Patrick Maillé

Bruno Tuffin

Yiping Xing

Rajarathnam Chandramouli

Decentralized Learning for Pricing a RED Buffer

We study a buffer that implements the Random Early Detect/Discard (RED) mechanism to cope with congestion, and offers service differentiation by proposing a finite number of slopes at different prices for the RED probability. As a characteristic, the smaller the slope, the better the resulting QoS. Users are sensitive to their average throughput and to the price they pay. Since the study of the noncooperative game played is rendered difficult by the discrete nature of the strategy sets, and since it is not likely that users have a perfect knowledge of the game but only know their experienced utility, we introduce a decentralized learning algorithm to progressively reach a Nash equilibrium over time. We examine the effect of prices on the final game outcomes.

I. INTRODUCTION

Random Early Discard/Detection (RED) is an active queue management technique advocated for telecommunication flows, when the goal is to avoid congestion, by controlling the average queue length at a router through rate adaptation. The basic idea behind RED is simple. It drops packets randomly, with dropping probability proportional to the buffer/queue length if the length is between a certain minimum and maximum threshold values [START_REF] Floyd | Random early detection gateways for congestion avoidance[END_REF]. Packets are dropped with probability 1 if the length exceeds the maximum threshold. For flows conforming to the Transmission Control Protocol (TCP), packet drops indicate to the source that its rate has to be reduced. As a consequence, TCP is a good candidate for the application of RED: when this mechanism is applied to several sources in the network, the overall input rate is controlled, thereby avoiding congestion. Notice that for real-time flows sent via the User Datagram Protocol (UDP), packet losses are not taken into account by the source since the sending rate of each user is fixed, despite the fact that more packets are dropped in case of higher congestion.

With respect to the traditional tail drop mechanism that drops only the packets arriving when the buffer is full, RED offers two advantages. First, the randomness introduced avoids situations where packet losses occur exactly at the same time, i.e. when the buffer is full, for several sources. The TCP sources experiencing a loss consequently reduce their sending rate, and the network may become underutilized for a while, then flooded again periodically. This problem is known as global synchronization [START_REF] Floyd | Random early detection gateways for congestion avoidance[END_REF], and is solved by RED since the continuous increase in packet loss probability spreads losses over time as the buffer fills in.

The second issue addressed by RED is related to fairness: RED is considered more fair than the tail drop scheme, as the flows with the largest sending rates are more likely to experience packet losses.

RED is therefore a tool for congestion avoidance but not for service differentiation. The idea of extending it to include service differentiation has already been proposed in the literature. For instance RIO deploys several drop probability functions depending on the service class (see for instance [START_REF] Pieda | A Network Simulator Differentiated Services Implementation[END_REF]). However, a disadvantage is that a there is (at least directly) no incentive for users to select classes providing lower quality of service (QoS), meaning that everyone will choose the one with the highest level, resulting in the classical RED. Weighted RED (WRED) [START_REF]WRED[END_REF] generally drops packets selectively based on IP precedence, so that packets with a higher IP precedence are less likely to be dropped than packets with a lower precedence. Thus, higher priority traffic is delivered with a higher probability than lower priority traffic. Again, no incentive is directly associated to enforce people to some classes. In [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF], the authors have made an alternative proposition where there is a mapping between the RED function used (more exactly the slope in the linear part of the loss probability) and the declared willingness to pay of users: the higher the price, the lower the slope, i.e. the less likely the packets will be dropped. Here again, the scheme has a drawback: it requires that the declared price, a real number in the aforementioned model, is carried by each packet when going through the router implementing RED. From a practical side, it means that a non negligible part of the packet has to be devoted to this price information in place of data to transmit. This implementation issue leads to a signalling burden.

In this paper, we consider the same kind of model as in [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF], offering service differentiation by proposing different slopes -at different prices -for the drop probability in the congestion avoidance mode. The main difference is that here a finite number of slopes are proposed to users instead of a free choice, so that the signalling issue can be addressed by using a very small number of bits in each packet, i.e., a very small overhead. Pricing is again used so that not all users do choose the best slope, and we study the user behaviour within the framework of noncooperative game theory [START_REF] Fudenberg | Game Theory[END_REF], where the strategy of each flow/user is the choice of a RED slope, and its utility depends on the experienced throughput and the price paid. Due to the discrete nature of the game now, the analysis is much trickier than in [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF]. We then suggest that users learn how to make their decision in a decentralized manner, following the principles developped in [START_REF] Narendra | Learning Automata: An Introduction[END_REF], [START_REF] Sastry | Decentralized earning of Nash equilibria in multi-person stochastic games with incomplete information[END_REF], [START_REF] Xing | Stochastic learning solution for distributed discrete power control game in wireless data networks[END_REF], so that a Nash equilibrium of the game is reached. The two cases, when the flows using the buffer are TCP flows and when they are UDP flows are considered. The case when a mix of those types of flows share the same RED buffer is left for future work.

Note that pricing telecommunication networks has recently been the subject of an extensive literature (see [START_REF] Courcoubetis | Pricing Communication Networks-Economics, Technology and Modelling[END_REF], [START_REF] Tuffin | Charging the Internet without bandwidth reservation: an overview and bibliography of mathematical approaches[END_REF] for surveys) and that to our knowledge, this paper is besides [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF] the only attempt to tackle the problem through RED.

The paper is organized as follows. Section II presents the basic model taken from [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF] and discusses results previously obtained for a continuous strategy set. The new discretized version of the game is then presented in Section III. The decentralized learning mechanism for the discrete game is described in Section IV, and simulation results are provided in Section V, enabling to illustrate the behavior of the algorithm and to discuss the influence of prices on the game outcomes. Some conclusions and directions for future work are given in Section VI.

II. MODEL AND PREVIOUS RESULTS

A very basic model of the RED algorithm that we will use throughout the paper is as follows. Consider a buffer (a router) represented by a (virtual) fluid queue with service rate µ. The average virtual queue length q is compared to two thresholds q min and q max , with q min < q max , in order to decide whether or not the incoming packet should be dropped. The drop probability is then as illustrated in Figure 1: it equals 0 if q ≤ q min , 1 if q ≥ q max , and p max (q -q min )/(q max -q min ) if q min < q < q max ; the latter being called the congestion avoidance mode of operation. The parameter p max denotes the value of the drop probability as the average queue length tends to q max (from the left). In a best-effort network, the value of p max is the same for all flows sharing the buffer. A simple principle to bring service differentiation to the network is to use different values of p max (or equivalently of the slope t in the linear region) according to the service class a user belongs to. The idea is therefore to provide a smaller value of p max (i.e., t) to users who pay more than the others so that, on average, they will experience less losses. We denote the set of users (players) by i ∈ I = {1, • • • , I}. Let t i be the strategy chosen by user i (here, the slope of the RED mechanism). We denote by (t i , [t] -i ) the strategy profile where user i plays t i and all other flows j = i use t j from vector [t] -i . The utility of a player i depends on the whole strategy profile, which determines the average throughput of each user i and the price user i will pay. We denote by U i (t i , [t] -i ) this utility, that user i will try to maximize.

In [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF], such a model was considered, where each user could operate for a continuous value of the slope t i in the linear region between q min and q max . For a TCP flow i, the resulting transmission rate λ i is then given by

λ i = 1 R i α p i , (1)
where R i and p i are TCP flow i's round trip time and drop probability, respectively. α is typically taken as 3/4 (when delayed ack option is enabled, i.e. an acknowledgement packet is sent by the TCP receiver every two incoming packets) or 3/2 (when it is disabled) [START_REF] Braden | Requirements for internet hosts -communication layers[END_REF]. For UDP flows, the rate λ i is uncontrolled. We assume in this paper that all UDP flows have the same sending rate (corresponding to a given service), that we denote by λ.

In general, since the bottleneck queue is seen as a fluid queue, we can write that at equilibrium

j∈I λ j (1 -p j ) = µ.
This leads to a system of I + 1 equations with I + 1 unknowns. The average throughput that each user experiences as a function of the strategy profile has been determined in [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF]. We quote here those results, that will be used in the remaining of the paper.

Case of only UDP users: In this case again, the sources do not take into account packet losses, so the sending rate λ for each user is fixed.

Proposition 1 ( [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF]): If all flows are UDP flows and the RED slope of each user i ∈ I is t i , then the packet loss probability p j for a player j is

• if n × λ ≤ µ (no congestion), then p j = 0, • else p j = t j nλ -µ λ × k t k .
Case of only TCP users: TCP flows take into account packet losses: they are considered as congestion signals that indicate the source to reduce its sending rate. Therefore the interaction between TCP transmission rates and the RED mechanism are more complex than for UDP. The following result gives the expression of the average transmission rate of each user.

Proposition 2 ( [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF]): If all flows are TCP flows and the RED slope of each user i ∈ I is t i , then the loss probability for a given player j is

p j = t j -µ + µ 2 + 4α j 1 Rj √ tj j √ tj Rj 2 4α j √ tj Rj 2 .
Other results obained in [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF] are:

• For some forms of utility functions, sufficient conditions were found under which the game converges to a strategy profile for which no user has an incentive to deviate, i.e. a Nash equilibrium. • Parameters optimizing the network revenue for specific families of price functions d were determined numerically.

