Nicolas Sanz

Bassirou Diop

A search-matching model of …sheries

Keywords: …sheries, search costs, bioeconomic matching model JEL Classi…cation: Q22

This paper studies the implications of …sh search activities on the working of …sheries. We develop a simple …shery matching model in which …rms' stochastic search for …sh requires speci…c resources. The equilibrium level of anthropic pressure (number of empty vessels/conserved …sh stock) exerted by …rms on the …sh stock is derived from their individual pro…t maximizing behavior. Anthropic pressure determines in turn the rate at which vessels …nd …sh i.e. catchability, which is therefore endogenous in our model, and the stock conservation (uncaught …sh) rate, through the dynamic equation of the latter. We then show that under open access, equilibrium anthropic pressure, …shing e¤ort and the conservation rate heavily depend on search costs. An empirical study of the French Guiana shrimp …shery con…rms the ability of the model to reproduce empirical data.

Introduction

A particular feature of …sh production is that the targets' search process, that it almost always includes, often represents the most time-consuming and costly activity, especially when compared with harvesting per se. Next to the "…shing e¤ort" cost, which encompasses notably input prices such as crew wages and catching material costs, …shing …rms must often spend additional resources to acquire some information about the location, size, and quality of …sh [START_REF] Mangel | Regulatory mechanisms and information processing in uncertain …sheries[END_REF]. The corresponding expenses in fuel consumption, investments into detection materials and so on, may be therefore non neglectable. There are numerous examples of …sheries in which signi…cant e¤ort is devoted to the search activity itself (see e.g. [START_REF] Mangel | Uncertainty, search, and information in …sheries[END_REF] 1 . Moreover, environmental variability implies that …sh harvesting does not correspond to a deterministic process. Since …sh distribution is often patchy within …shing areas, successful …sh catches are by de…nition uncertain and therefore potentially very costly for …sh producers. [START_REF] Mangel | Uncertainty, search, and information in …sheries[END_REF] underline the fact that unlike other natural ressources where search is also important (such as oil exploration), …sh harvest is even more complicated by the movement of …sh, which must still be found even if it is known to be present.

Nevertheless, a quite surprising fact is that the e¤ects of search costs on the stocks and harvest levels have rarely been deeply studied in the scien-ti…c literature devoted to this subject. Most canonical bioeconomic …shery models such as the Gordon-Schaefer model [START_REF] Gordon | The economic theory of a common property resource: the …shery[END_REF][START_REF] Schaefer | Some considerations of population dynamics and economics in relation to the management of marine …sheries[END_REF] or Clark's model (Clark, 1976) are of deterministic nature and voluntarily neglect search costs, thus preventing the analytical framework they rely on from giving an accurate and complete description of …sh harvesting2 . Only a few contributions have later dealt with the present topic. [START_REF] Mangel | Uncertainty, search, and information in …sheries[END_REF], [START_REF] Mangel | Search and stock depletion: theory and applications[END_REF], and [START_REF] Mangel | Regulatory mechanisms and information processing in uncertain …sheries[END_REF] explicitly model search activities as stochastic Poisson processes3 . Since matching between …sh and …shermen is not instantaneous, it mechanically generates some costs for the latter. The above authors then determine the optimal allocation of search e¤ort between several historical …shing grounds, taking into account the stochastic …sh distributions between them. According to these works, the most important source of uncertainty concerns the location of …sh.

In this paper, we start from the idea that search activities for …sh often imply the main costs for …shing …rms and must for this reason be explicitly taken into account in the economic and biological analysis of …sheries 4 . Moreover, the magnitude of search costs might mainly depend on the probability for a vessel to …nd …sh, which itself relies on the relative number of empty vessels and yet uncaught …sh that are simultaneously present within the …shery. In order to proceed, we adapt the search-matching framework developed by Sveriges Riksbank prize in economic sciences in memory of Alfred Nobel laureates Peter Diamond, Dale Mortensen, and Christopher A. Pissarides for the analysis of labor markets to the case of …sheries5 . In our simple bioeconomic matching model, …shing …rms explicitly allocate resources to search activities by sending empty vessels to sea. Once a …rm has decided to send an empty vessel to sea, the latter starts searching for …sh in a situation of imperfect information about the location of …sh. The …shing e¤ort function is thus microfounded and derived from …rms' pro…t maximizing behavior. Empty vessels and uncaught …sh then meet each other gradually according to a "matching technology". During the matching process and before all unmatched vessel-…sh pairs meet, some new …sh arrive into the …shery due to natural biological growth, providing a steady ‡ow into the uncaught …sh pool. With the above assumptions, there still remain some uncaught …sh in the steady state, i.e. the uncaught …sh rate, de…ned as the ratio between the uncaught …sh and the overall …sh stock available in the …shery, remains positive and is derived from the …sh's transitions in and out of the …shery. In our model, the uncaught …sh rate constitutes an indicator of the species conservation state. The transition probabilities are in ‡uenced by both individual …shing …rms'decisions and aggregate events that occur at the global …shery level. In particular, the rate at which vessels …nd …sh, which corresponds to the usual "catchability" coe¢ cient in the literature on …sheries, depends on the relative number of vessels with respect to the number of …sh units that evolve within the …shery, that we call "anthropic pressure". The fact that anthropic pressure is chosen by …rms through their pro…t maximizing behavior makes the catchability coe¢ cient endogenous, which constitutes an important departure of the present model from existing works that usually consider it as given. Here, harvest e¢ ciency evolves with respect to the ratio between both kinds of participating individuals to the …shing/meeting process. The model then allows for determining an equilibrium rate of uncaught …sh, and to highlight the factors that in ‡uence it, i.e. to develop a theory of natural resources conservation. Lastly, the introduction of a natural growth function into the dynamic equation of the uncaught …sh rate also makes it possible to capture some biological parameters usually retained in bioeconomics models, such as the intrinsic growth rate or the carrying capacity of the habitat of the studied species.

The paper is organized as follows. Section 2 presents a simple …shery search-matching model, that gives a detailed description of the meeting process between vessels and …sh and the determination of the steady state equilibrium uncaught …sh rate of a given species. Section 3 gives the equilibrium of the …shery when the latter is assumed to work under an open access environment, where …rms seek to maximize their pro…t in a decentralized manner. Section 4 provides an illustrative application of the model to the case of the French Guiana shrimp …shery. Lastly, Section 5 summarizes the main …ndings of the paper and gives some concluding remarks.

A …shery search-matching model

We assume that "our"…shery entails …shing …rms equiped with vessels, and …sh units (individuals of any size, schools...). Production is not modelled as a continuous process with varying intensity, but as instantaneous, with the search for …sh as time consuming.

The meeting process between vessels and …sh

The …shery is made up of a given …sh stock, X, and an endogenous number …rms, V , each equiped with a single vessel. We assume that once a unit of …sh has been found by a vessel, it is instantaneously successfully caught and removed from X. Thus, at any moment in time, the …sh stock X is equal to the sum of the F found and caught …sh and the U unmatched and thus, uncaught (conserved) …sh. We further assume that, after having found and caught it, …rms can only load their single vessel with a single unit of …sh. Thus the total number of vessels, V , equals the E empty vessels plus the F …lled vessels. We assume that only the U uncaught …sh and the E empty vessels engage in meeting. We use for convenience the corresponding density variables, u and e, respectively representing the uncaught …sh rate and the empty vessel rate, both with respect to the …sh stock, and thus de…ned by: u = U=X; e = E=X. Since it re ‡ects the conservation state of the stock, u represents the …rst main variable of the model. The following matching/catching function gives the number of met and thus caught …sh as a function of the U uncaught …sh and the E empty vessels at any moment in time: F (U; E) = F (uX; eX). We assume that F exhibits the same properties as most of the production functions used in the …sheries science literature, i.e. it is increasing and concave in its two arguments [START_REF] Morey | A Generalized Harvest Function for Fishing: Allocating E¤ort among Common Property Cod Stocks (a Generalized Harvest Function)[END_REF][START_REF] Clark | Mathematical Bioeconomics: The Optimal Management of Renewable Resources[END_REF]. Moreover, we further suppose that it exhibits constant returnsto-scale. Situations of either decreasing or increasing returns on both the harvest e¤ort and the stock levels may of course be observed in some …sheries and the matching function retained here would need further be empirically estimated. However, such a study is beyond the scope of the presentation of the basic model. As we shall see, the homogeneity property makes it possible to give a convenient formulation of the model and to present its main qualitative properties in a very simple manner. If F is homogenous of degree one, it can be expressed in the following way:

F (uX; eX) = f (u; e) X (1)
where f (u; e) denotes the matching rate, i.e. the fraction of the stock that is caught by vessels per unit time.

