Gerald Lang
email: gerald.lang@bem.edu

Fulfillment systems in Multi-Channel Retailing -Customer Expectations and Economic Performance

Keywords: Multi-Channel, Retail, Fulfillment, Distribution

Increasingly, store-based retailers are opening an additional online sales channel and becoming multi-channel retailers. The integration of these different channels raises the question how to redefine the strategic marketing elements and the operations, as the two channels have different constraints and require different competences. This multi-channel retailing has major impacts on the operations and the supply-chain management. Order fulfillment for the customers using the different sales channels is a key challenge, as customers are directly impacted from the performance of this process.

In this paper, we will highlight the different strategic elements and operational challenges to be addressed by a multi-channel retailer. In order to evaluate the match between a fulfillment system and a multi-channel business model, we propose a framework analyzing the overall performance of different fulfillment systems regarding the two dimensions customer expectations and economic performance.

INTRODUCTION

A growing number of traditional store-based retailers are becoming multi-channel retailers by opening an additional online-channel to offer their customers multiple ways of buying and interaction. At the same time, former pure online-retailers are opening stores or physical pickup points to allow their customers physical interaction and product pick-up.

Multi-channel retailing got primary attention principally in the field of multi-channel marketing (e.g. [START_REF] Wirtz | Multi-Channel Marketing -Grundlagen-Instrumente-Prozesse[END_REF]. Marketing literature and scholarly research address multichannel marketing mainly focusing on customer behavior on the one side (e.g. [START_REF] Venkatesan | Multichannel Shopping: Causes and Consequences[END_REF] and the conflicts between channels on the other side (e.g. [START_REF] Falk | Identifying Cross-Channel Dissynergies for Multichannel Service Providers[END_REF][START_REF] Avery | Adding Bricks to Clicks: The Contingencies Driving Cannibalization and Complementarity in Multichannel Retailing[END_REF]. Several authors (e.g. [START_REF] Schröder | Multichannel-Retailing -Marketing in Mehrkanalsystemen des Einzelhandels[END_REF][START_REF] Schobesberger | Multichannel-Retailing im Einzelhandel -Entwicklung, Motivation, Einflussfaktoren[END_REF][START_REF] Heinemann | Multi-Channel-Handel -Erfolgsfaktoren und Best Practices[END_REF][START_REF] Heinemann | Verkauf auf allen Kanälen -Multi-Channel-Systeme erfolgsorientiert ausrichten[END_REF] analyze different multi-channel strategies and implications on a retailers' marketing mix.

Whereas operational implications of online retailing have received attention since the end of the 1990s (e.g. [START_REF] Turban | Electronic Commerce[END_REF][START_REF] Swaminathan | Models for Supply Chains in E-Business[END_REF], specific analysis of operational implications of multi-channel retailing remains scarce. [START_REF] Agatz | E-fulfillment and Multi-Channel Distribution -A Review[END_REF] reviewed the specific supply chain management issues of internet fulfillment in a multichannel environment. They identified an important number of topics relevant for efulfillment, but recognized that there are only very few dedicated models on multi-channel retailing.

Multi-channel retailers with different business models are using different fulfillment systems to deliver the products to their customers: The online shop of the French retailer Fnac (cultural and electronics products), fnac.com, started its online activity completely separated from its store activity, with different warehouse locations and fulfillment organizations. They gradually integrated their logistics activities and are operating today a fully integrated warehouse, preparing deliveries for the stores as well as for the individual online customers.

Tesco.com, the online shop of the major UK retailer Tesco, prepares the customer's orders exclusively in the Tesco store next to the customer [START_REF] Smith | Tesco's supply chain management[END_REF]. So does monoprix.fr, the online shop of the French city-center supermarket chain Monoprix.

These observations suggest that a specific multi-channel business model may induce a specific, most suitable fulfillment system.

If this is the case, it should then be possible to evaluate how a fulfillment system fits a given business model.

From an operational point of view, fulfillment systems have to be mainly time-and costefficient. From a strategic marketing point of view, the fulfillment system has to be able to meet the customers' expectations created by the chosen business model.

We therefore propose to develop a qualitative framework to evaluate how a specific multichannel fulfillment system matches the business model of multi-channel retailers both in terms of the expectations of their customers on the one hand and the economic performances on the other hand. This framework should serve as basis for further empirical studies.

Our paper is organized in two parts:

In the first part, we will highlight the main characteristics of multi-channel retailing and present the major elements of the retailers' strategy, the retail mix, which have to be adapted to a multi-channel retailing strategy. 1 We then will explore the specific operational challenges to be faced when operating a multichannel activity.

This first part will be completed with the presentation of the five major multi-channel fulfillment systems used to deliver products to the customers.

In the second part, we will propose a qualitative analysis framework in order to evaluate the overall performance of a given fulfillment system, combining the two dimensions 'customer expectations' and 'economic performance'. The parameters of the framework as well as its applications will be based on the analysis of the strategic and operational challenges developed in the first part.

Applying the analysis framework to the five major fulfillment systems leads us to discuss how each of the systems impacts the different parameters.

We conclude our paper presenting managerial implications and suggested future research avenues to provide evidence if our framework is valid for our purpose.

Methodology

Our analysis is based on the one hand on our longitudinal industry observation of retailers' practices, completed with the analysis of company information and press articles. Moreover, 1 In our paper, we will concentrate on multi-channel retailers operating at least several physical stores and an online sales channel under the same brand and with the same types of products. Moreover, we focus on the distribution of physical products, as dematerialized products do not need a physical distribution. We exclude as well specific configurations (which may nonetheless be dominant in particular sectors), like delivery from manufacturers or 'complex assortments' (de [START_REF] Koster | The Logistics behind the Enter Click[END_REF] with bulky items and temperature sensitive goods.

we conducted a series of interviews with French and UK senior operations managers and consultants in the retail sector. This observation allowed us to identify five major multichannel fulfillment systems.

To be able to evaluate these fulfillment systems regarding the two dimensions 'economic performance' and 'customer expectations', we have to identify the relevant parameters to integrate into our proposed framework. To do so, we analyzed Marketing and Operations general literature and scholarly research. Exploration of the main strategy elements and operational challenges suggests number of parameters, indicating economic performance and customer expectations. We completed the identified parameters for evaluating customer expectations with dimensions from the ServQual and NetQual scales, measuring service quality perception from the customers. These parameters of our framework will be the dependent parameters, whereas the analyzed fulfillment systems will be the independent parameters. The exploration of the strategy factors and economic performances in the first part will allow us to apply the framework to the different fulfillment models by assessing the chosen performance parameters for each of the fulfillment systems. This shall provide a global evaluation for the each of the analyzed fulfillment systems.

STRATEGY FACTORS IN MULTI-CHANNEL RETAILING

The objective of this chapter is to summarize the context of multi-channel retailing as well as the main drivers for customer expectations and economic performances, which will enable us to identify the relevant parameters to use in our proposed framework. We will first present the main characteristics of multi-channel retailing. We then will investigate how the business model of a multi-channel retailer, characterized by different elements of the retail mix, will have to be adapted to a multi-channel strategy. An exploration of the major operational challenges for a multi-channel retailer will conclude this chapter.

Multi-Channel Retailing: Characteristics & Customer Behavior

Worldwide retailing is still dominated by the traditional store retailing, as 95% of food retailing and 80% of non-food retailing is realized through physical store-based retailers [START_REF] Dioux | La Distribution -Stratégies des Réseaux et Management des Enseignes[END_REF], and online channels still represent only a small part of the total sales of the retailers. Nevertheless, the motivation to open online sales-channels is strong due to the fact that e-commerce is realizing important annual turnover growth rates of about 30% for the last few years, whilst most other retail forms stay stable or even decline. It is therefore interesting for a store retailer to invest in the creation of an additional online sales channel.

To implement a multi-channel strategy, retailers have to decide how to adapt their business model by adjusting the elements of the retail mix. At the same time, they have to face and resolve numerous operational challenges.

In multi-channel retailing, customers expect at least the same level of service, counsel, and information on every channel (cf. [START_REF] Heinemann | Multi-Channel-Handel -Erfolgsfaktoren und Best Practices[END_REF]. They expect to have identical customer loyalty programs across all channels, to be able to pay and pick up in the shop a product ordered on the web site, or to be able to return and exchange a product bought on the web in a store [START_REF] Berman | A Guide to Developing and Managing a Well-Integrated Multi-Channel Retail Strategy[END_REF]). This channel hopping, which is a legitimate desire from a customer's point of view, may nevertheless prove to be very difficult to implement for the retailer [START_REF] Vanheems | Distribution Multicanal -Pourquoi les clients mixtes doivent faire l'objet d'une attention particulière ?[END_REF].

At the same time, online shoppers transfer their perception and experiences from a physical store to the website of the same brand and vice versa. Customers tend to transpose a bad experience in one channel to the other channels of the same distributor [START_REF] Heinemann | Multi-Channel-Handel -Erfolgsfaktoren und Best Practices[END_REF].

As soon as their expectations are not fulfilled, they may adapt their purchasing habits, in the worst case stop buying with this retailer at all. It is therefore vital for a multi-channel retailer to make sure that every channel always meets the same level of expectations of the customers.

Multi-Channel Retail Mix

The Retail Mix consists of the mix of variables, including geographical location, merchandising, communication, price, services, product range and the personnel, which form the overall strategic marketing components of a retailing activity. It is the right mix of all these factors which determines if a retail concept is viable and successful.

Compared to a store retailer, the online sales channel is not only a new channel, but rather a new business model, where experiences from the traditional (brick and mortar) store channel may not be valid any more [START_REF] Heinemann | Multi-Channel-Handel -Erfolgsfaktoren und Best Practices[END_REF]. Decisive success factors in a store like store location, sales personnel and material management in the shop are not the same success factors in an online sales channel.

Retailers which decide to open an e-commerce activity beside their existing store sales channel have to make sure that the co-existence of the two sales channels is working well.

They have to decide how to adapt the different elements of the retail mix to each channel, which product strategy to chose, which prices to apply, how to maintain a consistent image across the channels or what role to assign to each channel, to name a few (cf. Berman & Evans, 2009).

We will briefly describe the major issues of some of the most important variables of the Retail Mix, assessing their relevance and importance in the store, the online channel and the multichannel strategy.

Product Mix

The product mix is the major variable determining the activity of a retailer. It establishes if a retailer is specialized in selected product categories or a generalist.

