Predator-Prey Model

Jean-Marc Ginoux

Robert Malthus

THE PARADOX OF VITO VOLTERRA'S

Francis Verhulst and then Vito Volterra and, on the other hand, to present the main hypotheses of the very famous but very little known predator-prey model elaborated in the 1920s by Volterra in order to solve a problem posed by his son-in-law, Umberto D'Ancona. It is thus shown that, contrary to a widely-held notion, Volterras model is realistic and his seminal work laid the groundwork for modern population dynamics and mathematical ecology, including seasonality, migration, pollution and more.

1. A short history of modeling 1.1. The Malthusian model. If the rst scientic view of population growth seems to be that of Leonardo Fibonacci [2], also called Leonardo of Pisa, whose famous sequence of numbers was presented in his Liber abaci (1202) as a solution to a population growth problem, the modern foundations of population dynamics clearly date from Thomas Robert Malthus [START_REF] Malthus | An Essay on the Principle of Population, printed for J. Johnson[END_REF]. Considering an ideal population consisting of a single homogeneous animal species, that is, neglecting the variations in age, size and any periodicity for birth or mortality, and which lives alone in an invariable environment or coexists with other species without any direct or indirect inuence, he founded in 1798, with his celebrated claim Population, when unchecked, increases in a geometrical ratio, the paradigm of exponential growth. This consists in assuming that the increase of the number N (t) of individuals of this population, during a short interval of time, is proportional to N (t). This translates to the following dierential equation : (1)

dN (t) dt = εN (t)
where ε is a constant factor of proportionality that represents the growth coecient or growth rate. By integrating (1) we obtain the law of exponential growth or law of Malthusian growth (see Fig. 1). This law, which does not take into account the limits imposed by the environment on growth and which is in disagreement with the actual facts, had a profound inuence on Charles Darwin's work on natural selection. Indeed, Darwin [1] founded the idea of survival of the ttest on the 1. According to Frontier and Pichod-Viale [START_REF] Frontier | Ecosystèmes, structure, fonctionnement, évolution[END_REF] the correct terminology should be population kinetics, since the interaction between species cannot be represented by forces.

2. A population is dened as the set of individuals of the same species living on the same territory and able to reproduce among themselves. 1 impossibility of an indenite population growth. He illustrated this impossibility by a superb parabola describing the descendants of a pair of elephants that, under optimal conditions, would cover the surface of the Earth in a few centuries 3 . However, in laboratory experiments, the predictions of the Malthusian law remain correct on small numbers, while there is divergence for high values of the population. Thus, we are led to conclude that the exponential law remains valid as long as the density of the population does not saturate the environment. 1.2. P.F. Verhulst's model. It was on the basis of these considerations that the Belgian biologist Pierre-François Verhulst [START_REF] Verhulst | Notice sur la loi que suit la population dans son accroissement[END_REF] proposed in 1837 a model that took into account the limitation imposed by the increasing population size :

(2)

dN (t) dt = εN (t) -λN 2 (t) = εN (t) ( 1 - 1 K N (t)
)

where ε represents the growth rate. The second coecient λ = ε/K originates in a mechanistic interpretation of the phenomenon. Indeed, it is assumed that growth is limited by a kind of interior friction within the population, that is, the resources remaining the same, the higher the number of individuals, the more dicult it is for them to feed themselves and then to grow. This is a struggle between individuals for the existence, and so an intra specic competition for food. The factor K, called carrying capacity, corresponds to the capacity of the environment to support the population growth and represents the population limit beyond which it can no longer grow. This law, which Verhulst called logistic equation, is radically dierent from Malthus's, since it imposes a limiting value on the population (see Fig. 2, where exponential growth is represented in blue and logistic growth in red). It has been successfully applied to many real life situations, such as population growth in the United States between 1790 and 1950 (see Pearl and Reed [START_REF] Pearl | On the rate of growth of the population of the United States since 1790 and its mathematical representation[END_REF]) or in experiments conducted by the Russian biologist Georgii Frantsevich Gause [START_REF] Gause | The struggle for existence[END_REF] on the growth of a protozoan, Paramecium caudatum. Independently of these two archetypes, other growth models have been developed : let us mention, for instance, the model by B.

Gompertz [START_REF] Gompertz | On the Nature of the Function Expressive of the Law of Human Mortality, and on a New Mode of Determining the Value of Life Contingencies[END_REF] aimed at evaluating the growth rate of a tumour. 1.3. Volterra's predator-prey model. In the rst half of the twentieth century, the study of the dynamics of several interacting species developed considerably.

It was at this time, called the golden age of theoretical ecology [START_REF] Scudo | The Golden Age of Theoretical Ecology : 1923-1940[END_REF], that the rst models based on competition-type behaviours and predator-prey relationships were developed. The paternity of the rst model used to transcribe this kind of interactions was the subject of a quarrel between Alfred J. Lotka [START_REF] Lotka | Elements of physical biology[END_REF] and Vito Volterra 4 , which is described in the work of Giorgio Israel [START_REF] Israel | Volterra's Analytical Mechanics of Biological Associations[END_REF][START_REF] Israel | The Emergence of Biomathematics and the Case of Population Dynamics : A Revival of Mechanical Reductionism and Darwinism[END_REF][START_REF] Israel | La Mathématisation du réel : essai sur la modélisation mathématique[END_REF][START_REF] Israel | The Biology of Numbers[END_REF]. It has been established that the merit of the development of the rst model of predator-prey type applied to a biological situation is Volterra's. Indeed, Volterra's interest in problems of equilibria between animal species in ecosystems was prompted by his son-in-law, zoologist Umberto D'Ancona, who for some years had been dealing with statistics on shing in the northern Adriatic Sea. These data involved the percentage of predatory sh (Selachians) caught in three Italian ports, Trieste, Fiume (now Rijeka)

and Venice, during the period 1905-1923. They proved that during the period 1915-1920, when shing was less intense because of the war, there had been a relative 4. For more about the dispute over priority, see [START_REF] Guerraggio | Trans. Kim Williams[END_REF], pp. 146-147.

increase in the Selachians. According to D'Ancona's hypothesis, shing disturbed the natural balance between species. It favoured a relative increase in prey species, that is, sh that feed only on plankton, and a decrease in predatory species, that is, sh that feed on other sh. The decline in shing due to World War I had thus restored, at least in part, the natural balance. D'Ancona turned to Volterra, asking him to nd a mathematical proof of his hypothesis. In 1926 Volterra published a reply in Italian [START_REF] Volterra | Variazioni e uttuazioni del numero d'individui in specie animali conviventi[END_REF] which took the form of the famous predator-prey model, reproduced below. He then presented his results in English and, in more detailed and complete form, in French [START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF][START_REF] Volterra | Variation and uctuations of the number of individuals in animal species living together[END_REF][START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF]. In the rst chapter of his Leçons sur la théorie mathématique de la lutte pour la vie [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF], Volterra studies the coexistence of two species, one of which devours the other. Considering two species, the rst, the prey N 1 (t), would undergo a Malthusian growth if it were alone. The second, the predator N 2 (t), feeds exclusively on the rst and in the absence of prey will gradually wear out and disappear. The formulation of the equation representing the predation is based on the méthode des rencontres (method of encounters) and on the hypothèse des équivalents (hypothesis of equivalents) due to Volterra [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF].

The former assumes that for predation to occur between a predatory species and a prey species, it is rst necessary to have encounters between these two species and that the number of encounters between them is proportional to the product of the number of individuals composing them, that is, N 1 (t) N 2 (t), the coecient of proportionality being equal to the probability of an encounter. The second hypothesis consists in assuming that there is a constant ratio between the disappearances and appearances of individuals caused by the encounters, that is, that predation of the preys is equivalent to increase of the predators. At the beginning, Volterra considers this increase as immediate 5 . This led him to the system :

(3)

     dN 1 dt = ε 1 N 1 -γ 1 N 1 N 2 = N 1 (ε 1 -γ 1 N 2 ) dN 2 dt = -ε 2 N 2 + γ 2 N 1 N 2 = -N 2 (ε 2 -γ 2 N 1 )
where ε 1 represents the prey's growth rate in the absence of the predator ; γ 1 the predation rate of the predator on the prey ; ε 1 the predator's mortality rate in the absence of prey ; and γ 2 the growth rate of the predator due to its predation. From this model, Volterra was able to state the law of the disturbance of the averages : If an attempt is made to destroy the individuals of the two species uniformly and in proportion to their number, the average of the number of individuals of the species that is eaten increases and that of the individuals of the species feeding upon the other diminishes [31, p. 20] To establish this result, Volterra assumes that, for a time interval dt, we destroy αλN 1 dt preys and βλN 2 dt predators. He then proves that the average values of prey and predators, which were previously equal to ε 2 /γ 2 and ε 1 /γ 1 , respectively, become, after this destruction, (ε 2 + βλ)/γ 2 and (ε 1 -αλ)/γ 1 . The phenomenon observed by D'Ancona is thus explained : the increase in the number of predators 5. This means that predation is immediately transcribed in terms of growth of the predator species, whereas its eect naturally occurs with some delay. It will be seen below that Volterra also took this delay into account. and the reduction in the number of prey resulted from the decline in shing, which before the war had changed the natural balance of this biological association.