The steady state conservation rate

In this subsection, we focus the analysis on the conservation rate of the …sh stock. We assume that only empty vessels and uncaught …sh take part to the matching process and are thus are randomly selected from the sets eL and uL. According to the above matching function, the process that makes a unit of …sh meet a vessel is Poisson with parameter f (uX; eX) = (uX). Still from the homogeneity property of the matching function, f (uX; eX) =uX = (e=u) f (u=e; 1) = f (1= ; 1) q ( ), where = e=u and q 0 ( ) < 06 . By de…nition, constitutes an indicator of the anthropic pressure exerted by …rms on the …sh stock. As will be shown below, is derived from the …rms' pro…t maximizing behavior, which makes the catchability coe¢ cient, q ( ), endogenous in our model. By construction, q ( ) indicates the rate (frequency) at which …sh are being found by vessels7 . Hence, the mean number of uncaught …sh that are found by vessels during a small time interval is f X t = q ( ) uX t, where q ( ) t reads the …sh's transition probability, and 1= [ q ( ) t], the period at the end of which a …sh is being caught. The steady state uncaught …sh rate is equal to the di¤erence between the rate of new safe …sh that arrive into the …shery, which corresponds to the …sh stock's (relative) natural growth rate, b ( ) = B ( ) =X, and the rate of …sh that leave the uncaught …sh pool, q ( ) uX. In the steady state, b = q ( ) uX, which, isolating u, gives:

u = b q ( ) (2)
Equation ( 2) is the stationary condition of the model and corresponds to its …rst key equation. It indicates that, at the stationary equilibrium, the uncaught …sh rate logically increases with the stock natural growth rate and decreases with the rate at which …sh are being caught by vessels. Whereas u is clearly decreasing with q ( ), the e¤ect of itself on u could seem ambiguous at …rst sight. This is due to the simultaneous presences of two kinds of externalities that work in opposite directions. On the one hand, the participation externality (a higher number of vessels relative to the uncaught …sh stock increases the matching rate/the probability for a …sh to be caught), explains why the uncaught …sh rate partly increases with anthropic pressure through the term q ( ) (q 0 ( ) < 0). On the other hand, the search or congestion externality (a higher number of vessels reduces the probability that an isolated vessel …nds a …sh unit) is re ‡ected by the fact that a rise in leads directly to a fall in u. With the functional form of the matching technology that will be retained later for illustration purposes, the …rst e¤ect dominates the second one, making u all in all decreasing with .

Open access equilibrium

In this section, we assume that …shing activities are totally uncontrolled and that there is free entry and exit of …rms into and out of the …shery.

Firms'…shing e¤ort and equilibrium anthropic pressure

This subsection is devoted to the determination of the equilibrium level of anthropic pressure, , that …rms set on the basis of pure economic considerations. In our setting, the "…shing e¤ort" variable usually encountered in the literature on …sheries has its counterpart through the …rms'decision about whether or not to enter the …shery and to send empty vessels to sea for …shing. An empty vessel searching for …sh costs s per unit time, and after having found a …sh unit, immediately catches the latter which yields an instantaneous pro…t . We assume that s integrates all the costs that are necessary to equip a vessel for a …shing campaign (crew wages, fuel...).

It corresponds here to the usual overall cost of …shing e¤ort. Firms have a positive discount rate, r, and seek to maximize the present-discounted value of expected pro…ts. They have full knowledge of the matching process but do not coordinate their actions and take the matching probabilities as given.

As long as …shing …rms have not found any …sh yet, their vessel remains empty 8 . The process that changes the state of an empty vessel is Poisson with rate f (uX; eX) = (eX) = f (u=e; 1) q ( ). In other words, during a small time interval t, an empty vessel …nds a unit of …sh at a meeting rate (or frequency) q ( ), or with probability q ( ) t, which implies that its mean duration (or period) is equal to 1=q ( ). Let E and F be respectively the present-discounted value of the ‡ow of instantaneous pro…t earned by a …shing …rm with respectively an empty vessel and a …lled vessel. After some calculations, E and F can be written as (see Appendix):

r E = s + q ( ) ( E + F ) (3) r F = r ( E + ) (4)
In a perfect capital market, the valuation of their vessels by …shing …rms is such that, whatever the state of the vessel (empty or …lled), the capital cost, r , is exactly equal to the rate of return on the vessel. In Eq. ( 3), an empty vessel costs the …rm s per unit time. The empty vessel …nds a …sh unit and changes state according to a Poisson process with rate q ( ), yielding net return ( E + F ). At a stationary state, since the vessel's change in status from …lled to empty is instantaneous, the pro…t from a …lled vessel, F , is independant from the interest rate and equals the present-discounted value of an empty vessel plus the pro…t from the catch, E + 9 .

We assume that …rms maximize their pro…t by sending vessels to sea until the present-discounted value of the expected pro…t from the last engaged empty vessel is equal to zero ( E = 0). This assumption ensures that all …sh production opportunities are exploited. Since each …rm only owns a single vessel, it corresponds to a zero pro…t condition applied at the global …shery level, and thus represents an open access situation. Applying E = 0 to (3) and isolating F gives the present-discounted value of the expected pro…t from a …lled vessel under open access, F = s=q ( ): in equilibrium, the number of vessels sent to sea by …rms is such that the expected pro…t from a …lled vessel, F , is exactly equal to the expected cost of a vacant job, i.e. the per unit time search cost, s, weighted by the rate at which a …sh unit is found, q ( ). Now, also setting E = 0 in (4) leads to F = 8 As [START_REF] Mangel | Uncertainty, search, and information in …sheries[END_REF], we assume here that the search component of …shing operations is the most important stochastic consideration. What …shing …rms can do about the uncertainties concerning weather or stock size and quality is neglectable in comparison with what they can accomplish in the way of locating …sh (see also [START_REF] Mangel | Regulatory mechanisms and information processing in uncertain …sheries[END_REF], on this particular point). 9 Here, the vessels owned by …rms may be interpreted as assets. In …nancial economics, Eq. ( 3)-(4) are called "Capital Asset Pricing Market" (CAPM) equations. In a more general context, [START_REF] Scott | The Fishery: The Objectives of Sole Ownership[END_REF] already suggested to treat stocks of natural resources as assets as well.

. Replacing F by in (3), always with E = 0, dividing all terms by q ( ) and rearranging implicitly leads to:

s=q ( ) = 0. In equilibrium, …rms' instantaneous pro…t yielded by the production of one …sh unit, , net of average search cost, s=q ( ), equals zero. Lastly, isolating q ( ) gives implicitly the equilibrium value of anthropic pressure:

q ( ) = s (5)
Since q 0 ( ) < 0, Eq. ( 5) is the second key equation of the model. It indicates that equilibrium anthropic pressure increases with …rms'pro…t, , and decreases with the search cost, s. Since it is derived from …shing …rms' economic decisions, it constitutes the driving force of the model.

Global outcome

The model includes three unknowns, , u, and e and the …shery equilibrium is de…ned around three equations. Eq. ( 5) gives implicitly the equilibrium value of . Though being set at the individual level, is a global variable and states therefore a bridge between the …rms'microeconomic decisions and the global situation of the …shery. For a given , the uncaught …sh rate, u, is found with help of Eq. (2). Finally, the equilibrium rate of …rms with an empty vessel, e, follows from the de…nition of , that implies: e = u. Deriving from the above results and the properties of q ( ), the sign of the changes in the exogenous parameters on OA , u OA , and e OA are summarized in Table 1: Ta b le 1 : S e n sitiv ity a n a ly sis o f sta tio n n a ry e q u ilib riu m

s q b OA - + 0 u OA + -+ e OA -0 +
The equilibrium of the …shery can be represented by the intersection of two curves, the Species Conservation curve (SC) and the Firms' Entry curve (FE), respectively given by Eq. ( 2) and ( 5), and the de…nition of . The overall situation is depicted in Figure 1.As can be seen, the (SC) curve is decreasing and convex in the (u, e) space. As the empty vessel rate falls along the (SC) curve, the global harvest level decreases, which mechanically translates into a rise in the uncaught …sh rate. In other words, the simultaneous fall in e and rise u make anthropic pressure, and thus the probability for a …sh to be caught by an empty vessel, q ( ), decrease. Eq. (2) then indicates that for the uncaught …sh rate to remain constant at the stationnary state, the number of empty vessels engaged into search must decrease. Conversely, a higher empty vessel rate corresponds to a lower uncaught …sh rate. As far as the (FE) curve is concerned, Eq. ( 5) clearly indicates that it is linear in u, going through the origin in the (u, e) locus, with slope . An increase in the uncaught …sh rate, u, leads to a fall in anthropic pressure, , which makes the probability for an empty vessel to capture a …sh, q ( ), increase (q 0 ( ) < 0). This in turn makes the cost incured by an empty vessel, s=q ( ), decrease, which induces …rms to search for …sh more intensively, i.e. a rise in e.

