Eric Benoit

Expression of uncertainty in fuzzy scales based measurements

Keywords: uncertainty, fuzzy scale, measurement science

Expression of uncertainty in fuzzy scales based measurements

Introduction

The concept of fuzzy scale was introduced in order to add properties to nominal scales and ordinal scales [START_REF] Benoit | Towards fuzzy nominal scales[END_REF]. By the way, it establishes a link between strongly defined measurements used for example in physical sciences and weakly defined measurements used in behavioural sciences as for example in psychophysics [START_REF] Finkelstein | Widely, strongly and weakly defined measurement[END_REF]. Actually, except in Physics, the Measurement Science is more and more concerned by weakly defined measurements. As examples, Psychophysics or Economy and more globally the measurement of complex systems are concerned.

Since their first introduction, fuzzy scales were exclusively used to perform a precise representation of a unique property manifestation. In a measurement context, it is now admitted that the representation of a property manifestation needs to expose the imprecision and the uncertainty related to the definition of the property and to the measurement process. Within the current metrological approach, a measurement result is given by a probability density function (PDF) that expresses a knowledge about the measurand. More recent approaches extend such knowledge representation with a possibility function (PF). Considering that the source of this knowledge is a set of indications, the computation of the PDF or of the PF is performed on the representation space including these indications. In usual measurements, this space is a numerical one. Within the context of fuzzy scales, a second stage in the measurement process performs a translation from a numerical representation of indications to a fuzzy lexical representation. The computation of PDF or PF may not depend on the representation space. By the way, it may be possible to define a function on a space and to translate it into another space. The issue in which this paper is concerned is that such functions are actually approximated with parametric functions. The accuracy of the approximation depends on the capability for a family of parametric functions to fit an original PDF or PF.

Our hypothesis is to consider that, even if quantities are part of the concrete world, they are defined within a model, itself respecting a theory. Within the context of weakly defined measurement, we consider that the definitional uncertainty of a quantity has to be expressed on the representational space. Then, when a measurement is performed with a fuzzy scale, the definitional uncertainty has to be defined on the associated representation space, i.e. on a space of lexical fuzzy subsets.

Fuzzy scales

Fuzzy scale are characterized by their capability to represent a similarity relation between manifestations with a fuzzy equivalence relation between quantity values. The quantity values are expressed as Lexical Fuzzy Subset (LFS), i.e. as fuzzy subsets of lexical terms. See for example in Fig. 1 the description of a manifestation with a Lexical Fuzzy Subset. They are recommended for weakly defined measurements on quantities that support similarity relations. This new family presently includes the weakest fuzzy scales: the fuzzy nominal scales, and metrical fuzzy scales [START_REF] Benoit | An ordinal metrical scale built on a fuzzy nominal scale[END_REF]. The last ones are fuzzy nominal scales enhanced with the definition of a distance. The fuzzy ordinal scales are still under studies. Expressing quantity values with LFSs can be performed by measurements with physical sensors, or with psychophysics experiments. For the first case, quantity values are first expressed on a conventional numerical space with a ratio scale or at least with an interval scale. At this step, empirical relations are mapped into numerical relations. Then a second translation maps the numerical expression of quantity values into a fuzzy lexical expression. The second mapping, based on a fuzzy scale, transfers only the fuzzy equivalence relation from the empirical relational set to the representational relational set in the case of a fuzzy nominal scale, and transfer a distance operator in case of a metrical scale. In previous studies, we proposed a familly of fuzzy equivalence relations denoted (Σ-T) that holds the reflexivity property and a weak transitivity property [START_REF] Foulloy | Building a class of fuzzy equivalence relations[END_REF]. The general form is given by Eq. (2) where T is a t-norm.

(2)

The reflexivity condition induces a constraint on the set U

D: E FS S T ( ) → e E ∈ ∀ D e ( ) s S T ∈ eRs { } = x y , ( ) U 2 ∈ ∀ ( ) x y , ( ) , Σ T ( ) x y , ( ) T x s ( ) y s ( ) , (
)

s S T ∈ ∑ = = x U ∈ ∀ T x s ( ) x s ( ) , (
)

s S T ∈ ∑ , 1 = x y , ( ) U 2 ∈ ∀ ( ) x y , ( ) , Σ T m ( ) x y , ( ) min x s ( ) y s ( ) , (
)

s S T ∈ ∑ = = x y , ( ) U 2 ∈ ∀ ( ) x y , ( ) , Σ T p ( ) x y , ( ) x s ( )y s ( ) s S T ∈ ∑ = =
The constraint on U given by the reflexivity property of ~ become respectively: and From a mathematical point of view, the measurement values, expressed on FS(S T ), are elements of a subset of a |S T |dimensional unit cube where |S T | is the cardinality of S T .

Modelling uncertainty 3.1 Model defined on a numerical space

In a first trivial approach, the set of the numerical expression of the observations, i.e. the set of quantity values expressed in a numerical space, is used to produce a measurement result defined on this numerical set. The usual way is to define a probability density function (PDF) that models the set of quantity values (see GUM). An approximation of the probability density function by a parametric function allows analytical computation on PDFs.

Another way also compatible with GUM is to model the set of quantity values with a possibility function (PF) that supports also the approximation with parametric functions [START_REF] Dubois | Probability-Possibility Transformations Triangular Fuzzy Sets and Probabilistic Inequalities[END_REF]. The second stage is to perform a projection of this function on the set of LFSs. The result is a PDF (or a possibility function) on LFSs.

The advantage of this approach is the possibility to use the existing mathematical tools to create the PDFs or the possibility functions on a numerical space. A disadvantage is that the fuzzy scales need a lot of parameters to be defined [START_REF] Benoit | An ordinal metrical scale built on a fuzzy nominal scale[END_REF]. Then a function defined on a numerical space with few parameters is mapped on the fuzzy lexical space into a function with a lots of parameters. In this case, the parametrization which role is to simplify the expression of a function, becomes useless. Another disadvantage is the impossibility to use this method for psychophysical measurements due to the fact that the numerical stage is avoided.

Models defined on LFSs

In a second approach, each observation is first expressed as a Lexical Fuzzy Subset of terms. Then a parametric function is created on LFSs to represent the set of quantity values. As mentioned in previous studies, the set of relations and operators that can be used to define the parametric functions is limited by the nature of the scale. For fuzzy nominal scales, the available relation is a fuzzy equivalence relation ~. The metrical fuzzy scale defines a distance between LFSs. Let first define a general family of parametric functions on LFSs when the measurement

x U ∈ ∀ T m x s ( ) x s ( ) , (
)

s S T ∈ ∑ , x s ( ) s S T ∈ ∑ 1 = = x U ∈ ∀ T p x s ( ) x s ( ) , (
)

s S T ∈ ∑ , x s ( ) 2 s S T ∈ ∑ 1 = =
is performed with a fuzzy nominal scale. given by [START_REF] Smets | Data fusion in the transferable belief model[END_REF] For example, the simplest PDF family is given by f E a ,m where E a is the step function defined by Eq. ( 8). The PDF f E a ,m defines a uniform distribution on lexical fuzzy subsets.

m U ∈ f h m , x ( ) h 1 ( ) x m , ( )
(
)

h 1 ( ) y m , ( )
(
) -----------------------------------------------------------------------------------= We can simply verify that (1 -(~)(x,y)) is a distance with a saturation effect. The preceding definitions can be extended to the case of metrical fuzzy scales. A metrical fuzzy scale is defined by its symbolisation < E, S T , R, {~,d}, {=,d S }, {(~,=),(d, d S )}> such that < E, S T , R, {~}, {=}, {(~,=)}> is a fuzzy nominal scale, d S is a distance between terms s of S T , and d is a distance on manifestations homomorphic to a distance on U. At this step, we can conclude that with a distance, or at least a fuzzy equivalence between Lexical Fuzzy Subsets, it is possible to define parametric PDFs or PFs centred on a Lexical Fuzzy Subset.

y U ∈ ∑ --------------------------------------------------- = σ 1 ( ) x m , ( )
( ) 2 f h m , x ( ) x U ∈ ∑ = E a b ( ) 1 if b a ≤ 0 if b a > ⎩ ⎨ ⎧ = m U ∈ g h m , x ( ) h 1 ( ) x m , ( )
( ) max h 1 ( ) y m , ( ) ( ) y U ∈ ( ) { } ( )
The available methods to define the parameters are similar to the well known methods used on numerical spaces.

Concerning the translations between PDFs and possibility functions, a general approach based on the Dempter-Shafer evidence theory [START_REF] Yamada | Probability-possibility transformation based on evidence theory[END_REF] is directly applicable.

Construct method for PDF and PF

Evidence theory

The Dempster-Shafer evidence theory, also named the transferable belief model (TBM) is a general approach for

m U ∈ f h m , x ( ) h d x m , ( ) (
) ------------------------------------ ----------------------------------------------------------------------= uncertainty management based on the assignment of probabilities to sets. It generalises both the probability theory and the possibility theory [START_REF] Smets | Data fusion in the transferable belief model[END_REF]. The main element of the TBM is the frame of discernment denoted Ω and presented as a set of elementary events, or as a discourse set for the expression of the belief. The knowledge is then expressed with a mapping denoted m and called belief function from the set of subsets of Ω, denoted 2 Ω , to the set [0,1]. This mapping represents the distribution of a weight of belief over the elements of 2 Ω . It verifies:

h d y m , ( ) ( ) y U ∈ ∑ --
= m U ∈ g h m , x ( ) h d x m , ( ) ( ) max h d y m , ( ) ( ) y U ∈ ( ) { } ( )
(12)
The set {A| m(A) > 0} is called the set of focal elements of m, and the couple ({A| m(A) > 0}, m ) is called a body of evidence of a variable with values on Ω. Fig. 2 gives the graphical semantic chosen in this paper to represent the body of evidences. For a given body of evidence, the associated set of focal elements is simply represented by a set, and the associated belief function is represented by a surface which projection is the set of focal elements.

Two measures Bel and Pl are defined on the subsets of Ω. In [START_REF] Yamada | Probability-possibility transformation based on evidence theory[END_REF] Yamada proposed a probability-possibility transformation based on the evidence theory. The principle is to identify a set of disjoint subsets of Ω computed from a PDF, then to compute a new body of evidence made of

m ∅ ( ) 0 = m A ( ) A 2 Ω ∈ ∑ 1 = Bel A ( ) m B ( ) B A ⊆ ∑ = Pl A ( ) m B ( ) B A ∩ ∅ ≠ ∑ = focal elements {a} B C D m(D) m(B) m(C) m({a}) belief function
consonant subsets. We propose in this paper to build a parametric set of disjoint subsets in order to model a PDF within the framework of evidence theory. Then the Yamadas approach is applied to build the corresponding possibility distribution.

Within the context of metrology, the frame of discernment is the set all individual measurement values, and a measurement result is interpreted as a body of evidence of a quantity. It expresses the known representational information about a quantity.