A decrease in shing favours the more voracious species at the expense of the other. This deterministic model, which constitutes the archetype of the trophic network, aims at transcribing dierent types of animal behaviour into mathematical functions. Two types of behaviour are represented : those related to increase and those related to decrease. Natality and predation are related to increase, whereas natural mortality and mortality by predation correspond to a decrease in the number of individuals. Each of these behaviours has a mathematical form, called a functional response.

1.4. Dierent types of functional responses. Since the mid-1920s, these functional responses have been the subject of numerous studies and developments aimed at making the representation of animal behaviour by a mathematical function more realistic. Natural growth, that is by natality, of the prey, represented by a functional response of Malthus type [START_REF] Malthus | An Essay on the Principle of Population, printed for J. Johnson[END_REF] was later modied by Verhulst [START_REF] Verhulst | Notice sur la loi que suit la population dans son accroissement[END_REF] to account for its being bounded. The decrease by natural mortality was initially considered in a way symmetrical to that of natural growth, that is, by substituting in the equations ( 1) and ( 2) the growth rate ε with a natural mortality rate -ε. It is important to emphasise that this functional response, sometimes called closure relation, was later the subject of special studies aimed at transcribing the specic behaviour of certain species, for instance cannibalism. All these functional responses are summarized in Table 1.

natural increase natural mortalitity Malthus εN (t)

-εN (t)

Verhulst εN (t)

( 1 - N (t) K ) -εN (t) ( 1 + N (t) K ) Table 1.
Functional responses for natural increase and decrease.

The functional response proposed by Volterra [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] to describe predation and which was based on the principle of encounters was proportional to the product of the number of individuals of each species : N 1 (t) N 2 (t). In other words, the predation rate was a linear function of the prey, that is, N 1 (t). A few years later, Gause [START_REF] Gause | The struggle for existence[END_REF][START_REF] Gause | vérications expérimentales de la théorie mathématique de la lutte pour la vie[END_REF], who was one of the rst to make experimental verications of the mathematical theory of the struggle for life, proposed another type of functional response to describe predation, a nonlinear one, aimed at transcribing a certain satiety of the predator with respect to its prey : N g 1 (t) N 2 (t) with 0 < g 1.

In this case, the predation rate becomes indeed a nonlinear function of the prey, that is, N g 1 (t) (see Fig. 3).

In the late 1950s, entomologist Crawford Stanley Holling [START_REF] Holling | The components of predation as revealed by a study of small-mammal predation of the European pine sawy[END_REF][START_REF] Holling | Some characteristics of simple types of predation and parasitism[END_REF] developed, from the celebrated disc equation two new functional responses for predation, also intended to describe a certain satiety of the predator with respect to its prey : Holling (4)

N 1 (t) h + N 1 (t) N 2 (t)
where h represents half-saturation, that is, the value of the prey density N 1 (t) = h for which the predation level reaches a value equal to half its maximum.

Holling's type III (see Fig. 5 where Holling's type II is represented in blue and Holling's type III in red) is a functional response in which the attack rate of the predator rst increases when the prey number is low and then slows down when the predator reaches satiety. In other words, the predator increases its research activity when prey density increases. Thus, type III functional responses are typical of generalist predators moving from one species of prey to another and concentrating their activities in areas where resources are abundant. In this case, the mortality of the prey initially increases with their density and then decreases. Holling's type III is represented by :

(5)

N 2 1 (t) h 2 + N 2 1 (t) N 2 (t)
where h represents half-saturation, as above. Contrary to what one might think, Holling did not deduce his two functional responses from the observation of a natural environment, but, as he himself wrote in his article, by concocting an articial predatory-prey situation :

J.M. GINOUX

In the rst articial situation devised the prey were sandpaper discs four centimetres in diameter thumb-tacked to a three-foot square table. A blindfolded subject, the predator, stood in front of the table and searched for the discs for one minute by tapping with her nger. As each disc was found, it was removed, set to one side and searching continued. Each experiment was replicated eight times at densities of discs ranging from four to 256 per nine sq. ft.

The results of one such experiment are shown in Fig. 1 [see Fig. 4 here], where it can be seen that the number of discs picked up increased at a progressively decreasing rate as the density of discs rose. [12, p. 385] Mathematically, it has been proved by Real [24] that the functional responses of Holling type II and III are analogous to the function developed in 1913 by Leonor

Michaelis and Maud Menten [START_REF] Michaelis | Die Kinetik der Invertinwirkung[END_REF] to describe the kinetics of enzymatic reactions.

All these functional responses are summarized Table 2.

Volterra Gause

Holling type II Holling type III

N 1 (t) N 2 (t) N g 1 (t) N 2 (t) N 1 (t) h + N 1 (t) N 2 (t) N 2 1 (t) h 2 + N 2 1 (t) N 2 (t)
Table 2. Functional responses for predation

The origin of the paradox

According to Yuri A. Kuznetsov [START_REF] Kuznetsov | Elements of Applied Bifurcation Theory[END_REF], in the context of a Volterra predator-prey model, the functional responses that limit growth (see Table 1) have a stabilizing eect for the prey whereas the functional responses that limit predation (see Table 2) have a destabilizing eect for predators. Thus, in the Volterra predator-prey model 3, by restricting the growth of the prey N 1 (t) by a Verhulst functional response and the growth of the predator N 2 (t) by a functional response of Holling type II, we obtain the celebrated Rosenzweig-MacArthur model [START_REF] Rosenzweig | Graphical representation and stability conditions of predator-prey interaction[END_REF] :

(6)        dN 1 dt = ε 1 N 1 -λN 2 1 -γ 1 N 1 h + N 1 N 2 dN 2 dt = -ε 2 N 2 + γ 2 N 1 h + N 1 N 2
where ε 1 represents the preys growth rate in the absence of the predator, γ 1 the predators predation rate on the prey, ε 2 the predators mortality rate in the absence of prey, and γ 2 the predators growth rate due to its predation. The combination of these two eects then leads to the existence of a periodic solution.

Henri Poincaré [START_REF] Poincaré | Sur les courbes dénies par une équation diérentielle[END_REF] named this solution, or more exactly this periodic oscillation, limit cycle. This terminology derives from the fact that in the phase space (N 1 , N 2 ), in this case the prey-predator space, it takes the form of a cycle towards which every solution converges asymptotically. Thus, the solution of the Rosenzweig-MacArthur model ( 6) is periodic, like that of the Volterra model (3). Nevertheless, there is a fundamental dierence in that it is totally independent of the initial conditions, that is, of the initial density of prey and predators, which is consistent with reality. Indeed, in nature, the amplitude and period of the periodic oscillations of prey and predators cannot depend on their initial densities considered at a given time arbitrarily taken as the origin of the time [START_REF] Ginoux | Analyse mathématiques des phénomènes oscillatoires non linéaires[END_REF][START_REF] Ginoux | Histoire de la Théorie des Oscillations Non Linéaires[END_REF][START_REF] Ginoux | History of Nonlinear Oscillations Theory[END_REF].

Discussion

Volterra's model, which in the study of nonlinear dynamic systems has become a kind of paradigm, has been the subject of many books and articles criticizing it for its lack of realism. The main aws of this model are considered to be the absence of limitation in the growth of prey and predator, the fact of not having taken into account seasonality (that is, having considered the growth rate of prey and predator as constant), and nally having proposed an ideal, simplied model by limiting it to two species. The analysis of these aws will shed light on the paradox of Volterras model. In the rst place, it should be recalled that even today it is impossible to formulate a simple expression of the periodic solutions of Volterra's model [START_REF] Frontier | Ecosystèmes, structure, fonctionnement, évolution[END_REF], that is, an expression that uses only elementary functions. Consequently, criticizing Volterra for having proposed an ideal, simplied model is inadmissible. Indeed, it is precisely thanks to the simplicity of his model that Volterra was able to solve the problem posed by D'Ancona and explain the phenomenon by establishing the law of perturbation of means. In fact, Volterra was well aware of the limitations of the model and the assumptions attached to it, including the one assuming the environment as invariable and without inuence on the growth rate and the one that considers the homogeneity of the individuals of each species. Indeed, in [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] Volterra laid the foundations for what would be much later called population dynamics and considered all the aspects of the problem : the inuence of environment or pollution, the heterogeneity of individuals or age groups (on hereditary actions [32, p. 141]), diusion or migration (contribution of a small number of individuals [32, p. 118]), seasonality (variation of exterior conditions with time, [32, p. 131]). In the introduction, he writes : Certainly there exist periodic circumstances relating to environment, as would be those, for example, which depend upon the changing of the seasons, which produce forced oscillations of an external character in the number of individuals of the various species. These actions of external periodic nature were those which were specially studied from the statistical point of view, but are there others of internal character, having periods of their own which add their action to these external causes and would exist even if these were withdrawn ? [32, p. 5].

