arbitrary gas gain 6.1 Gain sensitivity to P, T, g and f variations

We investigated the dependence of the gain on various ambient and gas parameters. In particular the pressure and temperature dependence were measured in two independent ways. A good agreement was found between both measurements. In Ar/CO 2 80/20 at a grid voltage of 570 V, we obtained: with ∆P, ∆T, ∆g and ∆f in millibar, Kelvin, micron and %.

Impact on the detection efficiency

Considering a gain of 10 4 (grid voltage of 605 V in Ar/CO 2 80/20), NTP and taking realistic changes in pressure, temperature, gap size and CO 2 fraction of 10 mbar, -5 K, 5 µm and 0.5 %, Equations 15 and 21 yield negative gain variations of 6 %, 10 %, 17.5 % and 8.9 % respectively.

To calculate the impact on the efficiency an important input is the signal distribution in that gas mixture. It was not measured but can be inferred from that measured in Ar/iC 4 H 10 95/5 (cf. Figure 1 (a)) assuming that the two parameters of the Landau distribution both scale with the total charge at the anode pads (23) The trend of the detection efficiency with the Landau distribution MPV (in unit of threshold) calculated by numerical integration of Equation 1 is shown in Figure 13 (a). The ratio MPV/t and the efficiency for various thresholds (from 20 to 60 fC) are listed in Table 3. The efficiency drop due to the mentioned P, T, g and f variations also appear in this We observe that even with a threshold of 20 fC the efficiency is not so high (∼ 70 %) and that the calculated drops are mildly pronounced. In conclusion, the detection efficiency of a Micromegas DHCAL operated in Ar/CO 2 80/20 would be rather insensitive to small changes in pressure, temperature, amplification gap size and CO 2 fraction. However, with a maximum gas gain of 10 The assumption is made that the Landau distribution parameters scale with the gain. 20 fC is assumed). To increase the efficiency, other gas mixtures with higher maximum gains should be considered.

Q t = Q p G. Because Q p in

  (a). The calculated efficiency as a function of threshold is shown in Figure1(b).

Figure 1 :

 1 Figure 1: Landau distribution fitted to beam test data (a). Calculated detection efficiency as a function of threshold (b). These results are for 200 GeV/c muons traversing 3 mm of Ar/iC 4 H 10 95/5.

Figure 2 :

 2 Figure 2: Photograph showing the stack of four Micromegas chambers.

Figure 3 :

 3 Figure 3: 55 Fe spectrum in Ar/CO 2 80/20 at V mesh = 570 V.

Figure 4 :

 4 Figure 4: 55 Fe photopeak position and field ratio in Ar/CO 2 mixtures.

Figure 5 :

 5 Figure 5: Gas gain of a 128 µm gap Micromegas in Ar/CO 2 mixtures.

Figure 6 :

 6 Figure 6: Energy resolution at 5.9 keV and gain in Ar/CO 2 mixtures.

Figure 7 :

 7 Figure 7: 55 Fe photopeak position in Ar/iC 4 H 10 95/5 and gas flow through four chambers placed in parallel. The volume of the chamber of interest is equal to 0.03 l and the total volume of the four chambers is 0.2 l.

Figure 8 :

 8 Figure 8: 55 Fe photopeak position (a), pressure (b) and temperature (c) as a function of time. Peak position corrected for pressure variations as a function of time (d) (cf. section 5.4).

Figure 9 :

 9 Figure 9: 55 Fe photopeak position and pressure measured between 5.2 and 16 days.

Figure 10 :

 10 Figure 10: 55 Fe photopeak position corrected for pressure variations as a function of temperature measured between 4.0 and 5.1 days.

Figure 12 :

 12 Figure 12: Gas gain at a grid voltage of 550 V in various Ar/CO 2 mixtures.

  ∆G G ∼ -(0.5 -0.6) % ∆P + (1.5 -2.0) % ∆T -3.5 % ∆g -17.7 % ∆f (22)

4

 4 the efficiency should reach at most 70 % (a minimum threshold of HARDROC and DIRAC chips of

Figure 13 :

 13 Figure 13: Calculated detection efficiency as a function of the Landau MPV in unit of threshold (a) and its derivative (b). The assumption is made that the Landau distribution parameters scale with the gain.

Table 1 :

 1 Parameters A and B of Equation 5 adjusted to the measured gain curves in Ar/CO 2 mixtures.

	µm)

Table 2 :

 2 Gain

relative variation in % for changes in pressure, temperature and gap size. The quantity (Pg/T) * = V/B is calculated for a grid voltage yielding a gain of 5600 (e.g. 570 V in the 80/20 mixture). The gap g * is calculated at 963 mbar and 298 K.

  Gain curves measured in various Ar/CO 2 mixtures at 958 mbar and 298 K.

	4 10		2 Ar/CO	88/12
	gas gain		2 Ar/CO 2 Ar/CO 2 Ar/CO	86/14 84/16 82/18
			2 Ar/CO	80/20
	500 520 540 560 580 600 620 640 660
	V	mesh	(V)
	2 CO 14 2 χ Constant 16 / ndf / ndf 2 χ Constant Figure 11: fraction (%) 10 12 18 20 gas gain 3 10 2.709e+04 / 3 0.04857 ± 11.44 2.709e+04 / 3 0.04857 ± 11.44 10 4 Slope 0.003517 ± -0.1768 Slope 0.003517 ± -0.1768	22
				21)
	Accordingly, in Ar/CO 2 mixtures with a quencher fraction between 12-20 %, an absolute variation
	of 1 % of the relative concentrations results in a gain variation of about 18 %. During the two

  Ar/iC 4 H 10 95/5 and Ar/CO 2 80/20 are similar, we have: MPV(Ar/CO 2 80/20) ∼ MPV(Ar/iC 4 H 10 95/5)• G(Ar/CO 2 80/20) G(Ar/iC 4 H 10 95/5) ∼ 29 fC• 10 4 1.5 • 10 4 ∼ 19 fC

Table .