Construct method

Let S ⊂ U a finite set of measurement values, being the frame of discernment. Let s 0 ∈ S being a measurement value that will play the role of mean value.

Let F = {S i | 1 ≤ i ≤ n} being an ordered family of n non-empty subsets of S defined by

S i = {s | d i ≤ d(s,s 0 ) < d i+1 }.
Where d is a distance on U as defined in Chap. 3.2. and d 1 = 0.

An analogous definition on an Euclidean plane will define a family of rings, all centred on s 0 . Fig. 3 and Fig. 4 give some examples of an ordered family of rings respectively in the case of a non-Euclidean metric space, and in the case of a mono-dimensional space. We can remark that (F P ,m p ) is a body of evidence defining a probability distribution (see Fig. 5). By definition these sets are nested. It is then possible to define a belief function m π which plausibility measure Pl is a possibility measure Π (see Fig. 6).

(18) Fig. 6. Belief function m π defined on the set

F π h d i ( ) S i ⋅ i 1 = n ∑ 1 = s 0 d 6 m p ({s}) F π F π k 1 k n ≤ ≤ { } = F π k S h h 1 = k ∪ = m π F π k ( ) F π k h d k ( ) h d k 1 + ( ) ( ) = S h h 1 = k ∑ ⎝ ⎠ ⎜ ⎟ ⎜ ⎟ ⎛ ⎞ h d k ( ) h d k 1 + ( ) ( ) = s 0 d 6 S F π 4 m π F π 4 ( ) h(d 4 ) -h(d 5 )
The possibility distribution is directly deduced from the belief function m π.

(19) (20)

Definition of the parameters

With the chosen approach, all information about the probabilistic or the possibilistic representation of uncertainty on the space of LFSs is included into the ordered family S i = {s | d i ≤ d(s,s 0 ) < d i+1 } and on the mapping h. The procedure to define this information is to acquire a set of statistical data to translate this data into LFSs to extract the mean s 0 from the set of data to build a set family F to create an histogram from data mapped into F to compute the mapping h

The value s 0 has the same semantic than the mean in usual PDFs defined on interval scales or ratio scales. Due to the scales used in our case, the addition operator is not allowed for computations in the set of LFSs. So the mean cannot be computed with the usual formula. The value s 0 is then obtain by minimization of the cumulative distance to the data.

The set family F is also characterized by a distance d on LFSs and a set {d i | 1 ≤ i ≤ n+1}. The distance d is induced by the scale and cannot be considered as a parameter except for a calibration process.

The set {d i | 1 ≤ i ≤ n+1} is related to discrete aspect of the method. The influence of this information the PDF or the PF is small and, in a first approximation, it can be reduced to the parameters Δd and n. The set become {d i = (i -1)Δd | 1 ≤ i ≤ n+1}.

Finally, h is the main source of parametrization. It can be defined as a continuous parametric function, or as a discrete set of n values. The usual statistical methods are then applied to estimate h parameters.

Conclusion

This paper shows that, with fuzzy nominal scale and metrical fuzzy scales, the definitional uncertainty can be managed in the case of weakly defined measurements. It provides a method to represent uncertainty on fuzzy lexical sets with the two main frameworks that are the probability theory and the possibility theory. The Dempster-Shafer theory used as a mainframe for these two theories, allows to compute simultaneously a possibility distribution and an associated a probability distribution.

The capability to manage uncertainty directly on fuzzy lexical sets is an important advance for behavioural sciences that cannot express their measurement result on numerical spaces.

Fig. 1 .

 1 Fig. 1. Example of lexical fuzzy subset (LFS).

Fuzzy

  scales are defined by fuzzy symbolisation < E, S T , R, {~}, {=}, {(~,=)}> where: E is the set of the manifestations of a quantity, S T is a set of terms used to qualify measurements, R is a fuzzy relation on ExS T . Let denote FS(S T ) the set of fuzzy subsets of S T , and D the injective mapping defined as (1) The relation ~ is a fuzzy equivalence relation on E that characterizes the scale. In this paper, (~)(x,y) will denote the membership function of the relation x ~ y. This relation is homomorphic to a fuzzy equivalence relation ~ on a subset U ⊂ FS(S T ) such that D(E) ⊂ U and ~ is reflexive on U: (~)(x,x) = 1.

( 3 )

 3 The special cases with T m (a,b) = min(a,b), see Eq. (4), and T p (a,b) = a.b, see Eq. (5), were successfully applied to build fuzzy nominal scales.

Definition 1 :

 1 The function f h,m defined bellow is a probability density function on U with mean m.Let < E, S T , R, {~}, {=}, {(~,=)}> be a fuzzy nominal scale. let be a lexical fuzzy subset called mean, let h be a monotonic decreasing function with domain [0,1] and codomain [0, +∞[. f h,m is a parametric function with domain U, codomain [0, +∞[ and defined by (6) A PDF f h,m is defined with respect of the constraints of fuzzy nominal scales. The only relation involved into the definition is the fuzzy equivalence relation. The concept of standard deviation is based on an aggregation of the differences with the mean. Into the set of LFSs, such difference is given by the equivalence relation: (1 -(~)(x,m)). Then an expression of the standard deviation is

Definition 2 :

 2 The function g h,m defined bellow is a possibility distribution on U which kernel includes m.Let < E, S T , R, {~}, {=}, {(~,=)}>be a fuzzy nominal scale, let be a lexical fuzzy subset called mean, let h be a monotonous decreasing function with domain [0,1] and codomain [0, +∞[. g h,m is a parametric function with domain U, codomain [0, +∞[ and defined by (9)

Definition 3 :Definition 4 :

 34 The function f h,m defined bellow is a probability density function on U with mean m. Let < E, S T , R, {~,d}, {=,d S }, {(~,=),(d, d S )}> be a metrical fuzzy scale Let be a parameter. Let h be a monotonous decreasing function with domain [0,+∞] and codomain [0, +∞[. Let d be the distance on U that characterizes the metrical fuzzy scale. f h,m is a parametric function with domain U, codomain [0, +∞[ and defined by (10) The function g h,m defined bellow is a possibility distribution on U which kernel includes m. Let < E, S T , R, {~,d}, {=,d S }, {(~,=),(d, d S )}> be a fuzzy nominal scale, let be a lexical fuzzy subset called mean, let h be a monotonous decreasing function with domain [0,1] and codomain [0, +∞[. g h,m is a parametric function with domain U, codomain [0, +∞[ and defined by (11)

  elements are singletons, then the belief function is qualified as Bayesian. In this case, the two measures Bel and Pl are equal to a probabilistic measure P. The belief function m is then equivalent to an assignment of probabilities to individual values, i.e. equivalent to a PDF.When the focal elements are sets ordered by the inclusion operator, called nested sets or consonant sets, Bel and Pl are respectively the necessity measure N and the possibility measure Π of a possibility distribution derived from m.

Fig. 2 .

 2 Fig. 2. Semantic used in this paper to represent belief functions. As a mass function, the belief function is represented by a surface

Fig. 3 .Fig. 4 . 1 F

 341 Fig. 3. Example of an ordered set family centred on s 0 and built with a non-Euclidean distance.

Fig. 5 .

 5 Fig. 5. Belief function m p defined on the singleton set F P Let now follow the Yamadas approach. A new family of sets is built from the family F.


Fabio Akashi Hernandes

Philippe Auger

Maria Marta Lima

Alves Moro

Eric Carvalho Waquim

Luanna Soares De Melo Evangelista

Feather mites (Acari: Astigmata) of the Zoobotanical State Park in Teresina, Brazil

Keywords: birds, ectoparasitism, feather mites, record, captivity, Psoroptidia

 on the harpy eagle (Harpia harpyja) and Hieracolichus sp. on the southern caracara (Caracara plancus). These findings increase the known ranges of the mite species, with four of them reported for the first time in Brazil.

Introduction

Feather mites (Arthropoda: Chelicerata: Arachnida: Acariformes: Astigmata) are the most diverse and abundant permanent avian ectosymbionts, associated with nearly all bird orders [START_REF] Gaud | Feather mites of the World (Acarina, Astigmata): the supraspecific taxa[END_REF]. Most feather mites exhibit morphological adaptations to inhabit specific microhabitats on their hosts' body -flight feathers of the wings, tail, down, and contour feathers, quills and skin; some members of the Pyroglyphidae (Analgoidea) also inhabit nests [START_REF] Dabert | Origin and evolution of feather mites (Astigmata)[END_REF].

While a number of feather mite lineages quite certainly evolved by coevolution with their hosts, others have successfully colonized new hosts by horizontal transfer [START_REF] Doña | Host specificity, infrequent major host switching and the diversification of highly host-specific symbionts: The case of vane-dwelling feather mites[END_REF]. Therefore, the understanding of the origin of mite-bird associations is intrinsically tied to the knowledge about bird phylogeny, ecology and behavior (Proctor and Owen, 2000). Also, mites can affect bird reproductive success and exposure them to diseases, especially in birds held in captivity [START_REF] Atyeo | Feather mites (Acarina) of the parakeet, Melopsittacus undulatus (Shaw) (Aves: Psittacidae)[END_REF][START_REF] Mironov | Allopsoroptoides galli n. g., n. sp., a new genus and species of feather mites (Acari: Analgoidea: Psoroptoididae) causing mange in commercially raised domestic chicken in Brazil[END_REF]pers. obs. by FAH).

Given the exuberant diversity of birds found in the Neotropical region, it is not surprising that the study of feather mites on birds of Brazil is still incomplete, which renders a huge potential for taxonomic discoveries in this country. Researches involving these organisms, as well as training of students, ornithologists and parasitologists, have been improving the understanding of the ecological and biological associations of these mites with their hosts in Brazil [START_REF] Roda | Ácaros plumícolas em beija-flores no município de Vicência, Pernambuco, Brasil[END_REF][START_REF] Pedroso | New records of feather mites (Acariformes: Astigmata) from nonpasserine birds (Aves) in Brazil[END_REF].

The aim of this work was to investigate the feather mites associated with some wild birds of the Zoobotanical State Park of Teresina, Piauí State, Brazil.

Materials and Methods

This work was evaluated and approved by the Ethics Committee in Animal Experimentation of the Federal University of Piauí (UFPI), (protocol 242/2016). The material was collected at the Zoobotanical State Park, in an urban area of the municipality of Teresina, Piauí. The park was created in 1973 and has approximately 137 hectares, being an area designated for the protection, conservation, and researches of fauna and flora, governed by the public administration of the State Secretariat for the Environment and Water Resources of Piauí.

