Ivana Botica

Josip Grgić

Vinko Muštra

Slađana Pavlinović

A BASELINE ASSESSMENT A thematic network on High Value Farming Learning, INnovation & Knowledge

Keywords: Selca, Brač (source: Jakšić, K.)

A portrait of the Dalmatian islands Learning Area (LA)

The learning area of Dalmatian islands covers the territory of three local action groups LAG Škoji, LAG brač and LAG 5 situated in central and south Dalmatia in the Split-Dalmatia and Dubrovnik-Neretva region. Specifically islands of Brač, Hvar, Šolta, Vis, Mljet, Lastovo, Korčula, Pelješac peninsula and the Dubrovnik West Coast municipality. More then 80% of the area is Natura 2000 sites, encompassing one National Park ''Mljet and Nature park ''Lastovo islands'' as well as many nature protected landscapes. The area is highly touristic due to its natural and cultural heritage. This territory ( 185 824 ha) is characterised with a predominant HNV type 2, which consists of a mosaic of low intensity farming namely composed of small fields of arable plots, vegetable gardens, orchards and vineyards, usually on stone wall (AgroParistech, 2017:13). Therefore the whole description of the LA we here provide, is structured around the mosaic agricultural landscape that despite the fact that today it represents only 5,75% of the officially cultivated agricultural land, it has a significantly higher importance in terms of biodiversity, preservation of natural habitats and long term sustainable tourism activity as the major economic activity of the entire LA.

Limits and the key characteristics of the LA ''Dalmatian islands''

The LA ''Dalmatian islands'' is formally encompassed by official borders of 27 local governmental units (LGUs). 15

LGUs are belonging to Split-Dalmatia County while remaining 11 LGUs belong to Dubrovnik-Neretva County. In the LA there is a total of 57 566 inhabitants and the population density is 31 inh./km2 that is well below the national average that is 76 inh./km2. The average size of a parcel is 0,27 ha and there are more than 6 295 small scale family farms in the area. The area is characterised by a continuously decreasing population, small-scale agriculture and on some islands micro and on some islands small-extensive livestock farming.

The diverse landscape is composed of different types of land cover that are quite differently spatially organized. The majority of the islands is covered in guarrigues, macchias and Pinus halepensis. Agricultural landscapes differ from island to island with the overall typology being internal and coastal hills, coastal slopes and karstic plains. The HNV type 2 to in the LA is seen through the presence of semi-natural vegetation and understorey, the land use and farming practices that foster biodiversity, and the presence of a landscape mosaic at a broader scale (Abdessater, 2017:50). The landscape is defined by its components and their position in relation to the landscape (ager: cropped land; sylva: woodland; hortus: gardened land; saltus:uncultivated land) that are elements characteristic in all of the learning area islands. Its specificities on the scale of the island are seen in the distribution of agricultural practices where some of the island have predominant olive groves semi-natural habitats, some vineyards, some small scale livestock (Beug, 1967;Krklec et al, 2011).

The LA area according to the Koppen climatic classification belongs to several basic types of the Mediterranean climate. Open islands such as the archipelagos of the island of Vis and Lastovo and the southernmost part of Korčula belong to the semi arid or ''olive'' climate covers that are characterised by pronounced heat and dryness, while precipitation comes only in the coldest parts of the year. True Mediterranean climate is one the nearer islands and the Pelješac peninsula. It is similar to the semi arid but has much more rainfall (Ozimec et al, 2015:33). In line with climate conditions, dominant agriculture consists of citrus fruits, olive trees and vineyards.

Three types of vegetation (Defilippis, 1997:28): Ø "makija" -impassable dense holm communities Ø garigue -evergreen shrubs, low trees, aromatic herbs, and bunchgrasses found in poor or dry soil Ø Pinus halepensis Until deagrarisation of these areas in 50-ies, impact of man on pasture and forest area was very significant. Wood was the only energy source. It was used for heating, cooking, and for construction. Huge amounts of wood area used in production of lime (high temperatures). Wood is cut to feed animals, and goats eat small trees preventing their growth. Energy and construction substitutes, as well as deagrarisation, led to autochthon forest regeneration (Defilippis, 1997).