If a store based retailer opens an additional online channel, he has to decide which product mix he offers on this new channel. He can offer the identical product mix, a combination or part of his existing product mix or a completely different additional product mix on the new channel.

Location of the stores

A physical store's success is to a great extend influenced by its location. In order to determine if the location can attract the store's target customers, a store based retailer has to analyze socioeconomic data, transport access, mix of competitors, costs and regulation issues.

These parameters, decisive for a physical store, have generally no influence on a pure online sales channel, as access to the websites is independent from a geographical location.

In a multi-channel strategy, however, the store location remains important; the density of the store network can be an important factor for offering store pick-up, returns and after sales services in the stores for their online customers.

Pricing

The pricing policy applied in a store may take into consideration the environment of the store (socioeconomic characteristics of the geographical catchment area) and the price levels of the local competitors. Such a local pricing policy which can be applied in a store is incompatible with the global price transparency in an online channel and the exposure to price comparisons.

The pricing policy of a multi-channel retailer encounters specific difficulties: the global price transparency on the internet, the specific cost structures of the store and the online channel, the existence of specific offers and the different means of payment render it difficult to apply a consistent pricing policy across all channels.

Especially the price transparency makes it difficult for a retailer to offer different prices for the same product on different sales channels.

Service, store pick-up and return

A multi-channel service offer can be composed of special delivery options (delivery speed, delivery timing, information on order status), the exchange policy, return policy, refunding, after sales service, payment options and consumer credits.

Multi-Channel retailers can offer either to deliver the order to the customer's home or to let the customer pick-up the order in one of the stores. This service allows the customer to pick up the order whenever he likes, even sooner than with home delivery. 2Allowing customers in-store return or exchange can avoid the cost and time of returning via mail. At the same time, this offers an important possibility for the retailer to create a personal contact between the client and the sales personnel to increase customer loyalty and to generate additional sales.

Personnel

The sales personnel play a decisive role in stores. They welcome clients, provide counsel and advice and can influence sales. Their behavior and attitude contribute in an important way to the atmosphere and ambiance of the store and its concept.

This important parameter in a store is irrelevant in a pure online channel, except for telephone hotline contacts or the delivery service personnel.

In a multi-channel strategy, however, the store personnel have to be able to orient and counsel the client using different channels. It is very important for the client not to perceive any differences or barriers between the channels.

Operational Challenges of Multi-Channel Retailing

In general, the operational organization of a retailer is mainly focused on efficient store replenishment, lowering costs and shortening the product turnover time.

To offer an online channel to its customers, a retailer has to be able to manage its operations in a very different way from its physical store activity. E-fulfillment may be considered as one of the most expensive and critical operations of Internet retailers (de [START_REF] Koster | The Logistics behind the Enter Click[END_REF], logistics and stock management becoming core competencies [START_REF] Benghozi | Relations interentreprises et nouveaux modèles d'affaires[END_REF]. Keeping stock of a high number of different references, making it available, picking and packing, sending and delivering quickly at a low cost are the numerous constraints to respect.

In the following, we will present and discuss the main operational issues, which are decisive for the overall performance of the multi-channel retailing.

Product Availability

Product availability is one of the most important parameters for customer satisfaction (Browne, 2009). A store customer usually can compare the products present in the store.

Availability is indicated through the simple presence of a product in the store shelve. A customer can ask the sales personnel if there is any stock of the product or when it will be available again.

An online client needs to have all this information indicated on the website together with the product information. The online channel offers the possibility to display the remaining stock of a chosen product, or even, in a multi-channel context, the remaining stock and the location of the store where the product is available for testing, purchasing or withdrawal.

Displaying the availability of products or the delay of their replacement is an important marketing tool impacting the sales and influencing customer satisfaction.

Stock keeping

Stores usually receive deliveries either directly from the manufacturer or from one or several logistics platforms with either warehousing or cross-docking function.

As online orders may consist of different products, composing the orders from different warehouse locations may prove to be difficult and inefficient in terms of cost and delivery time. A central warehouse and fulfillment center may therefore, at a first view, be more adapted to the online channel.

However, multi-channel retailers may as well pick the items for an online order directly in the stores.

For an online order, the retailer has some flexibility where from he delivers the products to his customers, as he can use multiple stock locations and therefore offer a wider product range than the one contained in a single warehouse, as multiple channels may share their inventories. In-store inventories can therefore be available for online customers' orders [START_REF] Agatz | E-fulfillment and Multi-Channel Distribution -A Review[END_REF].

Centralizing stock allows easier monitoring of the stock levels for different products. If a retailer is serving its online clients (totally or partially) from stock in the stores, the overall stock level on these different locations is difficult to monitor. This is the same problem for multi-channel retailers who offer the possibility to their customers to order online and withdraw the product in a nearby store.

As information on product availability is a crucial parameter for the customers buying decision, the accuracy of the stock level information is very important.

Order taking and fulfillment

In an online channel, the customer enters the type and number of products he wants to acquire and enters the payment information (usually done by credit cards), the online channel therefore becomes a self-service channel [START_REF] Falk | Identifying Cross-Channel Dissynergies for Multichannel Service Providers[END_REF]. All this information entered by the customer triggers the order preparation and fulfillment process.

Retailers once delivering their stores from centralized distribution centers (often with complete boxes or pallets) have to re-think the way orders are prepared, as their organizations are not designed for small individual customers' orders. They must therefore be reorganized to treat these orders (de [START_REF] Koster | The Logistics behind the Enter Click[END_REF], implementing a process of single item picking and packing.

As mentioned above, another option for the retailer is to pick the items in the stores, where the products are available on the shelves, and to prepare the order in the shops.

Furthermore, orders may be consolidated from multiple stock locations (for example from a warehouse and a store), which needs a specific process. The more complex the fulfilment process is organized, the more difficult it becomes to respect the quality and efficiency of this process.

Delivery to the customer

Besides the availability of the product, the different delivery options for a product are an important parameter for the online channel client. Different delivery service levels offered like delivery in 24 hours, delivery to a pick-up point, national post service etc. are becoming a product differentiation factor of the retailer.

Delivery to the customers' homes, which is a distinctive element of an online channel, implies to deal with cost and organizational issues. The usually small transactions have to be managed and delivered in an efficient way, while respecting the promised delivery timing.

The costs of the home delivery can be totally or partially paid by the customer. This becomes an important parameter of choice for the client, as no direct delivery costs appear in the store channel. On the other hand, the costs of delivery can be used as a marketing tool for special offers ("your next order free of charge") or minimum order amounts ("delivery free of charge for every order above 100 Euros").

Delivery becomes therefore a service parameter, a price parameter and a promotional parameter.

Reverse logistic

Store customers can usually see, touch and try the goods before purchasing. Returns are limited and very often linked to an exchange of the product, where the returned product is put back into the shelves. Defect products are reimbursed.

As online customers cannot see, touch and try the products, online retailers usually offer good conditions for the customer to return the products they don't like (due to legal and regulatory obligations, but as well as part of their customer service offer).

Return handling must be quick in the eyes of the customers, either to get a product exchange or to get reimbursed and the customer account re-credited. The process must as well be efficient as it is a very manual process and impacts the stock management of the returned products [START_REF] Ko | A Genetic Algorithm Approach to Developing the Multi-Echelon Reverse Logistics Network for Product Returns[END_REF].

A physical store channel can be an easy way in the customer's eyes to return the products.

MULTI-CHANNEL FULFILLMENT SYSTEMS

To operate the multi-channel strategy, retailers have to set up a fulfillment system to deliver products to the customers. This fulfillment system, organizing the process from order intake to delivery to the customer, has to match the business model and to reply to the operational challenges. Industry players are using different fulfillment systems, and several authors suggest typologies of fulfillment schemes, focusing on the locus of order preparation (de [START_REF] Koster | The Logistics behind the Enter Click[END_REF][START_REF] Durand | Les modèles logistiques du "B to C" en France, 7e journée nantaise de recherche sur le e-marketing[END_REF].

Multi-channel retailing per se intends to offer the clients the choice where and how to order and where and how to get the products delivered.

The major choice for the client is to shop in the store or shop online. If shopping online, the client can furthermore opt for delivery at home or pick up his products in one of the retailer's stores.

The retailer has to set up a fulfillment system able to respect the client's choice of the delivery mode, either by creating a new logistics infrastructure, or using the existing one, by adapting the existing processes.

In order to be able to analyze the fulfillment systems from a customer expectation's perspective, we decided to distinguish the fulfillment systems by the delivery option chosen by the customer:

Clients may either choose the direct home delivery for their purchases completed on the online channel or choose to pick up their purchased items in a physical store of the retailer. This in-store pick-up is a delivery option only multi-channel retailers can offer and which may be chosen to 30-40% of the customers at some multi-channel retailers [START_REF] Berman | A Guide to Developing and Managing a Well-Integrated Multi-Channel Retail Strategy[END_REF].

A client going into a store in order to buy a product, searches, compares, picks the product out of the shelve and pays the product. Single-channel store shoppers may therefore not be affected by the multi-channel fulfillment organization.

Direct Home Delivery

Direct home delivery can be completed either by the retailer's own delivery organization, a delivery logistics provider or via an external pick-up point network 3 . Before entering in this 'delivery pipeline', products have to be picked, prepared, packed and expedited, which can be done either in a central warehouse or distribution center, in a store, or in multiple locations (central warehouse and one or more stores). 3 We consider a delivery via an external pick-up-point network (usually neighborhood shops linked to a network which offers the retreat of parcels for the clients of different retailers) like a home delivery, as the fulfillment process is quite identical, with a delivery of (sometimes pre-sorted) parcels to the pick-up point network's injection hub. Although this is influencing the ease of access, this way of delivery cannot provide a contact with the retailer as in his own store.

Direct Delivery from central warehouse

Picking, preparation and packing of ordered items can be done from a central warehouse stocking the products (Figure 1.1). This central warehouse can either be a purpose-built entity to serve exclusively the online channel or the existing warehouse for the store operations (in which case the processes in the warehouse have to be adapted).

Direct Delivery from a store

Picking, preparation and packing of the ordered items can be done inside a physical store out of the shelves. In this case, operators pick the products into a specific order preparation cart from the shelves and then pack and send the completed order to the customer (Figure 1.2).