III. GAME FOR A FINITE NUMBER OF SLOPES

A. Discretized RED Game

Again, for reasons such as signalling or implementability, because not too many overhead informations can be carried on each packet, it is unlikely that a user could ask for a slope value in a continuous set. It seems indeed more realistic to choose from a finite set of predefined values (discretized RED). The strategy set available to users then becomes discrete, and the study carried out in [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF] for the game played among users does not hold anymore. In particular, the determination of the Nash equilibria becomes analytically intractable, and we therefore use different tools to reach those equilibria.

We suggest in this paper to use a decentralized learning mechanism [START_REF] Narendra | Learning Automata: An Introduction[END_REF], [START_REF] Sastry | Decentralized earning of Nash equilibria in multi-person stochastic games with incomplete information[END_REF] to reach a Nash equilibrium of the game, similarly to the approach taken in [START_REF] Xing | Stochastic learning solution for distributed discrete power control game in wireless data networks[END_REF] in the context of wireless power control (such an approach has also been considered in [START_REF] Tuffin | How many parallel TCP sessions to open: a pricing perspective[END_REF] for the game where players select a number of TCP sessions to open). We consider that time is discretized, and suggest that users select their strategy according to a vector of probabilities that is updated at each time period of the game in order to privilege "good" strategies over "bad" ones.

Let T denote the (common) finite strategy set of each user j. User j has a probability vector P j (k) = (P j,t (k)) t∈T (over T ) for the strategy choice at k-th time step. The expected utility of user j at k-th time step given that ∀j, player j chooses its strategy from the probability vector P j (k) is then

E [U j (t j , [t] -j )] = t1,••• ,tI ∈T U j (t 1 , • • • , t I ) I i=1 P i,ti (k).

B. Utility model for users

We assume here that each user j is sensitive to his mean throughput λ j (1 -p j ) a logarithmic function, i.e., user j has utility

U j (t j , [t] -j )) = log(1 + λ j (1 -p j (t j , [t] -j ))) -d(t j ) (2)
where d(t j ) the price for slope t j .

C. RED slopes and prices

Since there is a finite number N of different RED slopes, we number them by 1, ..., N , such that t 1 > t 2 > ... > t N . (The superscript indicates the order of the slope while the subscript is for the user index: if user j choose the kth slope then we write t j = t k ). In our numerical experiments, the minimum slope t N is determined using Theorem 1 of [START_REF] Altman | Pricing Differentiated Services: A Game-Theoretic Approach[END_REF], in order to ensure that the system operates in the linear region of RED. We choose the maximum slope t to be such that the discard probability tends to 1 as the virtual queue size tends to q max . The slopes between are equally spaced.

Prices are chosen to be of the form

π k = d(t k ) = price factor =π × N N + 1 -k .
For example with N = 4, λ = 20s -1 , q min = 10 and q max = 40, we obtain the RED curves and prices of Figure 2.

D. Bound for the Maximum Price Factor

We assume that the network users are rational in the sense that they will not try to access the network when their payoff U i < 0. This imposes a maximum price factor π max for the network operator. When this price is exceeded no network users will have any incentive to access the network. That is, when the network operator sets π > π max the network revenue will be 0, therefore an optimal price factor in terms of network revenue is smaller than π max . We give a formula for π max in the next proposition.

Proposition 3: Let all the users have the same round-trip time R and there maximum sending rate be bounded by λ max . Then, with a discrete strategy set T , π max ≤ log(1 + λ max ).

Proof: A rational user j will be willing to access the network choosing slope t j = t k only when U j (t j , [t] -j )) ≥ 0. Then from [START_REF] Pieda | A Network Simulator Differentiated Services Implementation[END_REF] this implies that

d(t j ) ≤ log(1 + λ j (1 -p(t j , [t] -j ))), i.e. π ≤ log(1 + λ j (1 -p(t j , [t] -j ))) N N +1-k . ( 3
)
Hence π max cannot exceed the maximum value of the righthand side of (3), which is obtained when the numerator is maximized and the denominator is minimized. Since N N +1-k is monotonically increasing in k, its minimum is attained at k = 1. The obvious majoration log(1 + λ j (1p(t j , [t] -j ))) ≤ log(1 + λ max ) then gives the result.

IV. LEARNING MECHANISM

We introduce here the user strategy decision and learning mechanism over time. The consists for each user i in updating a probability vector P i (s) at each time slot P ik (s) being the probability that user i chooses strategy (slope) t k at instant s (the strategy t 0 corresponds to the case where the user does not establish any connection). Similarly to [START_REF] Xing | Stochastic learning solution for distributed discrete power control game in wireless data networks[END_REF], [START_REF] Tuffin | How many parallel TCP sessions to open: a pricing perspective[END_REF], we assume that the following discrete learning algorithm is used by each player:

1) Set the initial probability vector P i (0) for each user i. In this paper we will (arbitrarily) choose uniform initial distributions over {0, . . . , N }. 2) At each time step s, the RED slope t i,s is chosen by user i according to probability vector P i (s). 3) User i then monitors his throughput and determines his utility U i,s at time step s. 4) User i updates his probability vector according to the rule

P ik (s + 1) =        P ik (s) -bu i,s P ik (s) if t i,s = t k P ik (s) + bu i,s ℓ =ti,s P iℓ (s) otherwise.
In words, this step consists in adjusting the probability of choosing one's strategy in the next step, considering the utility brought by the current strategy: if that utility is high (u i,s is large, see below) then the probability of the current strategy is increased, otherwise it is lowered. 5) If the algorithm has not converged goto step 2), otherwise stop.

In the algorithm, parameter b is the step size of the updating rule, and u i,s is a normalized utility

u i,s = U i,s -A i,s B i,s -A i,s
with A i,s = min s-m≤l≤s U i,l and B i,s = max s-m≤l≤s U i,l for a parameter value m that represents the memory of the user: A i,s (resp. B i,s ) is the minimum (resp. maximum) utility that user i obtained in the last m time slots. When A i,s = B i,s then we set u i,s := 0 and no update is done on the probability vector P i . Note that no knowledge of the number of players I is required, nor any specific knowledge of other users' strategies. Each user just needs to keep track of the best and worse utilities he recently obtained.

Again, the above game is difficult to study analytically. Note nevertheless that this is a finite game [START_REF] Fudenberg | Game Theory[END_REF]. From a classical result in game theory, we know that there always exists a Nash equilibrium in mixed strategies for such a game. For general utility functions, no results of uniqueness can be given, but it is proven in [START_REF] Sastry | Decentralized earning of Nash equilibria in multi-person stochastic games with incomplete information[END_REF] that when b tends to 0, the above algorithm converges to a Nash equilibrium. We therefore suggest to use that learning algorithm to discover an equilibrium of the game.

V. SIMULATION RESULTS

All our simulations use the setting described in Figure 2. The memory parameter is set to 20 time slots.

A. Convergence of the algorithm

We use the above algorithm to discover some equilibria of the game in the two settings TCP users only and UDP users only.

For all the simulations we have run, it has turned out that the probability vectors for the strategy choice of each user converged to a Dirac, i.e. a strategy where the player plays a single strategy with probability 1 (pure strategy). However, having checked the resulting situation, we have observed that when the value of the parameter b in Step 4 of the learning algorithm is too large, the learned situation may not be a Nash equilibrium. This a posteriori verification illustrates the trade-off that relies in the choice of b, between the insurance of converging to a Nash equilibrium (when b tends to 0) and the rapidity of convergence. For our setting, a value of b around 0.01 seems to be satisfying, since it has always led to an equilibrium in our simulations.

1) Case of UDP sessions: Figure 3 shows the evolution with time of the strategies (the number of the selected slope, strategy 0 denoting the fact that the user prefers not to connect at all) chosen by the players, with the slopes and prices of Figure 2 and a price factor equal to 1. Remark that users try all the strategies at the beginning, and then rapidly tend to favour strategy 1, trying out the other ones less and less often. Figure 4 shows the evolution of the virtual queue size during the convergence phase. Since the strategies played converge, the corresponding virtual queue size also rapidly converges. 2) Case of TCP sessions: Figures 5 and6 show the evolution of the strategies played and the virtual queue size in the case of TCP sessions (price factor=1), with similar conclusions.

B. Effect of the price factor

Due to the discontinuity of the available RED prices, the equilibria of the game are also not continuous in terms of the price factor. When the Nash equilibria are the same for different values of the price factor, then the network revenue is simply proportional to that price factor. Figures 7 and8 show respectively for I = 4 UDP users and I = 4 TCP users the influence of the price factor on the network revenue.