Three types of landscapes can be found in the Dalmatian Islands: a) karstic fields (flooded due to eroded surrounding hills) b) terraces on slopes (typical man-made landscape; erosion prevention measure which was a result of the labour surplus and cultivable land shortage) c) karstic pastures (extensive pasture; scarce xerophyte vegetation) Due to the shortage of natural arable land in this karstic area, very sensitive landscape in Dalmatian Islands is a result of a longlasting efforts to create additional arable land and prevent erosion by cleaning stones and building stonewalls.

Erosion prevention is very important aspect of water management in a karstic area. It is necessary to apply measures which will prevent soil erosion from the bare, stony slopes and its relocation to the karstic fields. If slopes are covered with vegetation, water drains capillary and gradually. In this way floods in karstic fields and further slope erosion are avoided, and soil and vegetation on slopes are preserved. The HNV farming systems in the LA Dalmatian islands are linked to the adaptation of the agricultural practices to the landscape. Based on the geomorphology and agricultural activities in the islands the HNV farming consists mostly of a mosaic of low intensity farming mainly composed of small fields of arable plots, vegetable gardens, olive groves, orchards and vineyards, usually on terraces or surrounded by stone walls (Abdessater et al, 2017:49). Orchards and olive groves with large, old trees and a semi-permanent understorey are characteristic HNV type 2 in the LA.

The coastal villages are organized as ''huerats'' with ager that is characterised with a highly diverse small scale agriculture (Abdessater, 2017:111). Several municipalities had some legal characteristics of towns (e.g. Korčula). Since municipalities have more people employed in public services with lower risk of losing their job they are demographically more resilience than villages.

The lowest depopulation has been detected in the towns on the cost with the tourism as a dominant sector.

The economy of LA Dalmatian islands has been defined by natural environment and therefore its structure was very simple with large number of areas of economic activity missing or weakly present (islands were highly depended on agriculture production) (Defilippis, 1997).

The largest share of revenues belongs to business entities (which do not include crafts and family farms) is realised in service economy (accommodation, food, trade, transport and finance). Processing industry plays very important role on Korčula and Brač.

*FINA collects data from the business legal entities such as companies. This dataset does not include business data about family farm, public entities, crafts or NGO-s. Later in the presentation it is mentioned that family farms are the main organisational units in agriculture. Thus, business data from FINA does not capture agricultural activities adequately.

80% of employed population of age above 65 is in agriculture, while the most of young population (15-29) is employed in service sector. Agriculture is job intensive sector since it creates 6% of the local revenue and employs 15% of labour force.

Threats:

Agriculture in Croatian Islands is particularly sensitive to drought periods and extreme temperature shocks. Rainfalls transmit pollutants (e.g. sulphur) to the agricultural areas on islands (which for now does not represent a significant environmental threat). On the other hand, intensive agriculture on islands pollutes environment, such as vineyards on Vis, Korčula, Hvar and Pelješac) (Kružić and Povh Škugor, 2015:41). Furthermore, forest fires which are very often in summer in the Coastal Croatia degrade natural habitats, improve soil erosion, and contribute to extinction of various species.

Dingač is a specific location on island Pelješac at very steep and sunny slopes. It is adequate for cultivation of high quality black grape plavac mali. Wine produced at this specific location is also cold Dingač. Steep slopes and sun enable very fast runoff and created very dry microclimate which prevents development of grape illnesses. Weed growth is also limited due to this specific terrain. Due to very limited accessibility, grapes are cultivated by hand using safety ropes (to prevent falls down the slope). There are other similar sites within the LA.