Direct Delivery after multi-stock consolidation

If the desired articles of an order are not available in one place, picking of the ordered items has to be done in a central warehouse as well as inside a physical store from the shelves. The picked items have then to be consolidated, prepared and packed in one site (Figure 1.3). In some cases, the multichannel

Client Pick-up in the store

A customer ordering online may choose to pick up his products in one of the retailer's stores.

In this case, products have to be picked, prepared, packed and kept ready for pick-up in the store.

Direct in-store order preparation

If all the products of the customer's order are available in the store chosen by the customer to pick-up, the picking, preparation and packing of the order can be done immediately in the store (Figure 2.1).

In-store order consolidation from multiple stocks

If some of the products of the customer's order are not available in the store chosen by the customer for pick-up, the unavailable items must be ordered from the central warehouse (or other stores) to complete the order. The preparation and packing of the order can be done only after consolidation of all items in the store (Figure 2.2).

(A specific case of this in-store order consolidation can be a complete order preparation in a central warehouse sent to the store only to be picked up by the customer, without further adding any product in the store). -Ease of access and return of products (indicating the ease for a client either to get the products delivered/to pick them up or to return/exchange the products received)

In order to assess more precisely how a fulfillment system corresponds to the customer expectations, we completed these parameters with parameters from the ServQual framework, developed by [START_REF] Parasuraman | SERVQUAL: A Multiple-Item Scale for Measuring Consumer Perceptions of Service Quality[END_REF] to measure service quality. Nevertheless, this traditional service quality may not be applicable in an online environment [START_REF] Parasuraman | E-S-Qual: A Multi-Item Scale for Assessing Electronic Service Quality[END_REF]. [START_REF] Bressolles | La qualité de service électronique: NetQu@l Proposition d'une échelle de mesure appliqué aux sites marchands et effets modérateurs[END_REF] developed the NetQual framework to specifically assess electronic service quality.

As multi-channel retailing provides both 'traditional' and 'online' service experiences, we have to combine these two approaches in order to identify all the relevant dimensions to customer expectations and retain only those which may be directly impacted by the fulfilment model. We did not retain those dimensions having only an indirect link with the fulfillment systems.

These elements constitute the analysis framework we propose to analyze the overall performance of specific fulfillment systems.

Application of the Analysis Framework

We are now able to apply this proposed framework to the five identified fulfillment systems according to our objective to evaluate the fit between a multi-channel retailer's business model and a specific fulfillment system.

To be able to analyze each of the abovementioned fulfillment systems, we will explore the sensitivity of each of the parameters regarding the main characteristics of the different fulfillment systems. 4 We apply numerical scores to each parameter, depending if the efficiency of this parameter is high (2), medium (1) or low (0). This qualitative evaluation, based on the discussion of the different elements of the retail mix and the operational challenges, aims to compare the relative efficiency of one parameter in different fulfillment systems. We do not intend to provide an absolute performance test.

Quality and Quantity of Information

Customers expect correct information, especially on pricing and delivery costs of the different channels, but as well on other parameters like product availability, speed of delivery and product characteristics. The retailer's challenge is always to have the right information available on every channel.

4 Each of the above identified parameters expressing customer expectations and economic performance of a specific fulfillment system has multiple implications on other parameters. It is to retain that these parameters may be interdependent (e.g. the width of the product offer impacts the product availability, the picking efficiency and the fulfillment infrastructure costs).

Using a centralized stock provides the highest chance to have the right information available (score: 2), whereas an order preparation from stores induces the problem of obtaining the right information on the store stock level (score: 1). If the order preparation has to be completed on several stock locations (central warehouse and stores), it is very difficult to obtain correct information and the quality of information may be low (score: 0).

Reliability and Respect of commitments

Customers expect retailers to keep their promises, especially in terms of delivery time and delivery service. Central warehouse order preparation should facilitate this, as the quality of information is high and the fulfillment process can be optimized (score: 2). Order preparation in the stores suffers from difficulties in keeping a real-time stock level of all the products.

Therefore, store preparation may guarantee only a medium level of reliability and respect of commitments (score: 1), whereas multi-stock order preparation may only provide a low level (score: 0).

Offer

Customers expect to find the offer they expect on every channel, a large or narrow product range with the appropriate depth of assortment. The retailer has to make sure to be able to offer the products customers expect on every channel.

Order fulfillment at a central warehouse or from multiple stocks should be capable to provide the entire range of the products (both score: 2), order preparation at the store is limited to the only product assortment in stock at the store (score: 0).

Product availability

The customer expects that the products he intends to buy are available in the channel he desires to use.

A central warehouse order preparation may provide only a medium level of product availability (score: 1) compared to the other options, as the availability depends on the stock keeping and forecasting precision of this single warehouse. Stores can only guarantee a low level of product availability because of the small number of stock for every referenced product and the continuous buying from customers in the store (score: 0).

Multi-stock order preparation may use the stock of different entities (stores and central warehouse) and therefore guarantee a very high level of product availability (score: 2), especially in case of a high number of articles per customer order.

Speed of delivery

Customers usually expect a rapid delivery of the products they bought and to be informed about the delivery timing. Retailers therefore have to implement a fast and reliable delivery process, which may imply an external delivery service provider.

A central warehouse usually needs to bridge the distance to the final customer, which may be long and therefore time-consuming. A medium level of speed of delivery may therefore be guaranteed (score: 1). As stores are closer to the final customer, the speed of delivery can be high when orders are prepared in the store (score: 2). Orders prepared from multiple stock locations need a long time to be consolidated and can therefore only provide a low relative delivery speed (score: 0).

Ease of access and return of products

Customers expect easy access to the retailer to either pick up the products ordered or to return or exchange a product in case of dissatisfaction.

Home delivery from a central warehouse may make returns and exchange difficult for customers (score: 0), who have to use a postal service for returning the products (except in the case the retailer owns a quite dense network of stores and the store channel is fully integrated to handle returns and exchange from the online channel). Order preparation in the store is usually done in the store closest to the customer. This store is therefore easy to access for the customer (score: 2). Multi-stock preparation can only provide a medium level of ease of access and return (score: 1). Although the customer can return a product to the closest store, exchange can be impossible if the products to be exchanged are only available at another stock location.

Assurance

The customers expect competent and friendly personnel in the store for buying, picking up their orders, return or exchange. The same expectation may apply for the personnel on telephone customer hotlines or the delivery personnel. These functions being very often outsourced (especially the latter), retailers have to focus on their own sales personnel and make sure they provide the level of assurance the customer is expecting by providing the same information to all channels. The assurance of the personnel can only be provided in the case of order pick-up of the customer in the stores (score: 2; other option scores: 0).

Stock efficiency

For the retailer, the efficiency of stock management is crucial for numerous reasons. Stock keeping and monitoring of stock levels, balancing between different stock locations and the forecasting of sales and product reorders influence the availability of the products and therefore the sales. Efficient stock handling is as well an important cost element.

Central stock management can provide a high level of efficiency (score: 2), whereas a store can only provide a medium efficiency (score: 1) because of the low number of products per article and the continuous stock movement due to customers purchases out of the shelves. The handling of multiple stocks in the fulfillment process is the least efficient (score: 0).

Picking efficiency

The picking of products out of stock to prepare the customer's order is the key process of the overall fulfillment process, representing a very important part of the warehouse operating costs. This process is nevertheless transparent to the customer.

A central warehouse can be organized in a way to optimize the picking process by planning and reducing the way picking operators have to go during the process (score: 2). Depending on the volumes and the type of articles, it is possible to split the stock for whole cartons or open cartons picking or separate zones for stocking and picking. This optimization cannot be provided in a store or in multiple stock sites (scores: 0). Stores' layouts and product locations are planned to make the customer spend the maximum time in the store and not to optimize the picking (de [START_REF] Koster | The Logistics behind the Enter Click[END_REF].

Delivery cost efficiency

Customers expect to pay the lowest extra cost possible for the delivery of their online orders and the retailer has to reduce the overall cost of the delivery.

Delivery of orders prepared in a central warehouse has always to include the transport to the customer's home and therefore only achieve a medium efficiency (score: 1) compared to the delivery from a store (score: 2), where the distance to the final customer is short. Pick-up by the customer from the store where the order was prepared is the less costly way of delivery for the retailer. Any delivery from an order prepared in multiple stock locations (for home delivery or to a store for pick-up) is least efficient (score: 0), as the consolidation process of the order implies multiple costs of handling and transportation. This indicates several results:

-Home delivery from a central warehouse (System 1.1) is the fulfillment system best meeting the combination of the two dimensions customer expectations and economic performance, providing a total score of 16. This system provides the highest score in economic performance (9) and the second highest score in customer expectations (7).

-Customer pick-up after preparation in the store (system 2.1) provides the highest score in customer expectation (8) and a medium score in economic performance (6). The overall performance of this system is second best (total score of 14).

-Home delivery after store preparation (system 1.2) provides medium scores in customer expectations (6) and economic performance (6), ranking third best in the overall score (12).

-Any fulfillment system implying the order preparation out of multiple stocks appears to be less efficient: If the order is picked up by the customer (system 2.2), the total score is 9, if the order is delivered at home (system 1.3), the total score is only 7. In terms of economic performance, these two systems provide the lowest score (2).

-Economic performances of order preparation in the store or from multiple stocks are identical for both home delivery or in-store pick up by the customer. The difference in customer expectations (difference of 2) results from the possibility to have a personal contact with the customer, therefore responding to the customer expectation of 'Assurance', which can only be provided in a store.

DISCUSSION, MANAGERIAL IMPLICATIONS AND RESEARCH AVENUES

The qualitative analysis of the different fulfillment systems and their impact on customer expectations and economic performance reveals that different fulfillment systems can provide high overall performance levels in terms of customer expectations and economic performance.

The framework shows that orders for home delivery prepared in a central warehouse present only a slightly higher overall performance than an order prepared in a store for customer store-pick-up.

A retailer has to assess the importance of each of the analyzed parameters for his specific retail mix and to optimize the trade-offs between channel fulfillment synergies and specific needs on different channels. Depending on the product mix between food and non-food products, the width of product assortment and the characteristics of the existing store network, specific solutions may be more appropriate than others to fit with a specific business model.

Whereas proximity to customers is best achieved through a store network, stores add complexity on different levels in the fulfillment process which affect timing, costs and reliability.