 .

	t (fC)	MPV/t	η	η(+10 mbar)	η(-5 K)	η(+5 µm)	η(+0.5 % CO 2 )
	20	0.95	0.69	0.65	0.63	0.58	0.64
	40	0.48	0.33	0.30	0.28	0.24	0.29
	60	0.32	0.16	0.14	0.12	0.10	0.13

Table 3 :

 3 Effect of pressure, temperature, gap size and CO 2 fraction variation on the efficiency in Ar/CO 2 80/20 at a gain of 10 4 (MPV = 19 fC) for various thresholds.

With a mesh voltage of 880 V, we have (Pg/T ) * ∼ 415 which yields an optimum gap of 128 µm at NTP. This voltage, however, is not reachable in practice because of the onset of sparks.


Vincent Martinet
email: vincent.martinet@u-paris10.fr

Olivier Thébaud
email: olivier.thebaud@ifremer.fr

Luc Doyen
email: lucdoyen@mnhn.fr

Olivier Guyader

Claire Macher

Recovering viable fisheries

Keywords: sustainable fishing, recovery, fishery policies, bio-economic modeling. JEL Classifications: Q22, C61

This paper develops a formal analysis of the recovery processes for a fishery, from undesired to desired levels of sustainable exploitation, using the theoretical framework of viability control. We define sustainability in terms of biological, economic and social constraints which need to be met for a viable fishery to exist. Biological constraints are based on the definition of a minimal resource stock to be preserved. Economic constraints relate to the existence of a minimum profit per vessel. Social constraints refer to the maintenance of a minimum size of the fleet, and to the maximum speed at which fleet adjustment can take place. Using fleet size and fishing effort per vessel as control variables, we identify the states of this bioeconomic system for which sustainable exploitation is possible, i.e. for which all constraints are dynamically met. Such favorable states are called viable states. We then examine possible transition phases, from non-viable to viable states. We characterize recovery paths, with respect to the economic and social costs of limiting catches during the recovery period, and to the duration of this transition period. Sensitivity of each of the constraints to transition costs and time are analyzed. The analysis is applied to a single stock fishery; preliminary results of an empirical application to the bay of Biscay nephrops fishery are presented.

Introduction

According to recent studies, the maximum production potential of marine fisheries worldwide was reached at least two decades ago; since then, due to the widespread development of excess harvesting capacity, there has been an increase in the proportion of marine fish stocks which are exploited beyond levels at which they can produce their maximum [START_REF] Garcia | Gloom and doom? The future of marine capture fisheries[END_REF]FAO, 2004). Hence, the problem of managing fisheries is increasingly cast in terms of restoring them to both higher and sustainable levels of fish stocks, catches, and revenues from fishing. Examples are the restoration plans discussed and/or adopted by the European Commission in recent years for several collapsed stocks in E.U. waters, or the international commitment to return fisheries to levels allowing their maximum sustainable yield to be extracted by 2015, taken by countries present at the Johannesburg Summit on Sustainable Development in 2002.

The problems posed by fisheries restoration are dynamic in nature: beyond the issue of selecting adequate objective levels for restored fisheries, a key question is the identification and the selection of the possible paths towards these objective levels. In practical situations, this question is crucial as it relates to the feasibility (technical, economic, biological) and to the social and political acceptability of the adjustments required for fisheries to be restored, hence to the actual possibilities to drive fisheries back towards decided sustainability objectives.

The definition of optimal strategies for the harvesting of marine fish stocks has been widely studied in the literature on renewable resource management. While most of the initial work focused on the comparative statics of the problem, analysis of the dynamics of bio-economic systems has developed as a substantial body of literature. Different approaches have been proposed. In the domain of fisheries, [START_REF] Clark | Bioeconomic Modelling and Fisheries Management[END_REF] described how to optimally drive a dynamic bioeconomic system towards a stationary state, based on a single command variable (fishing effort), and looking at a single optimization criteria (net present value of the expected benefits derived from harvesting). Alternative approaches have been based on simulations of specific adjustment trajectories for given bioeconomic systems, according to predetermined scenarii, and on their a posteriori evaluation with respect to various criteria (see e.g. [START_REF] Smith | On Models of Commercial Fishing[END_REF]; [START_REF] Mardle | Modelling the effects of trade-offs between long run and short run objectives in the North Sea[END_REF]; [START_REF] Holland | Modeling a Rights Based Approach to the Management of Habitat Impacts of Fisheries[END_REF]).

In other areas of natural resource management, the maximin approach [START_REF] Solow | Intertemporal use of exhaustible resources and intergenerational equity[END_REF][START_REF] Cairns | Maximin: a direct approach[END_REF] has been used. Instead of maximizing the net present value criteria, this approach considers the maximal level of profit (or utility) that can be sustain forever, given the initial state of the economy. It thus strongly includes an intergenerational equity concern. For example, [START_REF] Tian | Sustainable Development of Easter Island?[END_REF] examine the sustainable management of a renewable resource: the forest on Easter Island. They show that, depending on the initial configuration of the bioeconomic system, a sustainable exploitation pattern would have been possible on the island, with higher population levels and welfare than actually observed [START_REF] Brander | The simple economics of Easter Island: A Ricardo-Malthus model of renewable resource use[END_REF]. But this approach still considers only an economic criterion, when sustainable management of renewable resources may require to consider economic, social and environmental criteria together.