Photo credit: LAG5 (left); EFST (right) Dalmatian islands are a very important area of biodiversity. Land use abandonment is recognized as the major threat for biodiversity due to succession caused by lack of mowing and pasture (Nikolić et al, 2009:23) The greates biodiversity found

The timeline consists of four periods:

1. Heritage from the past (-1945) • ends with WWII 2. Extraction of capital accumulation from agricultural, collectivisation and agrarian reform • end with abolishment of mechanics purchase prohibition in agriculture 3. Associated labor organisations for cooperation and economic crises (1967( -1990( ) 4. Croatian independency (1991-present) -present) Population: The highest number of people has been identified at the beginning of 20 century, (high fertility rates has been overcoming long-lasting depopulation process that has been started in 19 century till 1921) Cities: Several municipalities had some legal charateristcs of towns (e.g. Korčula) Transport: weak connection with mainland (islands were focal point in transport and trade networks by the end of 19th century) Governance: local self-government, top-down decision-making, one municipality per island Main economic activities: The economy of LA Dalmatian islands had been defined by natural environment and therefore its structure was very simple with large number of areas of economic activity missing or weakly present (islands were highly depended on agriculture production). Cooperatives: The first agricultural cooperative in Croatian Islands was established in 1864 in Korčula (Mataga, 2005) • several attempts during 20th century to substitute agricultural dependency by industry: shipbuilding (Korčula), navigation (Orebić), food production (Postira-Sardina), stone extraction (Brač) • since agriculture on island was not very productive, a need for development of other sectors was present. • Dalmatia in general is relatively poor in terms of livestock compared to Croatian regions. Domestic animals are even less present in Dalmatian Islands. Based on data from 1900 and 1935 it can be speculated that up to one hundred cattles, between 5 000 and 10 000 sheeps and goats are expected to be found on a Dalmatian island before WWII. • Special breeds of sheep pramenka evolved in islands due to isolation, for example in island Brač. Pramenka gives low returns but it is very resistent and adaptive to extreme conditions such as on islands. Defilippis (1997:224) mentions that breeding of sheep pramenka is mostly oriented on lamb production. Common way of production is in cattles of 20-30 animals which are kept in enclosed pastures and olive groves. They are given additional feeding and their return is 1,5-2 youngsters per year which is considered as a cost-effective production.

Ø Erosion -a consequences of wine crises and previous deforestation for the landscape:

• "The naked mountains do not give sufficient pasture for the livestock. The unorganised water and mountain streams disperse the juices of the land" (from newspaper Hrvatska riječ, in Antić, 1995:298). • There is a vicious circle: Insufficient pasture forces shepherds to let their animals (goats and sheeps) eat young plants. In this way pastures regenerate even more slowly. The landscape remains bare and erosion progresses.

Ø Large human pressures on pastures and forests till 50-ies (Defilippis, 1997:27-28):

• heating • wood as construction material • grazing (goats in particular)

• heating for lime production • Period between collectivization and agrarian reform (1953)(1954)(1955)(1956)(1957)(1958)(1959)(1960)(1961)(1962)(1963)(1964)(1965)(1966)(1967) • Family farms became involved in social reproduction by cooperation between family farms and cooperatives or social enterprises • Cooperatives became important economic and social institutions in rural areas.

• Small farm unites (less than 2 ha) represent more than half of all farming unites (Defilippis, 1997:155).

Trends in agriculture:

Ø decrease in arable land size Ø livestock:

Ø Sheeps dominate in number but it sharply decreases in this period. Numbers of horses, pigs and sheeps fall as well. Ø 1954 Law on prohibition of goat keeping

• Since goat keeping in public pastures was considered as an important obstacle to forest regeneration, goat keeping was restricted in several attempts dating back from the end of 19th century. The most rigorous was the law from 1954 when goat keeping was prohibited and goats had not been registered any more in official statistics. The law was abolished in 1982.

★ REMARK: Available agricultural data that refer to Dalmatian islands for period 1945-1990 include municipalities from that period of Brač, Hvar, Vis Korčula and Lastovo (Defilipis, 1969:2), and do not include islands Šolta, Pelješac and Dubrovačko primorje. For this reason, presented data serve for obtaining general insights, but not for exact comparison with other periods.

Ø Depopulation and deagrarianization trends decrease pressures on land. However, this processes are at the beginning in this period and traditional way of living still requires use of wood for construction and heating while that wood can hardly be regenerated due to animal grazing (especially goats which are prohibited but still present).

Ø Japjenica • a traditional technique lime preparation for building. Lime had to "baked" for several days. Thus, this process required cutting off a lot of wood/bush on nearby surfaces (Source: LAG Brač).