Then, he adds : Later, it will be observed that it is closer to reality to suppose that the growth coecients depend not only, at each moment, on the current values of the quantities N i (characterizing the species i), but also on past values up to a more or less remote period. It will no longer be sucient to consider them as functions of the N i , but as functionals, and this will lead us to integro-dierential equations that we will approach from the equations we are led to in the so-called hereditary mechanics [32, p. 5].

It is in this context that he invents the famous Volterra equations of the rst and second type. With regard to the limitation to a two-species model of predatorprey type, in chapter II of [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] Volterra presents the study of the coexistence of any number of species. Moreover, the chapter ends with the analysis of a very remarkable case : that of three species of which the rst feeds on the second and this on the third. Dierent types of interactions between species are also considered outside of predation, including competition, cooperation and migration. Regarding the absence of limitation in the growth of prey and predators, Volterra proposes in chapter III of [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] to study a model of n coexisting species having reciprocal actions, in which he replaces the Malthusian growth by a Verhulst logistic growth. Thus, Italian or French authors who claim that Volterra's model is unrealistic have obviously not read Volterra's texts [START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF][START_REF] Volterra | Variation and uctuations of the number of individuals in animal species living together[END_REF][START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] and this is clearly one of the origins of this paradox. Indeed, it seems that Volterra's model and criticisms of it are known worldwide, but his work has not even been read by his detractors ! For authors who only read English, this paradox is rooted in another problem. While the original Italian version of Volterra's study [START_REF] Volterra | Variazioni e uttuazioni del numero d'individui in specie animali conviventi[END_REF] comprised 84 pages, the very rst English translation published in the prestigious journal Nature [START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF] was only two pages long 6 . It is easy to understand that for many Volterra's study may have seemed very limited given the conciseness of this summary, which did not allow the exposition of analytical developments and considerably reduced the scope of Volterra's results. Nevertheless, and this is again a paradox, even though by 1928 there was an integral English translation of the original Italian text by Volterra [START_REF] Volterra | Variation and uctuations of the number of individuals in animal species living together[END_REF], it seems to have been almost totally ignored by English-speaking authors. Thus, Volterra's work is perhaps the most quoted and the most criticized in the world while being the least read and studied.

From 1926, Volterra's publications intensied in the eld of population dynamics.

It was in 1931 that his work entitled Leçons sur la théorie mathématique de la lutte pour la vie [START_REF] Volterra | Leçons sur la Théorie Mathématique de la Lutte pour la Vie[END_REF] was published in French 7 , following a series of lectures at the Institut Henri Poincaré where Volterra had been invited by Borel. Compiled by Marcel Brelot, this book contains Volterra's entire memoir on biological uctuations [START_REF] Volterra | Variazioni e uttuazioni del numero d'individui in specie animali conviventi[END_REF] as well as a part about the case where heredity plays a role. From 1936, Volterra questioned the validity of his model and the possibility of experimental verication.

The experiments of Gause (1910Gause ( -1989) ) seemed to conrm the rst law, and he showed Volterra his acceptance of the model's forecasts. But what followed was disappointing because they could not nd irrefutable cases of a cyclic behaviour in predator-prey ecosystems. Moreover, voiced by biologists Karl Pearson and Friedrich Simon Bodenheimer (1897-1959), objections multiplied to the point of questioning the interpretation of D'Ancona's statistics, which led to D'Ancona himself coming to doubt their merits. At present, the only example of a predatorprey ecosystem showing a cyclic evolution is the famous set of statistics by Hudson's Bay Company about hares and lynxes in Canada. According to Giorgio Israel, it is thus important to note that unlike Van der Pol's model [START_REF] Van Der Pol | On 'Relaxation-Oscillations[END_REF] : 6. Under the same title, in a later issue of the same journal, two letters were published, one by Lotka and one by Volterra himself [START_REF] Lotka | Fluctuations in the abundance of a species considered mathematically[END_REF][START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF]. 7. To the best of our knowledge, this book has never been translated into English.

Volterra's model is not deduced from an analogy but from a more traditional approach that consists in starting from the analysis of a real phenomenon, making some abstraction of the accessory aspects such as friction, determining the state variables, formulate a mathematical hypothesis concerning the pace of the phenomenon. . . . For Volterra's model, the problem of the experimental verication consists in nding empirical evidence directly verifying the laws deduced from the model and not justifying its validity in an indirect way, that is, from the eectiveness of some of its consequences. Thus, for Volterra, the justication of D'Ancona's hypothesis on the eects of shing was not sucient to prove the empirical validity of the model. It is because of this conviction that he sought for the rest of his life a direct empirical proof of the validity of the rst law according to which the evolution of populations presents periodic oscillations [START_REF] Van Der Pol | On 'Relaxation-Oscillations[END_REF].

Thus, if Volterra sought to make his mathematical research work at the service of biological sciences, it seems that it always was with the aim of explaining phenomena and describing reality as faithfully as possible :

. . . the hypotheses will be seen to become more and more complex in order to get closer to reality . . .

Figure 1 .

 1 Figure 1. Malthus's model of exponential growth.

3 .

 3 An example of this impossibility is also shown in the lm by V.A. Kostitzin and J. Painlevé titled : Images mathématiques de la lutte pour la vie, 1937, Médiatèque du Palais de la Découverte, Paris.

Figure 2 .

 2 Figure 2. Verhulst's model of logistic growth (in red).

1 N 1 gFigure 3 .

 113 Figure 3. Gause's functional response

Figure 4 .

 4 Figure 4. Functional response of Holling's type II.

h 2 N 1 2Figure 5 .

 215 Figure 5. Functional response of Hollings type III (in red).

  Translated from the French by Daniele A. Gewurz

Acknowledgments I wish to express my sincerest thanks to my friend Christian Gérini, Agrégé de Mathématiques and Docteur en Histoire des Sciences, who encouraged and supported me in my work. Références


H Al Majou

B Hassani

A Bruand

Transferability of continuous-and class-pedotransfer functions to predict water retention properties of semiarid Syrian soils

Keywords: Pedotransfer functions, Soil water retention, Syrian soils, Field capacity, Texture, Bulk density

Hydraulic properties of soils, particularly water retention, are key for appropriate management of semiarid soils. Very few pedotransfer functions (PTFs) have been developed to predict these properties for soils of Mediterranean regions, where data are particularly scarce. We investigated the transferability of PTFs to semiarid soils. The quality of the prediction was compared to that for soils originating from temperate regions for which most PTFs were developed. We used two soil datasets, one from the Paris basin (French dataset, n = 30), and a Syrian dataset (n = 30). Soil samples were collected in winter when the water content was near field capacity. Composition and water content of the samples were determined at seven water potentials. Continuous-and class-PTFs developed using different predictors were tested using the two datasets and their performance compared to those developed using artificial neural networks (ANN). The best performance and transferability of the PTFs for both datasets used soil water content at field capacity as predictor after stratification by texture.

The quality of prediction was similar to that for ANN-PTFs. Continuous-and class-PTFs may be transferable to other countries with performances that vary according to their ability to account for variation in soil composition and structure. Taking into account predictors of composition (particle size distribution, texture, organic carbon content) and structure (bulk density, porosity, field capacity) did not lead to a better performance and best transferability potential.

Introduction

The hydraulic properties of soils, particularly soil water retention properties, are key data for the appropriate management of soils. These properties, which are generally unavailable and expensive to measure, can be predicted from other more easily measured properties using prediction tools called "pedotransfer functions (PTFs)" [START_REF] Bouma | Using soil survey data for quantitative land evaluation[END_REF]. PTFs have been applied in a large number of studies in recent decades (e.g. [START_REF] Cornelis | Evaluation of pedotransfer functions for predicting the soil moisture retention curve[END_REF][START_REF] Wösten | Pedotransfer functions: bridging the gap between available basic soil data missing soil hydraulic characteristics[END_REF][START_REF] Al Majou | Variation of the water retention properties of soils: validity of class-pedotransfer functions[END_REF][START_REF] Nasta | Scaling soil water retention functions using particle-size distribution[END_REF][START_REF] Haghverdi | A simple nearest-neighbor technique to predict the soil water retention curve[END_REF][START_REF] Nemes | Why do they keep rejecting my manuscript-do's and don'ts and new horizons in pedotransfer studies?[END_REF][START_REF] Khlosi | Enhanced pedotransfer functions with support vector machines to predict water retention of calcareous soil[END_REF][START_REF] Nguyen | Comparison of statistical regression and data-mining techniques in estimating soil water retention of tropical delta soils[END_REF].