Feather mites were collected from wild birds held in captivity, from January to July of 2017. Mites were collected manually from the feathers using fine brushes and tweezers; a few feathers were partially cut and only one post-mortem bird -an osprey, Pandion haliaetus (Linnaeus, 1758) -was defrosted and washed for ectosymbionts. This was done by washing the bird with water and detergent [START_REF] Mironov | New feather mite taxa (Acari: Analgoidea) and mites collected from native and introduced birds of New Zealand[END_REF], and the resulting liquid was filtered in paper filters; the mites were collected with a fine brush from under a dissecting scope. The number of examined feathers for each bird species is shown in Table 1. After collecting, the mites were counted and stored in vials with 70% ethanol, labeled with locality, date and host. The sorting was done in the Laboratory of Parasitology of the Department of Parasitology and Microbiology of UFPI, and the material was sent to FAH for mounting and identification. Mites were mounted in slides with Hoyer's medium, and subsequently analyzed and photographed on a Leica DM3000 microscope equipped with a DFC295 digital camera. Debris surrounding the specimens were digitally removed using Adobe Photoshop CS6. Voucher specimens were deposited at the Acari collection of the Universidade Federal de Santa Catarina. The classification of birds follows BirdLife International (2018).

Results

All examined bird species were shown to harbor feather mites. We found feather mites on 7 of the 11 bird specimens that we examined. The following mite species were recovered from their respective hosts: Pandionacarus fuscus (Nitzsch, 1818) (Avenzoariidae) from the osprey Pandion haliaetus (Pandionidae); Freyana dendrocygni Dubinin, 1950 (Freyanidae) and Heterobrephosceles megathrix Peterson & Atyeo, 1977 (Alloptidae) on the white-faced whistling duck (Dendrocygna viduata) (Linnaeus, 1766) (Anatidae); Nyctibiolichus megamerus Atyeo, 1979 (Pterolichidae) on the common potoo (Nyctibius griseus) (Gmelin, 1789) (Nyctibiidae); Hieracolichus hirundo [START_REF] Mégnin | Les sarcoptides plumicoles. Révision du groupe des Analgesinae, et description des espèces et genres nouveux de la collection du Musée d'Angers (Suite 1)[END_REF] on the harpy eagle (Harpia harpyja) (Linnaeus, 1758) (Accipitridae), and Hieracolichus sp. (Gabuciniidae) on the southern caracara (Caracara plancus) (Miller, 1777) (Falconidae) (Table 1).

Discussion

Besides reporting feather mites for the first time from the Northern Brazilian state of Piauí, this study also reports four species for the first time in Brazil, and two mites constitute new locality records. Pandionacarus fuscus (Figure 1), one of the largest feather mites at about 1.2-1.3 mm in length, and commonly associated with the osprey [START_REF] Miller | Records of ectoparasites collected on ospreys from Ontario[END_REF][START_REF] Philips | A review and checklist of the parasitic mites (Acarina) of the Falconiformes and Strigiformes[END_REF], was recently reported on a toucan -Ramphastos citreolaemus Gould, 1844 (Piciformes: Ramphastidae) -in Colombia [START_REF] Parra-Henao | Detection of ectoparasites in wild birds evaluated in Medellin (Colombia)[END_REF], that is definitely a contamination. The osprey is a worldwide distributed bird, including in several parts of Brazil (BirdLife International, 2016), and we herein report Pandionacarus fuscus for the first time in this country.

Dendrocygna viduata was reported as a host of Freyana dendrocygni (Figure 2), along with D. javanica, D. arcuata, D. eytoni, and D. bicolor (McDaniel et al., 1966). Mites of this genus are normally associated with aquatic birds of the family Anatidae (ducks, geese, swans) [START_REF] Dubinin | Systematical analysis of species of feather mites (Sarcoptiformes, Analgesoidea) parasitic on Anseriformes[END_REF][START_REF] Aksin | Freyana anatina (Koch, 1844) Feather Mites (Acarina, Freyanoidea) Recorded for the First Time on Wild Ducks (Subfamily, Anatinae) in Turkey[END_REF]. There are records of the mite Heterobrephosceles megathrix (Figures 3A-C) from lesser whistling ducks in Malaysia and Thailand, and white-faced whistling ducks in Uganda [START_REF] Peterson | The feather mite family Alloptidae Gaud IV -A new genus of the Alloptinae (Acarina, Analgoidea)[END_REF]. Gaud (1952: 87) reported undetermined alloptid females on the latter host, but he asserted that the absence of males could not allow a reliable identification of the genus. The females of H. megathrix remained unknown until this date, and were found herein for the first time. Freyana dendrocygni and Heterobrephosceles megathrix, both found on Dendrocygna viduata, are recorded for the first time in Brazil.

Nyctibiolichus megamerus (Figure 3D), one of the two known species of the genus Nyctibiolichus Atyeo, was described from Nyctibius griseus in Panama and Jamaica [START_REF] Atyeo | New feather mite taxa (Acarina: Pterolichoidea) from the avian order caprimulgiformes)[END_REF]. Up to this date, this mite had not been reported from hosts or countries other than those in the original publication. In our study, only a few females and immatures, and no males of this mite were recovered, that can be due to the low number of feathers collected. Despite about 25 caprimulgid species occurring in Brazil, feather mites had been previously reported from only three host species in that country, which suggests the existence of a number of undescribed taxa [START_REF] Mironov | New feather mites of the family Ascouracaridae (Astigmata: Pterolichoidea) from some parrots and nightjars[END_REF][START_REF] Enout | Chewing lice (Insecta, Phthiraptera) and feather mites (Acari, Astigmata) associated with birds of the Cerrado in Central Brazil[END_REF][START_REF] Hernandes | The feather mites of nightjars (Aves: Caprimulgidae), with descriptions of two new species from Brazil (Acari: Xolalgidae, Gabuciniidae)[END_REF][START_REF] Pedroso | New records of feather mites (Acariformes: Astigmata) from nonpasserine birds (Aves) in Brazil[END_REF]. The genus Hieracolichus includes nine described species primarily associated with birds of the family Accipitridae (Accipitriformes) [START_REF] Gaud | Gabuciniidae, famille nouvelle de Sarcoptiformes plumicoles[END_REF][START_REF] Mironov | New genera and species of feather mites of the family Gabuciniidae (Astigmata: Pterolichoidea) from New World raptors (Aves: Falconiformes)[END_REF]. Nevertheless, a few studies [START_REF] Barreto | Feather mites (Acariformes: Psoroptidia) from Colombia: preliminary list with new records[END_REF]Pereira, 2015, present study) indicate that this genus is also associated with falcons (Falconiformes), which are presently considered to be distant relatives to hawks and eagles [START_REF] Prum | A comprehensive phylogeny of birds (Aves) using targeted next-generation DNA sequencing[END_REF].

Feather mites studied from birds held in captivity can constitute an important resource for understanding patterns of bird-mite associations. For instance, they can help to confirm poorly known associations. Until recently, Hieracolichus hirundo (Figure 4) was represented only by a few specimens in a single slide at the Museum National d'Histoire Naturelle, Paris. Based on those and also on freshly collected specimens from a harpy eagle in captivity, this species was recently redescribed (Hernandes, 2017a).

However, a caveat must be made to unsuspecting students or researchers studying feather mites from birds held in captivity: it is not uncommon to recover unnatural mite-bird associations, because birds can acquire mites from other birds held in the same or neighboring cages. Thus, rheas -Rhea americana (Linnaeus, 1758) (Rheiformes: Rheidae) -were once reported as hosts of the same mite species commonly associated with ostriches -Struthio camelus (Linnaeus, 1758) (Struthioniformes: Struthionidae) [START_REF] Mégnin | Les sarcoptides plumicoles. Révision du groupe des Analgesinae, et description des espèces et genres nouveux de la collection du Musée d'Angers (Suite 1)[END_REF][START_REF] André | Sarcoptides plumicoles parasites des autruches[END_REF], and flamingos (Phoenicopteriformes: Phoenicopteridae) may eventually show mites commonly found on ducks (Anseriformes: Anatidae) (pers. obs. by FAH). These are obvious contaminations. All mites reported herein, however, are natural associates of their respective hosts, for they have been originally reported from the same hosts in other localities. This fact may help to reinforce the knowledge about these mite-bird associations. [START_REF] Mironov | Captive individuals of endangered Philippine raptors maintain native feather mites (Acariformes: Pterolichoidea) species[END_REF] demonstrated that feather mites may persist on a host in captivity for more than 40 years after being captured from the wild, even after undergoing through antiparasitic treatments. Likewise, cockatiels, Nymphicus hollandicus (Kerr, 1792) (Psittaciformes: Cacatuidae), bred and kept in captivity for decades in Brazil -and often treated for parasites in pet stores, were also shown to contain feather mites from its place of origin, Australia (Hernandes, 2017b).

The studies on feather mites collected from birds kept in captivity are still very scarce in Brazil [START_REF] Valim | Duas espécies de cisnes (Aves: Anatidae) de cativeiro como novos hospedeiros para Brephosceles discidicus Peterson, 1971 (Acaridida: Pterolichidae)[END_REF][START_REF] Pereira | Feather mites (Acari: Astigmata) of captive Psittaciformes in Brazil[END_REF]. We hope to stimulate further studies since valuable results may be obtained from these birds, including confirmation of parasite-host associations, discovery of new species, recollecting of rare species, new records and hosts, information on microhabitats, and also provide a better understanding on epidemiological case studies.

Figure 1 Figure 2

 12 Figure 1 Pandionacarus fuscus found on the osprey (Pandion haliaetus): A -male; B -female. Scalebar A, B = 100 μm.

Figure 3

 3 Figure 3 Heterobrephosceles megathrix (A-C) found on the white-faced whistling duck (Dendrocygna viduata): A -male, B -tritonymph (A and B attached in a precopulatory guarding position), C -female; Nyctibiolichus megamerus (D) found on the common potoo Nyctibius griseus: D -female. Scalebar A-D = 100 μm.

Figure 4

 4 Figure 4 Hieracolichus hirundo found on the harpy eagle (Harpia harpyja): A -male; B -female. Scalebar A-B = 100 μm.

Table 1

 1 Birds examined for feather mites at the Zoobotanical State Park, Teresina, PI, Brazil, 2017.

	Birds	EB	IB	EF Feather mites
	Pandion haliaetus (Accipitriformes: Pandionidae)	1	1	12	Pandionacarus fuscus * (Avenzoariidae)
	Harpia harpyja (Accipitriformes: Accipitridae)	1	1	2	Hieracolichus hirundo** (Gabuciniidae)
	Caracara plancus (Falconiformes: Falconidae)	2	2	8	Hieracolichus sp.** (Gabuciniidae)
	Dendrocygna viduata Anatidae) (Anseriformes:	5	2	17	Freyana dendrocygni* (Freyanidae) megathrix* (Alloptidae) Heterobrephosceles
	Nyctibius griseus				Nyctibiolichus
	(Nyctibiiformes:	2	1	5	megamerus*
	Nyctibiidae)				(Pterolichidae)
	Total	11	7	44

Legend: EB -number of examined birds; IB -number of infested birds; EF -number of examined feathers for each bird specimen; *First record of the mite species in Brazil; **species recorded in a new locality in Brazil.

Lima Alves Moro M. M. et al. (2019), Acarologia 59(4): 424-432; DOI 10.24349/acarologia/20194345

Acknowledgements

The authors are thankful to the administration staff of the Zoobotanical State Park of Teresina, Piauí, for granting the opportunity to perform this study.