An increase in the search cost, s, lowers expected pro…t, which logically leads to a fall in the number of empty vessels within the …shery in equilibrium, e OA , making the (FE) curve rotate clockwise. The move of the (FE) curve along the decreasing (SC) curve leads to a rise in the uncaught …sh rate, u OA . All in all, anthropic pressure increases within the …shery (since = e=u). An exogenous increase in the …sh's natural growth rate, b, corresponds obviously to a larger available stock which translates into a higher conservation rate. The rise in b also allows a higher level of exploitation, since it rises the probability of …nding a …sh unit and thus expected pro…t. The resulting changes in u OA and e OA are of the same range so that a change in b has no e¤ect on OA . An overall improvement of the matching process, illustrated by an exogenous rise in q, increases other things equal the probability of …nding …sh, and therefore lowers the average production cost. This translates into a counterclockwise move of the (FE) curve, and into a rise in the number of vessels searching for …sh, which makes the conservation rate fall and results into a rise in anthropic pressure.

The model can be further entirely closed by calculating the equilibrium number of vessels in the …shery. Since by assumption, each vessel can only load a single …sh unit, we have:

V OA = e OA X + (1 u OA ) X (6)
Beyond the technical closing of the model, the expression of V OA is of particular interest since it indicates that it results from two endogenous ‡ows of distinct natures: the ‡ow of empty vessels (…rst term), which derives from …rms'economic decisions, and the ‡ow of harvested …sh (second term), which depends on the conservation rate, thus on biological, technological and economic conditions. Furthermore, Eq. ( 6) will reveal useful for the empirical analysis undertaken in Section 5.

An example: the French Guiana shrimp …shery

The French Guiana waters entail numerous species and the shrimp constitutes a good example of a species that is exploited for economic purposes. Two shrimp species are mainly exploited in this …shery, the brown shrimp and the pink shrimp (resp. Farfantepenaeus Subtilis and Frafantepenaeus Brasiliensis). The Subtilis represents more than 85% of shrimp landings.

The French Guiana shrimp …shery started in the late 60's with the US ‡eet activity. Since 1992 the whole ‡eet is only composed with french trawlers targeting shrimp on the continental shell. One particular interest this …shery presents is that its stock has steadily decreased over years though the e¤ort and harvest levels have followed quite the same trend and some management in compliance with European legislation has been implemented. First a total allowable catch (TAC) of 4108 tons/year was adopted in 1983 and never changed until 2011. Furthermore, a license system was introduced in 1991 to limit the number of vessels. However it did not seem to be, in fact, an e¢ cient resource management tool. Finally there were also some spatial restrictions in order to limit the impact of trawling on juvenile shrimp. Trawling is nowadays forbidden in inshore water less than 30 meters deep. In addition, the pro…tability has declined in this industry due to growing aquaculture, which results in a lower shrimp price. The economic dynamics of the …shery has been characterized by a diminution of the ‡eet size, to concentrate the …shing activities on a reduced number of pro…table vessels. All in all, the ‡eet size is expected to be reduced to about 20 active vessels in year 2016 [START_REF] Chaboud | Bioeconomic model of the French Guyana shrimp …shery[END_REF].

The results obtained to this point enable to highlight some important economic, technological, and biological factors that may in ‡uence the conservation of the stock of a given species. The aim of this last section is to illustrate the use of the above search-matching model through the study of the French Guiana shrimp …shery, which might also help to more deeply understand the properties of the model.

Functional forms

In order to close the model explicitly and to be able to calibrate it, we retain the following functional form for the matching function (1):

f (u; e) = !u e 1 (7)
with 0 < < 1, and ! the harvest technology parameter. [START_REF] Sanz | On the in ‡uence of environmental factors on harvest : the French Guiana shrimp …shery paradox[END_REF] give some empirical support to the constant-returns to scale hypothesis in the case of the production of shrimp in French Guiana. Following [START_REF] Martinet | Fishery externalities and biodiversity: Trade-o¤s between the viability of shrimp trawling and the conservation of Frigatebirds in French Guiana[END_REF], we retain the logistic type function proposed by [START_REF] Verhulst | Notice sur la loi que la population suit dans son accroissement[END_REF] and [START_REF] Pearl | The Biology of Population Growth[END_REF] to describe the natural growth rate of the shrimp stock, that we express in stock units:

b (X; i; K) = i 1 X K (8)
where i represents the maximum relative growth or intrinsic growth rate, which is related to the studied species. Parameter K denotes the carrying capacity, that depends on the caracteristics of the natural environment in which the species evolves, such as the size and the biological productivity of the habitat. Both parameters are assumed to be …xed and for a biological equilibrium to exist, K > X and i > 0 must be veri…ed. Using (1) to isolate in (5), and assuming that search costs are proportional to production (i.e. s = sY ), one …nds10 :

OA = ! s 1 (9)
Eq. ( 9) gives an explicit expression of equilibrium anthropic pressure in the …shery under open access. Now, inserting (9) into the uncaught …sh rate equation (2) yields the expression of the equilibrium uncaught …sh rate in the …shery under open access:

u OA = i ! 1 1 X K s 1 (10)
Lastly, recalling that e = u, while using ( 9) and ( 10), leads to the equilibrium (relative) number of empty vessels within the …shery under open access11 :

e OA = i 1 X K s (11)

Calibration

Table 2 summarizes the values used for the parameters of the model. In line with the literature on production economics, we assume that the matching function exhibits constant returns-to-scale over its two inputs, i.e.

= 0:5. The intrinsic growth rate and the carrying capacity of the habitat are both taken from [START_REF] Martinet | Fishery externalities and biodiversity: Trade-o¤s between the viability of shrimp trawling and the conservation of Frigatebirds in French Guiana[END_REF]. The orders of magnitude for the biomass size are available at the French Guiana o¢ ce of the Institut Français de Recherche pour l'Exploitation de la Mer (Ifremer). Search cost s correspond to a mean value of fuel costs and are collected from the French Institut National de la Statistique et des Etudes Economiques (Insee; see Table 3). Lastly, the value of !, about which we do not hold a reliable order of magnitude, is chosen to calibrate the model, i.e. in such a way as to obtain conservation rates that are compatible with the data.

Simulations and results

Table 3 shows the available data that have been collected from the Ifremer and the Insee for the period 2000-2005, and the equilibrium and empirical values of the endogenous variables of the model. Firstly, Table 3 indicates that the stock has increased over the considered period while the harvest has decreased. Secondly, anthropic pressure on the shrimp stock decreases steadily because the rate of conservation (uncaught …sh rate), u, remains almost constant whereas the empty vessel rate falls. Now, comparison between u and u OA shows that …shing …rms have chosen an insu¢ cient level of anthropic pressure which led to an in ‡ated rate of conservation over the period. This statement is not surprising since the French Guiana shrimp biomass is known to be su¢ ciently abundant with respect to its exploitation, and quotas have never been reached when they were in force. This suggests that the French Guiana shrimp could be more intensively exploited under the economic viewpoint. As far as the total number of vessels is concerned, direct comparison between V and V OA suggests that there were too many vessels in the …shery over the period, some of them being non-pro…table. This result could explain the steady decline in the number of vessels evolving within the …shery during the considered period12 . Lastly, Table 3 shows that fuel costs per harvested unit have decreased from 0:56 to 0:51 between 2000 and 2005, while the shrimp uncaught …sh rate has risen from 0:59 to 0:64 over the same period, thus exhibiting an absolute rise of 0:05.

Conclusion

As [START_REF] Bjorndal | The economics of …sheries management: a survey[END_REF] point out, "the economics of …sheries management under is currently underdeveloped. Without question, much remains to be done". The search-matching …shery model proposed in this paper introduces quite directly pure biological features into a stochastic economic framework, while adapting search theory to the speci…c economic behavior of …shing …rms and being based on precise microeconomic foundations. The model therefore exhibits the attractive properties of its original source, making it possible to study the e¤ects of changes in search costs on several variables speci…c to …sheries. The application of the model to the French Guiana shrimp …shery suggests that catch e¢ ciency, and the costs that are associated with it, play a major role in explaining the evolution of the conservation of some species. It suggests that changes in search costs per se (mainly in fuel consumption) heavily explain the variations in anthropic pressure, …shing e¤ort and the state of shrimp conservation during the period, as well as the evolution of the …shing ‡eet engaged into the …shery.