PTFs can be grouped into two categories, continuous-PTFs and class-PTFs [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF]. Continuous-PTFs enable the continuous prediction of the water retention curve (WRC) over the whole soil water content range, i.e. from water saturation to residual water content (e.g. [START_REF] Vereecken | Estimating the soil moisture retention characteristics from texture, bulk density and carbon content[END_REF][START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF][START_REF] Hodnett | Marked differences between van Genuchten soil waterretention parameters for temperate and tropical soils: A new water-retention pedotransfer function developed for tropical soils[END_REF]Al Majou et al., 2008a;[START_REF] Ghorbani | Derivation and validation of pedotransfer functions for estimating soil water retention curve using a variety of soil data[END_REF][START_REF] Haghverdi | A pseudo-continuous neural network approach for developing water retention pedotransfer functions with limited data[END_REF][START_REF] Medrado | Development of pedotransfer functions for estimating water retention curve for tropical soils of the Brazilian savanna[END_REF] (Table 1).

The class-PTFs predict the water retention curve discontinuously by grouping the data according to the functional behaviour of different horizons but may also be grouped according to some other criteria, such as texture or bulk density [START_REF] Wösten | Functional sensitivity analysis of four methods to generate soil hydraulic functions[END_REF][START_REF] Baker | Development of class pedotransfer functions of soil water retention -A refinement[END_REF].

Thus, a single mean value or several mean values of the hydraulic properties are selected to represent each class [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF]Schaap et al., 2001;[START_REF] Hodnett | Marked differences between van Genuchten soil waterretention parameters for temperate and tropical soils: A new water-retention pedotransfer function developed for tropical soils[END_REF][START_REF] Nemes | Unsaturated soil hydraulic database of Hungary: HUNSODA[END_REF][START_REF] Bruand | Use of class pedotransfer functions based on texture and bulk density of clods to generate water retention curves[END_REF][START_REF] Pachepsky | Hydropedology and pedotransfer functions[END_REF][START_REF] Al Majou | Variation of the water retention properties of soils: validity of class-pedotransfer functions[END_REF]. Class-PTFs are often easy to use because they usually require little information about the soil compared with most continuous-PTFs that are more demanding of soil characteristics [START_REF] Lilly | The development and use of the HYPRES database in Europe[END_REF][START_REF] Nemes | Functional evaluation of pedotransfer functions derived from different scales of data collection[END_REF]. However, they are often regarded as leading to poorer quality predictions than continuous-PTFs [START_REF] Wösten | Comparison of class and continuous pedotransfer functions to generate soil hydraulic characteristics[END_REF]. The issue of whether WRC prediction is best carried out with continuous-PTFs or with class-PTFs is still debated [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF][START_REF] Medeiros | Assessment of pedotransfer functions for estimating soil water retention curves for the amazon region[END_REF] as is the issue of their transferability to other regions than those from which the soils originated to establish the PTFs considered [START_REF] Cresswell | The transferability of Australian pedotransfer functions for predicting water retention characteristics of French soils[END_REF][START_REF] Touil | Transposability of pedotransfer functions for estimating water retention of Algerian soils[END_REF]. Few studies relate to the prediction of the WRC of soils in the Mediterranean basin [START_REF] Dridi | Poids des différentes caractéristiques des sols dans l'estimation de leur rétention en eau[END_REF][START_REF] Mohawesh | Assessment of pedotransfer functions (PTFs) in predicting soil hydraulic properties under arid and semi-arid environments[END_REF][START_REF] Wösten | Soil hydraulic information for river basin studies in semi-arid regions[END_REF]. Thus, in Syria and in other countries of the Mediterranean basin, there is still little information available on hydraulic properties of soils and establishment of national soil databases is only just beginning [START_REF] Sommer | Simulating the effects of zero tillage and crop residue retention on water relations and yield of wheat under rainfed semiarid Mediterranean conditions[END_REF][START_REF] Khlosi | Exploration of the Interaction between Hydraulic and Physicochemical Properties of Syrian Soils[END_REF].

Other approaches, such as artificial neural networks (ANN) [START_REF] Haykin | Neural networks, a comprehensive foundation[END_REF]Schaap et al., 2001), support vector machines (SVM) [START_REF] Twarakavi | Development of pedotransfer functions for estimation of soil hydraulic parameters using support vector machines[END_REF] or the k-nearest neighbor technique (k-NN) (Nemes et al., 2006a) have also been developed in recent decades to predict the water retention properties of soils. The approach using artificial neural networks (ANN) use one or more hidden layers or hidden units and are based on a self-learning process by using a set of soils for which the water retention properties and the basic properties are known (Table 1). Artificial neural network-based PTFs were introduced by [START_REF] Pachepsky | Artificial neural networks to estimate soil water retention from easily measurable data[END_REF] and [START_REF] Schaap | Modelling water retention curves of sandy soils using neural networks[END_REF]. Unlike continuous-and class-PTFs, they do not require an a priori model (e.g. linear or exponential functions) [START_REF] Nemes | Validation of international scale soil hydraulic pedotransfer functions for national scale applications[END_REF]. During the last two decades, ANNs have been used extensively to predict soil water retention and their performance has been compared with PTFs based on other approaches [START_REF] Minasny | Neural networks prediction of soil hydraulic functions for alluvial soils using multistep outflow data[END_REF][START_REF] Twarakavi | Development of pedotransfer functions for estimation of soil hydraulic parameters using support vector machines[END_REF][START_REF] Nguyen | Comparison of statistical regression and data-mining techniques in estimating soil water retention of tropical delta soils[END_REF].

The objective of the present study was to select continuous-and class-PTFs from the literature and to analyze the quality of prediction when used for a set of French soils and a set of Syrian soils originating from temperate and Mediterranean regions, respectively. Prediction performance results for the selected continuous-and class-PTFs were also compared with those for ANN-PTFs. Additionally, their transferability to soil in France and Syria were assessed, given that the compared PTFs were established with soil samples from different origins.

Materials and methods

Soil data sets

A first set of soil samples was collected in France. Thus, 30 horizons (12 horizons A or Ap and 18 horizons E, B or C) originating from soils developed on sedimentary rocks in the Paris basin were sampled in winter. The climate of the Paris basin is temperate and influenced by the Atlantic Ocean, according to the distance from the coast. The soils were Cambisols, Luvisols and Fluvisols (ISSS Working Group R.B., 1998) [START_REF] Bruand | Water retention properties of the clay in soils developed on clayey sediments: Significance of parent material and soil history[END_REF].

Another set of soil samples was collected in Syria. The soil samples were also collected in winter between latitudes 32 and 37° N and longitudes 35 to 42° E, an area with a Mediterranean or degraded Mediterranean climate [START_REF] Rigot | L'évolution ralentie du milieu naturel dans la steppe aride du nord de la Syrie à l'Holocène[END_REF]. Along the coast, the climate is Mediterranean but continental influences and aridity contribute to a rapid degradation of the Mediterranean climate as the distance from the coast increases. A set of 30 horizons (16 horizons A or Ap and 14 horizons E, B or C) resulting from four sites were chosen as being representative of the main soil types. They were developed on calcareous and volcanosedimentary (basaltic) parent materials and collected in Aridisols, Inceptisols and Vertisols [START_REF] Ilaiwi | ACSAD report on soil of Syria and Lebanon scale[END_REF][START_REF] Yuksel | Land classification/soil survey project of the Syrian Arab Republic[END_REF]Land Classification and Soil Survey of the Syrian Arab Republic, 1982) or Calcisols, Gypsisols, Inceptisols and Vertisols [START_REF] Van Liere | Classification and rational utilization of soils[END_REF].

For each horizon, the particle size distribution without decarbonation [START_REF] Robert | Méthode de préparation des argiles des sols pour les études minéralogiques[END_REF], the bulk density of clods and horizons [START_REF] Bruand | Water retention properties of the clay in soils developed on clayey sediments: Significance of parent material and soil history[END_REF], the organic carbon content by oxidation using an excess amount of potassium dichromate in a sulphuric acid controlled at 135°C [START_REF] Baize | Guide des analyses en pédologie[END_REF], the CaCO 3 content [START_REF] Dupuis | Dosage des carbonates dans les fractions granulométriques de quelques sols calcaires et dolomitiques[END_REF] and cation exchange capacity (CEC) [START_REF] Ciesielski | Determination of cation exchange capacity and exchangeable cations in soils by means of cobalt hexamine trichloride. Effects of experimental conditions[END_REF] were determined. The volumetric water content was determined for the 60 soil samples from the two datasets at the water potential values -10 hPa ( 1.0 ), -33 hPa ( 1.5 ), -100 hPa ( 2.0 ), -330 hPa ( 2.5 ), -1000 hPa ( 3.0 ), -3300 hPa ( 3.5 ), -15000 hPa ( 4.2 ) using the pressure plate extractor method [START_REF] Bruand | Water retention properties of the clay in soils developed on clayey sediments: Significance of parent material and soil history[END_REF].