In this paper, we develop a formal analysis of the recovery paths for a fishery, based on viable control theory. This allows us to characterize of the dynamics of a fishery in terms of its capacity to remain within pre-defined constraints, beyond which its continued long-term existence would be jeopardized. The constraints considered in the analysis relate to micro-economic, biological and social factors. Following [START_REF] Béné | A viability analysis for a bio-economic model[END_REF], we use the mathematical concept of viability kernel to identify the set of states of the fishery for which it is possible to satisfy these constraints dynamically. This kernel represents the "target" states for a perennial fishery. Our analysis focuses on the ways by which the fishery can recover from states outside the kernel to viable states in general, and to specific target states in particular. We use the concept of minimal time of crisis [START_REF] Doyen | Scale of viability and minimal time of crisis[END_REF] to consider the horizon at which such targets can be reached, and examine transition paths considering transition time and transition costs defined as the discounted sum of fleet profits during the transition phase toward target states.

The analysis is applied to the case of the bay of Biscay (ICES area VIII) nephrops fishery, and focuses on the implications of restoring this fishery to levels allowing maximum sustainable yield to be extracted. We examine the relationship between the viability of the fishery on the one hand, and the possible transition phases towards this maximum yield objective on the other hand.

The paper is organized as follows. The simplified model of the bay of Biscay nephrops fishery used for the analysis is presented in section 2, as well as the definition of the economic, biological and social constraints determining the viability of the fishery. Section 3.2 then presents the analysis of the conditions under which these constraints can be satisfied throughout time, and of the possible transition paths towards these viable states from initially non-viable situations 3.3. The specific issue of recovering MSY production levels without jeopardizing the viability of the fishery is addressed in Section . Section 5 concludes.

2 Defining a sustainable fishery

A bio-economic model of the fishery

In this paper, we consider a single stock fishery, characterized by the size of the fleet X t ∈ [0, X] ⊂ N + , which can evolve in time. The exploited resource is represented by its stock S t ∈ [0, K], where K is the carrying capacity of the ecosystem.

The dynamics of the bio-economic system is controlled by the effort e t ∈ [0, ē] ⊂ N + (day of sea per period and per vessel1 ) and the change in the fleet size ξ t ∈ [-α 1 , α 2 ] ⊂ N + , the number of boats entering or exiting the fleet. We assume that the admissible controls belong to a set (e, ξ) ∈ U, where

U = [0, ē] × [-α 1 , α 2 ].
We use a discreet time version of the "logistic model" of [START_REF] Schaefer | Some aspects of the dynamics of populations[END_REF] to represent the fish stocks renewal function. Hence, the regeneration of the resource stock is given by

R t (S t ) = rS t 1 - S t K . ( 1
)
The fleet is assumed homogenous. Each vessel has the same access to the resource and the same harvesting characteristics. Global catches are defined by

C t = qS t e t X t (2)
where q represents the catchability of the resource.

We thus get the dynamics of the resource combining eq. ( 1) and (2), following [START_REF] Gordon | The economic theory of a common property resource: the fishery[END_REF])

S t+1 = S t + R t -C t = S t + rS t 1 - S t K -qS t e t X t (3)
The economic dynamics are characterized by the per vessel profit. This profit depends on the landings L t of the resource defined with respect to the per vessel catches c t = C t /X t = qS t e t and a discard rate τ d

L t = (1 -τ d )qS t e t . (4)
These landings give the gross return for the targeted species which is a part λ of the vessel's total gross return.2 Vessel profit thus reads

π t = p(1 -τ d )qS t e t 1 λ -(β 1 + β 2 e t ) ( 5
)
where p is an exogenous resource price that is considered constant. β 1 represents fixed costs and β 2 a per effort unit cost.

We consider that the production structure is (slowly) flexible, in terms of both capital and labour. The size of the fleet evolves according to a decision control ξ t ,

X t+1 = X t + ξ t . (6)
To take into account the inertia of capital, the change of the fleet size is limited. A maximum number α 2 of vessels can enter the fishery in any time period, due to technical constraints. The number of vessels exiting the fleet in any time period can not exceed α 1 , due to social and political constraints (see below). Its reads

-α 1 ≤ ξ t ≤ α 2 . ( 7
)
This means that levels of capital in the fishery (number of vessels) cannot change quickly. On the other hand, fleet activity (effort per period e t ) can change, and even be set to nil.

Sustainable exploitation patterns

We define sustainability of the exploitation with respect to a set of biological, economic and social constraints that have to be respected throughout time for a viable fishery to exist.

Biological constraints: In order to preserve the natural renewable resource, we consider a minimal resource stock S min which is the minimal biomass ensuring the regeneration of the stock:

S t ≥ S min (8)
Economic constraints: We consider an individual economic constraint on the vessel performance: profit per vessel is required to be greater than a threshold π min for economic units to be viable.

π t ≥ π min (9)
This minimal profit is defined such as to ensure remuneration of both capital and labour, at least at their opportunity costs. It can also be set as a sustainability goal ensuring level of economic performance greater than those ensuring strict economic viability.

Social constraints: To take into account social concerns, the viability of the fishery is described by a constraint on the fleet size. We require the number of vessels to be greater than a threshold X min :

X t ≥ X min (10)
ensuring a minimal employment and activity in the fishery.

In addition to this minimum fleet size, we assume that the speed at which fleet size can be reduced is also limited. The constraint on the adjustment possibilities regarding the fleet size (eq. 7) can be interpreted as a social and political constraint limiting the number of vessels (and employment) leaving the fleet.3

We call this set of constraint K, in the sense that states and controls respecting the entire set of constraints belong to K, which reads Viability constraint (8), ( 9) and ( 10) are respected ⇔ (S t , X t , ξ t , π t ) ∈ K

Considering the state constraints ( 8) and (10), viability requires as a necessary condition (S t , K t ) ∈ [S min ; K] × [X min ; X]. Nevertheless, it does not mean that this whole state constraint set make it possible to satisfy the profitability constraint.