Beyond the results obtained here, our framework may be further extended into several directions in order to capture other speci…c features of the working of …sheries, and seems quite easily tractable for direct empirical analysis. Among the numerous theoretical extensions of the search-matching model that have already been undertaken in the past, some of them, like the endogenization of search e¤ort, could also be considered in our framework to analyze the consequences of the interactions between …shing …rms13 . Additionally, a deeper analysis of the transitional dynamics of the model should also be undertaken in order to improve the explaining power of the latter. Lastly, the integration of bycatch and discard behaviors into the present basic model may also constitute other relevant avenues for future research 14 .

All in all, the use of a search-matching model for the analysis of …sheries seems at …rst sight at least as natural as its application to e.g. labor markets. We thus hope that our framework will help to improve the understanding of the working of …sheries by notably providing a convenient tool suitable for the design of management policies.

Appendix (in continuous time)

Method 1 (formal method)

The present-discounted values of expected pro…t from an empty and a …lled vessels are respectively given by: where " is the mathematical expectation. An empty vessel searching for …sh costs s per unit time between 0 and T , where T is the date at which the vessel …nds a …sh unit. T is thus a random variable that follows a Poisson process, whose parameter is given by the rate at which vessels …nd …sh units, q ( ). Still at date T , the vessel instantaneously yields pro…t from the catch, , and the present-discounted of expected pro…t from an empty vessel, E , to the …rm. A …lled vessel instantaneously yields pro…t from the catch, , becomes empty and search for …sh between 0 and T , where T is a random variable that follows a Poisson process of the same parameter as above, i.e. that is given by the rate at which vessels …nd …sh units, q ( ). At date T , the empty vessel …nds a …sh unit and recovers the status of a …lled vessel, earning the corresonding present-discounted value of expected pro…t, E .

Let us start from calculating the integral entailed in Eq. ( A1)-(A2), that become: Now, if T is a random variable that follows a Poisson process of parameter q, we have: , " e rT = R 1 0 qe (r+q)T dT . Thus, Eq. ( A3)-(A4) become:

E = s Z 1 0 ( 1 r e rt

Figure 1 :

 1 Figure 1: Fish search and equilibrium conservation

  Ta b le 3 : E n d o g e n o u s va ria b le s e q u ilib riu m a n d e m p iric a l va lu e s ove r th e p e rio d 2

  Figure 2: Sensitivity of the shrimp conservation rate with respect to the search cost

e

  rt dt + F " e rT (A2)

e

  rt dt + F " e rT (A4)

Table 2 :

 2 Parameters values used in the model

Ceregmia-Université des Antilles, Campus de Schoelcher, BP 7209, 97275 Schoelcher Cedex, France. Tel.: +596.596.727.400; Fax: +596.596.727.403; Email: nicolas.sanz@univ-guyane.fr, bassirou.diop@etu.univ-ag.fr.

The …sheries retained in[START_REF] Mangel | Uncertainty, search, and information in …sheries[END_REF] are the eastern tropical Paci…c tuna-purse-seine …shery, the British Colombia salmon-purse-seine …shery, the Australian Gulf of Carpentaria prawn …shery, and the California-Oregon-Washington shrimp …shery. However, in aquaculture or for some easily locatable species, search activities and their associated costs may be much lower (see e.g.[START_REF] Bjorndal | Stock Size, Harvesting Costs, and the Potential for Extinction: The Case of Sealing[END_REF], for a discussion on this point in the case of seals or schooling species).

See e.g.[START_REF] Munro | Mathematical bioeconomics and the evolution of modern …sheries economics[END_REF] for a survey on mathematical bioeconomic models of …sheries.

[START_REF] Clark | Foraging and ‡ocking strategies: Information in an uncertain environment[END_REF] also use a Poisson stochastic process to describe the behavior of animals searching for forage and the role of information in their foraging strategies. See also[START_REF] Mcconnell | Random utility models of recreational …shing: catching …sh using a poisson process[END_REF] for a use of Poisson processes in the modelling of sport…shing.

For a study of …sheries management costs, see[START_REF] Arnason | Costs of Fisheries Management: Theoretical and Practical Implications[END_REF].

For a general presentation on search-matching models applied to the labor market, see[START_REF] Pissarides | Equilibrium Unemployment Theory[END_REF].

It can be easily shown that the assumptions adopted above for the matching technology imply q 0 ( ) < 0, and that the elasticity of q ( ) with respect to is entailed between 0 and 1. Hence, empty vessels …nd …sh more easily when there are more …sh relative to empty vessels within the …shery.

For the sake of clarity, we assume here that found …sh are systematically caught and therefore de…nitely eliminated from the stock. The possibility of discarding behavior by …shermen, although interesting, is left to later work. More broadly, bycatch phenomena are also voluntarily ignored here.

[START_REF] Clark | The economics of …shing and modern capital theory: A simpli…ed approach[END_REF] provide some theoretical foundations to the linear dependence of costs on the harvest level.

In the present example, the equations of the (SC) and (FE) curves can be respectively obtained by replacing by e=u and isolating e in Eq. (2) and (5), which leads to e SC = (b=!)1=(1 ) u =(1 ) , and e F E = ! s 1= u.

Several …shing …rms have closed over the same period, mainly because of pro…tability de…ciency.

Variable search e¤ort must be related to the question of information sharing (de…ciency) among …shermen about the location of productive …shing sites, as inEvans and Weninger (2014).

 14 The present model may be also applied to e.g. terrestrial animals hunting.

After calculating the integrals and simplifying, one obtains:

Lastly, solving for E and F leads to respectively Eq. ( 3)-(4).

Method 2 (intuitive method)

According to the assumptions retained in our model, the present-discounted values of expected pro…t from an empty vessel and a …lled vessel can be respectively written as:

At a discount rate r, an empty vessel costs s per unit time and go on yielding the expected value of an empty vessel, E , as long as it remains in this state with probability 1 q ( ) dt. With the complementary probability q ( ) dt, it …nds a …sh unit, yielding the expected value from a …lled vessel, F . Besides, a …lled vessel generates instantaneously a pro…t . It then becomes also instantaneously empty and yields the expected value associated with the empty vessel state, E . Multiplying all terms by (1 + rdt) and rearranging terms if dt tends to zero leads to Eq. ( 3)-(4) in the text.


Statistical Properties of Fracture Precursors

A. Garcimartı ´n,* A. Guarino, L. Bellon, and S. Ciliberto † Laboratoire de Physique, Ecole Normale Supérieure de Lyon, 46, allée d'Italie, 69364 Lyon Cedex 07, France (Received 4 June 1997) We present the data of a mode-I fracture experiment. The samples are broken under imposed pressure. The acoustic emission of microfractures before the breakup of the sample is registered. From the acoustic signals, the position of microfractures and the energy released are calculated. A measure of the clustering of microfractures yields information about the critical load. The statistics from energy measurements strongly suggest that the fracture can be viewed as a critical phenomenon; energy events are distributed in magnitude as a power law, and a critical exponent is found for the energy near fracture.

[S0031-9007(97)04346 -9] PACS numbers: 62.20.Mk, 46.30.Nz

Fracture is a problem which has recently received a lot of attention in the physics community [1][2][3]. It is troublesome to calculate the force needed to break a heterogeneous material. Instead, it is customary to resort to tests involving the destruction of the sample. Therefore it is interesting to provide additional knowledge about cracks by studying the events that occur prior to the fracture. Besides, despite great experimental and numerical efforts [1-6], many aspects still remain unclear about the fracture process itself. Conceptually simple models, such as percolation [START_REF]Statistical Models for the Fracture of Disordered Media[END_REF] and self-organized criticality [START_REF] Bak | [END_REF], are attractive but often fail to convey the complex phenomenology observed. The main motivation of this work is to understand if these models can reproduce the main features of crack formation.

We report here some experimental results that may help to gain valuable information in that direction. Our main tool is the monitoring of the microfractures, which occur before the final breakup, by recording their acoustic emissions (AE). Because of its ability to pinpoint the emission source, this technique has been widely used in seismography and to map the nucleation of fractures [8]. From these signals, we have also obtained the acoustic energy of each microfracture, which is a fraction of the total energy released. The behavior of the energy just before fracture is a good parameter to compare with the above mentioned models.