Multi-stock consolidation seems to be always by far the less efficient model, regarding the customer expectations and the economic performance. This corresponds to the three initially mentioned multi-channel retailers: Fnac abandoned the multi-stock preparation and completely integrated their fulfillment for the different channels, and neither Tesco nor Monoprix apply a multi-stock solution, whilst optimizing the in-store order preparation.

Our proposed framework could prove useful for multi-channel retailers in various ways: It provides a tool to evaluate the chosen fulfillment system and therefore identifying improvement potentials on specific parameters.

It could also provide helpful to measure customer satisfaction. Moreover, it should clearly be useful in anticipating the needs for fulfillment systems when planning to implement a multichannel strategy.

The proposed qualitative framework should be further validated by conducting supplementary empirical analysis, although it may be difficult to compare economic performances and customer expectations of different fulfillment systems in a same company. Another difficulty may be to isolate the impacts of the supply chain from other impacts (marketing, pricing, product policy etc.).

A limitation of our framework is the fact that it does not assess the interdependencies between the different dependent parameters, but only focuses on one major impact at a time.

A first step to further back this framework could be to conduct a dynamic analysis through a simulation, testing values for the different parameters of the analysis framework.

Research could aim to find evidence if there is an 'optimum' fulfillment system for every multi-channel retailer. Such an optimal model could be defined depending on different independent variables like product assortment, average number of items per shopping basket, value of products, product characteristics, degree of channel integration etc. This would also include an analysis of the trade-off between cost-and customer service level optimization.

Such a research could as well reveal that there is no optimum model, but just a set of independent and non stabilized models evolving quickly as the retailers' strategy mix evolves.

ACKNOWLEDGEMENT

The work on this paper was partially supported by the Innovation and Regulation Chair of Orange, Ecole Polytechnique, Telecom Paris.

Figure 1 . 1 :

 11 Figure 1.1: Direct delivery from central warehouse

Figure 2 . 1 :

 21 Figure 2.1: Direct in-store order-preparation

  Therefore, we selected the following dimensions out of the ServQual and NetQual scales for our analysis: -Quality and quantity of information (NetQual) (indicating the availability and correctness of the information provided in terms of product availability, stock level, delivery times etc. either on the website or in the stores) -Reliability and respect of commitments (ServQual & NetQual) (indicating the ability to deliver the products in terms of quantity, price, description and timing) -Offer (NetQual) (indicating the lengths and the width of the product range available on the different channels) -Assurance (ServQual) (indicating the competence and friendliness of the personnel usually in the store for buying, pick-up and exchange transactions of the client).

In case of unavailable items, the retailer can chose to send the missing items directly to the customer and only reserve the available items in the store for pick-up. In this case, the customer experience will be negative, as the customer's decision to pick up his order in the store can't be fulfilled.

Medium (1) Low ( 0) High ( 2) Low (0) High (2)

Product availability

Medium (1) Low (0) High ( 2) Low (0) High (2)

Speed of delivery

Medium (1) High ( 2)

Ease of access and return of products

Economic performance

Stock efficiency

High (2) Medium ( 1) Low ( 0) Medium ( 1) Low (0)

Picking efficiency

High (2) Low ( 0) Low ( 0) Low ( 0) Low (0)

THE ANALYSIS FRAMEWORK

In this part, we will first determine the parameters to integrate in our proposed framework.

We then will apply this framework to the five abovementioned fulfillment systems and discuss the results. We will conclude this part discussing the limitations of the framework, suggestions for further research and managerial implications.

Customer expectations and economic performance

Having analyzed the main strategy factors and operational challenges, and identified the five major fulfillment systems, we now have to determine the parameters to include into our proposed framework in order to evaluate these fulfillment systems.

The different fulfillment systems lead to different levels of complexity, of costs and of product availability and therefore different levels of customer satisfaction and economic performance.

To explore how a fulfillment system fits a given business model, we have to evaluate the overall performance of the different multi-channel fulfillment systems, combining the two angles of analysis: On the one hand, we analyze to which extend a given fulfillment system is able to meet the customer expectations. On the other hand, we analyze the economic performance of each model.

In our proposed framework, we will confront the different fulfillment systems to several parameters which are relevant for the customer expectations and the economic performance.

The fulfillment systems may be considered as the independent variables of our analysis, whereas the different parameters of our framework represent the dependent variables.

The identification of these parameters results from the exploration of the main strategy elements and operational challenges.

Channel organization efficiency

As stated above, customers want to be able to use different channels simultaneously. A retailer has therefore to manage all the channels in an integrated manner.

Preparing orders for home delivery from a central warehouse is the simplest and most efficient organization (score: 2). Order preparation in the store involves the online and the store channel in the fulfillment process, therefore adding a level of complexity (score: 1).

Multi-stock preparation adds further complexity, being therefore the most difficult solution to achieve an efficient channel organization (score: 0).

Fulfillment infrastructure costs

The fulfillment process may generate important investment costs, when a specific site is built as the stock and the distribution center, or if the design and the organization of an existing facility have to be modified. Renting or outsourcing this infrastructure generates additional costs as well.

A central warehouse order preparation generates a high infrastructure cost (score: 0), whereas the store preparation generates the lowest additional infrastructure costs (score: 2). Order preparation from multiple stocks may generate medium additional infrastructure costs (score:

1) to organize the consolidation of the orders.

Return handling efficiency

Customers expect their returns to be processed quickly, to receive an exchange product or to get reimbursed the purchasing price. The retailer must therefore provide a quick and efficient return handling process.

A central warehouse enables creation of optimized return processing (score: 2) to check the returns, repack and restock them and send the exchange items to the customers. Multi-stock fulfillment can only provide a lower rate of return efficiency (score: 1), as the returned items arrive in different locations and the exchange items may only be available on different stock locations. For a store it is very difficult to offer an optimized return process (score: 0), especially when returned products are not part of the store's product offer.

We can therefore summarize the impacts of the different fulfillment systems on the different parameters as follows (Figure 3):


Oscillations Conditions in Homogenous Systems

 is extended to local homogeneity. Next, sufficient conditions of oscillations existence for locally homogeneous systems are formulated. The proposed approach allows one to estimate the number of limit cycles and the regions of their location. Efficiency of the technique is demonstrated on two academic examples.

INTRODUCTION

The homogeneity is a propriety of nonlinear dynamical systems introduced more than thirty years ago [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]) meaning that the state vector rescaling does not change the system behavior. Thus, the behavior of the system trajectories on a suitably defined sphere around the origin can be extended to the whole state space. This property is used for stability analysis [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF][START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF]Hermes, 1991a;[START_REF] Hermes | Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF], systems approximation (Hermes, 1991a), stabilization [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF][START_REF] Kawski | Homogeneous feedback stabilization[END_REF][START_REF] Moulay | Finite time stability and stabilization of a class of continuous systems[END_REF][START_REF] Sepulchre | Stabilizability does not imply homogeneous stabilizability for controllable systems[END_REF], and observation [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. In the work [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] the homogeneity in the bi-limit is introduced, that is homogeneity with different weights and approximating functions at a vicinity of the origin and far outside. Despite all these works are devoted to studying stability-like behavior of the origin during recent years an interest to more complex dynamical systems behavior including oscillatory (periodical or chaotic) has grown significantly [START_REF] Fradkov | Introduction to oscillations and chaos[END_REF][START_REF] Leonov | Frequency Methods in Oscillation Theory[END_REF][START_REF] Martinez | Analysis and design of oscillatory control systems[END_REF][START_REF] Meglio | A first principle model for multiphase slugging flow in vertical risers[END_REF][START_REF] Sinègre | Casing-heading phenomenon in gas-lifted well as a limit cycle of a 2D model with switches[END_REF]. An important and useful concept for studying irregular oscillations is the theory introduced by V.A. Yakubovich [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF][START_REF] Yakubovich | Oscillations in systems with discontinuous and hysteresis nonlinearities[END_REF][START_REF] Yakubovich | Conditions for selfinduced oscillations in nonlinear systems[END_REF]. Recently proposed conditions of oscillations in the sense of Yakubovich [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF] are based on two Lyapunov functions existence. The first Lyapunov function ensures local instability of the origin, while the second provides global boundedness of the system trajectories, that under some mild conditions implies existence of oscillations.

Such existence of two Lyapunov functions nicely interacts with homogeneity in the bi-limit: in both cases two subspaces of the system operation are considered separately. The pecu-liarity of oscillating systems is that the instability around the origin is required. In the present work we are going to develop the results from [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF][START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF] proposing conditions of oscillations for homogeneous systems, relaxing the [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF] conditions conservatism and extending them to the case of several limit cycles existence, when the system may be asymptotically stable around the origin and at infinity with instability regions among them. An extension of the instability Lyapunov theorems for homogeneous systems is also presented.

The outline of the paper is as follows. The homogeneity and the oscillatory properties are introduced in Section 2. Main results are presented in Section 3. Some applications of the proposed results are discussed in Section 4.

PRELIMINARIES

Consider the nonlinear dynamical system:

( )

= x f x , (1)
where

n R ∈ x
is the state vector, :

n n R R → f , (0) 0 = f
is a nonlinear function ensuring existence and uniqueness of the system (1) solutions (for any initial conditions n R ∈ 0 x the solution 0 ( , ) t x x of the system (1) is defined at least locally for T t ≤ , further we will simply write ) (t x if origin of initial conditions is clear from the context). If for all initial con-

ditions n R ∈ 0 x
the solutions are defined for all 0 t ≥ then the system (1) is called forward complete. The system (1) is called locally or globally asymptotically stable (at the origin) if the standard conditions are satisfied [START_REF] Khalil | Nonlinear Systems[END_REF]. The asymptotic stability of the system (1) with respect to an invariant set is treated in the sense of [START_REF] Lin | A Smooth Converse Lyapunov Theorem for Robust Stability[END_REF].

Recall that a continuous function : R R + + α → belongs to the class K if (0) 0 α = and the function is strictly increasing.

The function : R R

+ + α → belongs to the class ∞ K if α ∈K
and it is increasing to infinity.

Homogeneity

For any 0 For any 0

i r > , 1, i n =
the homogeneous norm can be defined as follows

1/ 1 | | | | i r n i i x = = ∑ r x .
There exist two functions ,

∞ σ σ ∈ r r K such that (| | ) | | (| | ) σ ≤ ≤σ r r
| | | = λ r r r Λ x x . Define { : | | 1 } n S R = ∈ = r r x x . D e f i n i t i o n 1 . The function : n g R R → is called r - homogeneous ( 0 i r > , 1, i n = ) if for any n R ∈ x ( ) ( ) d g g = λ r Λ x x
for some 0 d ≥ and all 0 λ ≥ .