In particular, the biological configuration for which it is possible to have a profitable fishing activity can be determined. It appears that the profit constraint (9) induces stronger limitations on stock size than the biological constraint (8). This result is stated in proposition 1 below, which is proven in the appendix A.2.

Proposition 1 The minimal resource stock for fishing activity to respect the per vessel profit constraint ( 9) is

S = π min + (β 1 + β 2 ē) pqē . ( 11
)
This induced constraint comes from the fact that a minimal effort is required to satisfy the profitability constraint (9), as proved in the appendix A.2 by the lemma 2. This minimal effort depends on the resource stock as the catches increase with the resource stock.

e(S t ) = π min + β 1 p λ (1 -τ d )qS t -β 2 (12)
This minimum effort is represented in Figure 1 with respect to the resource stock. The vertical asymptote represents the limit S from which fishing effort has a positive marginal value (Appendix A.2, Lemma 1). The fishing effort required to satisfy the profit constraint is a decreasing function of the stock size. 4 The more fishes there are, the less fishing effort to catch them is needed.

The horizontal line represents the maximal days of sea per period. One can see that there is a stock threshold S for which the number of days required to satisfy the profit constraint is greater than the possible number of days per period.

Minimal effort (day of sea per year) to satisfy profit constraint We thus have an induced constraint for the fishing activity to generate sufficient profits. This constraint level is greater than the initial resource constraint S min , which means that ensuring the economic profitability of the fishery in the long run implies that the resource constraint is also satisfied. This result was also derived by [START_REF] Béné | A viability analysis for a bio-economic model[END_REF].

Model parameters

The analysis is applied to a case study: the Bay of Biscay Nephrops fishery (ICES area VIII). Parameters estimation procedures are detailed in the appendix A.1.

Parameters values and constraints levels are given in the following table.

Parameter value Constraint level r = 0.78 S min = 5,000 tons K = 30800 tons X min = 100 vessels q = 72.10 -7 j -1 π min = 130,000 euros p = 8,500 euros per tons α 1 = 10

β 1 = 70,
τ d = 33% λ = 43%
In 2003, the fleet was composed by 235 vessels, of an average profit of 165 000 euros. The resource stock was about 18 600 tons. The average number of days of sea was 203. The catches were about 5769 tons.

Dynamic analysis of the fishery

In this section, we examine the consistency between the dynamics of the exploited system and the viability constraints, focusing on conditions for fishing activity to be viable in the long run.

Open access versus optimal harvesting strategies

Following [START_REF] Clark | Mathematical Bio-economics: the optimal management of renewable resources[END_REF], the dynamics of the system can be compared under two exploitation patterns: open access, and a policy guided exploitation maximizing the intertemporal profit derived from the fishery, which we call Cost-Benefit Analysis (CBA). We analyze the possible dynamics of the fishery under these two scenarios. The dynamics are illustrated in Figure 2.

Figure 2 represents the intertemporal dynamics of resource stock, fleet size and per vessel profit for the two exploitation patterns on a 50 years period. We compute 16 intretemporal paths representing various representative initial states.

Open Access The open access case corresponds to situations in which vessels can freely enter and exit the fishery, subject to the inertia constraints described above, and choose their individual effort level. In that case, as claimed in lemma 1, the individual effort will be maximum.

We consider that, if individual profit is greater than the minimal profit π min , the fleet size increases as new vessels enter the fishery. On the contrary, if profit are less than 90% of the π min level, vessels leave the fleet. This represents the fact that negative profits often occur transitionally in fisheries: some negative profits may be supported for short periods.

It appears that in this open access case, whatever the initial stock configuration, the system reaches a limit cycle in both the resource stock and the fleet size.5 In the case considered, the stationary state is characterized by 263 vessels and a resource stock of 14 400 tons. The individual profit at this stationary state is minimum (π min ).

Cost-Benefit Analysis Considering that the fishing fleet is regulated in order to maximize profit derived from the fishery, it is possible to establish the sets of optimal decisions concerning effort levels and changes in fleet size, for different initial conditions in the fishery.

At fleet level, the optimal behaviour is determined by maximizing the intertemporal sum of discounted fleet profits, with respect to the allocation of the fishing effort through time, which reads max e(.)

∞ t=t 0 1 1 + δ(t -t 0 ) X t pqS t e t -(β 1 + β 2 e t ) (13)
where δ represents the social discount rate or, from a microeconomic perspective, the opportunity cost of capital. 6 In the general framework, the optimal solution of such a problem [START_REF] Clark | Mathematical Bio-economics: the optimal management of renewable resources[END_REF]) is to reach an optimal steady state following a "bang-bang" strategy (or most rapid approach). In our model, we can see that, whatever the initial state, the system reaches such a stationary state, with a high stock level and a lower fleet size, allowing to maximize the catches while minimizing harvesting costs. The stationary state is reached as quickly as possible. In the case considered, this stationary state is characterized by 190 vessels and a resource stock of 19 000 tons. The individual profit is greater than the minimum viability profit π min . It is of 222 000 euros per vessel.