In order to avoid noise, we have designed a setup in which there are no moving parts, the force being exerted by pressurized air (see Fig. 1). A circular sample having a diameter of 22 cm and a thickness of 5 mm is placed between two chambers between which a pressure difference P ≠ P 2 2 P 1 is imposed. The deformation of the plate at the center is bigger than its thickness, then the load is mainly radial [START_REF] Landau | Theory of Elasticity[END_REF][START_REF] Timoshenko | Theory of Plates and Shells, Engineering Mechanics Series[END_REF]. Therefore, the experience can be thought of as a mode-I test with circular symmetry. The pressure difference P supported by the sample is slowly increased and it is monitored by a differential transducer. This measure has a stability of 0.002 atm. The fracture pressure for the different tested materials ranges from 0.7 to 2 atm. We regulate P by means of a feedback loop and an electronically controlled valve which connects one of the two chambers to a pressurized air reservoir. The time taken to correct pressure variations (about 0.1 s) is smaller than the characteristic time of the strain rate. An inductive displacement sensor gives the deformation at the center of the plate with a precision of about 10 mm (the deformation just before fracture is of the order of 1 cm). The apparatus is placed inside a copper box covered with a thick foam layer to avoid both electrical and acoustical noise. Four wide-band piezoelectric microphones are placed on the side of the sample (see Fig. 1). The signal is amplified, low-pass filtered at 70 kHz, and sent to a digitizing oscilloscope and to an electronic device which measures the acoustic energy detected by the microphones. The signal captured by the oscilloscope is sent to a computer where a program automatically detects the arrival time of the AE at each microphone. Afterwards, a calculation yields the position of the source inside the sample. A fraction of the detected events is rejected, either as a result of a large uncertainty of the location, or because they are regarded as noise. The mean standard error for the calculated positions is about 6 mm, which results mainly from the uncertainty of the arrival time. The electronic device that measures the energy performs the square of the AE amplitude and then integrates it over a time window of 30 ms, which is the maximum duration of one acoustic event. The output signal is proportional to the energy of the events [11], and its value is sent to the computer. The dynamic range for the energy measurement is four decades, and the device is adjusted in such a way that only the strong sound emitted by the final crack saturates it. The global results of the measurements are the following: a list of the positions of microfractures, the strain of the samples, and the energy released as a function of P. Further details of the setup will be described elsewhere [START_REF] Guarino | [END_REF].

The loading can be applied in several ways. In order to allow comparison with some numerical models (specially those involving fuse networks, see, for example, [3,[START_REF] Sornette | [END_REF]), we load the sample in such a way that P is imposed and slowly increased. The analogy with a fuse network goes like this: the electric current is formally equivalent to the stress, the voltage across the lattice to the strain, and the conductivity to the Young modulus. Our experiment corresponds to a situation where current is imposed and voltage is the dependent variable, which is the standard situation in numerical simulations. This is important because the features of energy near fracture are different if displacement, instead of pressure, is imposed, as is often done in fracture experiments. Energy from AE released by fractures has been studied in many different situations: in granite [8], in volcanoes [14], in chemically induced fracture [15], in plaster samples cracked by piercing through them [16], and in the explosion of a spherical tank [17]. In our experiment we pay special attention to the following points. First, we place ourselves in a clearly defined situation, namely, mode-I fracture. Second, we follow a load procedure in which we impose the control parameter. Third, we take care to ensure that the energy detected, which is the "order parameter," is not contaminated from noise.

The samples are made of composite inhomogeneous materials, such as plaster, wood, or fiberglass. The experimental results are similar for all the materials. In this paper we present results for chipboard wood plates, which are most representative of the observed behavior. The chipboard we employed is made of glued short fibers randomly oriented, and cracks when stressed without defoliating. Each run is carried out in the following way. A sample is placed in the apparatus and a pressure ramp is applied. The rate of pressure increase, which is adjusted so that the succession of microfractures does not proceed too fast, is about 0.5 atmyh. In this way, the run lasts for about 2 h. In several runs, we have changed the pressure increase rate so that the sample cracks after a time spanning from 30 min to 5 h without noticing significant differences. However, if the sample is loaded, then unloaded before fracture, and loaded again, only a small number of microfractures are detected before attaining the previous load (Kaiser effect). In all the runs reported here, we have increased the pressure monotonically. Several hundreds, even thousands, of microfractures are detected prior to the destruction of the sample.

With the data acquired we are able to replay a "movie" of the run, plotting the location of the microfractures as the pressure is increased (Fig. 2). Figures 2(a)-2(e) show the localization of microfractures at pressure intervals sm 2 1dy5 , p , my5, where m ≠ 1, . . . , 5 and p ≠ PyP c is the normalized pressure, P c being the fracture pressure. Figure 2(f) shows all the microfractures registered. The final distribution of microfractures agrees well with the observed crack pattern. It has been noticed [8] that as the applied stress is increased, microfractures tend to nucleate and form a major fault, eventually causing the failure of the material. If microfractures are tightly clustered, it can be suspected that the material is severely damaged. It is therefore natural to try to quantify the extent to which microfractures are grouped together. Several measures of disorder have been proposed [18,[START_REF] Hero | Proceedings of the International Symposium on Information Theory[END_REF] that can be applied to this case. We have proceeded as follows. We look for the distribution of microfractures that happen within a given pressure interval. The sample surface is divided in squares and the number of microfractures n i inside each square is evaluated. The entropy S is then computed: S ≠ 2 P i q i ln q i , where q i ≠ n i yN, N being the total number of microfractures in the pressure interval. To compare between different pressure intervals, each of them having a different number of microfractures, the entropy for each interval is normalized to the equipartition entropy S e (the entropy of N events evenly distributed through the grid), so a normalized entropy s ≠ SyS e is obtained. The values s ≠ 1 and s ≠ 0 correspond, respectively, to total disorder and extreme concentration. We have used a grid containing a number of squares of the same order as the average N. In this way, the smallest structure that can be detected has a size of about 2 cm, which is slightly larger than the typical size of the clusters of microfractures. The value of s slightly depends on the grid size; the aspect of the curve, however, remains the same for the different grids we have used.

In Fig. 3 s is plotted versus p. It can be seen that s monotonically decreases as the control parameter is increased. In this way, the extent to which faults have developed can be grasped. This measure may provide a nondestructive, albeit damaging test. While this is valid for the other materials we have tested, we are not in a position to state that the decrease of entropy follows in all cases the same law. More work is needed to develop this method, but with present results the point where half the critical pressure is reached can be already predicted with an error of 620% at s ¯0.8 (see Fig. 3).

We now study the behavior of the released energy as a function of P. The instantaneous energy ´released by a sample is shown in Fig. 4(a) as a function of p. The energy signal consists of bursts, which correspond to microcracks. In particular, we are interested in analyzing the dependence on P of the energy near fracture. In percolation models, the order parameter diverges following a power law as the control parameter is increased. A critical exponent is then found. This is valid only close to the phase transition, say within 5% of the critical value.

To do this, we first calculate the cumulated energy E, i.e., the total energy released up to a pressure P. We then search a law in the form E ≠ E 0 s

P c 2P P c d 2a
, where E 0 and a are constants to be fitted. In Fig. 4(b) we present such a fit for the normalized data of 11 wood samples. We obtain a ≠ 0.27 6 0.05. This value shows small variations between different samples of the same material but presents a little dependency on the material. For fiberglass, for instance, a ≠ 0.22 6 0.05. This is not surprising since the behavior near the critical point, even in numerical models [START_REF]Statistical Models for the Fracture of Disordered Media[END_REF], is expected to depend strongly on microscopic features such as the geometry and the type of the bonds. It would be tempting to compare this figure to other exponents obtained in percolation models [START_REF]Statistical Models for the Fracture of Disordered Media[END_REF][START_REF] Sornette | [END_REF] or in other experiments [17]. This would not be reasonable, given the sensitivity to the specific details of the model. Nevertheless, the power law behavior is clearly close to P c . This is a good indicator that fracture could be modeled using those frameworks, and described as a critical phenomenon.