The system (1) is called r -homogeneous (

0 i r > , 1, i n = ) if for any n R ∈ x ( ) ( ) d = λ r r f Λ x Λ f x
for some

1 min i n i d r ≤ ≤
≥and all 0 λ ≥ . □

The notation ( ) ( ) DV x f x stands for directional derivative of the differentiable function V with respect to the vector field f , and for Dini derivative in the direction of f ( )

0 ( ) ( ) ( ) ( ) lim inf t V t V DV t + → + - = x f x x x f x if the function V is Lipschitz continuous.
T h e o r e m 1 [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. For the system (1) with rhomogeneous and continuous function :

n n R R → f
the following properties are equivalent:

-the system (1) is (locally) asymptotically stable; -there exists continuously differentiable homogeneous Lyapunov function

: n V R R + → such that for all n R ∈ x , 1 2 (| |) ( ) (| |) V α ≤ ≤α x x x , ( ) ( ) (| |) DV ≤ -α x f x x , ( ) ( ) k V V = λ r Λ x x , 0 k ≥ , for some 1 2 , ∞ α α ∈K , α ∈K . □
Note, that the continuity of the function f is required for the necessary part only. The r -homogeneity property used in Definition 1 and Theorem 1 is introduced for some 0 > r and all 0 λ ≥ . Restricting the set of admissible values for λ we can introduce local homogeneity (in the 0-limit or in the ∞limit [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

D e f i n i t i o n 2 . The function

: n g R R → , (0) 0 g = is called ( r , 0 λ , 0 g )-homogeneous ( 0 i r > , 1, i n = ; 0 : n g R R → , 0 (0) 0 g = ) if for any S ∈ r x 0 0 0 lim ( ) ( ) 0 d g g - λ→λ λ - = r Λ x x
for some 0 0 d ≥ .

The system ( 1) is called ( r , 0 λ , 0 f )-homogeneous (

0 i r > , 1, i n = ; 0 : n n R R → f , 0 ( 0) 0 = f ) if for any S ∈ r x 0 0 1 0 lim ( ) ( ) 0 d - - λ→λ λ - = r r Λ f Λ x f x for some 0 1 min i n i d r ≤ ≤ ≥ - . □
In the paper [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF] this definition has been introduced for 0 0 λ = and 0 λ = +∞ (the function g is called homogeneous in the bi-limit if it is simultaneously ( 0 r , 0 , 0 g )-homogeneous and ( ∞ r , +∞ , g ∞ )-homogeneous), the case 0 0 λ = has been also treated in [START_REF] Bacciotti | Liapunov Functions and Stability in Control Theory[END_REF][START_REF] Hermes | Homogeneous coordinates and continuous asymptotically stabilizing feedback controls[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF][START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. Note, that the system (1) can be also homogeneous in more than two limits (some examples are considered in Section 4).

If the pairs of functions g , 0 g and f , 0 f are continuous, then for any compact set n X R ⊂ and any 0 ε > there exist

0 ε ε λ ≤ λ ≤ λ such that for all ( , ) ε ε λ ∈ λ λ : 0 0 sup | ( ) ( ) | d S g g - ∈ λ - ≤ε r x r Λ x x , 0 1 0 sup | ( ) ( ) | d S - - ∈ λ - ≤ ε r x r r Λ f Λ x f x .
The coefficients 0

i r > , 1, i n =
are called weights, 0 d is the degree of homogeneity (it may depend on 0 λ ), 0 f or 0 g are the approximating functions. Let us recall hereafter two useful results to be used later on.

T h e o r e m 2 (Hermes, 1991a;[START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. Let the system (1) be ( r , 0 , 0 f )-homogeneous with the continuous func-

tions : n n R R → f and 0 : n n R R → f
. If the system 0 ( ) = x f x is (locally) asymptotically stable, then the system (1) is also locally asymptotically stable. □

T h e o r e m 3 [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]. Let the system (1) be ( r , + ∞ , ∞ f )-homogeneous with the continuous functions

: n n R R → f and : n n R R ∞ → f . If the system ( ) ∞ = x f x is globally asymptotically stable, then there exists a compact invariant set n X R ∞ ⊂
such that the system ( 1) is globally asymptotically stable with respect to the set X ∞ . □

The theorems 2 and 3 present results on the system (1) stabil-ity derived from the corresponding properties of the approximating systems for 0 0 λ = or 0 λ = +∞ . The converse Lyapunov theorem similar to Theorem 1 for the homogeneous in the bi-limit systems can be also found in [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

Conditions of oscillations

The function

: n g R R → is called monotone if the condition 1 1 x x′ ≤ ,…, n n x x′ ≤
implies that everywhere either 1 1 ( ,..., ) ( ,..., )

n n g x x g x x ′ ′ ≤ or 1 1 ( ,..., ) ( ,..., ) n n g x x g x x ′ ′ ≥
.

D e f i n i t i o n 3 [START_REF] Yakubovich | Frequency oscillations conditions in nonlinear systems with stationary single nonlinearity[END_REF][START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF]. The solution

0 ( , ) t x x with n R ∈ 0 x of the system (1) is called ] , [ + -π π
-oscillation with respect to the output

) ( x η = ψ (where R R n → η :
is a continuous monotone function) if the solution is defined for all 0 ≥ t and

- ∞ + → π = ψ ) ( lim t t ; + ∞ + → π = ψ ) ( lim t t ; ∞ + < π < π < ∞ - + - . The solution 0 ( , ) t x x with n R ∈ 0 x
of the system ( 1) is called oscillating, if there exist some output ψ and constants

- π , + π such that 0 ( , ) t x x is ] , [ + -π π
-oscillation with respect to the output ψ . The forward complete system ( 1 there exist output ψ and constants - π , + π not depending on initial conditions. □

In other words the solution 0 ( , )

t x x is oscillating if the out- put 0 ( ) ( ( , )) t t ψ = η x x
is bounded and there is no single limit value of ( ) t ψ for t → + ∞ . Note that the term "almost all solutions" is used to emphasize that generally the system (1) has a nonempty set of equilibrium points, thus, there exists a set of initial conditions with zero measure such that the corresponding solutions are not oscillating.

T h e o r e m 4 [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF]. Let the system (1) have two locally Lipschitz continuous Lyapunov functions 1 V and 2

V possessing the following inequalities for all

n R ∈ x ( ∞ ∈ υ υ υ υ K 4 3 2 1 , , ,
):

1 1 2 (| |) ( ) (| |) V υ ≤ ≤υ x x x , 3 2 4 (| |) ( ) (| |) V υ ≤ ≤υ x x x ,
and for some

1 1 1 1 2 3 4 2 0 ( ) X X - - < <υ υ υ υ <+∞ : 1 ( ) ( ) 0 DV > x f x for all 1 0 | | X < < x and Ξ ∉ x ; 2 ( ) ( ) 0 DV < x f x for all 2 | | X > x and Ξ ∉ x , where n R ⊂ Ξ
is a set with zero Lebesgue measure contained all equilibriums of the system, and

Ω ∩ Ξ = ∅ , 1 1 2 1 1 3 4 2 { : ( ) | | ( )} X X - - Ω = υ υ < < υ υ x
x .

Then the system (1) is oscillatory. □

The Lyapunov function for the linearised system (1) at the origin is a candidate for the function 1 V [START_REF] Yakubovich | Oscillations in systems with discontinuous and hysteresis nonlinearities[END_REF]. Instead of existence of the function 2

V one can require just boundedness of the system (1) solutions with known upper bound (this fact can be verified using another approach not dealing with Lyapunov functions analysis).

T h e o r e m 5 [START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF]. Let the system (1) be uniformly oscillatory with respect to the output

) ( x η = ψ (where R R n → η :
is a continuous monotone function), and for all n R ∈ x the following relations are satisfied:

) | | ( ) ( ) | | ( 2 1 x x x χ ≤ η ≤ χ , ∞ ∈ χ χ K 2 1 , ;
the set of initial conditions for which the system is not oscillating consists in just one point

} 0 : { = = Ξ x x
. Then there exist two locally Lipschitz continuous Lyapunov functions

1 : n V R R + → and 2 : n V R R + → such that for all n R ∈ x the inequalities hold ( ∞ ∈ υ υ υ υ K 4 3 2 1 , , ,
):

1 1 2 (| |) ( ) (| |) V υ ≤ ≤υ x x x , 3 2 4 (| |) ( ) (| |) V υ ≤ ≤υ x x x ; 1 ( ) ( ) 0 DV > x f x for all ) ( | | 0 1 2 - -π χ < < x ; 2 ( ) ( ) 0 DV < x f x for all 1 1 | | ( ) - + > χ π x . □
The theorems 4 and 5 present the sufficient and necessary conditions for the system (1) to be oscillatory. Being rather simple these conditions can be useful in different applications [START_REF] Efimov | Yakubovich's Oscillatority of Circadian Oscillations Models[END_REF][START_REF] Efimov | Oscillatority of Nonlinear Systems with Static Feedback[END_REF]. Unfortunately, in some situations these conditions could be restrictive. For example, in sufficient part they need the knowledge of two Lyapunov functions for the system (1), that can be an ambiguous requirement. Additionally, the conditions are oriented to the locally unstable origin case, however, a system with several limit cycles can have a locally stable origin. These shortages can be resolved applying homogeneity approach as it is shown below.

MAIN RESULTS

For the homogeneous systems the functions 1 V and 2 V can be chosen according to the corresponding approximations at the origin or at infinity. For this purpose we are going to develop the Lyapunov theorems for locally homogeneous unstable/stable systems and apply them to detect the oscillations presence next.

Unstable homogeneous systems

At the first we present an analogue of Theorem 1 for local instability of the homogeneous system (1) at the origin.