Nevertheless, there is no "bang-bang" strategy as there is an inertia in the capital (fleet size) adjustement. When the fleet size is smaller than the targeted size, the stock size and the profit level evolve smoothly, with increasing or decreasing profit, depending on the resource stock (the larger the stock, the more harvesting and profit). On the contrary, when the fleet size is large and require a long time period for adjustement, we observe an alternance of nil harvesting and maximal one. The larger the fleet, the larger the variations of profit and stock size during the transition phase.7

In both exploitation scenarii, a minimum profit is guaranteed after a transition phase. When a steady state is achieved, both the minimum profit per vessel and the resource stock are lower in the open access case then in the regulated case, while the fleet in the open access regime is larger. Open access profit is characterized by periodical oscillations around the profitability constraint, ensuring a minimum profit lower than the economic viability constraint.

The time of transition can be long : it is between 10 to 25 years in the open access configuration, and between 3 to 25 years in the regulated case. During this transition phase there is no guaranteed profit for the fleet.

The two harvesting scenarios considered here thus lead to paths that do not respect the viability constraints, as defined in the previous section, at least until the transition phase is achieved (and periodically for the open access strategy from then on). If these constraints apply, it is possible that some of the trajectories represented above may actually lead to situations of crisis due to a collapse of the stock, the economic extinction of the fishery, or to social unrest associated with the adjustment paths considered. We propose to analyse the viability of the fishery by defining intertemporal paths of harvesting that satisfy all the constraints defined in the previous section simultaneously.

Viable harvesting strategies

The aim of this section is to define state configurations (resource stock and fleet size) which are compatible with our viability constraints. The question is to determine whether the dynamics (Eq. 3 and 6) is compatible with the set of constraints. For this purpose, we use the viable control approach and study the consistence between dynamics (3) and ( 6) and the constraints ( 7), ( 8), ( 9) and ( 10).

The set of bioeconomic states from which there exist intertemporal paths respecting the whole constraint is called the viability kernel of the problem. A description of the viability theory and viable control approach is provided in the appendix A.3, along with a formal definition of this viability kernel.

Viable stationary states A first analysis relies on the definition of viable stationary states. These states are characterized by

S t+1 = S t X t+1 = X t
We thus have ξ t = 0 and

R t = C t ⇔ e t X t = r q 1 - S t K
We can determine admissible pairs (X ss , e ss ) with respect to resource stock S.

Extreme cases correspond to maximun effort ē on the one hand (which leads to a linear relationship between the fleet size and the resource stock), and minimum effort e(S t ) on the other hand. These two frontiers are represented on Figure 3. The inner area corresponds to possible stationary states that satisfy all the constraints, including the profitability constraint. maximal effort e minimal effort e min Fig. 3. Viable stationary states (the upper limit corresponds to minimal effort e(S), the lower limit to maximal effort ē.

We can see on Figure 3 that there is a maximal sustainable size for the fleet, when the effort is minimum. It is interesting to note that the stock level associated with the maximal sustainable size of the fleet is not equal to the stock producing the MSY. This is due to the fact that a greater stock produces less yield but ensures less costly catches.

In our illustrative case, the maximum sustained fleet size is 279 vessels. It is associated with a resource stock of 17 600 tons and a per vessel effort of 166 days of sea.

Viable states In our problem, there are some states that do not belong to the stationary set described above, but that however make it possible to satisfy the constraints. We now describe the set of all states that satisfy all of the constraints in a dynamic perspective, including non stationary trajectories. General statements on this viability kernel and the way it is computed are presented in appendix A.3.

The viability kernel for the nephrops fishery, using parameter values presented in section 4 is represented in Figure 4. This viability kernel represents the "goal" of the recovery paths, i.e. the set of states the system must reach to ensure a viable exploitation. For any given initial state (S 0 , X 0 ) in the viability kernel, there exists at least one intertemporal decision series (e(.), ξ(.)) for which the associated trajectory starting from (S 0 , X 0 ) respects all of the constraints forever. Note that there may exist several viable decisions. Another important point is that all admissible deci-sions are not necessarily viable and may lead the system outside the viability kernel.

The stationary states described in the previous section are particular cases of viable trajectories (that are stationary trajectories, associated to ad hoc decisions). If the initial state belongs to the left-bottom hand-side of the viability kernel the resource stock will increase for any viable decision. On the contrary, if the initial state is on the right-up hand-side, the resource stock will decrease whatever viable decision applies.

From the very definition of this viability kernel, for any outside initial state, there are no decisions that make it possible to satisfy the constraints in the long run. For example, any trajectory strating from the upper area of the state's constraint set, i.e. for any (S 0 , X 0 ) ∈ [S min ; K] × [X min ; X], at least one of the constraint will be violated in a finite time, wathever decisions apply.

The system thus faces a crisis situation if the bioeconoic state is outside the kernel or if the intertemporal path leaves it.

Recovery paths

In this section, we use the above framework of analysis to characterize recovery processes, from situations outside the viability kernel which we call crisis situations, to viable situations. A crisis situation corresponds to configurations that do not make it possible to respect the viability constraints, namely to satisfy (S t , X t , ξ t , π t ) / ∈ K, in the long run. We define the characteristic function

X K as X K (S, X, e, ξ) =      0 If (S t , X t , ξ t , π t ) ∈ K 1 If (S t , X t , ξ t , π t ) / ∈ K (14)

Achieving viability in the future: the concept of minimal time of crisis

In the context of this analysis, the issue is to reach the viability kernel in the future. From a theoretical point of view, the characteristic function ( 14) counts the number of period when viability constraints do not hold true. It can be interpreted as the time spent outside the kernel. A transition phase is then characterised by a time of transition, corresponding to this time. Starting from a given bioeconomic state, various transition phases exist, that reach the kernel more or less quickly.

We define the minimal time of crisis as the time spent outside the kernel by the fastest transition phase starting from a given bioeconomic state (the minimal time to reach the target).