Christophe Voisin

Maira Alejandra

Romero Guzmán

Aurore Réfloch

Marco Taruselli

Stéphane Garambois

Journal of Water Resource and Protection Groundwater Monitoring with Passive Seismic Interferometry

Keywords: Groundwater, Water Table, Seismic Interferometry, Piezometer

Passive seismic interferometry takes advantage of natural ambient seismic noise generated by the wind, the storms and the human activities (e.g. cars, trains and hereafter pumps) to recover the slight variations of the seismic wave velocity induced by changes in the groundwater level. Here we compare the seismic measurements with actual piezometric data acquired on the Crépieux-Charmy (Lyon, France) groundwater exploitation field. We show the excellent correspondance between variations in the groundwater level and seismic velocity variations. We present hereafter the time and space monitoring of an hydraulic dome formed to prevent biological and chemical pollutions to enter the exploitation field. The horizontal resolution is solely limited by the number of seismic stations used, and is about 30 m in the present study. The vertical resolution of seismic measurement is impaired by spurious artifacts linked to the intermittent sources of noise. In average, the sensitivity of the seismic velocity change corresponds to a 50 cm change of waterlevel. This study confirms the possibility of groundwater monitoring in an industrial context with ambient seismic noise.

Introduction 1.Area of Study

The Crépieux-Charmy exploitation field (Figure 1) is located in the alluvial plain of the Rhône River (North-East of Lyon, France). With up to 375 ha it is Figure 1. The Crépieux-Charmy groundwater exploitation field is limited by the Miribel channel to the North, and the Vieux Rhône River to the South. The area of water uptake is bounded by infiltration basins used to produce hydraulic domes in case of biological or chemical pollutions in either of the limiting channels. The image focuses on basin 5.2 with positions of the piezometers installed by the exploitant (red dots). The seismic stations are deployed around basin 5.2. The red lines correspond to the direct paths between pairs of stations of interest. The blue dots labeled from A to H represent the locations of the virtual piezometers, at the center of each red path. one of the largest groundwater exploitation fields in France. Started in 1957, the Crépieux-Charmy field now covers the water needs of the city of Lyon (2 million inhabitants).

The Crépieux-Charmy field is localized in the well-known Eastern aquifer of Lyon, a free aquifer characterized by glacio-fluvial Würmien's deposits. In the particular zone of study, the glacio-fluvial deposits are covered by recent alluvial deposits related to the Rhône River. They describe a sedimentary sequence with mostly sandy-gravel size grain with a heterogeneous distribution classically found in such sedimentary environment. The recent alluvial deposits have a variable thickness between 10 and 25 m. The depth of the basement, represented by Pliocene glacial claystones and molasses is around 158 m.a.s.l [START_REF] Loizeau | Amélioration de la compréhension des fonctionnements hydrodynamiques du champ captant de Crépieux-Charmy[END_REF]. The sedimentary environment and the texture of this formation also contribute to the complexity of the hydrogeological regime in the area of study.

In order to prevent biological or chemical pollutions to enter the groundwater uptake area, different series of infiltration basins have been created along the path of the Miribel and Old Rhône channels (Figure 1). In case of upstream incoming pollution, these basins are filled up with water and form a series of hydraulic domes below them. Each dome represents a local increase of a few meters of the water table, supposed to invert the local water flow from the channel to the uptake area. In the present study, we will focus on the hydraulic dome formed under the infiltration basin 5.2, used for different scientific and experimental studies by the research platform of "Eau du Grand Lyon". representation of the functioning of the infiltration basin. Some questions arise about the shape of the hydraulic dome, the location of its maximum and the speed of infiltration that is known to be variable. The scarcity of piezometers installed in and around the basin 5.2 prevents to answer those questions. In the following, we explore the potential of passive seismic monitoring techniques to retrieve pertinent and additional information about the hydraulic dome.

The Hydraulic

Passive Seismic Monitoring of Groundwater

Passive seismic monitoring leans on the recording of the mechanical waves that propagate into the subsurface (Figure 3(a)). These waves originate from natural sources such as the wind, the storms, the rivers, and from anthropogenic sources such as cars, trains and hereafter pumps. The mechanical seismic waves are classified as body waves (P and S waves) and surface waves (Rayleigh and Love waves). All of them are characterized by a given velocity that depends on different parameters of the medium such as the shear modulus, the stiffness and the density for elastic parameters; fluid composition, porosity and saturation for poroelastic parameters. The waves velocity is accurately determined by cross-correlating the recordings of the ambient seismic wave field at two seismic stations. The result is formed by a series of waves arriving at different times, depending on their velocity and on the path they followed. We differentiate be- where local noise sources are dominated by the pumping and injection of water, highly variable both in time and space. We address the following issues: is it possible to retrieve the water

tween

Comparison between Piezometers and Seismic Velocity Changes

Methodology

Seismic Velocity Changes

For each seismic station and each one-hour trace, the signal processing operations are the following: the data is filtered between 3 and 20 Hz and is whitened in the spectral domain to correct for the differences of spectrum of the different noise sources. On a field experiment with standalone stations, it may happen that from time to time the seismic stations stop acquiring data; the missing data are replaced by zeros in order to complete the time series. Unfortunately in our experiment, station 1 stopped acquiring data early in the experiment. For this reason this station is not included in the analysis. For the pairs of stations of interest (linked by red paths in Figure 1) the one-hour data records are correlated to produce a correlogram similar to the one presented in Figure 3(b). The slight velocity changes are observable as slight changes of arrival time of the ballistic waves. Windowed crosscorrelation (or moving-window cross spectral method in the frequency domain) and trace stretching are two techniques commonly used to estimate local time shifts in seismic signals [START_REF] Mikesell | A Comparison of Methods to Estimate Seismic Phase Delays: Numerical Examples for Coda Wave Interferometry[END_REF]. Within this study, we use the stretching technique [START_REF] Sens-Schonfelder | Passive Image Interferometry and Seasonal Variations of Seismic Velocities at Merapi Volcano, Indonesia[END_REF] to measure seismic velocity variations between pairs of seismic stations. This technique requires the definition of a reference correlation that is computed by averaging all of the 1 h correlations over the duration of the experiment. In a second step, a time window is selected within the lapse time range of interest (where the strongest seismic phases arrived) to compute the velocity variations. For each hour, the time vector t of the correlation is multiplied by a stretching factor ε, i.e., t ← t × (1 ± ε) to align the arrival time of the seismic phases within the selected time window to their arrival time of reference.

Many different values of the stretching factor ε are tested. The one retained is the one that maximizes the zero-lag crosscorrelation coefficient between the reference and the velocity variation of each hour. The optimal stretching factor ε is either positive (i.e., the seismic phases arrive later than their reference time) or negative (i.e., the seismic phases arrive before their reference time). Finally, the seismic velocity change curve is defined as the opposite of the vector of the hourly stretching factor.

Piezometric Changes

The velocity changes retrieved following the processing described in the previous section are relative to the reference, chosen here as the average of all 1 h correlations. In order to compare the results with the piezometric data, we simply need to remove the average of each piezometric curve provided for the instruments B10, P96, S20 and M25. However, for some pairs of seismic stations, the details of the seismic velocity change do not match the evolution of the closest piezometer. The reasons for these discrepancies are partly found in the nonstationary noise source distribution. The activation and rest of the different pumps create spurious artifacts in the velocity change curves that are not related to changes in the medium.

Results

Broadband Velocity Changes Versus Piezometric Changes

Depth Dependence of the Velocity Changes

The seismic velocity changes presented in Figure 5 are measured on the ballistic waves that correspond to surface Rayleigh waves propagating directly from a station to the other. These Rayleigh waves have the interesting property to be dispersive, i.e. their velocity depends on the frequency [START_REF] Beaty | Repeatability of Multimode Rayleigh-Wave Dispersion Studies[END_REF]. This property is related to the sensitivity kernel of these waves that depend on the frequency: the lower the frequency, the deeper the kernel. By filtering in different frequency 6, 6 -8, 8 -10 and 10 -12 Hz seismic variation curves, respectively. Figure 6 shows the results for the station pair 5 -6. First, it is important to note that all limited spectral bandwidths exhibit velocity variations. The amplitude of velocity change in those frequency bands is of similar order (i.e. in the range of ±2%).

However only the velocity variations computed within the 8 -10 Hz frequency range match the reference computed over the 3 -20 Hz band. The 6 -8 Hz frequency band provides an acceptable fit to the broadband velocity changes, too.

The other frequency bands (5 -6 and 10 -12 Hz) provide seismic velocity variations that do not match the reference seismic velocity variations and/or are not in phase with the broadband pattern. This implies that the velocity variations measured in the broadband frequency range are borne by a limited spectral content. In the present case of the pair 5 -6, the frequency band that senses the environmental changes is 6 -10 Hz.

A similar analysis is performed for the different pairs of seismic stations. Table 1 summarizes the results for the different pairs. The results of the column "best frequency range" are coherent at the exception of the last pair of stations (3 -4) that shows a large frequency range. This is possibly due to the path between the two stations mostly formed by the eastern dam that limits the basin 5.2.

The relationship that links the frequency bandwidth with the "equivalent depth" is depending on the local shear wave velocity profile with depth. In the context of this study, different active seismic experiments have been performed

around the basin 5.2 to retrieve the velocity profile in the first tens of meters. The outcome of these profiles is that the shear wave velocity in the subsurface is quite low but highly variable. For the sake of simplicity the shear wave velocity is considered as constant over the first 5 meters and equal to 300 m•s -1 . The depth z is computed according to the relation z = 0.15*c/f, where c is the local shear wave velocity and f is the frequency [START_REF] Graff | Wave Motion in Elastic Solids[END_REF]. This depth corresponds to the maximum of energy of the fundamental mode of the surface Rayleigh waves with the given frequency. With a constant shear wave velocity profile with depth; this maximum of energy forms a good proxy for the sensitivity kernel of the surface waves that indicates the maximum sensitivity of the waves to a change of shear wave velocity. The 6 -10 Hz bandwidth has a maximum of energy at a depth of 4.5 to 7.5 meters below the surface. This range overestimates the piezometric values of P96 (Figure 4) that range from 2 to 5 meters below the bottom of basin imposes a supernumerary layer that modifies the free surface conditions. For these reasons, the simple approach of the maximum energy and the results of equivalent depth presented in Table 1 must be considered as gross approximations.

Additional Constraints on the Hydraulic Dome

3D Representation of the Hydraulic Dome

For practical reasons the seismic stations were installed around the basin 5. cation of piezometer B10 situated on the western bank of the basin 5.2. Figure 7 presents a 3D representation of the results obtained in this study. It includes on the same scale the relative piezometric changes (in meters) and the seismic velocity changes (in percentage) as a function of the distance to B10 and as a function of the time of the experiment. As for Figure 5, the overall agreement between the two independent measurements is acceptable. There seems to exist a difference in the behaviors of the virtual piezometers. Points A, B, C, D and E follow more or less the same variations, similar to the one measured by piezometers B10, P96 and S20. The draining stage of the basin is clearly visible in all virtual piezometers although apparently delayed for point A (see section 4.2 for an explanation). The filling stage is also extremely marked for all these virtual piezometers. The fastest filling rate is observed for the virtual piezometer B, located very close to piezometer P96.