L e m m a 1 . For the system (1) with r -homogeneous and continuous function :

n n R R → f
the following properties are equivalent:

-the system ( 1) is (locally) strongly unstable, i.e. there exists 0 δ > such that for any

0 0 | | < <δ x there exists 0 T x such that 0 | ( , )| t > δ x x
for all 0 t T ≥ x ;

-there exists continuously differentiable homogeneous

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Lyapunov function

: n V R R + → such that for all n R ∈ x , (i) 1 2 (| |) ( ) (| |) V α ≤ ≤α x x x , ( ) ( ) (| |) DV ≥ α x f x x , (ii) ( ) ( ) k V V = λ r Λ x x , 0 k ≥ , for some 1 2 , ∞ α α ∈K , α ∈K . ■
All proofs are excluded due to space limitations.

Stability/instability conditions for locally homogeneous systems

An advantage of the r -homogeneous system (1) is that the global behavior of the system can be completely characterized by the behavior of the system on the sphere with the unit radius S r . To explain this property let us introduce the coor-

dinate transformation = r x Λ y that connects any n R ∈ x with S ∈ r y for suitably chosen 0 λ ≥ . Let : n V R R + →
be a continuously differentiable homogeneous Lyapunov function (as used in Theorem 1, Lemma 1), then For the ( r , 0 λ , 0 f )-homogeneous system (1) this technique establishes the relation between the global stability properties of the approximating dynamics 0 ( )

( ) ( ) ( ) ( ) ( ) ( ) | | ( ) ( ),
= x f x (2)
and the local ones of the original system (1). L e m m a 2 . Let the system (1) be ( r , 0 λ , 0 f )-homogeneous, the functions :

n n R R → f and 0 : n n R R → f
be continuous and the approximating dynamics (2) have r -homogeneous and continuously differentiable Lyapunov function

0 : n V R R + → , 1 0 2 (| |) ( ) (| |) V α ≤ ≤α x x x , 1 2 , ∞ α α ∈K for all n R ∈ x . (i) Let 0 0 sup ( ) ( ) S a D V ∈ = - r y y f y , 0 a > , then 1) if 0 0 λ = , then there exists 0 ε < λ such that the system
(1) is locally asymptotically stable with the domain of asymptotic stability containing the set

1 0 1 2 { :| | ( )} n X R - ε = ∈ ≤α α σ λ r x x ;
2) if 0 λ = +∞ , then there exists

0 ε < λ < + ∞ such that
(1) is globally asymptotically stable with respect to the forward invariant set 1) is finite time asymptotically stable with respect to the forward invariant set X ∞ with the region of attraction

1 1 2 { :| | ( )} n X R - ∞ ε = ∈ ≤α α σ λ r x x ; 3) if 0 0 < λ < + ∞ , then there exist 0 0 ε ε < λ ≤ λ ≤ λ < +∞ such that the system (
1 1 1 2 1 2 { : ( ) | | ( )} n X R - - ε ε = ∈ α α σ λ < <α α σ λ r r x x provided that the set X is connected and X ≠ ∅ . (ii) Let 0 0 inf ( ) ( ) S a D V ∈ = r y y f y , 0 a > , then 1) if 0 0 λ = , then there exists 0 ε < λ such that the system
(1) is asymptotically stable with respect to the forward in- 1) is finite time asymptotically stable with respect to the forward invariant set 0 \ n R X with the region of attraction X provided that the set X is connected and X ≠ ∅ . ■ In other words the result of Lemma 2 means that the behavior of the system (1) is inherited after (2) into the set

variant set 0 \ n R X with the region of attraction 0 \{0} X ; 2) if 0 λ = +∞ , then there exists 0 ε < λ < + ∞ such that the set \ n R X ∞ is forward invariant for the system (1); 3) if 0 0 < λ < + ∞ , then there exist 0 0 ε ε < λ ≤ λ ≤ λ < +∞ such that the system (
{ : | | } n X R ε ε = ∈ λ < <λ r r
x x provided that it contains an equilevel set of the function 0 V (the set X is connected and not empty). The first two parts of the case (i) correspond to theorems 2 and 3. If we assume that the approximating vector field 0 f is r -homogeneous, then the requirement on existence of the r -homogeneous Lyapunov function 0 V follows by Theorem 1 and Lemma 1 results. The conditions of the lemma can be relaxed skipping homogeneity and f , 0 f continuity assumptions as follows.

C o r o l l a r y 1 .

Let 0 > r , 0 0 λ ≥ and 0 : n n R R → f
be given and the approximating dynamics (2) have rhomogeneous and continuously differentiable Lyapunov func-

tion 0 : n V R R + → , 1 0 2 (| |) ( ) (| |) V α ≤ ≤α x x x , 1 2 , ∞ α α ∈K for all n R ∈ x
. Let one of the following properties hold

(i) 0 0 sup ( ) ( ) S a D V ∈ = - r y y f y , 0 a > ; (ii) 0 0 inf ( ) ( ) S a D V ∈ = r y y f y , 0 a > ,
and there exist

0 ε ε λ ≤ λ ≤ λ such that 1 0 0 sup | ( )[ ( ) ( )] | d S DV a - - ∈ λ - < r y r r y Λ f Λ y f y for all ( , ) ε ε λ ∈ λ λ , then all claims (i),1-(i),3 and
(ii),1-(ii),3 of Lemma 2 are valid. ■ Let us apply these results for oscillations detection. T h e o r e m 6 . Let the system (1) be ( j r , j λ , j f )-

Oscillations in locally homogeneous systems Let

homogeneous for 1, j N =
, the functions :

n n R R → f and : n n j R R → f , 1, j N =
be continuous and the locally approximating dynamical systems ( )

j = x f x , 1, j N = have j r -homogeneous and continuously differentiable Lyapunov functions : n j V R R + → , 1, 2, (| |) ( ) (| |) j j j V α ≤ ≤α x x x , 1, 2, , j j ∞ α α ∈K for all n R ∈ x and 1, j N = . Let n R Ξ ⊂
be the set containing all equilibriums of the system (1). Then (1) has oscillating trajectories into the set Ω . ■

Let one of the following conditions hold. (i)

The result of the last theorem says that if the system (1) is locally homogeneous and unstable in an inner (outer) subset and locally homogeneous stable in an outer (inner) subset, then between these subsets should exist an invariant set containing oscillating trajectory providing that the equilibriums are excluded from this region. The conditions of Theorem 6 can be relaxed taking in mind the result of Corollary 1 (this reformulation is omitted here for brevity of presentation). The set Ω can be used to estimate the constants - π , + π , i.e. to estimate the amplitude of oscillation.

EXAMPLES Consider the system

1 1 2 1 2 tanh( ) x x x x = -+ + ; 2 1 3 2 x x x = - + ; (3) 3 1 1 1.5 2 tanh( ) x x x = - + , where 3 1 2 3 [ ] T x x x R = ∈ x
is the state vector, and the system (3) is in Lurie form (linear asymptotically stable system closed by nonlinear feedback). The system (3) has the single equilibrium at the origin, it is homogeneous in the bi-limit, namely ( j r , j λ , j f )-homogeneous with 1, 2 j = and 1 [ 0.5 0.5 0

.5] = r , 1 0 λ = , 1 1 ( ) = f x A x ; 2 [ 0.5 0.5 0.5] = r , 2 λ = +∞ , 2 2 ( ) = f x A x; 1 1 1 0 2 0 1 0.5 0 0 ⎡ ⎤ ⎢ ⎥ = - ⎢ ⎥ ⎣ ⎦ A , 2 1 1 0 2 0 1 1.5 0 0 - ⎡ ⎤ ⎢ ⎥ = - ⎢ ⎥ - ⎣ ⎦
A with zero degree. All eigenvalues of the matrix 1 A have positive real parts, and all eigenvalues of the matrix 2 A have negative real parts. Therefore, the conditions of Theorem 6 hold and the system (3) is oscillating, actually the results of its simulation presented in Fig. 1 show, that it has the stable limit cycle. . The origin is the only equilibrium of the system. This system is homogeneous in three limits:

-+ ⎡ ⎤ = ⎢ ⎥ - - ⎣ ⎦ f x , 3 2 2 2 1 2 1 2 3 2 ( ) ( ) x x x x ⎡ ⎤ = - + ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ f x , 1 3 3 2 2 2 2 1 2 2 2 ( ) {1 0.005 [ / ( ) ]} x x x x x ⎡ ⎤ = ⎢ ⎥ - π + ⎣ ⎦ f x
with 0 d = for all cases. In all three modes the approximating dynamics have Lyapunov function 2 2 1 2 ( ) 0.5(

) V x x = + x
, and it is easy to verify that the vector fields 1 f , 2 f are asymptotically stable and 3 f is unstable. Thus, the conditions of Theorem 6 can be verified twice signalizing that the system has two limit cycles (the system (4) is planar, thus any isolated oscillating trajectory is a limit cycle). The results of simulation presented in Fig. 2 confirm this conclusion.

CONCLUSION

The paper presents conditions of oscillations existence for locally homogeneous systems. The conditions allows us to detect multiple limit cycles presence for homogeneous vector fields in several limits. Efficiency of the proposed conditions demonstrated on several examples. Partly the proposed conditions interact with the second part of Hilbert's 16th problem that addresses the issue of evaluation of an upper bound for the number of limit cycles in polynomial vector fields of degree n and investigation their relative positions (for the planar system (1)). Homogeneity is naturally satisfied for the polynomial vector fields, verifying local homogeneity conditions proposed in this work and checking conditions of Theorem 6 it is possible to estimate the maximum number of limit cycles and localize their positions. That is rest for future works.

  the system are oscillating. The oscillatory system (1) is called uniformly oscillatory, if for almost all n

  where d is the homogeneity degree of the function f and k is the degree of the Lyapunov function V . Therefore, sign definiteness of the function V derivative can be checked on the sphere S r only.

  ≤ λ are defined as for the case (i), and

  Fig. 1. The results of the system (3) simulation.

Fig. 2 .

 2 Fig.2. The results of the system (4) simulation.

Copyright by IFAC


Olivier Boutron
email: oboutron@yahoo.fr

Christelle Margoum

Jean-Marc Chovelon

Céline Guillemain

Véronique Gouy

Laboratory studies of the adsorption of two pesticides (diuron and tebuconazole) using a batch design and an experimental flume: influence of contact conditions

Keywords: Pesticides, Diuron, Tebuconazole, Adsorption, Contact conditions, Agricultural Ditches, Experimental flume, Hemp

Introduction

The intensification of agriculture has led to a diffuse contamination source for surface and ground waters by various pesticides [1][2][3][4]. Once pesticides are applied to a plot, they can be transferred through the soil and different landscape components before reaching the water courses. Of particular importance are farm ditches, which play a major role in the collection and transfer of water from agricultural plots. The fluxes of pesticides through ditches can be modified by a number of physical, chemical or biological processes which can contribute to the reduction of the amount of pesticides reaching the rivers [5][6][7][8][9]. Pesticides adsorption onto ditch substrates is a very interesting process for the mitigation of surface water contamination, the efficiency of which is poorly understood. The pesticides adsorption that occurs during water flows in natural ditches is not easy to predict, as it depends upon the adsorption capacities of the substrates which are present at the bottom of the ditches and the capacity of the pesticides to reach the reactive sites of the substrate, mainly by hydraulic transport.