Based on this notion of minimal time of crisis we are able to define the notion of viability at time T , which is the set of states that make it possible to belong to the viability kernel after T . For example, the set of states that are viable at time 2 is composed of all states for which the minimal time of crisis is lower than or equal to 2. In particular, the viability kernel defined in the previous section corresponds to viability at time t 0 . The formal link between the viability at scale T and the minimal time of crisis is developed in [START_REF] Doyen | Scale of viability and minimal time of crisis[END_REF].

More formally, the minimal time of crisis, i.e. the minimal time spent outside of K by trajectories starting at (S, X), is defined by the map

C K = inf ∈(e(.),ξ(.))U ∞ t 0 X K (S, X, e, ξ) (15)
This map is represented for the nephrops fishery by figure 5. By the very definition of the viability kernel, any state outside the kernel (crisis situation) does not make it possible to respect the constraints. All viability constraints thus cannot be respected during the transition phase.

In particular, the recovery strategy associated with the minimal time of crisis may require to close the fishery for a while (there is no effort, i.e. no fishing activity: the capital is not used), along with reducing the fleet size as quickly as possible (given the inertia constraint 7). This entails a strong violation of the minimum profit constraint. As noted before, due to economic and social requirements, transition phases may need to ensure a minimum level of revenue to vessels, even if it is lower than the minimum viable profit.

Recovery paths under constraint

Even if the optimal recovery strategy requires closing the fishery for a while [START_REF] Clark | Bioeconomic Modelling and Fisheries Management[END_REF], this is not always possible because it neglects fisher's needs to cover some fixed costs or to ensure a minimal activity and revenue. One may thus require a minimum activity during the transition phase, or more specifically, a minimum remuneration of labour and capital.

To take such requirements into account leads to "softening" one or several viability constraints during the transition phase. In particular, it is possible to accept that the fishery can face periods where profits from the activity in excess of the opportunity costs of capital and labour are negative, without inducing the definitive shutdown of the activity.

In our model, this possibility is defined by introducing constraints on transition decisions, i.e. by restricting the set of admissible choices such that e(t) ensures a minimal profit constraint during the transition phase. We define this constraint π.

The map representing the transition phases under constraint for the nephrops fishery is represented in figure 6. We can compare the various areas with respect to the minimal time of crisis without constraint defined in the previous section. As the admissible deci-sion set is restricted during the transition phase under constraint, it is longer to reach the target (the viability kernel) from any given crisis situation. This means that a same initial state will stand in a farther area of the map (characterized by a greater minimal time of crisis) with the π constraint on transition decisions.

Moreover, with this constraint, an area appears on the map, from which it is not possible to achieve recovery (the white area on the left hand side on Figure 6). Thus, for any given initial state, there is a maximum profit constraint on the transition phase for which it is possible to reach the viability kernel in a finite time.

Recovering MSY production

In this section, we apply the general framework previously developed to address a particular issue. We examine how to reach a sustainable goal defined as a production in the neighborhood of the MSY, starting from the situation of the Nephrops fishery in 2003.

In particular, we examine the consequences of this production objective on the fleet configuration (number of vessels and profit), and the time needed to reach it. First, we define the viability kernel associated to the MSY production objective, and the associated time of crisis. We then examine how to minimize the transition cost towards this objective state of the fishery.

MSY and viability analysis

We first determine the bioeconomic states that are compatible with MSY.

We then examine if these configurations are viable, i.e. respect the viability constraints defined in previous sections.

The maximum sustainable Yield is a particular stationary state where the resource regeneration is maximum, such that the sustainable harvesting is maximal too. We consider a production constraint defined as

C(t) ≥ C M SY (16)
The production of the stock at MSY level is

R M SY = rK 4 (17)
For the application case, we set C M SY = 6000 tons.

Assuming such a minimal production constraint induces a posteriori constraints on the resource stock, and on the minimal fleet size. 8 The induced constraint on the resource stock is

S(t) ≥ S M SY = K 2 . ( 18
)
In a static perspective, the minimal size of the fleet to ensure MSY production is derived from the defintion of catches (eq. 2). We get

qS t X t e t = rK 4 ⇒ X M SY (S) = rK 4qē 1 S . (19)
Thus, at S M SY we have X M SY = r 2qē . In our application, we get a minimum number of vessels equal to 247 boats at the MSY stationary state. With such a fleet size, the per vessel profit is limited to 150 000 euros. It is lower than the observed average profit in 2003 (about 165 000 euros). Having the MSY production as an objective requires to increase the fleet size and to reduce the per vessel profit with respect to the present situation.

Viability kernel of the MSY production

The first step of the analysis is to define bio-economic states that make it possible to respect the production constraint ( 16). For this purpose, we apply the viability approach to the system described by dynamics (3) and ( 6) and constraints (7) and we use the production constraint (16) instead of the economic and social constraints (10) and ( 9) on the minimal profit and minimal fleet size. We apply the framework proposed in section 3.3.

The viability kernel associated with this viability problem is given in Figure 7. It represents the combinations of stock and fleet size for which there exist viable decisions compatible with the minimum production level.

We see in figure 7 that a minimum size of the fleet is required to produce at this level. As an increasing stock ensures increasing catches for a given effort level, the bigger the stock is, the less boats are needed.

We now turn to analyse the economic viability of such a production objective. We consider once again the profit constraint π min defined by equation ( 9). We consider the viability kernel associated with all of the constraints, including the minimal profit per vessel and the minimal production of the ecosystem. This viability kernel and the associated times of crisis are presented in Figure 8. Comparing Figure 7 and Figure 8, we can see that the profit constraint reduces the set of viable states that make it possible to produce MSY. Stationary states correpond to the resource stock producing MSY. The minimal effort satisfying profit constraint (eq. 12) at the MSY stock is

e(S M SY ) = π min + β 1 p λ (1 -τ d )q K 2 -β 2 (20)
Using relationship (19), we get the maximal number of vessels sustainable at MSY.