On the contrary, points F, G and H exhibit a different behavior than the previous virtual piezometers. For these three virtual piezometers, the draining stage is not pronounced or is not even present in the data. This has to be related to the initial water level conditions of the basin 5.2 at the beginning of the seismic recordings (Figure 4). It is very likely that the hydraulic dome is locally much less expressed (as seen in piezometer M25 during the filling stage) and vanishes quite rapidly, hence the small relative changes of seismic velocity mostly governed by the regional groundwater flow. The filling stage of basin 5.2 turns into a small

Limitations of the Approach

If the overall agreement between piezometric measurements and seismological measurements is acceptable, some discrepancies appear when it comes to the fine details of the time evolution of the water table. As illustrated in Figure 3(b), the monitoring of the water table based on seismological measurements requires a stable distribution of the sources of noise. This requirement is hardly achieved in the industrial context of the exploitation, where water pumping and injection happens at different times and locations in the 11 hectares of the field. The nonstationary distribution of the sources of noise creates spurious artifacts in the velocity changes. This might become an issue when trying to identify the depth dependence of the velocity change, because most of the industrial noise sources have a frequency content of a few Hertz. Activation of such sources creates strong energetic seismic body waves that are refracted inside the medium and may arrive simultaneously with the surface waves of interest for the monitoring of the water table. An example of this is provided in Figure 3(b).

Conclusion

We have shown here the potential of the seismic noise for the monitoring of the water table in an industrial context. The seismic velocity changes presented in this study integrates the information along the path between the pairs of stations, i.e. horizontally. They also integrate information vertically because of the finite frequency band of analysis that scans the medium at different depths that include fully and partially saturated horizons. However, the velocity change is largely governed by the position of the saturation front (i.e. the water table): an analysis in different frequency bands helps to constrain the depth of this front through time. The nonstationary source of noise distribution impacts the retrieval of the details of the water table evolution. The interpretation of the seismic velocity changes might be impaired by spurious artifacts. Nonetheless, the seismic noise monitoring allows retrieving the major features of the water table evolution. The lateral resolution of the technique is limited by the distance between the seismic stations, with a minimum in this study of 30 meters. The spurious artifacts limit the vertical resolution of the retrieved changes.

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  Voisin et al. DOI: 10.4236/jwarp.2017.912091 1416 Journal of Water Resource and Protection

Dome of Basin 5. 2 Figure 2 .

 22 Figure 2. Schematic functioning of the hydraulic barrier in Crépieux-Charmy field [1].

Figure 3 .

 3 Figure 3. (a) One hour of seismic noise recording at the 9 operating seismic stations.Each trace is normalized by its maximum to facilitate the reading of the plot. Note the arrival of large wave packets between 1000 and 1500 s and between 2500 and 3000 s. They correspond to the noise of pumping and injection of water in the basin 5.2. The largest arrival at 3100 s corresponds to a train passing to the north of basin 5.2. The rest of the traces is formed by numerous small amplitude wave propagating inside the medium and carrying information on its state; (b) Normalized correlation of the records of stations 5 and 6 over the duration of the experiment (vertical axis from 0 to 210 hours). The horizontal axis gives the time lapse of the different waves propagating between the two stations. We distinguish the ballistic waves arriving between -1 and +1 s and the coda waves arriving at later times. The positive side corresponds to waves travelling from station 5 to station 6; the negative side corresponds to waves travelling from station 6 to station 5. Note the strong assymetry of the correlogram dominated by the strong arrivals between 0.4 and 1 s. This indicates that most of the energy carried by the waves is travelling from station 5 towards station 6. Note the slight changes of the arrival time of the ballistic waves over the duration of the experiment. These changes are related to changes of the velocity of the seismic waves inside the medium induced by changes in the water level below the basin 5.2. The black rectangle shows very strong waves arriving near zero time lapse, i.e. almost simultaneously to stations 5 and 6. They occur between hours 120 and 140 and correspond to the infilling of basin 5.2. The origin of these waves is found in the water pumping and injection into basin 5.2.

2. 1 .

 1 Figure 4. Piezometric data (B10, P96, S20 and M25) and water level in basin 5.2. The red arrow indicates the period of seismic noise recording. Note the formation of the hydraulic dome when the basin is filled with water.

Figure 5

 5 Figure 5 summarizes the results obtained for 6 different pairs of stations, orga-

Figure 5 .

 5 Figure 5. Comparison of broadband seismic velocity changes (black curve, in %) and relative pieozmetric changes (blue curve, in m) for 6 different pairs of stations (7 -8; 6 -8; 5 -6; 2 -8; 2 -5; 3 -4). Computations were made in the 3 -20 Hz frequency band. Note the excellent agrrement between both curves at the different locations. The letters (A, B, C, E, H) corresponds to the location of the virtual piezometers given in Figure1.

  bands, it becomes possible to track the depth range of the measured changes. To address the problem of localization of the change in the medium, the crosscorrelation of the seismic records is computed for several frequency bandwidths (5 -6, 6 -8, 8 -10 and 10 -12 Hz). The broadband correlations are filtered with a Butterworth filter of order 4. The seismic velocity variations derived for limited frequency bandwidths are compared with those obtained from the broadest frequency bandwidth (3 -20 Hz), which is used as a reference. The comparison is achieved by computing the autocorrelation of the 3 -20 Hz seismic velocity curve and the crosscorrelation of the 3 -20 Hz seismic velocity curve with the 5 -

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  Voisin et al. DOI: 10.4236/jwarp.2017.912091 1423 Journal of Water Resource and Protection

Figure 6 .

 6 Figure 6. Comparison of the autocorrelation of broadband velocity variations (reference in red) with the crosscorrelation of broadband velocity variation and band limited velocity variation (5 -6 Hz in light gray; 6 -8 Hz in black; 8 -10 Hz in blue and 10 -12 Hz in dark gray). The best match is obtained for the 8 -10 Hz bandwidth that bears most of the environmental signal, and it is in-phase with the broadband velocity variations (i.e., with the piezometric measurements of P96). The other bandwidths have little in common with the broadband velocity variation and are apparently out of phase with the reference, except the 6 -8 Hz bandwidth. The retained frequency band for this pair will be 6 -10 Hz.

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  Voisin et al. DOI: 10.4236/jwarp.2017.912091 1424 Journal of Water Resource and Protection

  to the values of piezometer M25 that indicates a small oscillation of the water table between 4 and 5 meters below the bottom of basin 5.2. The local path between the stations 3 and 4 is radically different from the path of the other stations, dominated by the structure of the eastern dam of the basin. This might explain the difficulty to reduce the frequency bandwidth to a few hertz, due to a strong contrast of velocity between the dam and the underground. In a general manner, the depths of water table inferred from seismic measurements overestimate the true values measured by the piezometers. This discrepancy has to be related to the approximation of a unique medium with constant velocity. A more complete description of the velocity profile below basin 5.2 would be required to properly compute the sensitivity kernels and the equivalent depth of the velocity changes. Of course, taking into consideration a more realistic velocity profile might enhance the importance of the higher modes of the surface waves and mislead the interpretation in terms of depth. Similarly, the filling of basin 5.2

2 .

 2 The seismic velocity change curves obtained in Figure5integrate the information on the changes of the medium below basin 5.2 all along the path between the seismic stations. The virtual piezometers lettered from A to H are conveniently placed at the midpoint of each path. A distance is computed from the lo-C. Voisin et al. DOI: 10.4236/jwarp.2017.912091 1425 Journal of Water Resource and Protection

Figure 7 .

 7 Figure 7. 3D representation of relative piezometric changes (thick ribbons) and of relative velocity changes (thin ribbons) as a function of distance (in meters from B10) and of time (in hours). Note the good agreement between the independent measurements. The virtual piezometers help to constrain the shape of the hydraulic dome in areas deprived from piezometers. Points A, B, C, D, and E have a similar behaviour to the piezometers B10 P96 and S20. Points G and H have smaller amplitudes responses. They do not seem affected by the draining of the basin in the first stage and respond only at the end of the experiment when the hydraulic dome is well developed.

Table 1 .

 1 Depth dependence of seismic velocity changes for the different virtual piezometers.

	Pair of stations	Virtual piezometer	Distance (m)	Closest piezometer	Best frequency range (Hz)	Equivalent depth (m)
	7 -8	A	75	B10	6 -8	5.6 -7.5
	6 -8	B	150	B10	6 -10	4.5 -7.5
	5 -6	C	170	P96	6 -10	4.5 -7.5
	2 -8	D	220	S20	6 -8	5.6 -7.5
	2 -5	E	115	S20	5 -8	5.6 -9.0
	2 -10	F	95	--	5 -8	5.6 -9.0
	2 -4	G	140	--	4 -7	6.4 -11.0
	3 -4	H	60	M25	8 -20	2.25 -5.6

  5.2. We note a similar behavior for the virtual piezometers A, B, C, D. They form a first group with a minimum depth of 4.5 meters obtained for points B and C, i.e. around the piezometer P96. The virtual piezometers E, F, G indicate deeper levels for the water table. Finally the virtual piezometer H forms a group on its own, indicating a water table oscillating between 2.25 and5.6 meters, to compare

  of the velocity, more pronounced for point F than for points G and H. Point F appears as the transition between the western side of basin 5.2, characterized by large and rapid variations of the water table and of seismic velocity, and the eastern side of the basin, characterized by limited changes both in water table level and seismic velocity.