As an example, the equilibria of the game in the case of 4 UDP sessions, corresponding to Figure 7, are given in Table I (when the equilibria are not symmetric, we write the equilibria strategies -number of the slope chosen -in an ascending order). It appears that the effect of the price factor on the equilibrium is not easy to study, because the discrete nature of the game implies that the user equilibria do not change while the price factor is between two thresholds over which changes of equilibrium occur. It is therefore beneficial to the network to fix the price factor as close as possible to one of those thresholds in order to maximize his revenue. However determining analytically the optimal value of the price factor in terms of network revenue appears to be complicated since no regularity properties such as concavity (0, 0, 1, 1) π ∈ [3.49, 4.17] (0, 0, 0, 1) π ∈ [4.17, +∞] (0, 0, 0, 0) hold, as illustrated in Figures 7 and8. Though, a numerical evaluation can be performed as done here.

VI. CONCLUSIONS AND PERSPECTIVES

In this paper, we have presented a pricing scheme where service differentiation is provided by proposing different slopes for the RED Active Queue Management (AQM) algorithm at different prices. A finite number of choices is proposed in order to limit the signalling overhead in each packet. Analyzing the resulting game has been proved to be difficult and we have proposed a decentralized learning algorithm that converges to a Nash equilibrium among users.

We plan to work towards several directions in the future. First, we would like to study through ns-2 simulations if our fluid model is a good approximation of the packet-level and how the presented scheme can be implemented. Second, we would like to provide a more theoretical insight on the convergence of the decentralized learning algorithm. Third, we plan to investigate how such a pricing scheme could also be provided to other AQM schemes, such as gentle RED or RIO for instance.

Fig. 1 .

 1 Fig. 1. Drop probability in RED as a function of q.

Fig. 2 .

 2 Fig. 2. RED curves and prices.

Fig. 3 .

 3 Fig. 3. Evolution of the strategies played (4 UDP users)

Fig. 4 .

 4 Fig. 4. Evolution of the virtual queue size (4 UDP users)

Fig. 5 .

 5 Fig. 5. Evolution of the strategies played (4 TCP users)

Fig. 6 .

 6 Fig.6. Evolution of the virtual queue size (4 TCP users)

Fig. 7 .Fig. 8 .

 78 Fig. 7. Network revenue as a function of the price factor (4 UDP users)

TABLE I EQUILIBRIA

 I OF THE GAME AS A FUNCTION OF THE PRICE FACTOR π, FOR 4 UDP USERS (EACH STRATEGY PERMUTATION AMONG USERS IS ALSO AN EQUILIBRIUM).


domain: sdv scco shs.phil

text:
Ph Lacorre

Predation and generation processes through a new representation of the cusp catastrophe

A new formulation of the cusp catastrophe is used to model the fundamental biological functions of predation and reproduction. This new representation lies on the decomposition of the overall cusp potential in two component potentials individualising the conflicting pregnances. It results in a more accurate and less problematic description than the original proposition by R. Thom, mostly due to the use of parameters with strong physical and evocative power. For instance, it gives a very suggestive account for such biologically significant processes as digestion and assimilation, or pregnancy and delivery. Two couples of parameters are used in the new representation of the cusp: one defines the exact shape of the conflicting attractors which symbolises genetic coding, the other controls the deformation of -and interaction between-the two attractors and represent hormonal effects. This new description strikingly echoes Heidegger's philosophical thought on ontological difference, by giving a representation of the distinct concepts of entity and Being. It thus precludes a previous criticism concerning the impossibility for catastrophic models to depict the entity level. Such a type of representation might be useful in translating syntactical-semantic models of catastrophe theory into computer language.

Introduction

In a previous article [START_REF] Lacorre | Some remarks on correlations between conflict and instability[END_REF]), the conflict between two single-well, singlevariable, convex potentials has been studied through its effect on the stability of a system submitted to these potentials. Depending on the sign of the potential's third derivative, conflict generates either stability or instability, but it is always solved smoothly. The introduction of a concave segment in the otherwise convex potentials qualitatively changes the behaviour by generating discontinuities and bifurcations.

Such behaviour is reminiscent of what happens in catastrophe theory. As a matter of fact, it was shown in [START_REF] Lacorre | Some remarks on correlations between conflict and instability[END_REF]) that it is possible to define a new representation of the cusp catastrophe by using a new parameter set describing the conflict between two single-well potentials. This new set of control parameters, consisting in the relative strength and distance between individualised attractors, has a more physical significance than the original and purely mathematical control set of catastrophe theory. For this reason, it was anticipated that the new representation of the cusp has more evocative power, which could be of interest for some application of this catastrophe. The aim of this article is to give an illustration of the usefulness of such a representation by considering applications borrowed from biological sciences, concerning the vital functions of predation and reproduction. The core of this paper will be divided into five sections. In the first one, Thom's interpretation of animal psychism and his description of the predation loop within the cusp catastrophe model will be recalled. In section II, a new model for the predation loop will be presented, which takes advantage of the new representation of the cusp. The third section will be devoted to another illustration of the same model, applied to the reproduction loop. An application of the model to linguistics will be presented in section V. In the last main section, more philosophical questions will be tackled, and we will see how the new representation of the cusp can be used to give account of Heidegger's ontological difference within the scope of catastrophe theory.

I -René Thom's predation loop

Predation is an important biological function since it permits the survival of the predator by restoring its energy. It is ruled by the conflict between two individual animals, the predator and the prey, whose interaction end up by a fundamental singularity, the capture and ingestion of the prey by the predator. For this reason, R. Thom attempted to describe it by using catastrophe theory, more particularly the cusp catastrophe which seems best suited.

Although any description of animal psychism is conjectural, Thom advanced that animal consciousness could be described as a local map of the environment, which contains the images of the organism itself, and for which senses provide forms of the biologically significant animals such as preys, predators and mates [START_REF] Thom | Animal Psychism versus Human Psychism[END_REF]). Such forms, which trigger important hormonal and behavioural changes are called "pregnant" by Thom, and the associated quality is "pregnance". Thom wonders whether [START_REF] Thom | Morphologie du Sémiotique[END_REF]) the source forms of pregnances could be genetically coded. He argues that it seems difficult to understand how a visual, geometrical shape could be biochemically coded, but a biochemical coding of pregnance gradients is more easily conceivable, leading to a distortion of the environment map. Thom regards pregnances as "black holes" within the individual's local maps, representing the subject's self and the other biologically significant organisms. Such maps can conveniently be modelled by the use of potential functions, with the interaction of pregnances being described by catastrophe theory. For instance, the double-well potential in figure 1 represents the interaction between the predator and its prey in the predator's local map of predation. The bimodal process of predation is well described by the cusp catastrophe. The cyclic character of this fundamental biological function is represented by a closed loop within the control plane (u,v), centred on origin O, which has been called by Thom "predation loop" [START_REF] Thom | Stabilité Structurelle et Morphogénèse[END_REF], p.294). The potential distortion along the loop represents the interrelation between the predator and prey during the predation. Abrupt, catastrophic changes such as the capture of the prey, occur when the trajectory crosses the cusp bifurcation set and one of the minima disappears. The whole process [START_REF] Thom | Ambiguïtés de la complexité en biologie[END_REF]) is described as follows (see figure 1):

Fig. 1 : Catastrophic theoretical model of the predation loop according to Thom (adapted from [Thom, 1990] pp.223-224). The crossing of the cusp bifurcation set by the circular trajectory corresponds to the perception (J 1 ) and capture (K 1 ) catastrophe. See text for more details.

At point 1, the single minimum represents the famished predator (P), alienated by the prey image. At J 1 (perception catastrophe) the predator (P) meets its prey (p) associated with the emergence of a new minimum. Then, the prey (p) drives the predator out of the global minimum it used to occupy toward the other, metastable minimum. The remaining twominima sequence (2-3-4) represents the chase of the prey, which ends up with the capture of the prey by the predator (K 1 , capture catastrophe) where the prey's minimum disappears. At 5 the predator, after having swallowed the prey, is in the digestion phase. The semicircle u>0 corresponds to the sleeping period, during which the predator (6, single minimum) passes from satiety to hunger.

The previous description poses two problems: 1) One wonders why the predator, alienated by the prey image, occupies the minimum of the prey, before being driven out by the prey during the perception catastrophe (J 1 ). Thom himself considers such a situation as paradoxical, and has made attempts to solve the problem in several different ways. For instance he argued that the alienated predator actually considers itself as prey [START_REF] Thom | Stabilité Structurelle et Morphogénèse[END_REF], p.299). He also proposed a tunnelling effect [START_REF] Thom | Esquisse d'une sémiophysique[END_REF], pp.81-86) or a double cusp (Thom [1988], pp.72-73). None of these explanations seems to be really convincing, even to Thom himself [START_REF] Thom | [END_REF]).