The probability density function for ´is plotted in Fig. 4(c). The upper and lower energy limits are, respec- tively, the strongest event recorded and the noise level. A power law is obtained spanning through more than two decades. This has also been observed in numerical simulations of the fracture of a bundle of fibers [START_REF] Hemmer | [END_REF]. Again the exponent shows little variation between different samples of the same material. For wood, the exponent is g ≠ 21.51 6 0.05. This value compares well to the one given in [START_REF] Bak | [END_REF] and [START_REF] Caldarelli | [END_REF], but it differs a little from one material to another. For fiberglass, for example, g ≠ 22.0 6 0.1. Even if the exponent agreement is coincidental or if it is not universal, the power law behavior strongly suggests a critical dynamics. An important remark should be made here. This feature may depend on the experimental procedure. In our experiment P is imposed and slowly increased, and this may easily trigger an avalanche, as in the scenario of self-organized criticality. If the loading is instead carried out by slowly increasing the strain, the power law might not be found. A detailed discussion about the dependence of the results on the loading methods will be treated in a forthcoming article [START_REF] Guarino | [END_REF].

In summary, we have presented experimental data showing a strong analogy between the formation of a crack in composite materials and percolation in a fuse network. On a qualitative level the microcracks clusterize around the final crack. On a more quantitative level, if the system is driven at imposed pressure, the data show the existence of a critical exponent for the energy released by microfractures near the critical point, supporting the view that fracture can be thought of as a critical phenomenon. We have also shown that the probability density function of the energy can be fitted by a power law. Finally, we propose that the localization entropy can provide some information about the critical load before the fracture is reached.

This work has been partially funded by Contracts No. ERBCHRXT940546 and No. ERBFMBICT950126 from the European Community. We acknowledge useful discussions with R. Livi, A. Politi, and S. Roux, and technical support by J. L. DeMarinis, M. Moulin, and F. Vittoz.

FIG. 1 .

 1 FIG. 1. Sketch of the setup. (a) Top view. S: sample; M: microphones; DS: displacement sensor. (b) Side view. EV: electronic valve. HPR: high pressure reservoir.

FIG. 2 .FIG. 3 .

 23 FIG. 2. Localization of microfractures for one sample as P is increased. The microfractures occurring at five equal pressure intervals are represented in (a)-(e). Pressure grows from (a) to (e). In (f) all the microfractures occurring during the run are plotted.

FIG. 4 .

 4 FIG. 4. (a) The energy released ´as a function of p for one of the samples. Energy bursts correspond to microfractures. (b) Cumulated normalized energy EyE max versus reduced pressure sP c 2 PdyP c , where E max is the total energy. A power fit has been done near P c (solid line). The inset is a zoom near P c . (c) Histogram of ´from the events registered in 11 wood plates. Energy intervals have a width of 5 times the measurement precision. The fit ln N ≠ ln ´0 2 g ln ´is shown.

0031-9007y97y79(17)y3202(4)$10.00 © 1997 The American Physical Society


Brahim Azaoui
email: brahim.azaoui@univ-amu.fr

Polyfocal classroom interactions and teaching gestures. An analysis of nonverbal orchestration

Keywords: teaching gestures, two-handedness, coenunciative ubiquity, context, nonverbal orchestration

While a growing body of research suggests that gestures have an impact in the teaching/learning process, few have explored gestures produced by teachers to understand how instructors cope with the intrinsically polyfocal dimension of class interactions. This paper reports on an empirically grounded account of both how and in what circumstances teachers conduct multimodal orchestration, and the interactional issues it raises. Because it is based on videorecorded corpora of two instructors each teaching both French to native and to non-native students, my study also tackles the issue of the context-sensitivity of teaching gestures.

Theoretical framework 1.1 Teaching gestures

A growing body of research has tackled the topic of teaching gestures in instructional and non-instructional contexts. These studies have mostly shown the impact of teaching gestures in different areas of the learning process. For example, we can consider the role of gesturing in the comprehension of math instructions or math problems ( [START_REF] Goldin-Meadow | Explaining math: Gesturing lightens the load[END_REF], [START_REF] Alibali | Students learn more when their teacher has learned to gesture effectively[END_REF], [START_REF] Hostetter | Don't just tell them, show them! Teachers can intentionally alter their instructional gestures[END_REF], [START_REF] Brookes | Polysigns and information density in teachers' gestures[END_REF]). Alibali et al. [START_REF] Hostetter | Don't just tell them, show them! Teachers can intentionally alter their instructional gestures[END_REF] for instance provided a math teacher with a tutorial about ways to use gestures in connecting ideas in instruction. The results demonstrate that students benefit more from the teacher who expresses linked ideas using both gestures and speech than from a teacher who does not. In language teaching contexts, a range of research has examined the impact of gestures in L1 or L2 teaching and learning ( [START_REF] Stam | Second language acquisition from a McNeillian perspective[END_REF], [START_REF] Colletta | Le développement de la parole chez l'enfant âgé de 6 à 11 ans. Corps, langage et cognition[END_REF], [START_REF] Lazaraton | Gestures and speech in the vocabulary explanations of one ESL teacher: a microanalytic inquiry[END_REF], [START_REF] Gullberg | Gesture as a communication strategy in second language discourse[END_REF], [START_REF] Gullberg | Introduction to gesture and SLA: toward an integrated approach[END_REF], [START_REF] Sime | Because of her gesture, it's easy to understand'. Learners' perception of teachers' gestures in the foreign language class[END_REF], [START_REF] Tellier | Faire un geste pour l'apprentissage: le geste pédagogique dans l'enseignement précoce[END_REF], [START_REF] Goodrich | Co-speech gesture as input in verb learning[END_REF]). In an empirical study Sime [START_REF] Sime | Because of her gesture, it's easy to understand'. Learners' perception of teachers' gestures in the foreign language class[END_REF] sought to understand what learners made of their teachers' gestures. She showed that they made a distinction between relevant and irrelevant gestures among those that their teachers produced, and they were able to attribute the relevance of these nonverbal actions within the learning process as they enhanced comprehension and provide feedbacks. Others have considered more specific aspects, like the role of gestures in memorization ( [START_REF] Tellier | The effect of gestures on second language memorisation by young children[END_REF], [START_REF] Allen | The effects of emblematic gestures on the development and access of mental representations of French expressions[END_REF], [START_REF] Kelly | Brief training with cospeech gesture lends a hand to word learning in a foreign language[END_REF]) or error correction ( [START_REF] Muramoto | Gesture in Japanese language instruction : the case of error correction[END_REF], [START_REF] Taleghani-Nikazm | Gestures in foreign language classrooms: an empirical analysis of their organization and function[END_REF]). For example, Tellier [START_REF] Tellier | The effect of gestures on second language memorisation by young children[END_REF] experimentally examined the impact of gesture on second language memorization in teaching vocabulary to 5 year-old learners. She showed how the teacher's gestures, and especially their reproduction by the learners helped the latter remember the words they were taught. Muramoto [START_REF] Muramoto | Gesture in Japanese language instruction : the case of error correction[END_REF] considered the role of gestures in providing error correction so as to contribute to students' successful self-correction. He analyzed the gestures of three instructors in a university Japanese second language classroom and distinguished two sorts of gestures in class: specific language error correction gestures and general foreign language classroom gestures. Yet, despite this impressive body of research, it seems that few studies have been interested in considering the gestures as a way for teachers to organize class turn-taking and deal with overlapping talks ( [START_REF] Kååntå | Pointing, underlining and gaze as resources of instructional action in classroom interaction[END_REF], [START_REF] Foester | Et le non-verbal ?[END_REF], [START_REF] Azaoui | Déritualisation et normes interactionnelles. Entre stratégie pédagogique et revendication identitaire[END_REF]) rather than a means to enhance learning. Azaoui's empirical study [START_REF] Azaoui | Déritualisation et normes interactionnelles. Entre stratégie pédagogique et revendication identitaire[END_REF] is based on a mimo-gestural analysis of both a corpus of filmed classroom interactions led by the same teacher in two different instructional contexts (French to native students and to non-natives) and video-recordings of students confronted with extracts of lessons they participated in. He sought to understand how, when and why the teacher reacts to the students' disruption of the interactional norms, but also how and why the students break this conversational organization [START_REF] Sacks | A simplest systematics for the organization of turn-taking for conversation[END_REF]. The results show that the teacher's motivations are twofold: the instructor's verbal and nonverbal actions contribute both to the progress of the lesson plan and the prevention of threats to the students' face [START_REF] Goffman | Interaction ritual. Essays on face-to-face[END_REF].