The continuous-and class-PTFs selected

Among continuous-PTFs, some enable the direct prediction of the parameters of a WRC model (Table 1). In most studies, these continuous-PTFs are multiple linear regressions or non linear regressions providing the parameters of the van Genuchten model (1980) as output variables by using the particle size distribution, the organic carbon content and the bulk density as input soil properties (e.g. [START_REF] Tomasella | Comparison of Two Techniques to Develop Pedotransfer Functions for Water Retention[END_REF]Schaap et al., 2001) (Fig. 1a).

Among this group of continuous-PTFs, those established with a large number of European soils (Wösten-continuous-PTFs, see Table 5 in [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF], with Belgian soils (Vereecken-continuous-PTFs, see Table 7 in [START_REF] Vereecken | Estimating the soil moisture retention characteristics from texture, bulk density and carbon content[END_REF], and with French soils (VG-continuous-PTFs, see Table 6 in Al Majou et al., 2008a) were selected for this study (Table 1). They all estimate the parameters of the van Genuchten model (1980).

Other continuous-PTFs do not directly predict the parameters of a WRC model but rather the water content at several matric potentials as output variables, generally by using the particle size distribution, the organic carbon content and the bulk density as input data (e.g. [START_REF] Pachepsky | Accuracy and reliability of pedotranfer functions as affected by grouping soils[END_REF][START_REF] Reichert | Estimation of water retention and availability in soils of Rio Rrande Do Sul[END_REF][START_REF] Ghanbarian-Alavijeh | Point pedotransfer functions for estimating soil water retention curve[END_REF][START_REF] Minasny | Predicting soil properties in the tropics[END_REF]) (Fig. 1b). Then, knowing the water content at different water potentials, a WRC model is adjusted to the predicted water contents. Among this group of continuous-PTFs, those established with Brazilian soils (Reichert-continuous-PTFs, see Table 4 in [START_REF] Reichert | Estimation of water retention and availability in soils of Rio Rrande Do Sul[END_REF], with French soils (PSD-continuous-PTFs and FC-continuous-PTFs, see Table 4 &5 in Al Majou et al., 2008a), with soils originating from the USA (Ghanberian-continuous-PTFs, see Table 5 model 1 in [START_REF] Ghanbarian-Alavijeh | Point pedotransfer functions for estimating soil water retention curve[END_REF] and with soils originating from tropical regions (Minasny-continuous-PTFs, see section 5.1 in [START_REF] Minasny | Predicting soil properties in the tropics[END_REF] were selected for this study (Table 1). They estimate water contents at three to seven values of water potential and concern a large range of soil types.

On the other hand, some of the class-PTFs directly predict the parameters of a WRC model after stratification according to soil characteristics such as texture, bulk density, the type of horizon or type of soil, as input data (Table 1, Fig. 1c). Among this group of class-PTFs, those established after stratification by texture alone, and by both the type of horizon and texture with soils originating from North America (Schaap-class-PTFs, see Table 1 in Schaap et al., 2001), Europe (Wösten-class-PTFs, see Table 4 in [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF] and France (T-H-VGclass-PTFs, see Table 3 in Al Majou et al., 2008a), respectively, were selected for this study (Table 1).

Finally, other class-PTFs are sets of water contents at different values of water potential, these water contents being related also to classes of soil characteristics such as texture, bulk density, the type of horizon or type of soil, as input data (Table 1, Fig. 1d). Among this group of class-PTFs, those established with French soils after stratification by either texture alone, by both texture and bulk density or by all the three predictors texture, bulk density and type of horizon

The Artificial Neural Networks-PTFs

The type of ANN selected was the one most commonly used to predict the hydraulic properties of soils [START_REF] Børgesen | Point and parameter pedotransfer functions for water retention predictions for Danish soils[END_REF][START_REF] Fashi | Evaluation of artificial neural network and regression PTFs in estimating some soil hydraulic parameters[END_REF]. It usually consists of a three-layer feed forward back propagation network using, for each model, input layers (basic soil properties), hidden layers, and output layers (soil hydraulic properties), (Fig. 1e). Each neuron of the hidden layer calculates the sum s, of a weighted combination w i , of its input signals x i , and a bias term w 0 , and passes the result through the activation functions that were tangent hyperbolic and linear in the hidden and output layers, respectively.

0 1 w x w s i i n i     (1)
The Levenberg-Marquardt algorithm [START_REF] Demuth | Neural Network Toolbox for Use with MATLAB, User Guide[END_REF] was implemented to speed up the training of the multi-layer feed-forward neural network. The number of neurons in the hidden layer has to be found through trial and error; the number tested here varied from 1 to 10 neurons. The feed-forward process stops once the output is predicted. Back-propagation algorithms try to minimize the error (minimize the sum of squares of the residuals between the measured and predicted outputs) of the mathematical system represented by the neural network's weights. The error is estimated as the difference between the actual and computed outputs.

The French national database used by Al Majou et al. (2008b) to develop continuous-and class-PTFs, including those discussed in this study, and which consists of 456 samples was used to train and evaluate the predictive performance of the ANN developed (Table 2). To continue to test the ANN method, the ANN developed was then applied to the French and Syrian datasets. The ANN simulations were performed by using the neural network toolbox provided by Matlab (R2014b).

Criteria used to evaluate the performance of the continuous-and class-PTFs

To assess the continuous and class-PTFs performance, we used the mean error of prediction (MEP) and the standard deviation of prediction (SDP) which provide information on the estimation bias and precision, respectively, as follows:

       l i i j m i j p l j l l MEP 1 , , , , ' 1 ) ( ' 1     2 / 1 1 2 , , , , ' 1 ) ( ' 1               l i i j m i j p l j MEP l l SDP   (2) (3)
where  p,j,i is the predicted water content at matric potential i for the horizon j,  m,i,j is the measured water content at matric potential i for the horizon j, and l is the number of matric potentials for each horizon (l=7 in this study) and l' is the number of horizons studied.

The root mean square error (RMSE) which is commonly used to test PTFs (e.g. [START_REF] Wösten | Pedotransfer functions: bridging the gap between available basic soil data missing soil hydraulic characteristics[END_REF][START_REF] Schaap | Accuracy and uncertainty in PTF predictions[END_REF][START_REF] Lamorski | Using support vector machines to develop pedotransfer functions for water retention of soils in Poland[END_REF] and which varies according to both the overall prediction bias and the overall prediction precision was also computed:

  2 / 1 2 1 , , , , ' 1 ' 1                  l i i j m i j p l j l l RMSE  
Beside these three statistical criteria, the coefficient of determination (R²) was computed:

            l i i m i j m l j l i i j m i j p l j R 1 , , , ' 1 1 , , , , ' 1 )² ( )² ( 1 ²    
where  m,i represents the average of the measured water content at the matric potential i. The value of the coefficient of determination (R²) measures the strength of the linear relation between the predicted and measured values.

Results and discussion

Characteristics of the studied soils

The studied horizons showed Fine, Very Fine or Medium texture in the European triangle of texture (Commission of the European Communities, 1985) and no coarse elements (Table 2, Fig. 2a). Particle size distribution mean values for the datasets from France and Syria were close (Fig. 2bc). However, the Syrian dataset showed a smaller mean OC content because of the higher temperatures, limited precipitation and tillage systems that rapidly oxidize organic matter [START_REF] Mrabet | Climate change and carbon sequestration in the Mediterranean basin: Contributions of no-tillage systems. 4 ème Rencontres Méditerranéennes du Semis Direct[END_REF]. As for the smaller mean bulk density of the Syrian soils, it can be (4)

(5) attributed to a larger macroporosity in most horizons as observed in the field (data not shown). The mean CaCO 3 content and CEC were greater in the Syrian dataset.

Results also showed a greater water content for matric potential ranging from -10 hPa (θ 1.0 ) to -330 hPa (θ 2.5 ) for the Syrian dataset. These greater water contents are consistent with the smaller mean bulk density recorded for the Syrian dataset and a greater proportion of swelling clays in the Syrian dataset as indicated by the greater mean CEC (36.5 cmol c /kg), the OC content being smaller in Syrian soils and the clay content similar in the two data sets.