2) The second problem is more subtle, and representative of the ambiguities originating in a purely mathematical representation of catastrophe theory. It concerns a difficulty in the interpretation of the crossing of the bifurcation set. In a remarkable article on catastrophe theory applied to linguistics [START_REF] Thom | Topologie et linguistique[END_REF]), Thom describes action verbs as trajectories in the control plane of catastrophes, while subject and objects in the sentence are the potential attractors, represented by its minima. For instance, the two main sections of the cusp catastrophe at u=cst and v=cst correspond to the archetypal morphologies of verbs [TO BECOME] and [TO CAPTURE], respectively. In subsequent papers however, such as those referring to the predation loop, only the signification [TO CAPTURE] is kept whatever the section. The confusion is probably due to the strong similarity between the potential shapes along both main sections, as can be seen in figure 2. In the next section, it will be shown how the new representation of the cusp introduced in [START_REF] Lacorre | Some remarks on correlations between conflict and instability[END_REF]) permits a new description of the predation loop, which clears up the two difficulties above mentioned.

II -Proposition for a new representation of the predation cycle

II.1 -New formulation of the cusp

First it is necessary to come back to the new formulation of the cusp proposed in [START_REF] Lacorre | Some remarks on correlations between conflict and instability[END_REF]). It has been shown in this paper that it is possible to decompose the cusp potential x 4 +ux 2 +vx in the summation of two contributions, each of them being a single-well potential. The characteristics of these potentials lie in the presence of a concave segment in the otherwise convex well. The exact shape of the two conflicting potentials is defined by the coordinates (α,±β) of their representative points in the control plane (u,v). Two new control parameters a and c are defined as the relative strength of the two individual attractors (a) and the distance between them (c). They are related to the traditional control parameters (u,v). by the following relationships: When a and c are varied, the control point in the traditional set (u,v) belongs to the mapping region shown in figure 3a, which includes parts of all original four quadrants. Points A and B represent the pure archetypal shape of each individual conflicting potential and attractor at full strength. Their coordinates (α,±β) in the traditional control set, are chosen symmetrical about the u axis for convenience; they determine the exact nature of the concave bias. Points A and B could be located anywhere within each of the two quadrants u<0, outside the cusp bifurcation set in order to ensure the uniqueness of the attractor. Each point within the mapping region represents the conflict between two attractors of the type A and B, whose relative strength is determined by parameter a and distance between them by conflict parameter c, according to the relationship: P , where P represents the overall potential and p the individual archetypal potential of points A and B.

u = (1 -a 2 )(2m(α,β) -c) 2 +α v = ((1 -a 2 )(2m(α,β) -c) 3 +β)a ,
Within the mapping (see figure 3a), the lines joining points A and B are iso-c lines, with c varying from 0 and 2m (m being a real root of the equation 4m 3 +2αm-β=0), and the lines which do not pass through A and B are iso-a lines, with a varying between -1 and +1. Figure 3b represents the cusp bifurcation set within the new reference frame (a,c).

One of the interests of this new representation lies in the evocative power of physical concepts such as strength (parameter a) and distance (parameter c) compared to the much more abstract and purely mathematical parameters u and v. It allows an easier description of the cusp catastrophe by using a kind of "natural language" as advocated by Thom, thus enabling a clearer representation in some applications, such as the predation cycle.

II.2 -A new representation of the predation

The new representation of the predation cycle naturally takes advantage of the salient features of the new cusp mapping shown in figure 3. For instance, the two points A and B represent the two conflicting pregnances, namely predator and prey.

The great power of the catastrophe formalism lies in its independence of any substrate which makes it usable, through analogy, in a number of similar situations belonging to different domains. Even in a unique narrow domain such as predation, interpretation can be carried out at four different levels:

-biological level: the conflicting pregnances are the vital pregnances of predator and prey. Since we will adopt the predator point of view, the vital pregnance of the prey corresponds to a food or nutritive pregnance; -psychic level: the two conflicting pregnances are here what could be called the ego or self pregnance (self-consciousness) and the alter-ego pregnance (consciousness of the other, the desired being, here the prey); -physical level: the bodies of the predator and prey (saliencies in Thom's terminology) occupy and materialise the attractors defined by the corresponding biological pregnances. They move and interact in the usual tridimensional space; -semantic level : the conflicting pregnances define words and the trajectory in the control space define verbs which characterise the interaction between words.

In this section, we will concentrate on the three first levels; the semantic level will be considered later on since its application domain is much wider. The whole new mapping region for the couple predator/prey is shown in figure 4. It will be considered from the side of the predator, which survives to the predation process. Point A represents the vital pole of the predator, the point at which it experiences total fullness, plenitude. Any move away from this point, for instance under the influence of hunger, corresponds to a weakening of its vital pregnance. The predator will try to compensate it and restore its vital energy by chasing and ingesting its prey, which will tend to move its representative point back toward point A. In figure 4, the vital zones of predator and prey are highlighted, hatched differently. At the biological level, the "falling-off" of an animal from its vital zone (the zone where its vital pregnance defines an attractor) corresponds to the death of the animal. Note that it does not mean that the animal's body and "nutritive energy" have disappeared, but only that the animal is dead. The frontiers defined by the semi-cubic parabola (the folds of the cusp) determine the extent of the viability zone of the predator and prey. An encounter between the two living animals can only take place inside the shared region, within the parabola. Predation then consists in an excursion of the predator in the common part of their viability zones, from which it will be able to "extract" the prey.

Before detailing the new proposition for the predation loop, it is important to make some remarks on the nature of the different parameters used in the new representation of the cusp. Since point A determines the intrinsic characteristics of the predator, independent from its environment, it seems natural to ("see") interpret the coordinates of this point (parameters α and β, which determine the exact shape of the archetypal potential and gradient of pregnance) as resulting from a biochemical genetic coding. On the other hand, fluctuating parameters a and c, which control the relative strength and distance between pregnances, directly related to variables such as hunger or desire, are more likely governed by hormonal factors. Let us now give a more detailed description of the new predation cycle. In this interpretation, the trajectory in the control plane (u,v) is no longer a circle around the organising centre 0, but a closed circuit with 3 segments along iso-a and iso-c curves of the mapping. As mentioned above, the starting point is point A, representing the fulfilled predator in all its plenitude. The cycle trajectory within the control plane (u,v) is shown in figure 5, and the same cycle within the new control plane (a,c) is displayed in figure 6. In this last presentation, the segments of the trajectory are straight lines (note also that points A and B become here facing segments). The potentials shape for selected points along the trajectory are displayed around the cycle on figure 6. The 3 segments of the trajectory are called AJ (c=cst=c max , a decreases), JR (a=cst, c decreases) and RA (c=0, a increases). Although all the levels of interpretation apply for all segments, one of them dominates each segment, namely the psychic level for the first segment, the physical level for the second, and the biological level for the last one. Each segment will now be examined successively.

Segment AJ

This segment concerns the predator only, and the psychic and biological transformation occurring under the influence of hunger, which moves the representative point away from plenitude. At A, the predator has full self-consciousness. Then slowly, under the influence of hormonal factors (parameter a decreases) of hunger, it progressively becomes aware of a lack, of a missing alter-ego. The consciousness of another self, symbolised by the appearance of the alter-ego pregnance, is first vague (point 1) then strengthens to the prejudice of selfpregnance (a decreases). At M, the alter-ego pregnance is strong enough to give rise to a second attractor in the local map. It is probably at this point (when the trajectory crosses the bifurcation set) that the predator becomes fully aware that what it misses is actually prey.

Point M represents what could be called the "awareness catastrophe", which will trigger a change of behaviour, the start of the chase phase, the search for prey. Although some movement might be triggered, this part of the chase phase is not very active, and mostly driven by sight and smell.

In the new representation of the cusp, point M is characterised by a specific, critical value of parameter a, a c = . As can be seen, this value is only a function of parameters α and β, which are supposedly defined by genetic coding. This gives a clear image of what happens during this process: under the action of hormonal factors controlled by hunger, parameter a diminishes, intensifying alter-ego pregnance up to a critical value, predefined genetically, which triggers the search for prey. This search part can last for a while, during which (point 2) alter-ego pregnance keeps on reinforcing to the prejudice of self-pregnance. It ends up when the predator actually meets a prey (point J).

At the biological level, the increase of hunger originates from a weakening of the predator's vital energy expressed by the decrease of parameter a. The lost energy will only be compensated by gaining some external energy in the form of nutritive pregnance, which for the moment exists only as a latency.

To finish with this portion, it should be noticed that the part of the predator organism which controls this segment is the nervous system, first through psychism, then through sensory organs. We shall go back to this point later on.

Segment JR

Point J corresponds to the perception of prey by the predator. From this point, two beings are now present and a direct action can be carried out by the predator in the physical space, which was impossible while it was alone. This new perspective finds expression in a change of direction of the trajectory (see fig. 5 and6): a now stays constant while c diminishes. At the physical level, the two animals occupy the two attractors of the overall (summation) potential. As c diminishes, the two attractors come closer. It is the second phase of the chase: the active part of the hunt. The predator goes after the prey, the distance between the two attractors symbolising the spatial distance between the two animals. This distance shortens (point 3). The capture of the prey by the predator occurs when the prey attractor disappears by collision with the predator attractor (point K, capture catastrophe). Only one animal survives the catastrophe: the predator. However the prey saliency persists all along the segment.