A large economic reform was introduced in 1964/65 which introduced some market mechanisms into socialist economy transforming it into so called "self-governed economy". Export was liberalised. Such changes boosted industrial development and cities growth, as well as labour flow from rural to urban areas.

In 1967 a ban on purchase of machinery was abolished. From this year family farms can freely buy tractors and other machinery equipment. A model of small family farm supported by additional income (employment in industry) become a dominant and successful.

In 1976, one quarter of family farms which increased their size mention an acquisition of the machinery equipment as the main reason, while 70% of family farms which diminished in size mention a decrease in a household labour force as the main reason.

The period ends with establishment of a new state Republic of Croatia (RH) and shift from socialism to capitalism.

• rural depopulation and migration of young population to cities Ø main reasons: feeling of isolation and detachment from the social dynamics in the other parts of the region as a consequence of weak transport connections (Filipić, 1985:117) Ø young and male populations emigrates => feminisation and aging of the island population • deagrarisation countinues • development of tourism on islands (annual growth rate of service sector on islands was 12,2% in period 1965-1980(Filipić, 1985:118):118)).

This period is characterised by agricultural decline and tourism growth. Industry grows at much smaller rates than tourism.

REMARK: According to estimations from National island development program (1997:9) estimated number of inhabitants (based on actually present population and natural population dynamics) in islands of Dalmatia in 1991 was smaller than numbers recorded in official statistics for years 1981 and 1991.

• Forest succession, instead of oak regeneration, dominantly refers to development of Pine forests which are usually linked to low level of biodiversity.

• Development of garigue and "makija"

• Introduction of chemicals in agriculture Photo credit: LAG Škoji (Komiža, island of Vis)

Governance: many small local-government units which do not have sufficient fiscal capacity for good governance and effective management.

The Island act: The situation in LA Dalmatian islands has been determined also by unique institutional characteristics of the islands. The institutional differentia specifica of the islands has been tried to implement in national programmes and regulatory frameworks such as: National Island Development Programme (1997) and the Island Act (1999,2002,2006). This programmes have been partially implemented (Stubs and Starc,2007) Relevant institutions: Ø 2 the most relevant ministries: Ø Agricultural land covers 10.802 ha (5,8%) of the LA surface with following dominant types: karst pastures (15%), vineyards (21%) and olive groves (53%) in 2015 (ARKOD).

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Ø Although different motivation and data collection methodology applies, graph on the left based on Agricultural census (CBS, 2003) reveals similar structure: vineyards occupy 21% of utilised agricultural land, orchards and pastures is consistent with sum of olive groves and karst pastures from ARKOD (2015). The olive groves are contained under orchards in CBS ( 2003) and one can speculate that from 2003 to 2015 registered olive groves surface increased, while pasture surface decreased. Motives might be linked to subsides as well as to actually transition processes. Revival of olive grove cultivations is taking place. The same structure is revealed in data from Population Census (2011).

Ø State subsidies were asked for 70% of cultivated land in 2015, mostly as a compensation for natural and other constraints, as well as for preservation of autochthon and protected species. 5,64% of surfaces with demanded subsidies refers to transition to ecological practices. There are no demands for HNV farming practices such as for maintenance of field strips or HNV pastures, except for maintenance of extensive olive groves. (Croatian system of subsidies for agriculture includes several subsidy types for HNV farming practices just mentioned).

Ø Two types of vineyards (photos on the left):

• on slopes -more labour intensive but require less chemicals • in the flat fields Ø Olive collecting process through time (photos on the right)

• traditional labour-intensive process of olive collection dominates since terrain is usually inaccessible Ø Bee-keeping is important link in agriculture Photo credit: LAG5 Katarina Jakšić

In LA region there was a total number of livestock of 32.652 heads in 2011 (Census) out of which around 15.881 was poultry and 14.685 sheep. However, collected data do not allow detailed spatial livestock analysis.