Evaluation of the continuous-and class-PTFs developed with the French national database

The performance of the continuous-and class-PTFs developed with the French national database showed that the best prediction was recorded for the French dataset with the class-PTFs using the water content at field capacity as predictor after stratification by texture (T-FC-class-PTFs). These class-PTFs are the only ones that perform best for three statistical criteria (Table 3): the root mean square error (RMSE = 0.023 cm 3 /cm 3 ) was the smallest and the precision (SDP = 0.023 cm 3 /cm 3 ) and coefficient of determination (R 2 = 0.81), the greatest (Table 3) (Fig. 3d). The worst prediction was recorded with the class-PTFs after successive stratification by texture and bulk density (T-BD-class-PTFs) with the smallest precision (SDP = 0.036 cm 3 /cm 3 ) and coefficient of correlation (R 2 = 0.52) and the largest root mean square error (RMSE = 0.036 cm 3 /cm 3 ) (Table 3) (Fig. 3f). The continuous-PTFs developed for the parameters of the van Genuchten model (1980) (VG-continuous-PTFs) led to similar results to those obtained with the T-BD-class-PTFs, with small precision (SDP = 0.036 cm 3 /cm 3 ) and high root mean square error (RMSE = 0.036 cm 3 /cm 3 ) (Table 3) (Fig. 3a).

With the Syrian dataset, the best performance was also recorded with the class-PTFs using water content at field capacity as predictor after stratification by texture (T-FC-class-PTFs) (Table 3). The estimation bias (MEP = -0.001 cm 3 /cm 3 ) and the root mean square error (RMSE = 0.029 cm 3 /cm 3 ) were the smallest and the precision (SDP = 0.029 cm 3 /cm 3 ) the greatest, the coefficient of determination (R 2 = 0.75) being the second largest with the T-FCclass-PTFs (Table 3) (Fig. 4d). The worst prediction was recorded with the continuous-PTFs developed for the parameters of the van Genuchten model (1980) (VG-continuous-PTFs) with the smallest precision (SDP = 0.046 cm 3 /cm 3 ), a high estimation bias (MEP = 0.022 cm 3 /cm 3 ) and the highest root mean square error (RMSE = 0.048 cm 3 /cm 3 ) (Table 3) (Fig. 4a). The class-PTFs after successive stratification by texture and bulk density (T-BD-class-PTFs) gave results that had rather poor precision and root mean square error, though the estimation bias was very small (MEP = 0.001 cm 3 /cm 3 ) and the coefficient of determination was much smaller (R 2 = 0.51) (Fig. 4f) than with the VG-continuous-PTFs.

In accordance with the results published by [START_REF] Cresswell | The transferability of Australian pedotransfer functions for predicting water retention characteristics of French soils[END_REF], the rather good results recorded with the class-PTFs that used the water content at field capacity as predictor are likely related to the fact that the field capacity corresponds to a point on the water retention curve corresponding to a water potential close to -100 hPa (Al Majou et al., 2008a).

Preliminary stratification by texture increases the prediction quality (Table 3) since the shape of the WRC and the relative location of the water content corresponding to field capacity on that curve vary according to texture. Comparison of the performance recorded with the French and Syrian datasets using the averaged criteria show that the continuous-and class-PTFs perform better when used for the French dataset (Table 3) which can be considered to be related to the origin of the soils used to develop the tested PTFs.

Evaluation of the selected continuous-and class-PTFs from the literature and not using

French soils exclusively

Results recorded with the French dataset show that there was no PTF that performed best for all four criteria (Table 4). The smallest estimation bias was recorded with the Wösten-continuous-PTFs (MEP = -0.002 cm 3 /cm 3 ), the best precision with the Vereeckencontinuous-PTFs (SDP = 0.040 cm 3 /cm 3 ), the smallest root mean square error with the Ghanbarian-continuous-PTFs (RMSE = 0.046 cm 3 /cm 3 ) and the greatest correlation coefficient with the Wösten-class-PTFs (R 2 = 0.83 cm 3 /cm 3 ) (Fig. 5a,b,d,g). However, the performance of the Ghanbarian-continuous-PTFs is among the best except for the correlation coefficient (R 2 = 0.26 cm 3 /cm 3 ) (Table 4) (Fig. 5d).

On the other hand, results recorded with the Syrian dataset show that the Wösten-class-PTFs led to the best results for precision (SDP = 0.045 cm 3 /cm 3 ), the smallest root mean squared error (RMSE = 0.047 cm 3 /cm 3 ), the greatest correlation coefficient (R 2 = 0.83) but a rather intermediate prediction bias (MEP = 0.019 cm 3 /cm 3 ) (Table 4) (Fig. 6g).

Comparison of the results recorded with the French and Syrian datasets using the averaged criteria show similar performances but much smaller than those recorded with the continuousand class-PTFs derived from the French national database (Tables 3 and4).

Evaluation of the performance of the Artificial Neural Networks

The ANN-PTFs developed with the French national database (Fig. 7a) led to a prediction quality for the French dataset used in this study (Table 5) that was close to the prediction quality recorded with the best PTFs tested (T-FC-class-PTFs, Table 3) (Fig. 7c). The RMSE which varies inversely with the overall prediction quality was slightly greater with the ANN than the one recorded with the T-FC-class-PTFs (0.030 and 0.023 cm 3 /cm 3 , respectively) (Tables 3 and5). The correlation coefficient, which measures the strength of the linear relation between the predicted and measured water content, was slightly greater with the ANN-PTFs than with the T-FC-class-PTFs (0.85 and 0.81, respectively). It should also be mentioned that the prediction quality recorded with the T-FC-class-PTFs was not so far from the one observed with the ANN-PTFs when applied to the test dataset (20% of the original data, i.e. 91 samples belonging to the French national database) since the RMSE were 0.023 and 0.029 cm 3 /cm 3 , respectively, and the R 2 , 0.81 and 0.89, respectively (Tables 3 and5).

On the other hand, the ANN-PTFs developed with the French national database led to a smaller prediction quality for the Syrian dataset than with the T-FC-class-PTFs developed by Al Majou et al. (2008a). The RMSE recorded with the ANN-PTFs and the T-FC-class-PTFs was much greater (0.049 and 0.029 cm 3 /cm 3 , respectively) (Tables 3 and5) and the correlation coefficients recorded with the ANN-PTFs slightly greater than with the T-FCclass-PTFs (0.78 and 0.75, respectively) (Fig. 7b).

The analysis of the results also showed that the PTFs selected in the literature and not using French soils performed less well than the ANN-PTFs, as already reported in the literature [START_REF] Nguyen | Comparison of statistical regression and data-mining techniques in estimating soil water retention of tropical delta soils[END_REF] (Tables 4 and5). The RMSE and R 2 were respectively greater and smaller than with the ANN-PTFs, using the PTFs selected in the literature and not using French soils. The Wösten-class-PTFs had however a much higher correlation coefficient (R 2 = 0.83 for the French and Syrian datasets) than that recorded for the other PTFs selected in the literature and fairly similar to the correlation coefficient recorded with the ANN (Tables 4 and5).

Conclusion

Our results show that PTFs developed for French soils are transferable to the Syrian soils selected for this study of semiarid Mediterranean soils. With the class-PTFs which use the soil water content at field capacity as predictor after stratification by texture, the quality of prediction is similar to that recorded with ANNs which are nowadays recognized as leading to a quality of prediction better or similar to those recorded with the PTFs published in the literature. The performance of these PTFs can be explained by the fact that a point on the water retention curve is actually used as a predictor even if the attribution of an accurate water potential value to the field capacity state is not possible. Stratification by texture prior to the development of these PTFs then increases their ability to predict the water retention properties properly, as the shape of the water retention curve is closely related to the soil texture. In other words, the performance of the class-PTFs which use the field water content as predictor after stratification by texture is likely related to using information about both the elementary particle size distribution (texture) and their assemblage (structure and related porosity) which are known to be major basic soil properties responsible for variability in water retention properties. When the soil water content at field capacity is not available, the best performance recorded remains less clear. If the coefficient of determination recorded with continuous-PTFs is large when using texture information as predictor, the bias is large when applied to Syrian soils. With a lower coefficient of determination but the other statistical criteria close to those recorded when using water content at field capacity as predictor, our results show that worthy results in terms of transferability potential were recorded with class-PTFs using bulk density as predictor after stratification by texture.

Analysis of the performance recorded with the continuous-and class-PTFs selected in the literature shows a poorer quality of prediction than PTFs developed with French soils. This probably has nothing to do with the origin of the soils but with a more appropriate consideration of soil characteristics related to soil composition and structure. The high correlation coefficients recorded with the class-PTFs developed by [START_REF] Wösten | Development and use of a database of hydraulic properties of European soils[END_REF] and Shaap et al. (2001) are therefore likely related to their use of stratification by texture, thus leading to a high consistency of the water content variation according to water potential even if the prediction bias and prediction accuracy are high and low, as shown by the MEP and SDP, respectively. Finally, our results show that continuous-or class-PTFs could be considered as transferable with satisfactory performance to other countries where database of hydraulic soil properties are not available to establish their own PTFs. The expected prediction quality of the PTFs transferred appeared to be mainly related to their ability to take into account predictors related to the characteristics of the elementary particles and their assemblage, since the best results were recorded with PTFs which combine predictors related to both soil composition and structure. Our results also showed that taking into account several predictors related to soil composition together (particle size distribution, texture, organic carbon content) and soil structure together (bulk density, porosity, field capacity) does not lead to a better performance and subsequently to the best transferability potential. It is noteworthy that the best performance were recorded by the class-PTFs that use texture as stratification criteria and then the water content at field capacity as structure information. , 10, 33, 100, 1500) volumetric water content at matric potential (kPa), log(h): logarithm of the absolute value of matric head in cm water, T: texture class, Hor: type of horizon (topsoil or subsoil).