At the biological level, perception (point J) corresponds to the appearance of the prey in the predator's field of vision. Instantly, the prey saliency is invested by the nutritive pregnance. From there, the predator identifies the nutritive pregnance with the prey vital pregnance. The pursuit phase corresponds to the closing up between the two vital pregnances, which will allow the predator to get the missing vital energy back and to restore its own archetypal vital pregnance. The capture point K corresponds to the death of the prey, since it leaves its vital domain.

An important and yet very interesting point is the following: at the capture (point K) the two vital pregnances do not coincide (the conflict parameter c is non-null); the real coincidence of the two component attractors only occurs at point R, the end of the segment. How can this part of the trajectory between point K and R be understood ? The most simple and natural interpretation is the following: this portion of the segment corresponds to the digestion phase, with nutritive pregnance passing through the alimentary canal of the predator (mouth, oesophagus, stomach) up to the biological centre of the predator's vital pregnance, the small intestine. This digestion phase includes an ingestion step and a dissolution step of the nutriment (stomach, small intestine).

At the psychic level, the previous biological phase probably couples with a similar psychic evolution, with a progressive merging of alter-ego and self pregnances, which fills the lack felt by the hungry predator.

As a final remark, it can be noted that segment JR is mostly governed by the muscular system of the predator, which enables first, the motion in order to catch the prey, then, the digestion by swallowing it.

Segment RA

The last part of the cycle, with return to point A, essentially concerns the biological level. At point R, the prey is already digested, liquefied by gastric juices into chyle. However the "biological energy" of the prey has not been transferred to the predator organism yet: it is the purpose of this last step. It takes place within the small intestine. Here, the chyle is absorbed by the intestinal villosities. The substance of the prey is transferred to the predator, and restores its vital energy: the prey "becomes" the predator, in the real sense this time.

This ultimate step, called assimilation, is perfectly described by the segment RA of the trajectory (see figures 5 and 6). Conflict is null (c=0) and parameter a increases: nutritive pregnance (originally prey vital pregnance) is progressively absorbed (points 5 and 6) by the predator vital pregnance, until the archetypal vital pregnance of the predator is reconstituted (point A). The cycle is now closed, with predator in a biological fullness state, satiety.

This last segment is thus controlled at the biological level by intestinal tissues.

As for the previous segment, this biological step probably couples with a similar psychic step with assimilation of the alter-ego pregnance by self pregnance, and fusion with the desired being, leading to psychic fullness. A new cycle can now start again, with a new prey.

II.3 -Comments

Compared to the original proposition by Thom, the new representation of the predation cycle seems to offer several advantages: i) it solves the paradoxical situation with confusion between predator and prey. If a comparison is made between the first segment AJ and its equivalent in Thom's predation loop, it can be seen that the crossing of the first branch of the bifurcation set has a different meaning: "perception catastrophe" in Thom's description, "awareness catastrophe" here. Moreover, Thom extends the cycle up to the second branch of the bifurcation set, leading to the paradox. This is hardly conceivable in the new representation, since it would mean the biological death of the predator. A reinforcing of the alter-ego pregnance to the point it overrides the self pregnance is here very unlikely, but it could occur, for instance when the predator starves to death.

ii) it restores the original characteristic of the capture catastrophe, whose trajectory should be parallel to the u axis, rather than to the v axis.

iii) it gives a clear and natural representation of the fundamental biological functions such as digestion and assimilation. iv) it suggests a distinction between parameters of genetic (α,β) and hormonal (a,c) origins. v) it is in perfect adequacy with Thom's thoughts, who expressly points out that "actants" regulation catastrophes express the survival manoeuvres of a system submitted to adversary constraints which force it to leave its characteristics (quoted in [START_REF] Petitot | Physique du Sens -De la théorie des singularités aux structures sémio-narratives[END_REF], p.378, as a private communication from R. Thom). Even better than the original proposition by Thom, the new representation of the predation exactly follows such a scheme: the system's characteristics are represented by point A and the adversary constraints which make it leave its characteristics by segment AJ, while the survival manoeuvres are described by segments JR and RA. Enlarging the original scope, J. [START_REF] Petitot | Physique du Sens -De la théorie des singularités aux structures sémio-narratives[END_REF] presents intentionality as schematised by an homotopy class of trajectories going from an excited multi-actantial state towards the basis mono-actantial state. Here, point J represents the excited multi-actantial state, while point A is the basis mono-actantial state.

vi) it provides a vivid illustration of Thom's fundamental hypothesis that actants catastrophes convert the functional significance of tissues differenciated by embryogenesis catastrophes [START_REF] Petitot | Physique du Sens -De la théorie des singularités aux structures sémio-narratives[END_REF], chapter VI). Hence the identification between triploblastic differenciation and predation through the cusp archetype. We have already pointed out that each segment of the new predation cycle is associated with some specific part of the predator organism, namely the nervous system for segment AJ, the muscular system for segment JR and the intestinal tissues for segment RA. Now during embryogenesis, these organs originate from the original triploblastic differenciation in the following way: the nervous system comes from the ectoderm, the muscular system from the mesoderm and the intestine from the endoderm. Thus the new representation of predation clearly shows that this fundamental regulation effectively realises, one after the other, three main functions associated with the original triple differenciation during gastrulation, also modelled by Thom using cusp catastrophe (Thom [1990], p.196-198).

Other interesting consequences of some of the points presented here will be developed later on.

III -The generation process: re-examination of the reproduction loop

René Thom used the circular trajectory of the predation loop to represent another fundamental biological function, that of reproduction [START_REF] Thom | Stabilité Structurelle et Morphogénèse[END_REF], pp.300-302). Such a reproduction cycle is looped backwards compared to the predation loop. The capture catastrophe thus becomes an emission catastrophe, which corresponds to birth.

III.1 -A new reproduction cycle

Here, the new model of the predation cycle presented above has been used to give a more precise account of the generation process, which uses the full potential of the new representation of the cusp. The trajectory used is basically of the same kind as that used for predation, except that the cycle is looped backwards.

The current interpretation concerns mostly the biological level, although interpretations at the psychic and physical levels are also possible. Here the couple predator/prey of the predation cycle is replaced by the couple woman/child, whose archetypal vital pregnances are represented by points A and B of the new representation, respectively. The vital zones of woman and child are identical to those of predator and prey, respectively, as shown in figure 4. In terms of local map, it is now that of generation in human being.

As in the case of predation with a predator, a part of the trajectory takes place within the woman's body, the predator's digestive tube (mouth, oesophagus, stomach, intestine) being replaced by the female genitalia (vulva, vagina, uterus). The whole trajectory is presented on figure 7, and special potentials configurations on figure 8. The three segments of the cycle are now called AT, TS and SA. As for the predation cycle, we will start the trajectory at point A, which represents the archetypal vital pregnance of the woman.

Segment AT

Point A symbolises a woman in her whole sexual fullness. We will consider her just after having been fertilized. Segment AT corresponds to the diminution of strength parameter a, with c remaining null. The conflict parameter c, representing the distance between vital pregnances of the mother and the child, is zero. It means that what is happening along this segment takes place at the biological centre of the generation process (c=0), namely the woman's uterus. Going from point A towards point T means that the child's vital pregnance strengthens to the detriment of the mother's: this is pregnancy, when embryo development "pumps" energy reserves from the woman. Thus parameter a is a development parameter. If the pregnant woman reaches full term, this step lasts nine months, until point T. Cases of premature delivery will be examined later on.

It is clear that such a simple, general model does not enter into the detail of embryogenesis. Thom has already described such an evolution within the scope of catastrophe theory, at a slightly lower organisational level [START_REF] Thom | Stabilité Structurelle et Morphogénèse[END_REF]).

Segment TS

At point T, the foetus has reached its maximum development, and development parameter a is at its optimum a T . It will remain constant while c increases, symbolising the separation between the mother and her child. The first part of the separation takes place within the woman's organism: this is confinement, delivery, while the baby passes through the uterus cervix, then the vagina. This step is the backwards equivalent of digestion in the predation cycle. Childbirth corresponds to the crossing of the semi-cubic parabola branch of the bifurcation set (emission catastrophe, point E). The subsequent part of the segment out of the woman's body up to point S represents the separation, at the biological, psychic and physical levels, between the mother and her child.

Segment SA

The interpretation of this segment seems less obvious than the previous ones. Two interpretations can be given, depending on the adopted viewpoint. This ambiguity comes from the fact that two actants are present at the origin S of the segment (woman and child) but only one at the end A (woman). One has to assume that only one of the two actants is concerned by this segment (as it is the case for predation). The double interpretation reflects the two possibilities.