C. Voisin et al. DOI: 10.4236/jwarp.2017.912091 1426 Journal of Water Resource and Protection relative change

Acknowledgements

The authors acknowledge the financial support of Labex OSUG@2020. The authors thank the research platform of "Eau du Grand Lyon" for authorizing entrance on the exploitation field of Crépieux-Charmy and BENEFICIARIO COLFUTURO 2017. C. Voisin et al. DOI: 10.4236/jwarp.2017.912091 1427 Journal of Water Resource and Protection


L Corrias

Benoît Perthame

Hatem Zaag

L p and L ∞ a priori estimates for some chemotaxis models and applications to the Cauchy problem

Keywords: Chemotaxis, angiogenesis, degenerate parabolic equations, global weak solutions, blow-up AMS subject classifications. 35B60, 35Q80, 92C17, 92C50

come   L'archive ouverte pluridisciplinaire

1 Introduction.

Chemotaxis is the directed movement of a living system in response to a chemical stimulus. The first mathematical model for chemotaxis has been proposed by Keller and Segel [20] to describe the aggregation of the slime mold amoebae Dictyostelium discoideum (see also Patlak [25] for an earlier model and [START_REF] Keller | Assessing the Keller-Segel model : how has it fared ? Biological growth and spread[END_REF], [START_REF] Keller | Model for chemotaxis[END_REF] for various assessments). Since then, the mathematical description and analysis of the biological phenomenon has attracted a lot of attention. Indeed, this biochemical process has been involved in so many medical and biological applications. Moreover, from a mathematical point of view, the model has a rich structure and remains very challenging. See Horstmann [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] and [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] for a review on the subject. This paper is concerned with a simplified version of the Keller-Segel model but however rather general, namely

               ∂ t n = ∇ • [κ∇n -n χ (c)∇c], t > 0, x ∈ Ω,
ε∂ t c = η∆c + g(n, c), t > 0, x ∈ Ω, n(0, x) = n 0 (x), c(0, x) = c 0 (x),

where Ω is R d , n(x, t) is the density of the population moving under chemotaxis and c(x, t) is the density of the chemical substance. The function χ is the chemotactic sensitivity of the species that 1 plays a crucial role in the chemotaxis process. Indeed, χ (c) > 0 leads to positive chemotaxis (the population is attracted by the chemical), while χ (c) < 0 leads to negative chemotaxis (the population is repelled by the chemical). Moreover, in a mathematical language, χ measures the balance between the diffusion effect of the Laplace term and the hyperbolic effect of the drift term and we tried to understand its impact on the dynamics. The function g(n, c) contains the production and decay rate terms of the chemical and is generally taken of the form

g(n, c) = β(n, c)n -γ(n, c)c . (2)
Finally, we take the diffusion coefficient κ of the population n(x, t) constant with respect to n and c.

The diffusion coefficient of the chemical η is non-negative and the parameter ε ∈ {0, 1}. Obviously, η and ε are not equal to zero at the same time. This means that system (1) contains the extreme case of a parabolic-elliptic system (ε = 0 and η > 0) and a parabolic-degenerate system (ε = 1 and η = 0). The goal of this paper is to summarize some ideas previously used by the authors to obtain global existence results for these two extreme cases of system (1) (see [START_REF] Corrias | A chemotaxis model motivated by angiogenesis[END_REF], [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]) and to give some hint to apply these ideas to the more general system (1) together with new results.

The parabolic-elliptic system has been extensively studied by many authors and a huge quantity of mathematical results on the existence of global in time solutions and on the blow-up of local in time solutions, have been produced. We refer to [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] and [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] for a quite complete bibliography. The parabolic-degenerate system has been much less studied than the previous one, except in one space dimension. One reason to be interested in this kind of system is that it arises in modelling the initiation of angiogenesis, a kind of chemotaxis process that occurs in the development of a new capillary network from a primary vascular network ( [START_REF] Chaplain | Avascular growth, angiogenesis and vascular growth in solid tumors : the mathematical modelling of the stages of tumor development[END_REF], [START_REF] Chaplain | Macroscopic modelling of the growth and development of tumor masses[END_REF], [START_REF] Corrias | A chemotaxis model motivated by angiogenesis[END_REF], [START_REF] Fontelos | Mathematical analysis of a model for the initiation of angiogenesis[END_REF], [START_REF] Levine | Mathematical modelling of the onset of capillary formation initiating angiogenesis[END_REF]). Moreover, it is now clear that the asymptotic behavior of n depends strongly on the coupling effects of the dynamics of c and the chemotactic sensitivity χ (c) and that is the reason why chemotaxis systems of parabolic-degenerate type have been analyzed in [START_REF] Levine | A system of reaction diffusion equations arising in the theory of reinforced random walks[END_REF], [START_REF] Othmer | Aggregation, blowup and collapse : the ABC's of taxis in reinforced random walks[END_REF], [START_REF] Rascle | On a system of non-linear strongly coupled partial differential equations arising in biology[END_REF], [START_REF] Yang | On existence of global solutions and blow-up to a system of reaction-diffusion equations modelling chemotaxis[END_REF].

Most of the analysis presented in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] about the parabolic-elliptic system was known before, unlike our analysis for the parabolic-degenerate system which we would like to put forward. The parabolicdegenerate system we are interested in is :

               ∂ ∂t n = κ∆n -∇ • [n χ (c)∇c], t > 0, x ∈ R d , ∂ ∂t c = -nc m , t > 0, x ∈ R d , n(0, x) = n 0 (x), c(0, x) = c 0 (x), x ∈ R d (3)
where m ≥ 1 and the sensitivity function χ (c) is a given positive function on R + . Our main result for this system is the following:

Theorem 1 (Existence for the angiogenesis system (3)). Assume d ≥ 2, m ≥ 1 and χ (c) a positive say continuous function on [0, ∞). Consider some

n 0 ∈ L 1 (R d ) and c 0 ∈ L ∞ (R d ) such that n 0 ≥ 0 and c 0 ≥ 0. There exists a constant K 0 (κ, χ , d, c 0 L ∞ (R d ) ) such that if n 0 L d 2 (R d ) ≤ K 0 , then system (3) has a global (in time) weak solution (n, c) such that n ∈ L ∞ (R + , L 1 ∩ L d 2 (R d )), c ∈ L ∞ (R + × R d ) and for any fixed max{1; d 2 -1} ≤ p * < ∞, n(t) L p (R d ) ≤ C(t, K 0 , p * , n 0 L p (R d ) ), ∀ max{1; d 2 -1} ≤ p ≤ p * . Moreover, if n 0 L d 2 (R d ) ≤ K 0 2 and n 0 ∈ L ∞ (R d ), then n(t) L ∞ ≤ C κ, χ , n 0 L d (R d ) max { n 0 L ∞ , n 0 L 1 } .
The same conclusion holds with less regular χ (c) at c = 0 (see [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]). Moreover, since the goal of this note is to give a general idea of the method we use to obtain the global existence of the solution of (1), we consider the system on the whole space R d together with ad hoc decay condition at infinity on the densities n and c. In this way, the conservation of the initial mass is assured,

R d n(t) = R d n 0 =: M, ∀t ≥ 0.
In the case of a bounded domain Ω, the boundary conditions for n and c can be homogeneous Neumann boundary conditions or the no-flux boundary condition, so that the conservation of the initial mass is again assured and the results given here can be applied.

All the results will be dimension dependent with d ≥ 2, since in one space dimension global existence of smooth solutions can be obtained by the classical methods of parabolic equations (see for example [START_REF] Fontelos | Mathematical analysis of a model for the initiation of angiogenesis[END_REF], [START_REF] Rascle | On a system of non-linear strongly coupled partial differential equations arising in biology[END_REF]).

The paper is organized as follows. In section 2, we present a priori estimate in weighted L p spaces for the general system (1). These estimates are valid for a large class of sub-systems, including the extreme cases mentioned before and analyzed in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]. In sections 3 and 4, we recall results from [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] in order to show how the a priori estimates of section 2 allow to get the global existence result for the parabolic-elliptic and the parabolic-degenerate cases respectively. Moreover, for both systems, we give also the L ∞ bound not mentioned in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] and, as a consequence, the uniqueness of the solution of the parabolic-elliptic system.

Acknowledgement. The third author presented this work during the conference "The mechanism of the spatio-temporal pattern arising in reaction diffusion system" which took place in Kyoto University in October 2004. He would like to thank T. Tsujikawa and Y. Kan-on for their kind invitation, as well as H. Matano, H. Ninomiya, D. Hilhorst and C. Oikonomides who made his stay in Japan very fruitful and pleasant.

2 Differential inequalities in weighted L p spaces.

In this section, we give general differential inequalities for the weighted or not L p norm of the solution n of (1). These inequalities include all the differential inequalities used in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] and most of the known in the literature for system [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF].

Let us define u = n φ(c) where the function φ is such that

for all c > 0, φ (c) = 1 κ φ(c) χ (c) and φ(0) = 1, (4)
This means that the sensitivity function χ (c) is necessarily taken to be integrable at c = 0. Then, the change of variable n → u puts the equation on n in the following divergence form

∂ t u = κ 1 φ(c) ∇ • [φ(c)∇u] -u 1 φ(c) ∂ t φ(c). (5)
The equation on u suggests naturally to consider the evolution of the quantity

R d G(φ)F (u) dx where
G is a differentiable function on R + and F is a twice differentiable function on R + . Therefore, using (5), we write

d dt R d G(φ)F (u) dx = -κ R d G(φ)F (u)|∇u| 2 dx -κ R d 1 φ G(φ) F (u)φ∇φ • ∇u dx + R d [G (φ)F (u) -G(φ)F (u) u φ ]∂ t φ dx = -κ R d G(φ)F (u)|∇u| 2 dx + κ R d F (u)∆ψ(φ) dx + R d [G (φ)F (u) -G(φ)F (u) u φ ]∂ t φ dx (6)
where ψ is such that ψ (φ) = φ 1 φ G(φ) . The main differential identity [START_REF] Brenner | Physical mechanisms for chemotactic pattern formation by bacteria[END_REF] shows how the evolution of the chemical of density c influences the evolution of the population of density n. In order to obtain useful estimates from ( 6), one has to handle carefully every term in [START_REF] Brenner | Physical mechanisms for chemotactic pattern formation by bacteria[END_REF].

For ε = 1 and η > 0, one has to balance the second term in the right hand side of ( 6) with the third term. For example, for G(φ) = φ q and F (u) = u p we have

R d F (u)∆ψ(φ) dx = q -1 q R d u p ∆φ q dx = (q -1) 2 R d u p φ q-2 |∇φ| 2 + (q -1) R d u p φ q-1 ∆φ dx (7)
and

R d [G (φ)F (u) -G(φ)F (u) u φ ]∂ t φ dx = (q -p) R d u p φ q-1 [η(∆φ -φ |∇c| 2 ) + φ g(n, c)] dx . (8)
Then, using ( 7) and ( 8) in ( 6) we obtain

d dt R d φ q u p dx = -κp(p -1) R d φ q u p-2 |∇u| 2 dx +[κ(q -1) + η(q -p)] R d u p φ q-1 ∆φ dx + R d u p φ q-2 [κ(q -1) 2 φ 2 -η(q -p)φφ ]|∇c| 2 dx +(q -p) R d u p φ q-1 φ g(n, c) dx. (9)
In order to eliminate the second term in the right-hand side of ( 9), we choose p and q such that κ(q -1) + η(q -p) = 0, i.e. q = p+γ γ+1 with γ = κ η , and (9) becomes

d dt R d φ p+γ γ+1 u p dx = -4κ (p-1) p R d |∇u p/2 | 2 dx + 1 γ+1 (p -1) R d u p φ p+γ γ+1 [ χ (c) + 1 κ p + γ γ + 1 χ 2 (c)]|∇c| 2 dx -1 κ γ γ+1 (p -1) R d u p φ p+γ γ+1 χ (c) g(n, c) dx. (10)
Next, let us suppose that for the initial density c 0 non-negative and bounded, the density c stays non-negative and bounded. Let us suppose also that there exists p ≥ 1 such that

χ (c) + 1 κ p + γ γ + 1 χ 2 (c) ≤ 0 for all p ∈ [1, p] and c ∈ [0, c L ∞ ]. (11)
Then, for any p ∈ [1, p], the following differential inequality for the weighted

L p norm of u d dt R d φ p+γ γ+1 u p dx ≤ -4κ (p -1) p R d |∇u p/2 | 2 dx - 1 κ γ γ + 1 (p -1) R d u p φ p+γ γ+1 χ (c) g(n, c) dx (12)
holds true. Finally, if the range [1, p] of p is sufficiently large with respect to to the dimension d, [START_REF] Evans | Partial Differential Equations[END_REF] gives us the estimate of some L p norms of n under a smallness condition on the initial density n 0 (see subsection 3.1). The dependence of g on n is obviously crucial.