  (T-FC-class-PTFs, see Tables 5 Al Majou et al., 2008a; T-class-PTFs, T-BD-class-PTFs, T-BD-H-class-PTFs, see Tables 2, 4 and 5 in Al Majou et al., 2008b) and providing sets of water content at seven values of water potential were selected for this study.

*:

  Computed using the absolute value of every MEP. VG-continuous-PTFs are the continuous pedotransfer functions developed for the parameters of the van Genuchten model using multiple regression equations, PSD-continuous-PTFs are the continuous pedotransfer functions developed by multiple regression equations using particle-size distribution (PSD), organic carbon and bulk density, FC-continuous-PTFs are the continuous pedotransfer functions developed by using the water content at field capacity, T-FC-class-PTFs are the class pedotransfer functions developed by using the water content at field capacity as predictor after stratification by texture,, T-class-PTFs are the pedotransfer functions developed after stratification by texture, T-BD-class-PTFs are the pedotransfer functions developed after stratification by texture and bulk density, T-BD-H-class-PTFs are the pedotransfer functions developed for texture and bulk density classes according to the type of horizon, T-H-VG-class-PTFs are the pedotransfer functions developed after stratification by texture and type of horizon (topsoil and subsoil) for the parameters of the van Genuchten model (1980).

Figure 1 :

 1 Figure 1: The typical topologies of the continuous and class-PTFs used in this study. (a): continuous-PTFs providing the parameters of the van Genuchten model (1980), (b): continuous-PTFs providing the water content at several matric potentials, (c): class-PTFs providing the parameters of the van Genuchten model (1980), (d): class-PTFs providing the water content at several matric potentials, and (e): artificial neural networks (ANN)-PTFs.  1 ,  1.5 ...,  4.2 : volumetric water content in cm 3 cm -3 at seven different matric potentials,  r ,  s ,  and n are the parameters of the van Genuchten equation, T: texture, BD: bulk density, OC: organic carbon, H: type of horizon (topsoil or subsoil).

Figure 2 :

 2 Figure 2: FAO triangle (FAO, 1990) of texture used (a), distribution of soil texture from samples used in this study to test the validity of the continuous-and class-PTFs selected, Syrian soil samples (b) and French soil samples (c).

Figure 7 :

 7 Figure 7: Measured and predicted water content (cm 3 /cm 3 ) recorded with the artificial neural networks using soil properties (clay, silt, sand, organic carbon and bulk density) for the French national database (Al Majou et al., 2008b) (a), Syrian dataset (b) and the French dataset (c).

Table 1 537

 1 List of continuous-and class-PTFs according to the output variables and relationship between input and output variables. 538 : multiple linear regression, LR: linear regression, ANN: artificial neural networks, SVM: support vector machine, k-NN: k nearest neighbor technique, PSD: particle size distribution, BD: bulk density, OC: organic carbon, OM: organic matter, CEC: cation exchange capacity, Me: moisture equivalent, dg: geometric mean particle size diameter, g: geometric standard deviation, PL: plastic limit, FC: volumetric water content at field capacity, (1

	Output variables	Relationship between input and	Input variables	Geographical domain	Reference
		output variables (continuous-PTFs)		or country
		and stratification type (class-PTFs)
			Continuous-PTF
	Parameters of a water	MLR	PSD, BD, OC
	retention curve model		PSD, BD, OM
			PSD, BD, OC, pH, CEC
			PSD, BD, Me
			PSD
			PSD, BD, OC
			PSD, OC, pH
			PSD, BD, d g ,  g
			PSD, BD, OC
			PSD, BD, OM

MLR

Table 2

 2 Descriptive statistics of main characteristics of the soil dataset used in this study.

		Coarse elements (%)	Particle size distribution (%)	OC g/kg	CaCO 3 g/kg	CEC Cmol c /kg	D b g/cm 3	Volumetric water content (cm 3 /cm 3 ) at matric potential h ( h ) in hPa
		>2000	<2	2-50	50-2000					 10	 33	 100	 330	 1000	 3300	 15000
		µm	µm	µm	µm
					French National database used to train and test the PTFs (n = 456)
	mean	<1	29.3	43.8	26.9	6.0	54.2	14.8	1.52 0.354	0.335	0.315	0.289	0.259	0.221	0.187
	s.d.		15.4	21.8	25.6	5.1	171.3	9.0	0.15 0.068	0.070	0.075	0.076	0.079	0.076	0.073
	min.		1.9	1.6	0.1	0.0	0.0	0.6	0.95 0.134	0.100	0.080	0.056	0.045	0.033	0.013
	max.		92.9	82.1	95.4	28.8	982	52.8	1.98 0.605	0.596	0.586	0.557	0.510	0.462	0.370
						French soil samples used to test the PTFs (n = 30)
	mean	<1	42.1	32.0	25.9	8.58	34.30	22.03	1.45 0.387	0.365	0.347	0.326	0.307	0.274	0.244
	s.d.		14.4	13.7	15.1	6.8	98.5	10.1	0.16 0.049	0.050	0.051	0.054	0.055	0.050	0.058
	min.		20.1	10.5	3.1	0.0	0.0	5.30	1.10 0.310	0.295	0.275	0.242	0.203	0.167	0.142
	max.		68.9	53.5	59.8	28.8	424	45.90	1.77 0.496	0.489	0.469	0.446	0.415	0.366	0.363
						Syrian soil samples used to test the PTFs (n = 30)
	mean	<1	41.4	32.8	27.8	1.18	63.7	36.5	1.22 0.436	0.417	0.388	0.344	0.307	0.277	0.239
	s.d.		16.0	9.8	11.7	0.96	77.9	7.3	0.10 0.061	0.058	0.055	0.050	0.051	0.062	0.067
	min.		12.2	11.6	8,0	0.36	7.2	23.7	1.04 0.352	0.341	0.324	0.285	0.242	0.198	0.149
	max.		69.1	53.3	53.0	3.9	310	49.2	1.41 0.577	0.559	0.507	0.474	0.444	0.435	0.404

s.d., min, max are the standard deviation, minimum and maximum of soil variables.

Table 3

 3 Validation criteria of the continuous-PTFs and class-PTFs developed with the French national database when applied to the French and Syrian datasets.

	PTFs type	Descriptive statistics of the relationship between measured and predicted water content
		MEP	SDP	RMSE	R²
		(cm 3 /cm 3 )	(cm 3 /cm 3 )	(cm 3 /cm 3 )
			French dataset (n=30)
	VG-continuous-PTFs	-3*10 -4	0.036	0.036	0.77
	PSD-continuous-PTFs	-0.003	0.033	0.033	0.81
	FC-continuous-PTFs	-0.012	0.028	0.031	0.66
	T-FC-class-PTFs	-0.001	0.023	0.023	0.81
	T-class-PTFs	-0.002	0.028	0.028	0.71
	T-BD-class-PTFs	0.003	0.036	0.036	0.52
	T-BD-H-class-PTFs	-0.002	0.029	0.029	0.69
	T-H-VG-class-PTFs	-0.003	0.030	0.030	0.68
	Average	0.003*	0.030	0.031	0.71
			Syrian dataset (n=30)
	VG-continuous-PTFs	0.022	0.046	0.048	0.74
	PSD-continuous-PTFs	-0.021	0.033	0.035	0.85
	FC-continuous-PTFs	-0.005	0.033	0.034	0.66
	T-FC-class-PTFs	-0.001	0.029	0.029	0.75
	T-class-PTFs	-0.025	0.039	0.047	0.36
	T-BD-class-PTFs	0.001	0.041	0.041	0.51
	T-BD-H-class-PTFs	0.002	0.039	0.039	0.56
	T-H-VG-class-PTFs	-0.024	0.037	0.044	0.43
	Average	0.013*	0.037	0.040	0.61

Table 4

 4 Criteria enabling assessment of the prediction quality for the continuous-and class-PTFs selected in the literature and not using the French national database.