The first interpretation is to consider that the reproduction cycle is pertaining to one individual only, as is the predation cycle. If such is the case, the remaining actant should be the woman. Here the cyclic character of the generation process is less stringent than for predation, since a woman only gives birth a few times during her life, and it is not essential to her survival. By analogy with segment AJ of the predation cycle, a psychic interpretation can be found to segment SA: it can represent the satisfaction of the desire to have a child. This desire, or lack, is equivalent to hunger in predation, and symbolised by the existence of an alter-ego pregnance, that of the child B (point S). This segment is proceeded backwards compared to AJ, thus corresponding to the satisfaction of the desire, like the prey assimilation step RA in predation. At the biological level, segment SA could thus represent the fecundation (crossing of the bifurcation set), the "biological" satisfaction of desire through sexual intercourse.

The other interpretation, closer to Thom's original idea, is to consider the cycle as relevant to the species, not to the individual. Then, the cyclic character is essential to the species' survival. In this case, the remaining actant has to be the child, not the mother (the child survives the mother), and the child has to be a girl. Segment SA would thus correspond to the girl's psychic and biological development, and her transformation into a woman. The crossing of the bifurcation set (a=a c ) symbolises puberty. Point A represents the new woman in all her sexual fullness, and fecundation triggers the first step of the next cycle.

III.2 -Comments

As can be seen, this type of cycle gives a relatively simple and clear representation of the generation process. Such a description can even be pushed a little bit further by considering cases of premature delivery.

First let us come back to birth itself. The emission catastrophe (point E) corresponds to the stabilisation of a second attractor (minimum) of the global potential, to the entering of the child's vital zone. This entering coincides with the exit from the mother's body, that is birth (point N). Besides, birth corresponds to a specific value of separation between the child and the mother, to a specific conflict rate materialised by the dashed line on figure 7. At the biological and physical levels, everything happening on one side of this line occurs within the female's body (genitalia), and on the other side, outside female's body.

Let us now assume that the foetus did not reach its optimal development rate a T at delivery. Two possibilities can occur: either parameter a reached a value such that the delivery trajectory crosses the bifurcation set (segment T'S'), or it does not (segment T"S"). In this last hypothesis, the foetus will never reach the child's vital zone: such a trajectory T"S" encompasses cases of abortion and miscarriage, and some cases of stillborn children. In the other hypothesis (segment T'S'), the child's vital zone will be reached (point E'), although later than the birth itself (point N'): this includes the cases of premature children. The separation between these two possibilities (chain-dotted line on figure 7) corresponds to a critical development rate a c of the foetus. Parameter a c can be computed from the coordinates of archetypal point A, α and β, in the same way as point M of the predation loop (note however that the values of parameters α and β have nothing in common with those of predation, since we are now considering the case of human reproduction), that is a c = . On a temporal scale, parameter a c corresponds approximately to the 6 th month of the foetus' development. We now have a somewhat clear overview of the situation: the newborn child will have a chance of survival only if the foetus reaches a critical rate of development whose value, under our assumption, is genetically coded, since it depends upon parameters α and β.

If the foetus has a development rate between a c and a T at delivery, the child will thus be premature. On segment T'S', as c increases while a remains constant, birth (point N', leaving the mother's body) no longer coincides with the emission catastrophe (point E', entering the child vital zone). This can be interpreted as follows: in general, premature babies require special medical care (incubator, ...), all the more intensive as they are premature. These treatments, symbolised by segment N'E', allow the new-born baby to reach his vital zone. Therefore, the real emission catastrophe no longer corresponds to birth (point N'), but to the end of medical care (point E') without which the baby could not survive most of the time. One can clearly see here the importance of medical assistance for the premature baby: it increases his vital zone (dark area on figure 7).

IV -Application to linguistics

IV.1 -Verbal production

The previous scheme of the reproduction cycle can be used to model other generation processes, such as for instance language production.

We can assume that deep and surface structures of linguists are related to the existence of preverbal thought, whose language is expression. Verbal production can be seen as a generation process identical to the reproduction cycle described above, with the couple preverb/verb replacing the couple woman/child. It is interesting to note how easily common language has assimilated the analogy between the biological reproduction function and the verbal production process, as can be seen for instance in the following expressions: "to conceive an idea, a plan", "a plan that miscarried", "to bring forth, to give birth to a concept", and so on.

The trajectory (see figure 9) is decomposed in three segments.

Segment SA

By analogy with the reproduction analogue, this segment can be seen as symbolising the "fecundation" of thought by speech, the assimilation of alien concepts.

Segment AT

This is the conception step: the formation and development of the concept within the fecundated preverbal thought. The biological equivalent of this maturation step is embryo development (pregnancy).

Segment TS

The concept will be emitted (will pass from the preverbal to the verbal stage) only if a critical development rate a c is reached. If not (segment T"S") the concept cannot be formulated, since the trajectory T"S" does not cross the bifurcation set corresponding to the emission catastrophe: aborted concept. If the critical development rate is overrun, the concept will be emitted, all the easier as it will have ripened longer (compare the relative length of

IV.2 -The cusp archetypal morphologies

As evoked earlier (section I), Thom envisions a sentence as represented by a bifurcation of a dynamic system describing neurologic activity, whose attractors represent names and separation surfaces between attraction basins represent the verb (see also comment by E.C. Zeeman in Thom [1990], pp.358-359). For instance [START_REF] Thom | Topologie et linguistique[END_REF]), he defined archetypal morphologies of verbs [TO BECOME], [TO CAPTURE] and [TO EMIT] by the main sections of the cusp catastrophe: u=negative constant and v=constant with increasing u or decreasing u, respectively. He called these sections 12α, 12β and 12γ, respectively. By using the new representation of the cusp and specific trajectories of the mapping (parameters a and c), it is possible to define sub-categories to these three main categories. Such a classification attempt is presented in Table I. In this table, the reference pregnance is supposed to be A (β>0). Critical values a c and c c correspond to those values of a and c at which the trajectory crosses the bifurcation set, for fixed values of c and a, respectively. In the most general case, these values are complex functions of parameters α, β and c for a c , of α, β and a for c c . In certain specific cases as those presented in the previous sections, parameter a c can be simply expressed as a function of α and β (explicitly, a c = ).

V -New representation of the cusp and ontological difference

Heidegger's ontological difference is the difference between "entity" or "being", in the sense "what is" (German "seiend", French "étant") and "Being" in the sense "the fact of being" or "to be" (German "Sein, "French "Etre"). In a recent article, J.M. Salanskis [1993] studied the representation of the philosophical concept of ontological difference in three theoretical constructions used in cognitive sciences, among which is Catastrophe Theory. Namely, he utilises the catastrophist representation of the verb [TO ENTER] in linguistics, as proposed by J. [START_REF] Petitot | Syntaxe topologique et grammaire cognitive[END_REF], to illustrate the "defect" of the thomian figuration compared to the heidegegerian model. Now it seems to me that the new cusp formulation, of which some applications have been presented here, can remove the objection.

In his article, J. Petitot uses the capture archetypal morphology of R. Thom (cusp main section 12β) to represent the verb [TO ENTER]. J.M. Salanskis notes the difference between the verbs [TO ENTER] and [TO CAPTURE] in the traditional representation of the cusp (figure 10a):

" at the end of the potential distortion, one has no more access to the "entered" minimum, this minimum has disappeared. One must merely admit that, if its basin would have subsisted, it would be within the other minimum basin, the one which remains. It follows that the beingin is not at all represented; and if it could be, it is only with a considerable interpretative complement from the model maker" (Salanskis [1993], translated by the author). The author considers that it is the same problem as that of ontological difference; he notices that the potential representation favours the Being level to the detriment of the entity level. Taking into account the exact signification of the verb [TO ENTER] would necessitate the "marking of the two protagonist entities", which is not the case in the traditional representation of the cusp. However, it is clear that such a "marking" is actually realised in the new representation of the cusp: by points A and B, which are the poles of the representation (see figure 3). One can thus consider that this new representation is the kind of "interpretative complement" or "device amelioration" evoked by Salanskis, which gives access to the "entered" minimum (see figure 10b).

According to Heidegger, entity has a rather static, inert character, while Being has a dynamic, eventful nature [START_REF] Heidegger | Qu'appelle-t-on penser ?[END_REF], as commented by Salanskis [1993]). The new representation of the cusp makes a clear distinction between these two aspects:

-the static, inert part is accounted for by parameters (α,β), the coordinates of points A and B which are the static poles of the representation; -the dynamic part is taken over by parameters (a,c) which control distortions and positioning. Such a difference is particularly well expressed in the biological applications of the new representation of the cusp presented in this article: a clear distinction is made between parameters which are biochemically coded (α,β) by essence originally fixed, and hormonal parameters (a,c) by nature evolutive.

The new representation of the cusp unravels the double level of entity and Being, with entities represented by archetypal conflicting potentials with coordinates (α,β) of points A and B, and Being expressed by the trajectory in the control plane, the way parameters a and c change thus reflecting entities interaction. By comparison, the traditional representation does not clearly make the distinction, since parameters (u,v) express the dynamical part only, that of Being, hence the difficulty to account for the concept of ontological difference. The new representation of the cusp seems closer to the heideggerian model.