Ø According to the latest Agricultural Census ( 2003) the total available land on LA Dalmatian Islands covers 26.286 hectares. Total available land comprises of total utilised agricultural land and other other land. The other land includes unutilized agricultural land and wooded area. The total utilized agricultural land includes arable land and gardens, kitchen gardens, meadows, pastures, orchards (total), vineyards (total) and nurseries. In 2003 the share of total utilized agricultural land in total available land for LA Dalmatian Islands was 37%, while 36% of utilized agricultural land was not cultivated. Brač, Vis and Lastovo are above the average with respect to share of utilized agricultural land in the total available land in LA Dalmatian Islands. Furthermore, they show the highest level of cultivation of utilized agricultural land as well.

Ø The more recent data do not provide the info on total available land in hectares although they show that agriculture covers 5,8% of total surface of the LA Dalmatian Islands (ARKOD, 2015). This corresponds to the percentage of arable land on islands (National island development plan, 1997:20). Brač continues to be the leading island in terms of agricultural land use, while Mljet (with majority of its surface under national park) has the lowest share of agricultural land use in both analysed years 2003 and 2015. • small-scale production • low capital-intensity agriculture • can be full-time, but also a part time agriculture (no further relevant data)

Ø Abandoned agricultural land under forest succession is prone to forest fires (left).

Ø Once bare stony hills are green today (right).

BAU combines over touristic development by the shore with land abandonment in the inland, compatible with scarce individual farms able to manage. Landscapes are closing but no one really matters

The main threats for the LA in the current situation is development of tourism infrastructure that will intensify the urbanization process and change the scarce agricultural land in the islands to building land. Increase in large scale infrastructure will put additional pressure to the environment in terms of communal infrastructure management and foster decline in biodiversity due to the reducing of the natural and semi-natural habitats.

If the current situation continues the major threats in the LA that are identified will further increase and cause permanent damage in terms of both tourist attractiveness of the area and biodiversity richness.

Photo Credit:

• Marija Roglić (Orebić, Pelješac peninsula)

• Pero Poljanić (Trstenik, Pelješac peninsula)

• Tourist demand for vocations on Dalmatian Islands is very high, but with a very short season, limited by weather conditions. There are drastic differences in number of tourists along the year. Very crowded coasts of islands in summer are abandoned out of this season. • Due to high demand for accommodation, accompanied by weak spatial planning and legal enforcement, process of so called "apartmanisation" takes place along the coast. This leads to development of mass tourism in coastal zones while island inland is abandoned. • tourism substitutes agriculture in terms of revenue sources. Locals orient to tourism as a source of revenues.

Production of wine and olive oil grows and it is mostly exported or sold in expensive touristic shops at very high prices. During the season there is very high surplus demand for food which needs to be imported to islands by trucks loaded on ferries. National food market chains open their shops on islands. • Local food marketplace and family farm production is marginalised due to dominance of large food retailers and food import to island. Hotels mostly offer imported food since adequate quantities at low prices are not available on islands. • Actors: investors, property owners and managers, tourists, local population, family farms • Development strategies recognised tourism and competitive advantage of islands, so that even public resources are directed for such use (development of touristic resorts, infrastructural investments, etd.). On the other hands, even at political stance, importance of agriculture on islands is not sufficiently recognised so that uniform agricultural measures are not sufficiently adapted for island conditions (small parcels, mosaic agricultures, emphasis on preservation of stone walls and other agricultural heritages). • There are various EU funding schemes which can be used for island development. LEADER program, and LAG-s on islands represent very important key for local development of islands. • National and supranational institutions develop various types of certification programs which are suitable for promotion of local agricultural products. Thanks to Island act and efforts at national level, a particular certificate, Hrvatski otočni proizvod, is developed for products composed mainly of local ingredients.

Actors: local, regional and national government, UNDP, EU bodies, LAG-s and cooperatives, Paying Agency for Agriculture, Fisheries and Rural Developmet, and Advisory Office for Agriculture (Poljoprivredna savjetodavna služba) -Due to market forces, agricultural island production which is competitive is transformed to large-scale, such as grape and medical plants. -Otherwise, if not attractive for market production at large scale, agricultural land is abandoned. Moreover, due to depopulation there is no labour force to cultivate small inaccessible parcels. -However, due to new consumer trends and policy efforts, islands are recognised as very important hotspots for eco-agriculture land is mostly preserved from pollution.