X(S

M SY ) = r p λ (1 -τ d )q K 2 -β 2 2q(π min + β 1 ) (21)
In our case study, we get X = 373 vessels, having an individual effort of 145 days of sea. Combining this result with the minimal fleet size to ensure MSY, we have several possible stationary states, with a number of vessels between 247 and 373 vessels.

Transition phases

As an application of the results of section 3.3, we consider in this section the following sustainability goals :

π min = 150000 (22) C min = C M SY (23)
Thus, viability constraints are associated with the MSY production (eq. 16) along with a constraint on the individual profit, in order to get it as close as possible to the observed profit.

The viability kernel and minimal time of crisis associated with this viability problem is represented in Figure 9. We can see on Figure 9 that the viability kernel is smaller than the one ensuring viable profit.

Here again, the minimal time of crisis is associated with a shutdown of fishing activity. To take into account social and economic constraints, we consider a minimal profit constraint during the transition phase. We set it at the level of opportunity costs (which is the level we used in the previous section as a minimal profit for economic viability): π = 130000 euros.

The problem is thus to reach viable states, where sustainability goals are to produce at MSY level, with a maximum individual profits, and limiting the loss during the transition phase (minimal profit covering opportunity costs of the producing factors).

The map on Figure 10 represents the time of crisis under transition constraints. Such a study show that the set of viable states is reduces and that transitions phases would be longer if a viability constraint on the profit is required during the transition phase.

Conclusion

In this paper, we examine the viability of a fishery with respect to economic, social and biological constraints. The main constraint is a minimal profit per vessel that must be guaranteed at each time period. We show that requiring such a minimal profit induces a minimal threshold for the natural resource, and thus a stronger constraint on the resource stock than the initial biological constraint. As has already been demonstrated, it is thus possible to reconcile economic and ecological objectives.

We use the viability approach to determine the set of bioeconomic states that make it possible to satisfy the constraints dynamically. This set is called the viability kernel of the problem. Any trajectory leaving this set will violate the constraints in a finite time, whatever decisions apply. The system then faces a crisis situation.

We then study transition phases from crisis situation, i.e. states outside the viability kernel, to viable exploitation configurations. These transitions phases are characterized by the time of transition on the one hand, and the cost of the transition on the other hand. This cost is defined as the difference between a minimum profit ensuring economic viability and the observed profit during the transition phase. We show that the shorter the transition phase is, the higher the transition costs are.

Using this general framework of analysis, we focus on a particular issue, examining how to ensure MSY production for the bay of Biscay nephrops fishery.

We show that such a production requirement implies increasing the fleet size while reducing the per vessel profit. We then characterize transition phases toward desired exploitation patterns. We define how to reach viable states without jeopardizing the economic viability of the fleet.

A Annexe

A.1 Parameters of the case study: the Bay of Biscay Nephrops fishery (ICES area VIII)

All along the paper, numerical illustrations are provided, based on an empirical application to the bay of Biscay nephrops fishery. The numerical model has been calibrated with commercial time-series.

Biological parameters are estimated using CPUE series (catches per unit of effort) as an index of abundance. We used nonlinear parameter estimation techniques to find the best fit of the predicted biomass, given the observed catches. The fitting criterion is the minimization of the squared deviation between observed and predicted CPUE [START_REF] Hilborn | Quantitative Fisheries Stock Assessment: Choice, Dynamics and Uncertainty[END_REF]. Economic parameters are estimated using costs and earnings data collected by the Fisheries Information System of Ifremer via surveys of individual vessel owners.

A.2 Individual economic behavior

In this appendix, we detail individual optimal behavior. We first determine the effort level that maximizes the profit of vessels.

Lemma 1 If the resource stock is greater than a level S =

β 2 p λ (1-τ d )q
, the optimal fishing effort of a vessel is its maximum possible effort e(t) = ē. Else, the optimal effort is 0.

Proof of Lemma 1 The profit, defined by eq. ( 5) is

π t = p(1 -τ d )qS t e t 1 λ -(β 1 + β 2 e t ).
At a given time t, and for the resource stock S t , taking the profit derivative with respect to the effort level e t leads to

∂π ∂e = p λ (1 -τ d )qS t -β 2
which is positive if the resource stock S t is greater than a threshold S such that

S = β 2 p λ (1 -τ d )q
.

The optimal individual effort thus follows a "bang-bang" strategy : no fishing if S t < S and a maximum activity ē if S t > S . In our illustrative case, this value is S = 4, 075 tonnes, which is lower than the resource constraint S min . We will thus consider that it is always optimal to fish as much as possible.

We then define the minimum effort level ensuring the minimum profit π min .

For this purpose, we examine instantaneous condition on the effort e t for constraint (9) to be satified at time t, given stock S t .

Lemma 2 The minimum effort e t insuring profit π min at a given level of stock S t is given by e

(S t ) = π min + β 1 p λ (1 -τ d )qS t -β 2 (A.1)
Proof of Lemma 2 At a given level of stock S t at time t, for constraint (9) to be satisfied, we must have

p(1 -τ d )qS t e t 1 λ -(β 1 + β 2 e t ) ≥ π min which leads to e t ≥ π min + β 1 p λ (1 -τ d )qS t -β 2 (A.2)
Hence the minimum effort e(S t ).

We are now able to prove proposition 1

Proof of Proposition 1 Given the profit equation

π t = pqS t e t -(β 1 + β 2 e t ) ≥ π min
and combinig the optimal effort from Lemma 1 along with the maximum effort bound ē, we get

S t ≥ π min + (β 1 + β 2 ē) pqē . (A.3) Hence S.
Note that at this stock level S, we have e(S) = ē, which means that the minimum effort to satisfy the constraint is the maximum effort.