Please note that [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] implies that χ (c) must be decreasing and in particular that χ (c) cannot be identically constant.

Let us also observe that, without any hypothesis on χ (c), letting η → 0 + so that γ → +∞ in [START_REF] Corrias | A chemotaxis model motivated by angiogenesis[END_REF], we obtain the differential identity below for the parabolic-degenerate case already used in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]

d dt R d φ(c)u p dx = -4κ (p -1) p R d |∇u p/2 | 2 dx - 1 κ (p -1) R d u p φ(c) χ (c)g(n, c) dx. (13)
For ε = 0, the third term in the right hand side of (6) has to be eliminated. This is possible for example taking G(φ) = φ p and F (u) = u p . With this choice and after some computation, (6) becomes nothing else than the evolution equation for the

L p norm of n d dt R d n p dx = -κp(p -1) R d φ p u p-2 |∇u| 2 dx -κp(p -1) R d n p-1 ∇φ • ∇u dx = -4κ (p-1) p R d |∇n p/2 | 2 dx + (p -1) R d χ (c)∇n p • ∇c dx. (14)
In the case of constant sensitivity, from [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF], one obtains an estimate on the L p norm of n. See Section 3 below.

Remark. Following [START_REF] Horstmann | Lyapunov functions and L p estimates for a class of reaction-diffusion systems[END_REF], one can also obtain Lyapunov functions for the system (1) with η > 0. Indeed, for G(φ) = φ and if [F (u) -uF (u)]φ = -g(n, c), then the function

H(n(t), c(t)) = R d φ(c)F n φ(c) dx + η 2 R d |∇c| 2 dx
is a Lyapunov function since from ( 6) we have

d dt H(n(t), c(t)) = -κ R d φF (u)|∇u| 2 dx - R d g(n, c)∂ t c dx -η R d ∆c ∂ t c dx = -κ R d φF ( n φ(c) )| ∇n φ - n φ 2 ∇φ| 2 dx -ε R d (∂ t c) 2 dx ≤ 0. ( 15
)
3 The parabolic-elliptic system.

In this paragraph, we consider the parabolic-elliptic system given by (1) when ε = 0, i.e.

               ∂ ∂t n = κ∆n -χ ∇ • [n∇c], t > 0, x ∈ Ω, -∆c = n -cf (c), t > 0, x ∈ Ω, n(0, x) = n 0 (x), x ∈ Ω. ( 16
)
where f is a non-negative function, χ (c) is constant with respect to c and for simplicity in the notations, we take η = 1. When f (c) ≡ 0, the validity of ( 16) in the framework of chemotaxis is supported by some experiments on the Escherichia Coli bacterium (see [START_REF] Brenner | Physical mechanisms for chemotactic pattern formation by bacteria[END_REF] and the references therein), even if this model does not seem to reproduce some of the observed chemotactic movement. Moreover, in this case (f ≡ 0), system [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] has other interesting physical interpretations. For example, it arises in astrophysics and in statistical mechanics (see [START_REF] Biler | Global solutions to some parabolic-elliptic systems of chemotaxis[END_REF], [START_REF] Biler | A class of nonlocal parabolic problems occurring in statistical mechanics[END_REF], [START_REF] Biler | Global and exploding solutions in a model of self-gravitating systems[END_REF] and the references therein).

Concerning the chemotaxis system ( 16), the situation is well established and we recall the following results.

Theorem 2 (Existence and uniqueness for the chemotaxis system ( 16)) Assume d ≥ 2 and consider some

n 0 ∈ L 1 (R d ) such that n 0 ≥ 0. There exists a constant K 0 (κ, χ , d) such that if n 0 L d 2 (R d ) ≤ K 0 ,
then system (16) has a global (in time) weak solution (n, c) such that for all t > 0

n(t) L 1 (R d ) = n 0 L 1 (R d ) , n(t) L p (R d ) ≤ n 0 L p (R d ) , max{1; d 2 -1} ≤ p ≤ d 2 ,
and

n(t) L p (R d ) ≤ C(t, K 0 , n 0 L p (R d ) ) d 2 < p < ∞. Moreover, if n 0 L d 2 (R d ) ≤ K 0 2 and if n 0 ∈ L ∞ (R d )
, then the solution is unique with

n(t) L p (R d ) ≤ n 0 L p (R d ) , max{1; d 2 -1} ≤ p ≤ d, and
n(t) L ∞ ≤ C κ, χ , n 0 L d (R d ) max { n 0 L 1 , n 0 L ∞ } .
The results of this theorem were known before the authors' paper [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF], except for three facts : (i) the smallness condition for the global existence is needed only in the critical norm

n 0 L d 2 (R d ) , (ii) the L ∞ bound of n, (iii)
the uniqueness of the solution in the case of bounded initial data n 0 .

We established (i) in the proof of Theorem 2 given in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]. Therefore, we will give only some comments about the existence issue (for the whole proof, please see [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]). Let us just remark that a smallness assumption is nevertheless necessary in the above theorem. Indeed, it is known since [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] that blow-up may occur in two space dimensions for large initial data. The L ∞ bound and the uniqueness result were not mentioned in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF], hence we will give the proof later in this section.

Finally, let us remark that in the case of the whole space R d under consideration and with the ad hoc decay conditions at infinity on n and c, the chemical density can be represented exactly

c(t, x) = R d E d (x -y)n(t, y)dy (17)
when f is constant, say f ≡ α, and where E d is the fundamental solution of (-∆ + αI). However, we prove Theorem 2 without using [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF]. That is the reason why our proof extends naturally to the case of a non constant f . Remark. System ( 16) has the well known conserved energy (see for example [START_REF] Brenner | Diffusion, attraction and collapse[END_REF], [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF] and [START_REF] Horstmann | Lyapunov functions and L p estimates for a class of reaction-diffusion systems[END_REF]) given by d dt κn(ln n -1) -χ

2 αc 2 + |∇c| 2 dx = -n|∇(κ ln n -χ c)| 2 dx ( 18
)
when f is constant, say f (c) = α. Let us just mention that this energy follows from (6) when we take G(φ) = φ, F (u) = u ln u and χ constant. Indeed, in this case φ defined in (4) is given by φ(c) = exp χ κ c . Therefore, (6) becomes

d dt n ln n - χ κ nc dx = -κ n|∇(ln n - χ κ c)| 2 dx - χ κ n ∂ t c dx. ( 19
)
Using the fact that n = -∆c + cf (c) (see ( 16)) in the second term of both the right and left hand side of ( 19), we obtain

d dt n ln n - χ 2κ |∇c| 2 + αc 2 + χ κ [ψ(c) -c 2 f (c)] dx = -κ n|∇(ln n - χ κ c)| 2 dx,
which is precisely [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF].

The existence issue in Theorem 2.

First of all, let us observe that by the maximum principle n > 0 and c ≥ 0 if n 0 ≥ 0, while with the decay at infinity condition on n and c, the initial mass M = R d n 0 (x) dx is conserved. Next, for constant χ the identity ( 14) becomes

d dt R d n p + 4κ p -1 p R d |∇n p/2 | 2 ≤ χ (p -1) R d n p+1 . ( 20
)
Using the Gagliardo-Nirenberg-Sobolev inequality (see [START_REF] Brezis | Analyse fonctionnelle. Théorie et applications[END_REF], [START_REF] Evans | Partial Differential Equations[END_REF]) in space dimension d > 2, we get for any p ≥ d 2 -1 (so that :

d 2 ≤ p + 1 ≤ dp d-2 ) R d n p+1 ≤ n p L pd d-2 (R d ) n L d 2 (R d ) = n p/2 2 L 2d d-2 (R d ) n L d 2 (R d ) ≤ C(d) ∇n p/2 2 L 2 (R d ) n L d 2 (R d )
.

(21) In dimension d = 2, following [START_REF] Jäger | On explosions of solutions to a system of partial differential equations modelling chemotaxis[END_REF] (see also [START_REF] Brezis | Analyse fonctionnelle. Théorie et applications[END_REF]), we have in a similar way for any p > 0

R 2 n p+1 ≤ C p + 1 p 2 ∇n p/2 2 L 2 (R 2 ) n L 1 (R 2 ) . (22)
Inequalities ( 21) and ( 22), together with [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF], lead to the main differential inequality in any dimension d ≥ 2

d dt R d n p ≤ (p -1) ∇n p/2 2 L 2 (R d ) χ C(d) n L d 2 (R d ) - 4κ p , ∀ max{1; d 2 -1} ≤ p < ∞ . ( 23
)
In dimension d = 2, ( 23) means that if the initial mass M is sufficiently small, in terms of p ≥ 1, then the n(t) L p (R 2 ) norm (for the same p) decreases for all times t ≥ 0.

In dimension d > 2 and for p = d 2 , [START_REF] Levine | Mathematical modelling of the onset of capillary formation initiating angiogenesis[END_REF] gives us that whenever we have initially

χ C(d) n 0 L d 2 (R d ) - 8κ d ≤ 0 , (24)
the n(t)

L d 2 (R d )
norm decreases for all times t ≥ 0. As a consequence, whenever [START_REF] Othmer | Aggregation, blowup and collapse : the ABC's of taxis in reinforced random walks[END_REF] holds true, all the n(t) L p (R d ) norms, with max{1; d 2 -1} ≤ p ≤ d 2 , decrease for all times t ≥ 0. In the same way, if the initial density

n 0 satisfies χ C(d) n 0 L d 2 (R d ) - 4κ p * ≤ 0 , (25)
for d 2 < p * < ∞, then [START_REF] Othmer | Aggregation, blowup and collapse : the ABC's of taxis in reinforced random walks[END_REF] follows again and all the n(t) L p (R d ) norms with max{1; d 2 -1} ≤ p ≤ p * decrease for all times t ≥ 0.

In conclusion, a smallness condition depending on 1 p with max{1; d 2 -1} ≤ p < ∞ on the initial density n 0 is sufficient to get a decreasing L p norm of n, in any dimension d ≥ 2. However, if one writes [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF] for (n -K) + with K > 0 large enough and uses the same techniques, then we get rid of the dependence of the smallness condition on p (see [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] for details). However, the L p bound we get this way depends also on t.

Finally, using a regularizing method, we derive from the a priori estimates the existence of solutions for system [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] and we get the first part of Theorem 2.

3.2

The L ∞ bound in Theorem 2.