Classroom polyfocal interactions

Coping with multiple simultaneous actions is the reality of many teachers in classroom. Thus, it seems more accurate to consider classroom interactions as typically "polylogal" [START_REF] Bouchard | L'interaction en classe comme polylogue praxéologique[END_REF] (i.e., more than three persons usually speak at the same time; consequently interventions may overlap) -rather than looking at them as if they followed a regular three-part pattern [START_REF] Sinclair | Towards an analysis of discourse: the English used by teachers and pupils[END_REF]. If "trilogues are potentially more conflicting organizations than dialogue" [25:6] because participants may struggle even more for the floor, one can easily imagine what the situation may be like during polylogues where intrusions and overlapping turns may occur more spontaneously and frequently. In addition, classroom interactions can be said to be polyfocal as several foci of interaction may simultaneously take place [26:66]. Consequently, there is barely a moment when teachers do not produce several gestures at the same time (head/hand gestures, right hand/left hand gestures). So, as much as we can say that students have a polyfocal attention, to the extent that they very rarely "direct their attention in a focal, concentrated way to any single text or medium" (Scallon et al, cited in [27:28]), teachers' attention can also be qualified as being polyfocal. Since they have to manage various actions at the same time, Kress proposed the term "orchestration" to name the "process of assembling/organizing/designing a plurality of signs in different modes into a particular configuration to form a coherent arrangement" [28:162]. If we pay attention to the way this orchestration is conducted, we can notice that it takes various forms and has implications for the interactional process. These are the issues this paper proposes to tackle. It sets out to provide an empirically grounded account of both how and in what circumstances teachers conduct this orchestration, and what the interactional issues are. I will first present the methodology of this research. Then, I will examine the results in two separate but complementary sections: I will explore the notion of two-handedness, understood as the production of two-handed independent gestures, and that of co-enunciative ubiquity, which refers here to the teacher's nonverbal ability to be the co-utterer with at least two students simultaneously.

Methodology

Participants

My research is based on the analysis of two native French secondary school teachers from the South of France (Toulouse and Montpellier). They both teach French to native learners (FL1) and French to non-native students (FL2). The initial idea was to analyze how these teachers dealt with school norms (i.e., linguistic and interactional norms) according to the contexts and students they taught. The Toulouse teacher's French students were aged 14 whereas the Montpellier teacher's were 11. Both had 28 students per class on average. As for their non-native students, the classes they teach gather students from different origins and ages. In Toulouse, the class consisted of 12 different nationalities. The average age of the nonnative students was 12.5 while Montpellier's FL2 class was composed of non natives aged 13 or so who came from 4 different countries.

The corpora and the coding

To carry out this study the data were gathered empirically ( [START_REF] Labov | Some principles of linguistic methodology[END_REF], [START_REF] Van Lier | From input to affordance: Social-interactive learning from an ecological perspective[END_REF], [START_REF] Cambra Giné | Une approche ethnographique de la classe de langue[END_REF]) by filming each teacher in action in her two classes. I recorded some 20 hours of classroom interactions among which 6h30 were fully transcribed and coded using ELAN [START_REF] Sloetjes | Annotation by category -ELAN and ISO DCR[END_REF]. It included the transcribing of the speech of the teachers and the students on separate tiers, the annotating of the teachers' gesture dimensions, and the annotating of their mimics. I designed my typology of gesture and mimic dimensions and functions based on various works ( [START_REF] Heylen | Searching for prototypical facial feedback signals[END_REF], [START_REF] Mcneill | Hands and mind: what gestures reveal about thought[END_REF], [START_REF] Tellier | L'impact du geste pédagogique sur l'enseignement-apprentissage des langues étrangères : Etude sur des enfants de 5 ans[END_REF]). As far as gestures were concerned, I annotated emblems, deictics, metaphorics, beats, and iconics. As for the facial mimics, I coded the following dimensions: orientation of the gaze, frown, raise eyebrows, smile, nod, tilt. Combinations of two or three of these facial movement dimensions were possible. Following Tellier's typology [START_REF] Tellier | L'impact du geste pédagogique sur l'enseignement-apprentissage des langues étrangères : Etude sur des enfants de 5 ans[END_REF], I considered three main teaching gestures functions: informing, managing and assessing. I adapted the latter considering that it also concerned assessing the way students took the floor in compliance or not with school rules [START_REF] Azaoui | Coconstruction de normes scolaires et contextes d'enseignement. Une étude multimodale de l'agir professoral[END_REF].

The analysis tools

I mostly draw my analysis tools from the talk-in-interaction framework espoused by Kerbrat-Orecchioni [START_REF] Kerbrat-Orecchioni | Le discours en interaction[END_REF]. The author emphasizes on the need to analyze interactions by merging theoretical tools proposed by discourse and conversational analysis, which implies calling upon Goffman's interactional approach, ethnography of communication and language act theory. This stance may seem to combine incompatible theories (e.g. language act theory and conversational analysis), yet according to the author only the combination of these approaches will facilitate a thorough understanding of the embodied (inter)actions. This approach generated the following results.

Results

It is possible to distinguish two aspects of nonverbal orchestration: two-handedness and co-enunciative ubiquity. Both will be studied in the following lines.

Two-handedness, one mode yet two functions

Two-handedness will not be understood here as the use of the two hands to produce a single gesture serving one of the three previously mentioned functions [START_REF] Tellier | Segmentation et annotation du geste : méthodologie pour travailler en équipe[END_REF]. Rather, as each hand may generate gestures occurring within separate gesture units, the two hands may produce two different dimensions to serve two independent and complementary functions. In the first example, the class is talking about the 2012 French elections for presidency. The word "debate" has come up during the discussion and non-native students are trying to define the word. This episode illustrates how, in less than 4 seconds, two-handedness can be used to assess a student's intervention and allocate the next turn to another student: Frames a to d illustrate the teacher producing an emblem with her right hand to assess the intervention of Antonio (turn 13), who is interrupted in turn 10 by Nolan at whom the teacher nevertheless points her left hand to give the floor (frames e-g). Interestingly, the teacher keeps her right hand oriented towards Antonio as if not to break the interaction initiated with him. This enables her both to build an interpersonal relationship with the two students and to accomplish shared understanding. She then retracts her right hand to mime the verbal explanation given by Nolan to whom she finally pays full attention as illustrated by the orientation of her head, gaze, body and hands (frames i-j).

Corpus M-FL2 1 T
The second example is extracted from the French to native instructional context. As the teacher is explaining the functioning of end-of-term school reports, a student (Loubna) interrupts her.

Corpus M-FL1 1 Youssef

Hum:::: are we not handed over the endof-term school report after the second term teachers' conference 2 T

No ++ during the meeting the teachers give their opinion + we talk about the student [and then// 3 Loubna there are the class reps, too 4 T we give 5 Serge there only is XX 6 T Hush, will you please not intervene (to Loubna) ++++ and the hea]d teacher, in other words me, writes down this + the decision + ok Frame a shows the teacher producing an iconic gesture that was meant to accompany her verbal explanation now postponed in turn 6 ("write down"). She is interrupted in her verbo-gestural explanation by Loubna, which accounts for the emblem she produces with both hands to ask the student to stop speaking (frames d to g). This pragmatic function is emphasized by the fixed gaze illustrated in frame i. She holds her left arm extended to literally keep the student at bay while she resumes her verbal-gestural explanation where she had previously left it. The two-handedness complementary functions are obvious in frames j and k: her right hand produces an iconic gesture to inform the students about the functioning of end-of-term school reports, and her left arm prevents Loubna from speaking. An interview I had with this teacher opens an enhanced window onto this gestural action. She explained how useful this two-handedness was both on a pedagogical level to organize simultaneous interactions and on a more personal psychological perspective since it helped her relieve her voice and the inner turmoil she felt.

Shift of attention and co-enunciative 1 ubiquity

Nonverbal orchestration is made even more evident when teachers' actions are analyzed in a combined approach of deictic gestures and gaze. In this paragraph I will examine how the interplay of these media enables the teacher to "multiply" herself so as to be the co-enunciator of several students almost simultaneously. This ability, which I termed co-enunciative ubiquity [START_REF] Azaoui | Multimodalité des signes et enjeux énonciatifs en classe de FL1/FLS[END_REF], is illustrated in the following examples. They will enable me to demonstrate that besides the interpersonal relationship it helps to build, this ability has an impact on the interaction level. This first extract of class interactions follows an excursion the FL2 class had to the theatre the previous week. The teacher is not pleased with the behavior her students had, and she wants them to reflect over their attitude. c). This is confirmed by the superimposed beat gesture (frames c and d). Finally, as she retracts her pointing gesture, she briefly looks at Maria, who is acknowledged as a co-participant of the interaction (frame d). This description aims to progressively unroll the multimodal teacher's action and to show how this teacher copes with the intrinsic polyfocal and polylogal dimensions of class interactions.