A.3 About the viability analysis

The viability approach and viable control framework [START_REF] Aubin | Viability theory[END_REF] focuses on intertemporal feasible paths. It consists in the definition of a set constraints that represents the "good health" or by extension the effectiveness of the system at any moment, and in the study of conditions which allow these constraints to be satisfied along time including both present and future. So, it leaves the optimality framework and focus on the respect of constraints of controlled dynamic systems; in this way, various criteria are considered instead of an unique optimization criterion.

More formally, the viability approach deals with dynamic systems under state and control constraints. The aim of the viability method is to analyse compatibility between the possibly uncertain dynamics of a system and state or control constraints, and then to determine the set of controls or decisions that would prevent this system from going into crises i.e. from violating these constraints. We refer for instance to [START_REF] Martinet | Sustainability of an economy with an exhaustible resource: a viable control approach[END_REF]; [START_REF] Béné | A viability analysis for a bio-economic model[END_REF]; [START_REF] Doyen | Sustainability of fisheries through marine reserves: a robust modeling analysis[END_REF] for some stylised models in other contexts.

More specifically, in the environmental context, viability may imply the satisfaction of both economic and environmental constraints. In this sense, it is a multi-criteria approach sometimes known as "co-viability". Moreover, since the viability constraints are the same at any moment and the term horizon is infinite, an intergenerational equity feature is naturally integrated within this framework.

The viability kernel: a definition The viability kernel is the set of bioeconomic states that make it possible to satifsy the constraints throughout time, given the dynamics. It is defined as the set of all states (S t , X t ) from which there is at least one feasable path (S(.), X(.)), associated with admissible decisions (e(.), ξ(.)), that satisfies all the constraints along time. It means that, starting from a state outside the viability kernel, there are no decisions that make it possible to satisfy the constraints forever (at least one of the constraints won't be respected after some finite time T , whatever decisions apply).

Formally, for our problem, the viability kernel is defined by

V iab =              (S 0 , X 0 )
∃(e(.), ξ(.)) and (S(.), X(.)), starting from (S 0 , X 0 ) satisfying dynamics (3) and ( 6)

and constraints ( 7), ( 8), ( 9) and ( 10) for any t

∈ N +              (A.4)
In the general mathematical framework, this set can alternatively be empty, the whole state constraint set K, or even a strict part of the initial state constraint domain. The viability kernel captures an irreversibility mechanism. Indeed, from the very definition of this kernel, every state lying outside the viability kernel violates the constraints in finite time, no matter what the decisions applied. This situation means that crisis is unavoidable. For instance, the extreme case where the viability kernel is empty corresponds to a hopeless configuration. An empty kernel means that there are no sustainable paths for the described system, i.e. no paths that respect the set of constraints representing sustainability. It is then necessary to "modify" the problem (to consider a different set of sustainability constraints) to be able to define some sustainable decisions or states.

This kernel is the set of initial fleet configurations and resource stocks from which it is possible to define acceptable regimes of exploitation satisfying all of the constraints throughout time. Therefore, the viability kernel provides the ex post viability constraints that are the "true" constraints to be satisfied by the bioeconomic system to be sustainable, in the sense that if a state is out this kernel, there are no decisions that make it possible to satisfy the constraints throughout time.

Fig. 1 .

 1 Fig. 1. Minimal effort e(S t ) required to satisfy profit constraint (9), with respect to the stock size.

  Fig. 4. Viability kernel.

Fig. 5 .

 5 Fig. 5. Scale of viability and minimal time of crisis.

Fig. 6 .

 6 Fig. 6. Transition phases under constraint.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Biological and economic states allowing to sustain the MSY production.

Fig. 9 .

 9 Fig. 9. Minimal time of crisis associated with MSY production level and maximal per vessel profit.

Fig. 10 .

 10 Fig. 10. Transition phases under economic viability constraint to reach MSY production level.

  Figure A.1 represents observed and predicted CPUE.

Fig. A. 1 .

 1 Fig. A.1. Fitting of observed and predicted CPUE in the biological parameters estimation model.

  Fleet size X t Open Acces CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size CBA_Fleet_Size

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		(e) Profit π t Open Acces				(f) Profit π t CBA
	Fig. 2. Initial conditions are combining four stock levels (5 000, 12 500, 20 000 and
	27 tons) and four fleet size (100, 200, 350 and 450 vessels).

The maximal day of sea per period ē represents a technical constraints. In any case, it cannot be more than 365 days per year.

Taking λ = 1 means that the studied species is the only one exploited by the fleet.

This interpretation is somewhat different from that encountered in the literature regarding capital inertia, which is assumed to result mainly from the lack of possibilities to quickly reallocate specific fishing assets to alternative uses, a technical, rather than social constraint.

It tends towards infinity in the neighborhood of S .

The periodic behavior observed here is linked to the choice of the adjustment possibilities in vessel numbers. We consider that, in a free access configuration, a maximal number of vessels will enter the fishery if there are some positive profits. A finer adjustment parameter (a slower entry) would lead to smaller variations. The extreme case of a continuous capital stock adjustment would lead to a steady state point. Note that the resource can not been exhausted as the catches per unit of effort become very small when the stock is reduced to low levels.

For the numerical application, we set an interest rate equals to 5%

The observed amplitude of variations does not depend on the size of the resource stock, just on that of the fleet.

A higher production level requires higher capital use.

This paper was prepared as part of the CHALOUPE research project, funded by the French National Research Agency under its Biodiversity program. The authors would like to thank