Let us prove now that it is sufficient to control the L d norm of n in order to obtain the L ∞ bound of n, whenever n 0 ∈ L ∞ . Indeed, using straightforward interpolation and the Gagliardo-Nirenberg-Sobolev inequality, we have in dimension d = 2

R d n p+1 ≤ n L 2 n p 2 2 L 4 ≤ C n L 2 n p 2 L 2 n p 2 H 1 , (26)
while in dimension d > 2 it holds true that

R d n p+1 ≤ n p 2 L 2d d-2 n p 2 +1 L 2d d+2 ≤ C ∇n p 2 L 2 n p 2 L p n L d . (27)
Then, putting ( 26) and ( 27) in [START_REF] Keller | Initiation of slime mold aggregation viewed as an instability[END_REF] and using the Young's inequality in an appropriate way, we obtain in any dimension d ≥ 2

d dt R d n p + 4κ p -1 p R d |∇n p/2 | 2 ≤ 2κ p -1 p R d |∇n p/2 | 2 + p(p -1)C( χ , κ, n 2 L d ) n p L p (28)
Finally, taking n 0 ∈ (L 1 ∩ L ∞ )(R d ) and the smallness condition [START_REF] Patlak | Random walk with persistence and external bias[END_REF] on

n 0 L d 2 (R d ) so that n(t) L d ≤ n 0 L d , we have the main differential inequality d dt R d n p + 2κ p -1 p R d |∇n p/2 | 2 ≤ Cp(p -1) R d n p (29)
giving the following

L ∞ bound of n sup t≥0 n(t) L ∞ ≤ C max{ n 0 L 1 ; n 0 L ∞ } (30)
by the iterative technique of Alikakos [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF], where the constant C in the estimate (30) depends on κ, χ , n 0 L d (R d ) .

For the sake of completeness, we present the argument of [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF]. The constants may change from line to line. Let define p k = 2 k , for k ∈ N, and v = n p k-1 . Then, inequality (29) becomes

d dt R d v 2 + 2κ p k -1 p k R d |∇v| 2 ≤ C 1 p 2 k R d v 2 . ( 31
)
Next, by the Gagliardo-Nirenberg-Sobolev inequality and Young's inequality, we have (without making any difference between d = 2 and d > 2)

v 2 L 2 ≤ C 2 v 2α L 1 v 2(α-1) H 1 ≤ C 2 1 ε 1 α -1 v 2 L 1 + ε v 2 H 1 α ∈ (0, 1) (32)
and for ε sufficiently small

v 2 L 2 ≤ C 2 1 ε 1 α -1 v 2 L 1 + ε ∇v 2 L 2 . (33)
Multiplying ( 33) by (C 1 p 2 k + ε) and taking ε small enough so that

C 2 (C 1 p 2 k + ε)ε ≤ 2κ p k -1 p k , (34)
we obtain from (31)

d dt R d v 2 ≤ -ε R d v 2 + C 2 1 ε 1 α -1 (C 1 p 2 k + ε) R d v 2 . (35)
i.e. using again the function n,

d dt R d n p k ≤ -ε R d n p k + C 2 1 ε 1 α -1 (C 1 p 2 k + ε) sup t≥0 R d n p k-1 2 (36)
and by Gronwall lemma

R d n p k ≤ max{ R d n p k 0 ; C 2 ε -1 α (C 1 p 2 k + ε) sup t≥0 R d n p k-1 2 }, ∀ t ≥ 0 . ( 37
)
The estimate of n L p k follows by (37) using an iterative argument and a careful estimate of the constants in (37). Indeed, (34) gives us that ε = ε k is of order 1 p 2 k . Then, let us define

δ k = C 2 ε -1 α k (C 1 p 2 k + ε k ) and L = max{ n 0 L 1 ; n 0 L ∞ }.
Inequality (37) now reads as

R d n p k ≤ max{L p k ; δ k sup t≥0 R d n p k-1 2 }, ∀ t ≥ 0 . ( 38
)
Since δ k ≥ 1 for any k ∈ N, from (38) we obtain recursively

R d n p k ≤ δ k δ p 1 k-1 δ p 2 k-2 • • • δ p k-1 1 L p k . ( 39
)
Finally, let C be a constant such that δ k ≤ Cp

2( 1 α +1) k . We have n L p k ≤ C (1-1 p k ) 2 2( 1 α +1) P k i=1 i 2 i L (40)
and estimate (30) follows by taking k → +∞.

3.3

The uniqueness issue in Theorem 2.

In the following, we give the proof of the uniqueness of the global solution in the case of bounded data n 0 . This proof follows an idea of [START_REF] Gajewski | Global behaviour of a reaction-diffusion system modelling chemotaxis[END_REF] and was not mentioned in [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF].

Let (n 1 , c 1 ) and (n 2 , c 2 ) be two solutions of ( 16) and let us consider the function

f (n 1 , n 2 ) = n 1 ln n 1 + n 2 ln n 2 -(n 1 + n 2 ) ln( n 1 + n 2 2 ),
which satisfies

f (n 1 , n 2 ) ≥ 1 4 ( √ n 1 - √ n 2 ) 2 .
Then, using the second equation in [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] to estimate the L 2 norm of ∇(c 1 -c 2 ), we have

d dt R d f (n 1 , n 2 ) = -κ R d n 1 n 2 n 1 + n 2 |∇ ln( n 1 n 2 )| 2 + R d n 1 n 2 n 1 + n 2 ∇ ln( n 1 n 2 ) • [∇c 1 -∇c 2 ] ≤ -κ 2 R d n 1 n 2 n 1 + n 2 |∇ ln( n 1 n 2 )| 2 + 1 8κ n 1 + n 2 L ∞ R d |∇(c 1 -c 2 )| 2 ≤ χ 2 16ακ n 1 + n 2 L ∞ R d (n 1 -n 2 ) 2 ≤ χ 2 8ακ n 1 + n 2 L ∞ √ n 1 - √ n 2 2 L 2 , (41)
and the Gronwall lemma gives us that f (n 1 (t), n 2 (t)) = 0 for any t > 0 whenever n 1 and n 2 have the same initial data.

4 The parabolic-degenerate system.

In this section, we deal with the parabolic-degenerate system

               ∂ ∂t n = κ∆n -∇ • [n χ (c)∇c], t > 0, x ∈ Ω, ∂ ∂t c = -nc m , t > 0, x ∈ Ω, n(0, x) = n 0 (x), c(0, x) = c 0 (x), x ∈ Ω. ( 42
)
with Ω = R d , m ≥ 1 and the sensitivity function χ (c) a given positive function on R + , generally chosen as a decreasing function since sensitivity is lower for higher concentrations of the chemical because of saturation effects.

It is interesting to analyze systems of type (42), where the equation on c is a simple ODE, in order to understand the mechanism of the competition between the chemotactic effects due to the sensitivity function χ (c) and the dynamics of c. In this spirit, system (42) with χ (c) a positive continuous function on [0, ∞) has been analyzed in [START_REF] Corrias | A chemotaxis model motivated by angiogenesis[END_REF] and [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF]. In that paper, we prove Theorem 1, except for the L ∞ bound which is new. Let us note that the analysis is similar to what we did in Theorem 2 for the chemotaxis system [START_REF] Horstmann | From 1970 until present: the Keller-Segel model in chemotaxis and its consequences[END_REF] and founded on the following equivalent of (20) :

d dt R d n φ(c) p φ(c)+4κ p -1 p R d φ(c) ∇ n φ(c) p/2 2 = 1 κ (p-1) R d φ 2 (c) χ (c)c m n φ(c) p+1 . (43)
This is exactly the identity (13) obtained in section 2. In particular, we get the L ∞ bound in Theorem 1 using (43) and the iterative argument of Alikakos [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF] (see subsection 3.2). However, since the Alikakos [START_REF] Alikakos | L p bounds of solutions of reaction-diffusion equations[END_REF] estimates handle norms with respect to the Lebesgue measure, one should use here the following norm-equivalence : for all p ≥ 1, α ∈ R and v ∈ L p (R d ),

C 1 (p, α, c 0 L ∞ ) v L p (R d ) ≤ v(x) p φ(c(x)) α dx 1 p , ≤ C 2 (p, α, c 0 L ∞ ) v L p (R d )
which comes from the fact that φ(c) = exp 1 κ c 0 χ (c )dc and 0 ≤ c(x, t) ≤ c 0 L ∞ (R d ) . This way, the conclusion of Theorem 1 follows without difficulty.

Finally, let us mention that when χ (c) is such that

µ := 1 2 inf c≥0 { c χ χ + m} > 0, (44)
it has been proved in [START_REF] Corrias | A chemotaxis model motivated by angiogenesis[END_REF] that system (42) has an energy structure given by d dt E(t) ≤ - Note that E(t) cannot be obtained from the general identity [START_REF] Brenner | Physical mechanisms for chemotactic pattern formation by bacteria[END_REF] with an appropriate choice of G and F .

We obtain E(t) directly from system (42) in the (n, c) variables using the function N (c) := - For initial data with finite energy and d ≥ 2, the energy (45) allows to prove the existence of weak solutions to (42) where the drift term n χ (c)∇c is well defined. This energy also provides equiintegrability for n, therefore solutions cannot exhibit concentrations. But the question of propagation of smoothness of the solutions is largely open. Our result in Theorem 1 is a contribution in this direction.

Note that the relaxation of the smallness condition in dimension d ≥ 3 is an open problem. However, in two space dimensions, we have the following: Theorem 3 (L p bound for the angiogenesis system (42) in two dimensions). Assume d = 2, m ≥ 1, χ a positive say continuous differentiable function on [0, ∞) and µ > 0 in (44). Consider some nonnegative initial data (n 0 , c 0 ) with finite energy i.e. E(0) < ∞, such that ln(1 + |x|)n 0 ∈ L 1 (R d ) and c 0 ∈ L ∞ (R d ). Then, there is a weak solution (n, c) of system (42) such that for any fixed 1 ≤ p * < ∞,

n(t) L p (R 2 ) ≤ C(t, p * , n 0 L p (R 2 ) ), ∀ 1 ≤ p ≤ p * .
See [START_REF] Corrias | Global solutions of some chemotaxis and angiogenesis systems in high space dimensions[END_REF] details on this theorem.

Ω

  n κ|∇ ln(n)| 2 + µc m-1 |∇Φ(c)| 2 ≤ 0 , )| 2 + n ln(n)] and Φ (c) = χ c m .

  n dx = -κ n|∇ ln n| 2 dx + χ (c)∇n • ∇c dx = -κ n|∇ ln n| 2 dx -χ (c)c m ∂ t n + χ (c)c m |∇N | 2 2 ] dx = -κ n|∇ ln n| 2 dx -n( χ (c)c m ) c m |∇N | 2 2 .

b Département de Mathématiques et Applications, École Normale Supérieure, CNRS UMR8553 , 45 rue d'Ulm, F 75230 Paris cedex 05 email: perthame@dma.ens.fr, zaag@dma.ens.fr