	PTFs type	Descriptive statistics of the relationship between measured and predicted water content
		MEP	SDP	RMSE	R²
		(cm 3 /cm 3 )	(cm 3 /cm 3 )	(cm 3 /cm 3 )
			French dataset
			(n=30)
	Wösten-continuous-PTFs	-0.002	0.051	0.051	0.30
	Vereecken-continuous-PTFs	0.038	0.040	0.055	0.21
	Reichert-continuous-PTFs	0.031	0.042	0.052	0.12
	Ghanbarian-continuous-PTFs	0.005	0.045	0.046	0.26
	Minasny-continuous-PTFs	0.016	0.051	0.054	0.20
	Schaap-class-PTFs	-0.019	0.072	0.075	0.60
	Wösten-class-PTFs	0.040	0.050	0.064	0.83
	Average	0.022*	0.050	0.057	0.36
			Syrian dataset
			(n=30)
	Wösten-continuous-PTFs	0.005	0.059	0.059	0.43
	Vereecken-continuous-PTFs	0.022	0.052	0.052	0.20
	Reichert-continuous-PTFs	-0.020	0.044	0.048	0.42
	Ghanbarian-continuous-PTFs	-0.006	0.051	0.051	0.21
	Minasny-continuous-PTFs	0.003	0.050	0.050	0.30
	Schaap-class-PTFs	-0.048	0.056	0.074	0.76
	Wösten-class-PTFs	0.019	0.043	0.047	0.83
	Average	0.018*	0.051	0.054	0.45

*: Computed using the absolute value of every MEP

Table 5

 5 Descriptive statistics of the relationship between measured and predicted water content developed by artificial neural networks (ANN).Soil properties used in predictive procedures are the sand, silt, clay and organic carbon contents, and the bulk density.

		Training data		Test data
		RMSE	R²	RMSE	R²
		(cm 3 /cm 3 )		(cm 3 /cm 3 )
	French National database	0.030	0.90	0.029	0.89
	Syrian dataset	-	-	0.049	0.78
	French dataset	-	-	0.030	0.85

Børgesen & Schaap (2005) Lamorski et al. (2008) Haghverdi et al. (2012) Patil et al. (2012) Khlosi et al. (2016) Nguyen et al. (2017) SVM PSD, BD Poland Lamorski et al. (2008)

Acknowledgement

The authors would like to thank the Embassy of France in Damascus, and the Francophone University Agency (AUF) for supporting scientific collaboration between France and Syria.

The first author also received funding support from the Labex Voltaire (ANR-10-LABEX-100-01) and the French program PAUSE. Belgium Europe Tropics Brazil India France India Iran India Brazil Vereecken et al. (1989) Wösten et al. (1999) Hodnett & Tomasella (2002) Tomasella et al. (2003) Adhikary et al. (2008) Al Majou et al. (2008a) Santra & Das (2008) Ghorbani et al. (2010) Patil et al. (2012) Medrado & Lima, (2014) ANN PSD, BD,  33 ,  1500 PSD, BD, OM,  1 ,  10 ,  100 ,  1500 PSD, BD, OC, log(h) North America, Europe Denmark Iran, Australia Schaap et al. (2001) Børgesen & Schaap (2005) Haghverdi et al. (2012) SVM PSD, BD, log(h) USA Twarakavi et al. (2009) Set of  at different water potentials MLR and LR PSD, BD, Me  FC PSD, BD, OM PSD, BD, d g ,  g , (h) PSD, BD, d g ,  g PSD, BD, OC PSD, BD, OM, PL PSD, BD, OC, log(h) Brazil France Brazil USA, France Iran Tropics, USA Syria Vietnam Tomasella et al. (2003) Al Majou et al. (2008a) Reichert et al. (2009) Ghanbarian & Millán (2010) Ghorbani et al. (2010) Minasny & Hartemink (2011) Khlosi et al. (2016) Nguyen et al. (2017) ANN PSD, BD, OM,  1 ,  10 ,  100 ,  1500 PSD, BD PSD, BD, OC, log(h) PSD, BD, OC PSD, BD, OM, PL PSD, BD, OC, log(h) Denmark Poland Iran, Australia India Syria Vietnamese

Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 )

Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 )

Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 )

Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm 3 ) Measured water content (cm 3 /cm


Jules Masson Mourey

Jules Masson

The anthropomorphic engravings of Azrou Iklane

teaching and research institutions in France or abroad, or from public or private research centers.

(Aix Marseille Univ, CNRS, Minist Culture, LAMPEA, Aix-en-Provence, France) julesmassonmourey@yahoo.fr

The anthropomorphic engravings of Azrou Iklane (Guelmim Region, Morocco)

Jules Masson Mourey

The site of Azrou Iklane (the « Tattooed Stone ») is perhaps the oldest rock art station reported in Morocco, at the end of the 19 th century [START_REF] Duveyrier | 1875 -De Mogador au Djebel Tabayoudt, par le Rabbin Mardochée Abi Serour, résumé du journal de voyage[END_REF]. The approximately 10,000 pecked engravings is attributed to the Bovidian style (ca. 3000-1000 BC), succeeded by the Libyco-Berber style, also known as the « stage of the horsemen » (ca. 500 BC-1500 AD), before the tradition was perpetuated into modern and contemporary times [START_REF] Graff | -La longue durée de l'Art rupestre au Sahara[END_REF]. Although each of the chronological phases has its own particular iconographic themes, the anthropomorphic image seems to be constantly present, in varying forms and numbers.

Contexts of the site and engravings

Azrou Iklane is located in south-west Morocco, about 200 km south of Agadir, on the southern edge of the Anti-Atlas. The site is a summer camp for various groups of the Aït Oussa Arabic tribe, which makes it an area for important social expressions. It is characterized by the presence, on the dry river bed of the oued, of a slab of brown quartzitic sandstone 140 m long by 20 m wide were the engravings were made.

At Azrou Iklane, most human representations (about 650) are highly codified and appear in the form of small horsemen and infantrymen as well as isolated body parts (mainly podomorphs). Others, 83 in number, are still in full-length and have more heterogeneous dimensions and iconography.

Situation map of Azrou Iklane (© Microsoft Corporation ; J. Masson Mourey)

Bronze Age anthropomorphs?

After having been the object, between 2013 and 2017, of four multidisciplinary franco-moroccan missions within the framework of the project « Paysages gravés », the engraved slab of Azrou Iklane will be published in full in the form of a monograph. The complete analysis on the long time of anthropomorphic engravings, hitherto briefly treated in short notes (Rodrigue 2007), will occupy an important place.

Several anthropomorphs could correspond to the very first phases of engraving of the slab. For example, we see the representation of a character with raised arms, brandishing an « hache-pelte » with a very curved edge and angled fitting. Similar images of weapons are represented in the High Atlas and their context appears generally of the Bronze Age (Chenorkian 1988). The isomorphism between the edge of the object and the head of the character is here an interesting detail.

Photography (by night) and recording of an anthropomorph of the Bovidian phase (© G. Graff ; J. Masson Mourey ; Mission « Paysages Gravés »)

Recurrences and differences within the corpus

For more recent phases, some anthropomorphs are strongly sexed (often male), others not at all. Some have two points represented on or below the shoulders. Many anthropomorphs for which there are few comparisons off the site display an enigmatic horizontal line crossing the body from side to side. Are they daggers or swords ? Sometimes, the hands of the characters are figured with outsized fingers and spread out.

The most schematic anthropomorphs correspond to the letter Z of the current tifinagh alphabet which means « men » (« imazighen »). This motif, composed of a vertical bar for the trunk and two opposite arcs for the upper and lower limbs, is since the 1960s a symbol of Berber resistance.

(

  © JMM ; Mission « Paysages Gravés ») (© JMM ; Mission « Paysages Gravés ») (© JMM ; Mission « Paysages Gravés »)


Anicet Kouomou-Choupo

Laure Berti-Équille

Annie Morin

Features Mining for Multimedia Indexing And Retrieval (Poster)

published or not. The documents may come

Features Mining for Multimedia Indexing And Retrieval

Anicet Kouomou-Choupo, Laure Berti-Équille, Annie Morin IRISA, University of Rennes I, FRANCE Problem F How to solve the following problem : Retrieve all the images similar to a given image using appropriate global visual features ? F What about query response time and quality of results ?

Our aim: Describe a new method applied to large still image databases that combines clustering and association rules mining in order to:

F better organize the image collections F improve the performance of query processing Our method:

Retrieval processing Indexing processing

Data descriptor

Files of fixed images

File of association rules Files of descriptors

Clustering process

Files of Clusters

Rules Generator

FF

  The method can be applied to every kind of very large image databases F Interesting perspectives for optimizing query plans 7727 of still images F 5 global MPEG-7 features of color, texture, and form F K-means algorithm for clusters computation and Apriori algorithm for association rules computation F minsup= 10% and minconf = 50% F C++ programming under Linux OS Reference: A. Kouomou et al. Multimedia Indexing and Retrieval with Features Association Rules Mining, IEEE ICME 2004. To appear .