This study is aimed at evaluating the influence of various contact conditions on the adsorption of two pesticides onto a standard of ditch substrate under controlled conditions. Special attention was given to the influence of the initial conditions (dry or water saturated substrate) and water dynamics. We studied the adsorption kinetics and the adsorption at equilibrium which are observed for three different contact schemes: 1) contact in a glass beaker using a modified batch method ; 2) contact in a recirculating experimental flume, in which the substrate covers the bottom part of the flume and is initially without water ; 3) same as 2) but with a substrate initially saturated with pesticide free water.

The comparison of the data obtained with schemes 1) (where the transfer of the pesticides onto and inside the substrate assisted by orbital shaking of the beaker) and 2) (where the transfer of the pesticides is linked to the flow in the flume) allowed us to evaluate the influence of the water dynamics.

The comparison of the data obtained by schemes 2) and 3) allowed us to test the influence of the presence of water inside the substrate at the start of the experiment (using initially dry or water saturated substrate). Schemes 2) and 3) were designed to mimic the contact conditions that are commonly encountered in ephemeral ditches, those in which the substrate dries out between successive water flow events, and those with a permanent water flow.

The use of an experimental flume was selected to overcome the difficulties encountered with field studies, which make it difficult to establish quantitative balances. In addition, when considering the wide diversity of substrates that are found at the bottom of agricultural ditches [10] (typically living and/ or dead plants, with various decomposition levels, or sediments), and any possible changes in their characteristics over time, we have decided to use a simplified organic substrate made of hemp fibres, as discussed in detail in refs [11][START_REF] Boutron | Geotextiles and Geomembranes[END_REF].

Experimental

Pesticides

Two pesticides, diuron (3-(3,4-dichlorophenyl)-1,1-dimethylurea) and tebuconazole ((RS)-1-pchlorophenyl-4,4-dimethyl-3-(1H-1,2,4-triazol-1-ylmethyl)pentan-3-ol) were selected because they have been extensively used in agricultural plots and are commonly found in surface waters. They are characterized by a wide range of physico-chemical properties as shown in Table 1, so that various behaviours can be expected for their adsorption onto the substrate. The commercial formulations Zodiac TX (diuron) and Folicur 250 EW (Tebuconazole) from Bayer Crop Science were used. In addition, potassium bromide was added as a tracer for water transport through the substrate.

Substrate

We have used hemp fibres as a standard for the natural vegetation based substrates that can be found in agricultural ditches. They were obtained as rolls (with a thickness of 10 cm) from Chanvre et

Modified batch adsorption experiments

Adsorption experiments were performed using the modified batch technique as described by Margoum et al. [START_REF] Margoum | [END_REF]. A piece of hemp roll (5 g) was placed in a glass beaker with 250 mL of an aqueous solution containing the two pesticides at a concentration of 20 μg/L and bromide at a concentration of 100 mg/L. The solution was prepared using deionized water containing NaCl (5 mM) and NaHCO3 (4 mM).

The pH was typically 7.5 (see ref [11]). The beaker was covered with a Petri dish to avoid evaporation and was gently shaken for 7 hours (50 rpm -orbital mixer STR, Stuart Scientific) at room temperature (Figure 2). Triplicate water samples were taken at different intervals during the 7 hour period, in order to determine the adsorption kinetics. Control samples without substrate, and pesticide free blanks were also analyzed. No pesticide degradation or sorption onto the glass beaker walls were observed.

The initial concentration of 20 μg/L for the two pesticides was chosen as a good compromise between the typical concentrations found at the edge of agricultural plots (see e.g. [14]) and the minimization of analytical uncertainties.

Experimental flume

The experimental flume (Figure 3) was 730 cm long and had a 40 cm wide channel. The flume and all the recirculation pipes were made of PVC. However, one lateral wall was made of Plexiglas to permit flow visualization. The hemp rolls were on the bottom of the flume, and covered the entire bottom surface. The flume allows the control of the water speed and depth, and recirculates the water to increase the contact time. As shown in Figure 3, a mixing tank was added to homogenise the concentration of the pesticides in the water solution. A flowmeter and a stabilization tank are necessary to prevent the occurrence of eddies at the entrance of the flume, and are placed upstream of the flume itself. The water depth in the flume is fixed using an impermeable end-plate. Preliminary experiments showed that the impermeable ends did not provide very good end conditions. A slim box was then installed at each end of the flume (see Figure 3). The boxes were closed except one vertical face of each boxe which was covered by a stainless steel mesh. The subsurface flow passed through the mesh and was recirculated from the end of the flume to the inlet by means of a peristaltic pump.

Two vertical cross sections of the hemp were equipped with capillary tubes to determine the penetration depths of the pesticides, see Figure 3.

The experimental duration was 7 hours. For experiments with the hemp initially saturated in water, the flume was filled at the beginning of the experiment with the deionized water in which 5 mM of NaCl and 4 mM of NaHCO3 had been added. For the experiments with hemp initially without water, the flume was left dry. In both cases, a solution containing the two pesticides and bromide was prepared in the mixing tank using deionized water with 5 mM of NaCl and 4 mM of NaHCO3 as a diluent, to obtain concentrations of 20 µg/L for the two pesticides and 100 mg/L for bromide in the surface water flow in the flume (after dilution in the case of the substrate initially saturated with water). Water samples were collected during the whole experiment at different time intervals downstream of the flume, at various points in the surface flow, and from each capillary tube to determine the pesticides and bromide concentrations. The water depth above the hemp surface and the speed of the surface water flow were also monitored. During the experiments, the surface water speed was kept to 3 cm/s, while the subsurface water speed was 0.03 cm/s [11] and the water depth was kept at 2.5 cm [11]. These three parameters were kept constant during the experiments. Additional detailled information about the experiments can be found in ref [11].

Pesticides and bromide determination

Bromide concentrations were determined by Ion Chromatography (Dionex DX-120 with a AS9-HC column) using direct injections (limit of detection: 0.2 mg/L). For diuron and tebuconazole, the water samples were filtered using 0.2 μm filters (Chromafil filters PET 20/15 MS, Macherey Nagel) and a 990µL aliquot was then transferred into an auto-sampler vial with 10 µL of deutered diuron D6 as an internal standard. The samples were then analyzed using a high performance liquid chromatography instrument (Series 1100, Agilent Technologies) equipped with a 125-2 RP18e 5 μm LichroCART column (Merck), coupled to a triple quadrupole mass spectrometer (API 4000, LC/MS/MS system, Applied Biosystems) with an ESI source. The injection volume was 100 µL (with triplicate determinations). The limit of detection was 0.02 µg/L for diuron and 0.2 µg/L for tebuconazole. The analytical uncertainties were about 10%.

Results and discussion

Comparison of the adsorption kinetics for schemes 1) and 2)

The comparison of the experiments with schemes 1) (modified batch method) and 2) (dry hemp) allowed an investigation of the influence of the water dynamics on the adsorption of the pesticides. In both cases the solution of pesticides is in contact with the whole hemp volume at the beginning of the experiment (at introduction of the pesticides solution into the beaker or into the flume, which were initially dry). In addition, the mass of hemp / volume of water ratio was kept the same for both experiments.

Figure 4 shows changes in diuron, tebuconazole and bromide concentrations in water as a function of time for the batch experiments (contact scheme 1)). It is interesting to compare these data with those obtained for the flume with dry hemp (Figure 5a). For the flume experiments, equilibrium was reached after 7 hours. Conversely, equilibrium for the batch experiments was not reached after 7 hours (additional experiments over 48 hours showed that it was reached after about 24 hours).

For the two schemes, the transfer of the pesticide molecules into the whole volume of hemp was relatively easy at the start of the experiments, as the hemp was initially dry (see steps 1 and 2 in Figures 6 and7), leading to the adsorption of a fraction of the pesticide molecules (see steps 2 and 3 in Figures 6 and7). The pesticide molecules that remained outside the hemp were then transferred into the hemp via various mechanisms.

Tentatively, it can be assumed that this transfer is mainly linked to turbulent diffusion in the batch experiments (see step 4 in Figure 6), and to advection for the flume experiments [15][16][17][18][19][20][21] (see step 4 in Figure 7). Turbulent diffusion, which returns to a combination of molecular diffusion coupled with turbulence, is a process which is slower and less efficient than advection for the transfer of contaminants, this is in good agreement with the observed differences for the two experiments (see Figures 4 and5a). It should however be kept in mind that for the batch experiments, turbulent diffusion is not homogeneous in the whole hemp volume because of orbital shaking, leading to stronger diffusion in the outer parts of the hemp and much weaker diffusion in the center (see step 4 in Figure 6). For the flume experiments, the situation is different, there is a rather homogeneous advection due to the use of two pumps (see step 4 in Figure 7).

Our experiments indicate that it is necessary to take into account the water dynamics when assessing the adsorption kinetics of the pesticides. So when investigating the adsorption kinetics of pesticides in agricultural ditches, it will then be necessary to design experiments which reproduce the water dynamics in agricultural ditches for realistic field conditions, as example obtained with flume experiments.