The problem discussed here is not only of philosophical relevance but can also have practical interest. It is undeniable that, beyond Heidegger's controversial positions in contemporary history, his philosophical thought directly or indirectly inspires a new generation of specialists in Artificial Intelligence (see [START_REF] Winograd | Heidegger et la conception des systèmes informatiques[END_REF] and some other articles in Intellectica 17 (1993)). The late recognition of Thom's avant-garde role on these topics should be added to that [START_REF] Visetti | Modèles connexionnistes et représentations structurées[END_REF], Salanskis [1993]). Indeed the stake is of consequence, since the matter is to find new computing structures which should be able to reproduce human cognitive capacities. Actually J.Y. [START_REF] Visetti | Modèles connexionnistes et représentations structurées[END_REF] evoked recently the difficulty to translate syntactical semantic models of Catastrophe Theory in computers language. He noticed that one way to get through it would be to give up "the principle one actant = one attractor, and represent several actants together under the form of one single complex attractor". This would necessitate the "beforehand disposing of attractors capable of coding a collection of actants, that is to say already structured themselves in a certain way". As a matter of fact, it is exactly this type of structuration which is proposed by the new representation developed in this article.

VI -Conclusion

In this article, a new interpretation of the fundamental processes of predation and generation have been presented, within the scope of catastrophe theory. A new representation of the cusp catastrophe has been used, which allows a more detailed description than the original proposition by R. Thom. This new representation also seems to be in better agreement with some philosophical ideas such as Heidegger's ontological difference. One can wonder whether the new formulation could be a useful tool for other biological applications of catastrophe theory. For instance, considering the problem of cellular differenciation in embryology, parameter c gives a very suggestive image of the differenciation concept, of a separation between pregnances. First the separation occurs softly, continuously, until a critical rate of separation beyond which differenciation is abrupt, with creation of a frontier which stabilises.

More generally, the success of some applications of the new representation of the cusp as those presented in the current article is an incitement to undertake the "marking" of more complex catastrophes. This should allow the definition of a general framework for a structuration of attractors within each member of the series of catastrophe potentials, which could be of interest in connection with the computer coding of syntacticalsemantic models of catastrophe theory.

Fig. 2 :

 2 Fig. 2 : Potential distortion while crossing the cusp bifurcation set by increasing u at v = cst (a) and by increasing v at u = cst (b).

Fig. 3 :

 3 Fig. 3 : Mapping of the new representation of the cusp as a function of the conflict c between two individual potentials A and B of relative strength a. Note that points A and B representing conflicting pregnances in the traditional (u,v) control set (a) become facing segments in the new (a,c) reference frame (b). Iso-c lines join A and B while iso-a lines do not.

Fig. 4 :

 4 Fig. 4 : The couple predator/prey as the two conflicting poles A and B of the new representation of the cusp. The vital zones of the two animals are hatched differently. An encounter between the two live animals can only take place within the cusp bifurcation set.

Fig. 5 :

 5 Fig. 5 : Proposition for a new representation of the predation loop, within the traditional control set (u,v). Organs of the predator's digestive tract are added in italics, as approximate indicators of the prey's progression inside the predator's organism. The potentials shape for the points of the trajectory are shown in figure 6.

Fig. 6 :

 6 Fig. 6 : Same as figure 5, within the new control set (a,c). The overall (full line) and component (predator=dashed, prey=dotted) potentials for a selection of points along the trajectory are shown. At the psychic (resp. biological) level, the component potentials represent the predator's self-pregnance (resp. vital pregnance) and alter-ego pregnance (resp. prey vital pregnance or nutritive pregnance). The star materialises the unoccupied attractor, while predator and prey saliencies are represented by full and open circles, respectively.

Fig. 7 :

 7 Fig. 7 : New representation of the generation process, within the traditional (u,v) reference frame (a) and the new (a,c) control set (b). Organs of the female genitalia, in italics, give an approximate indication of the baby's positioning during delivery. The potentials evolution along the trajectory is presented on fig. 8.

Fig. 8 :

 8 Fig. 8 : Evolution of the global potential (full lines) and relative strength and position of the component potentials (dashed=woman's vital pregnance, dotted=child's vital pregnance) along the trajectories presented on figure 7. The star symbolises the foetus in the womb, or outside when nonviable. The woman and child saliencies are represented by full and open circles, respectively.

Fig. 9 :

 9 Fig. 9 : Representation of the verbal production process, by analogy with that of generation: a) within the traditional control plane (u,v), b) within the control plane (a,c) of the new representation of the cusp.

Fig. 10 :

 10 Fig. 10 : Crossing the bifurcation set of the cusp catastrophe as a representation of the verb [TO ENTER]: a) potential evolution within the traditional cusp representation (section 12β,u increases at v=cst) b) component potentials in the new representation (c decreases at a=cst)

Table I main section constant parameter variable parameter variation domain actants number signification under-category main signification

 I

		c = 0	a increases	[-1,1]	2	to assimilate, satisfy
		c = 0	a decreases	[-1,1]	2	to conceive, ripen
	12α	c < c c	a decreases	[-1,1]	1	to change continuously	to become
		c > c c c ¹ 0	a decreases a decreases	around a c [-1,1]	1 1	to change abruptly to desire, want
		a < a c	c decreases	c > c c	2	to pursue, overtake
	12β	a < a c	c decreases	around c c	2	to capture, catch, enter to bring closer
		a < a c	c decreases	c < c c	2	to swallow,ingest,digest
		a < a c	c increases	c < c c	1 or 2	to deliver, express
	12γ	a < a c a < a c	c increases c increases	around c c c > c c	1 or 2 1 or 2	to emit, appear to separate,part,throw	to move away
		a > a c	c increases	[0, c m ]	1 or 2	abort, miscarry

Acknowledgements:

The author would like to express special thanks to Ms. M. Ratcliff for her help with English, and to Professor G. Ferey and Dr. J Pannetier for their constant support.


Tahani Abdelhakim

Report on "Agricultural cooperatives in Egypt

Introduction

During the first year of ENPARD's activities, topics as Good Agricultural Practices (GAP) and reform of the extension system were discussed as well as their possible reforms, through four workshops each attended by almost 40 members of the Think Tank group in addition to other participants as specialists and actors in the field. During these discussion sessions and evaluations, agricultural cooperatives have emerged as key actors that cannot be ignored or excluded from the framework of discussions and proposals that explore potential future policies. Accordingly, it was found necessary to address the topic of agricultural cooperatives in order to link the present subject to prior discussions and proposals.

In this context, a workshop was held on the 8 th and 9 th of January, 2017 in cooperation with the Department of External Agricultural Relations at the Ministry of Agriculture in the presence of the Think tank members and other participants including representatives of agricultural cooperatives from different governorates and officials from the Central Administration of Agricultural Cooperatives at the ministry of agriculture. And, as there is a project on the " Support to Cooperatives Reforms in Egypt ", supported by the Food and Agriculture Organization of the United Nations (FAO), representatives from the FAO branch in Cairo were contacted and invited to participate in ENPARD's workshop in order to present the project, its objectives, its activities and expected results.

Agricultural cooperatives in Egypt

Summary of debates and main conclusions of the workshop Cairo, 8 and 9 January 2017

The objective of the workshop was to display and discuss the current situation of agricultural cooperatives, the objectives of the Cooperatives' Act of 2014, major changes resulting from the Act, answers brought by the Act to the presented inquiries and future needs.

Day 1

As an introduction to the workshop, Ms. Tahani AbdelHakim -CHIEAM-IAMM expert for ENPARD program -introduced agricultural cooperatives experience in one of the European countries in order to define the types of agricultural cooperatives in terms of their role, scope of work, services provided to farmers, the evolution of its functions and services to follow-up with the new needs of producers and finally the changes of needs and requirements of the market and consumers.

Following the introduction, Dr. Mohammad Hassan Abed Alaal -FAO's coordinator for the "Support to Cooperatives Reforms in Egypt" project -presented the project, which mainly seeks to achieve the following four outcomes:

 Developing an environment that suits agricultural cooperatives and supports small farmers and agricultural households in Egypt  Assessing the needs and capabilities of stakeholders to be considered in the framework of the law's reform  Disseminating know-how of best practices related to the reform of agricultural cooperatives in Egypt in order to develop their efficiency, dynamicity and equality  Formulate a proposed law to support the reform of cooperatives (in partnership with other development partners such as the International Labour Organization -ILO)

Following the above two presentations, the floor was open for discussions and debates.

Day 2

Participants were divided in two groups and an analytical framework, which included a number of questions and discussion topics to be answered, was distributed. After that, findings of the groups were presented.

Summary of discussions 1. Elements on the current situation of agricultural cooperatives in Egypt Agricultural cooperatives were found to be one of the oldest and most important cooperation sectors in Egypt, first appearing in the beginning of the twentieth century. It was also found that agricultural cooperatives played an important role in the development of the agricultural sector and provided support and services to farmers since the 1960s, within the framework of state policies and agricultural reform.