The following example taken from the FL1 class enables us to pursue the demonstration of the teacher's co-enunciative ubiquity and its implications. Here, the teacher is working on a short story about totalitarianism. First, she asks her students to describe the they have of the characters in the story. She then overtly allocates the turn to one specific student, as confirmed by the use of the student's name and the orientation of her gaze (frame a). An overlapping intervention coming from the left side of the class draws her attention and makes her briefly shift her head and eye orientation towards another student, Albert (frames b and c).

Corpus T-FL1 The first analysis we can make is that this action exemplifies the instructor's ability to pay attention to (at least) two students at the same time. Additionally, the two channels have two separate functions: her gaze has a managing function (attributing the turn) while her pointing gesture assesses Albert's utterance. A second analysis concerns the instructional technique the teacher uses. It corroborates the divided attention we mentioned since the co-enunciative ubiquity she performs helps her select the utterance that best fits her lesson planning. Note that the hand gesture may also serve as a way to "provide the recipients with a 'forwardunderstanding', i.e., an anticipation, of what will come next" [44:226]. In other words, it anticipates the following exchange with Albert; and the other students are thus informed about the next locus of interest. This nonverbal action also has consequences on the interactional level. Indeed, research on interaction has often recognized the use of gaze as a means to indicate the ratified interlocutor ( [START_REF] Kerbrat-Orecchioni | Les interactions verbales[END_REF], [START_REF] Goodwin | Conversational organization. Interaction between speakers and hearers[END_REF], [START_REF] Goodwin | Transparent vision[END_REF]). It is here confirmed by the teacher's use of the name Pierre to overtly designate her privileged interlocutor. Yet, the combined analysis of the gesture/gaze disjunction and the teacher's utterance tells us what is really at stake in the extract. An interpretation that can be hypothesized is that this hand gesture/gaze action entails a "communicational trope" [45:92], i.e., the inversion of the hierarchy of the interlocutors. Pierre's utterance loses its interest, the teacher hardly paying attention to the end of his sentence (turn 4). Right from the beginning her attention is polarized by Albert's intervention which is more in compliance with what she wanted her students to understand and keep in mind.

Conclusion

To summarize, in this paper I have focused on how teachers resorted to multimodal resources to cope with polyfocal classroom interactions which require organizing turn-taking, informing, and assessing several students simultaneously. I first explored the production of two-handed independent gestures. The results show that they serve distinctive yet complementary teaching functions: assess verbal proposal and allocate turn, or inform and assess unauthorized intervention. By producing two independent gestures, the teacher is able both to build an interpersonal relationship and progress in her lesson plan. The teacher's comments that I collected during an interview enabled to expand this analysis. They draw our attention to the importance of twohandedness on a more intrapersonal and psychological level. Secondly, I have examined the nonverbal orchestration a step further by investigating the production of hand gestures in collaboration with gaze orientation. I have paid attention to what I termed co-enunciative ubiquity, i.e., the multimodal ability to manage polyfocal and polylogal class interactions. The interplay of gaze and deictic gestures also served the teacher's intention to have students anticipate the next focus of attention. Additionally, reference to attention theory enabled me to show how this ability attested the fact that the teacher selected intervention that best suited her pedagogical purpose. This was confirmed by the interactional consequence of this multimodal action, namely a reversal in the hierarchy of the addressed which follows a teaching goal: showing interest to the most appropriate answer.

Interestingly, the results also show that the instructional context has no impact on how the teacher handles this nonverbal orchestration. Two-handedness and coenunciative ubiquity compose each instructor's "teaching style" ( [START_REF] Cicurel | Les interactions dans l'enseignement des langues. Agir professoral et pratiques de classe[END_REF], [START_REF] Azaoui | Coconstruction de normes scolaires et contextes d'enseignement. Une étude multimodale de l'agir professoral[END_REF]). This term refers to the fact that while some teaching actions may be adapted to the specificity of a given context, others may be recurrent from one pedagogical context to another both in the form they take and in their pedagogical intent. These unvaried actions compose the "teaching style" of some teachers. In this perspective, and as far as our teachers are concerned, no matter the instructional context (FL1 or FL2), there is no difference neither in the way they conduct this orchestration nor in the motivations behind it. I believe these examples of orchestration are not specific to the language teaching classes and may be observed also in other instructional contexts.

Finally, this study corroborates the need to analyze teaching gestures in natural teaching contexts. It enables the opening of an enhanced window onto the complexity of teachers' nonverbal actions.

Figure 1 :

 1 Figure 1: Two-handedness in FL2 context, turns 12-15.

Figure 2 :

 2 Figure 2: Two-handedness in FL1 context, turns 2-6.

CorpusFigure 3 :

 3 Figure 3: Co-enunciative ubiquity in FL2 context, turns 2-6. Three students speak out almost simultaneously. The instructor's initial gaze orientation (frame a) informs us about the attention she pays to the utterance of a student (Omar) seated at the back of the class. At the same time, Ericka's overlapping turn makes the teacher orient her gaze towards her student and produce a deictic gesture to indicate the interest she gives to her idea (frames b andc). This is confirmed by the superimposed beat gesture (frames c and d). Finally, as she retracts her pointing gesture, she briefly looks at Maria, who is acknowledged as a co-participant of the interaction (frame d). This description aims to progressively unroll the multimodal teacher's action and to show how this teacher copes with the intrinsic polyfocal and polylogal dimensions of class interactions.The following example taken from the FL1 class enables us to pursue the demonstration of the teacher's co-enunciative ubiquity and its implications. Here, the teacher is working on a short story about totalitarianism. First, she asks her students to describe the they have of the characters in the story. She then overtly allocates the turn to one specific student, as confirmed by the use of the student's name and the orientation of her gaze (frame a). An overlapping intervention coming from the left side of the class draws her attention and makes her briefly shift her head and eye orientation towards another student, Albert (frames b and c).

Figure 4 :

 4 Figure 4: Shift of attention and co-enunciative ubiquity in FL1 context.While considering the frames, it is important to remember that "no one would dispute the close connection between movements of our eyes and shifts of attention"[41:5], no matter how restricted it may be. Posner[42:26] subdivided attention into three separate but interrelated functions: "(a) orienting to sensory events; (b) detecting signals for focal (conscious) processing, and (c) maintaining a vigilant or alert state". The first one is of some particular interest for our understanding of the interaction under study. Indeed, Lamargue-Hamel[43:10] explains that orienting to sensory events is implied in the selection and focalization of relevant pieces of information in a given task. Consequently, it is possible to give the teacher's re-orientation of her gaze and head an intentional purpose that serves her pedagogical interest. It also illustrates the ability to divide her auditory attention: she seems to be constantly filtering external stimuli according to their relevance for the current interaction. Additionally, frames d, e and f illustrate the almost simultaneous combined gesture/gaze disjunction. As her gaze comes back to focusing on Pierre she starts a pointing gesture with her right hand indicating Albert at the back of the class. The beat she produces on her deictic gesture (frame e) informs us about the relevance of his intervention.The first analysis we can make is that this action exemplifies the instructor's ability to pay attention to (at least) two students at the same time. Additionally, the two channels have two separate functions: her gaze has a managing function (attributing the turn) while her pointing gesture assesses Albert's utterance. A second analysis concerns the instructional technique the teacher uses. It corroborates the divided attention we mentioned since the co-enunciative ubiquity she performs helps her select the utterance that best fits her lesson planning. Note that the hand gesture may also serve as a way to "provide the recipients with a 'forwardunderstanding', i.e., an anticipation, of what will come next"[44:226]. In other words, it anticipates the following exchange with Albert; and the other students are thus informed about the next locus of interest. This nonverbal action also has consequences on the interactional level. Indeed, research on interaction has often recognized the use of gaze as a means to indicate the ratified interlocutor ([START_REF] Kerbrat-Orecchioni | Les interactions verbales[END_REF],[START_REF] Goodwin | Conversational organization. Interaction between speakers and hearers[END_REF],[START_REF] Goodwin | Transparent vision[END_REF]). It is here confirmed by the teacher's use of the name Pierre to overtly designate her privileged interlocutor. Yet, the combined analysis of the gesture/gaze disjunction and the teacher's utterance tells us what is really at stake in the extract. An interpretation that can be hypothesized is that this hand gesture/gaze action entails a "communicational trope" [45:92], i.e., the inversion of the hierarchy of the interlocutors. Pierre's utterance loses

The notion "co-enunciative" insists on the simultaneous work of both participants of the interaction[40:44].