Influence on the kinetics of adsorption of the presence of water in the hemp at the beginning of the experiments

The comparison of the data obtained with the experimental flume for schemes 2) (dry hemp) and 3) (saturated hemp) allows an evaluation of the influence on the adsorption kinetics of the presence of water at the beginning of the experiments. Figure 5 shows the changes in diuron, tebuconazole and bromide concentrations in water as a function of time for the contact schemes 2) (Figure 5a) and 3) (Figure 5b). It indicates that the adsorption kinetics is faster with initially dry hemp than for hemp initially saturated with water. For initially dry hemp, as soon as the water overflows above the downstream impermeable end-plate (after about 30 minutes), the C/C0 ratio (where C and C0 represent the concentration of diuron or tebuconazole in the surface water flow determined at the downstream end of the flume and in the water at the beginning of the experiment, respectively) is very close to the equilibrium ratio value for the two pesticides. The whole hemp cross section is rapidly involved in the adsorption of the two pesticides since no water was present in the hemp at the beginning of the experiment (see steps 1 and 2 in Figure 7). The situation is different for hemp initially saturated with water, with equilibrium still not reached even after 7 hours. In this case, the penetration of the pesticides into the hemp is determined by the capillary tubes data, and is shown as increasing over time as illustrated by the observed changes in the penetration of diuron and tebuconazole into the hemp (Figures 8a,b). It can be seen that diuron and tebuconazole have not penetrated uniformly into hemp after 7 hours since they are mainly present in the upper layer of the hemp. Similar observations have already been made by Elliott and Brooks [17] for a sand bed and two fluorescent dye tracers.

These data suggest that pesticides penetration into the lowest part of the hemp would probably continue to increase slowly after seven hours, but this would need to be confirmed by performing longer experiments.

Author-produced version of the article published in International Journal of Environmental Analytical Chemistry, 90, 3-6, 286-298, doi:10.1080/03067310903353487

Comparison of the adsorption after 7 hours for the different experiment schemes

As bromide is not adsorbed onto hemp, the mass of pesticides adsorbed at the end of the experiments has been estimated from the difference between the mass of bromide and the mass of diuron or tebuconazole remaining in the aqueous solution after 7 hours, as explained in detail in ref [11].

Table 2 shows the percentages of the mass of diuron and tebuconazole adsorbed onto hemp after 7 hours for the three experimental schemes (the results are expressed as percentages of the initial mass of pesticide in water at the beginning of the adsorption experiment), with the same hemp / water mass / volume ratio for the three schemes. The adsorption for tebuconazole is larger than for diuron in all cases, which is in good agreement with their respective Koc values (see Table 1).

Table 2 shows that after 7 hours, contact scheme 2) (flume with dry hemp) gives the highest adsorption both for diuron and tebuconazole. These high adsorption values can be explained by the lack of water in the hemp at the beginning of the experiment and a rapid transfer of the pesticides into hemp, probably mainly by advection. Intermediate adsorption values are observed after 7 hours for the batch experiments, which also allow full contact of the pesticides solution with hemp, but with a transfer of the pesticides into hemp probably mainly by turbulent diffusion.

When comparing the adsorption at equilibrium for schemes 1) (equilibrium reached after 24 hours) and 2) (equilibrium reached after 7 hours), it appears that total adsorption is similar for the two schemes for tebuconazole while adsorption is lower for diuron with the batch experiment (Table 2). For diuron, this suggests that advection could favour both the adsorption kinetics and the adsorption at equilibrium, compared to turbulent diffusion, while it may only favour the adsorption kinetics for tebuconazole. It would be interesting to perform similar experiments for other pesticides to better assess the effect of these two processes on their adsorption at equilibrium. Finally, the least favourable situation appears to be that for scheme 3) (flume initially saturated hemp), see Table 2. In this case, equilibrium is still not reached after 7 hours, and adsorption is linked to the penetration of pesticides into the hemp. It will be necessary to perform additional studies to investigate the influence of the water flow parameters on their penetration (especially the water depth and speed), the characteristics of the substrate (conductivity, porosity) and the shape of the interface between the substrate and the surface water flow. Similar studies have already been performed for other compounds and substrates, by Elliott and co-workers [15-17, 20, 21].

Conclusion

The aims of this study were to compare the adsorption of two pesticides onto a standard (hemp) for agricultural ditch substrate with different contact conditions, using three different experimental schemes. The first was adapted from a standard batch method (scheme 1) ). For this first scheme the hemp was initially dry and the transfer of the pesticides into hemp was tentatively attributed to turbulent diffusion. The second and third schemes, based on an experimental flume, were designed to approach the field conditions as encountered for dry (scheme 2) ) or wet agricultural ditches (scheme 3) ). In these last two schemes, the transfer of the pesticides into hemp was tentatively attributed to advection.

The highest adsorption was observed for schemes 1) and 2) with initially dry hemp, especially for scheme 2) which corresponds to the conditions typically encountered in the field in dry agricultural ditches. The highest adsorption observed for scheme 2) could tentatively be explained by the prevalence of advective transfer of the pesticides, compared to the predominance of turbulent diffusive transfer for scheme 1). Conversely, adsorption by hemp initially saturated with water was much less and is mainly linked to the limited penetration of the pesticides into the hemp during the experiments.

To improve the mitigation of pesticides in agricultural ditches, it seems important to favour ephemeral ditches in which the substrate remains dry between successive water flow events, and to favour the advective transfer of the pesticides into the bed substrate. discussions, and Warren Cairns from CNR-IDPA /University Ca' Foscari in Venice for his various constructive comments on the manuscript. Author-produced version of the article published in International Journal of Environmental Analytical Chemistry, 90, 3-6, 286-298, doi:10.1080/03067310903353487 Table 1. Selected properties of the two pesticides considered in this work [22][START_REF] Tomlin | The Pesticide Manual[END_REF][START_REF]FOOTPRINT Pesticide Properties Database[END_REF].

Figures Captions

  Laboratory studies of the adsorption of two pesticides (diuron and tebuconazole) using a batch design and an experimental flume: influence of contact conditions O. Boutron, C. Margoum, J.M. Chovelon, C. Guillemain, V. Gouy To cite this version: O. Boutron, C. Margoum, J.M. Chovelon, C. Guillemain, V. Gouy. Laboratory studies of the adsorption of two pesticides (diuron and tebuconazole) using a batch design and an experimental flume: influence of contact conditions. International Journal of Environmental Analytical Chemistry, 2010, 90 (3-6), p. 286 -p. 298. 10.1080/03067310903353487. hal-00561006

.

  Figure 1. The rolls of hemp used in this study.

Figure 2 .

 2 Figure 2. Modified batch adsorption experiment.

Figure 3 .

 3 Figure 3. Schematic diagram of the flume. 1: mixing tank; 2: flowmeter; 3: upstream stabilization tank; 4: cross sections equipped with capillary tubes; 5: downstream subsurface flow discharge box; 6: peristaltic pump to recirculate subsurface flow; 7: upstream subsurface flow discharge box; 8: centrifugal pump. Hemp is shown with a brown color. Depending upon the experimental scheme, it is initially with water or without water.

Figure 4 .

 4 Figure 4. Diuron, tebuconazole and bromide concentrations in water as a function of time for the batch experiments. The results are expressed as C/C0, where C represents the concentration of diuron, tebuconazole or bromide in the water at a given time, and C0 the concentration of diuron, tebuconazole or bromide in the water at the beginning of the experiment.

Figure 5 .

 5 Figure 5. Diuron, tebuconazole and bromide concentrations in water as a function of time for two contact schemes using an experimental flume. (a) initially dry hemp and (b) initially saturated hemp (same mass of hemp / water volume ratio for the two schemes). The results are expressed as C/C0. Where C represents the concentration of diuron, tebuconazole or bromide in the surface water flow at the downstream end of the flume at a given time. For the flume experiments with initially dry hemp, C0 represents the diuron, tebuconazole or bromide concentration in the water at the beginning of the experiment. For the flume experiments with hemp initially saturated with water, C0 represents the concentration after dilution in the surface water flow.

Figure 6 .

 6 Figure 6. Batch experiment. The diagrams show a top view of the beaker and the piece of hemp. Introduction of the pesticides solution into the beaker (initially dry hemp).  Some of the pesticides molecules have already penetrated inside hemp.  Adsorption of some of the pesticides molecules onto hemp (illustrated by the disappearence of some of the black dots).  The clockwise shaking of the beaker creates a water motion in the beaker, which results in a transfer of the molecules of pesticides from the outside water to the bulk hemp mainly by turbulent diffusion. Because of the orbitalshaking, turbulent diffusion is small in the center of hemp and greater in the outer parts of the hemp. Vertical motions of the water and molecules are not represented in the diagrams.

Figure 7 .

 7 Figure 7. Flume experiment with the initially dry hemp.  Introduction of the pesticides solution into the flume.  Some of the pesticides molecules have already penetrated inside hemp.  Adsorption of some of the pesticides molecules onto the hemp (illustrated by the disappearence of some of the black dots).  Because of the high surface water flow and the weaker lateral water flow inside the hemp, the molecules of pesticides that are present in the surface water flow penetrate into hemp mainly by advection. Inside the hemp, the transfer of these molecules is mainly by advection.

Figure 8 .

 8 Figure 8. Penetration of diuron (a) and tebuconazole (b) into hemp initially saturated with water as a function of time. The results are expressed as CC/CS, where CC represents the concentration of diuron or tebuconazole in the water sample collected with a capillary tube for a given depth while CS represents the concentration of diuron or tebuconazole measured in the surface water flow. Open squares: surface down to 3.3 cm; black triangles: 3.3 to 6.6 cm; open circles: 6.6 to 10 cm. The values shown on the figure are averages of the different CC/CS values measured at the two sections equipped with capillary tubes.

Figure 1 .

 1 Figure 1.

Figure 4 .(

 4 Figure 4.

Figure 5

 5 Figure 5.

Figure 6

 6 Figure 6.

Figure 7 .

 7 Figure 7.

Table 2 .

 2 Percentages of the mass of diuron and tebuconazole adsorbed onto hemp for the three experimental schemes after 7 or 24 hours (the results are expressed as percentages of the initial mass of pesticide in water at the beginning of the adsorption experiment).Author-produced version of the article published in International Journal of Environmental Analytical Chemistry, 90, 3-6, 286-298, doi:10.1080/03067310903353487

				Flume with initially dry	Flume with initially saturated
		Batch experiment	Batch experiment after 24
	Pesticide			hemp	hemp r
		after 7 hours	hours (at equilibrium)
				after 7 hours (at	after 7 hours
				equilibrium)
	Diuron	29	35	44	9
	Tebuconazole	34	47	48	20

Acknowledgements

We wish to thank Nathalie Touze-Foltz and Rolland Gallo from Cemagref Antony for the determination of the conductivity and porosity of hemp, Francis Bereiziat, Nathalie Ball, Anne-Laure Achard and Sébastien Chaumat from Cemagref Lyon for their contribution to the experimental work and fruitful