There are currently 5,435 and agricultural cooperatives on the local level, i.e. at the level of villages, subdivisions known as "marakez", governorates and some general cooperatives at the national level.

The Central Administration of Agricultural Cooperatives in the Ministry of Agriculture conducts the financial and management supervises on cooperatives as well as other tasks as planning and projecting policies customized for cooperatives.

The issued 2014 law introduces some amendments to the Cooperation law of 1980 and is conveyed as a beginning to the development of agricultural cooperatives. Major amendments introduced by the 2014 law includes the give agricultural cooperatives the possibility of establishing projects and legal personalities may contribute to the capital of projects initiated by the Cooperatives.

Through the discussion sessions, several problems emerged, which present an obstacle to the development and activation of the role of agricultural cooperatives and limit the effectiveness of the implementation of the 2014 Act. Major obstacles include:

 The lack in appropriate legal and legislative framework and the lack in its adaptation to the change of economic conditions and the orientation of agricultural policy towards market economy, for decades  Lack of coordination and cooperation between cooperatives at various levels  The intervention of administrative authorities in the activities of agricultural cooperatives which obstructs their ability to adapt with the changes, the provision of services needed by farmers and their ability in taking initiatives. Additionally, it was clear that the submission of cooperatives to the administrative system has greatly deprived their credibility as autonomous productive organizations that work for the benefit of producers. This led farmers to lose trust in cooperatives and to consider it as tools for the implementation of the state policy.


 Multiplicity of statutes. The classification of agricultural cooperatives to 1) credit cooperatives, 2) agricultural reform cooperatives and 3) reclaimed land cooperatives leads to the intervention of three different authorities; where credit cooperatives are subject to the supervision of the central administration for agricultural cooperation, the agricultural reform cooperatives are subject to the supervision of the central administration for cooperation and the reclaimed land cooperatives are subject to the supervision of the sector of land reclamation  The small size of cooperatives, where the law specifies a minimum area of 750 feddans to be covered, which is equivalent to 300 hectares. With land fragmentation and the small size of farms, this leads cooperatives to deal with small quantities of produce  Inadequacy of the definition used and the criteria for membership in agricultural cooperatives; where the law defines the difference between a "small farmer or peasant"who owns less than 10 feddans (4 hectares) and farmers -especially when the membership to agricultural cooperatives is based on the land ownership. This results in several problems including the deprivation of tenant farmers from the membership of cooperatives on one hand and allowing the possibility of improper practices regarding trade in agricultural inputs that are sold by the cooperative to its members.

 Multi-functionality of cooperatives, its change with time and the orientations of the agricultural policy, which led to the misperception and an unclear role of cooperatives at the level of farmers. At the workshop, participants estimated agricultural cooperatives to be currently conducting the following tasks:

-Cooperative Marketing by 3 to 5% -Agricultural credit by 2% -Productive projects by 10% -Sales of production supplies and agricultural inputs by 85%

Proposals and requirements for the development of agricultural cooperatives

Participants agreed that agricultural cooperatives will not be able to operate effectively and efficiently in the absence of a clear strategic vision of core functions. There are also important axes needed for the development of cooperatives and which need to be addressed especially when the 2014 law does not cover them all.

Axis 1: Administrative and Institutional Framework

Activating and developing the role of cooperatives require institutional adjustments consistent with the economic changes as well as with the new required tasks. The first of these amendments includes the restructuring of the relationship between administrative/official bodies and cooperatives in order to limit the intervention of public authorities and to increase the autonomy of agricultural cooperatives, allowing them to take initiatives and to quickly adapt to the needs of producers.

In this context, it is necessary to unify cooperatives and withdraw the prior-mentioned categorization system, ultimately ceasing the supervision of multiple authoroties, along with the modification of the role of the government administrative to only follow-up and evaluate tasks.

At the level of cooperatives, certain conditions should be supplemented and that ensure the renewal of the Board Members and Chairperson on a regular basis, while imposing that the same person may not Chair the Board on two consecutive periods; unlike the current situation which allows the same person to maintain a position for life. Moreover, it is seen imperative to mobilize financial and human resources that are necessary for training and thus developing the efficiency and capabilities of both workers and Board Members in cooperatives.

Finally, it is seen necessary to simplify regulations to alleviate bureaucratic complications and to increase transparency in order to restore the confidence of producers in cooperatives, where cooperatives are currently subject to two operational regulations which lead to a competitive instead of a complementary relationship between various institutions (such as between the Ministry of Agriculture and the Union of Cooperatives ... etc).

Axis 2: Organizational and Technical Framework

Cooperatives suffer from weakness in financial resources and from a continuous deficit in budget.

Although the value of one share was raised from 1 Egyptian pound to almost 50 to 100 pounds, this

is not yet sufficient to provide the required financial resources, especially if the goal is to develop cooperatives as independent and profitable economic entities. For this reason, a few propositions were presented to provide additional financial resources, such as: allocating a profit percentage from the market value of products, commission on marketing or on services as rental of agricultural machinery, or applying for grants or donations from members, agricultural companies or banks.

As the small size of some cooperatives creates a significant obstacle, it is suggested to raise the minimum agricultural area of each cooperative to 1,500 feddans, equivalent to 600 hectares. This relates to the problem caused by the small size of farms and the fragmentation of owned agricultural land, which leads to severe negative impacts on the management of water resources on one hand and on the implementation of a successful marketing policy of agricultural crops.

One of the proposals to alleviate the negative effects is to generalize the presence of cooperatives which are specialized in marketing of certain agricultural crops as well as cooperatives having multiple activities and functions. Accordingly, cooperatives may play a major role in convincing producers to apply proper and optional agricultural rotations to improve the use of irrigation water on one hand and in order to obtain sufficient quantities of marketable products under better prices and terms. The latter proposal raised numerous discussions within the groups indicating the need for continued communication between actors and decision-makers in order to form a consensual vision and strategy for cooperatives and their role in the development of the agricultural sector.

Axis 3: Political and Strategic Framework

This axis is related to the strategic vision that defines the expected role of cooperatives as a key actor in the development of the agricultural sector. Taking into account the structures of the agricultural production in Egypt as well as the current and future challenges facing agriculture, cooperatives become an important activator of the development process and which can perform different tasks as:

 At the level of the product: Since the agriculture sector in Egypt is characterized by the presence of a large number of small and marginal farms, the services provided by the cooperatives may help develop the efficiency of the production processes, increase production quantities and increase farmers' income. Services may be presented in the form of:

-Provision of know-how in marketing or marketing of products for small farmers in order to acquire suitable prices, thus contributing to an increase in farmers' incomes -Provision of services complementing the marketing process, as storage, sorting and packaging -Provision of agricultural inputs and rental of agricultural machinery -Provision of extension services and technical consultancies  At the regional level: cooperatives are one of the most important forms of collective action and coordination between actors. Consequently, agricultural cooperatives have a vital function in coordinating between partners and actors at the local level in order to improve the quality of production and maximize the value of local products, and thus contributing to local development mainly through:

-Attending to local products having a superior quality and working towards its promotion, and marketing while encouraging farmers to invest in such products -Coordinating with the actors in the value chain as industries and traders in order to identify market demands and contribute to improving the efficiency of the production chain holistically -Contributing to the activation of contract farming and its success through proper collection and storage of cops and through the provision of needed quantities for contractors

 At the national level: cooperatives may execute important tasks and form a strategy for the agricultural sector, such as: -Preserving the environment and resources from pollution and waste by promoting and disseminating know-how of good agricultural practices (GAP) amongst farmers. This requires strengthening the relationship between cooperatives and research/consulting institutions. -Contributing to the rational use of irrigation water mainly by convincing farmers to plant the same crop at once. -Contributing to the improvement of the quality of agricultural production and compliance to international safety standards in order to increase export opportunities, which may be achieved by adopting TORs followed by its dissemination and implementation within members of cooperatives. -Reducing losses in yield, which represent a burden and huge loss at the national level through the provision of proper storage and transportation infrastructures.

Finally, it seems evident that the development of agricultural cooperatives is a multi-dimensional process that requires a strategic vision and specific operational phases over the years, without which partial amendments will not lead to effective results.

Contact :

Tahani ABDELHAKIM, CIHEAM-IAMM senior expert: abdelhakim@iamm.fr

  Inconsistency and contradiction in the tasks entrusted to the cooperatives over time which led to the lack of a clear definition of the role of agricultural cooperatives as part of liberal economic policies  The management of agricultural cooperatives in a traditional way does not fit with the current changes  Lack of management and technical skills along with a weakness or lack in financial and human resources for training, which is necessary for developing the efficiency of workers in cooperatives  Limited financial capacity of agricultural cooperatives, thereby weakening their ability to invest and develop  Poor infrastructure in a large number of cooperatives that do not have the infrastructures needed for proper storage, separation and packaging…etc  The transfer of some operations that are at core to the role of cooperatives to the Principal Bank for Development & Agricultural Credit, such as the collection and marketing of certain crops