Industries which are welcome on islands are those which can be embedded into natural and other values of islands, such as: food processing, stone processing and shipbuilding. Island development cannot be only the responsibility of local leaders but specific set of selective measures has to be developed at the national level (Filipić, 1985:119).

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  Photo credit: Marija Roglić (LAG 5)

  Photo credit: Govedari (Mljet), postcard used in 1910, collection of Krunoslav Leko, source: Goleš, 2014

  Fishing and fish processing were lowering dependency on other agricultural activities. Query flourished due to high demand for local marble and other types of stone.• foundation of tourism • trade, and thus, shipbuilding, was fostered during the golden period of wine export. Capital accumulated in agriculture was directed toward trade.Source of photo on the bottom left: http://www.mdc.hr/hr/mdc/zbirke-fondovi/fototeka/muzej-opcine-jelsavinogradarski-muzej-pitve/ Causes and consequences: à phylloxera in France and Italy (after 1850) à Increased demand for wine from Dalmatian islands à Intensive plantation of new vineyards in Dalmatian islands àThere is sufficient labour force but limited agrarian surfaces. à Deforestation, construction of terraces, cutting olive trees à Maximum number of vineyards surfaces around 1890 à Entrepreneurship development, land ownership growth, maritime transport , trade, establishment of cooperatives à1892 Wine clause: Wine import from Italy to Austro-Hungarian empire is allowed à Decline of wine demand (wine prices drop from 25-30 forinters/hl to 6-12 forintes/hl) phylloxera in Dalmatia (from 1894) à Agricultural and economic crises and emigration at the beginning of 20. st. • Despite very high natural population growth, Brač and Hvar and suffered an absolute fall in population in 1910 due to emigration (Antić, 1995:294). Ø After 2nd Word War agriculture dominates in Dalmatian area • Share of agricultural in active population in 1953 is 61%, based on Population Census (Defilippis, 1997:143) Ø Vineyards dominate in Dalmatian Islands in 1931 • End of 19th st. and beginning of 20st is characterized by short period of growth and then decline of wine market • Due to phylloxera in France and Italy demand for wines from Croatian islands increased Ø Farmland fragmentation => no capital accumulation • Among the regions in A-H monarchy, Dalmatia was ranked first in percentage of farms of size less than 2ha, and last in percentage of farms larger than 10ha § 1902 61,52% of farms had size less than ha (Antić, 1995:292) • October 19 1930 -Law on liquidation of agrarian relationship § Feudal relationship is abolished § Peasants became land owners and landlords get reasonable reimbursement § land fragmentation Ø Livestock:

  Institutional context and main issues of development: After 1945 and Before 1990s, LA Dalmatian Islands had been part of socialist Yugoslavia that had a tradition of so-called socialist self-management, with a specific participation at the local authority or municipal level. LA Dalmatians islands were represented with usually one municipality by island. The decision-making in this system were top-down and non-transparent, with informal and impenetrable groups of enterprise managers and senior party officials in control, and with other bodies (e.g. the Workers' councils) existing as a largely powerless bodies(Stubs and Starc, 2007) Population: The number of people had been decreasing (fertility rates decreasing and long-lasting depopulation process -political and economic motivation) Cities: have massive differences regarding socio-economic characteristics That is to say? Transport: By losing the geostrategic and economic importance, transport networking between islands were weak and only development were between the islands and mainland. Main economic activities: Development of agriculture under cooperative framework and attempts to implement some other activities (e.g. shipbuilding, navigation, food production and tourism). • Agrarian policy measures (Law on agrarian reform and colonization, 23/8/1945): • Forced establishment of peasant labour cooperatives (similar to Russian kolhoz), abandoned in 1953 • Farmland maximum of 25-35ha, in 1953 diminished to 10 ha • A ban which prohibited purchase of farming mechanics which was partly relaxed so that purchase of secondhand mechanics was allowed in 1953 and it was completely relaxed in 1967. • Permanent consequences for cultivation of grape § Public good problem => vineyards become collective property and individual incentive was lacking § Vineyards were not cultivated carefully; the consequences were permanent • Compulsory redemption of farming products • Free work obligation • The result is cheap labor and inputs necessary for fast industrialization. Due to tremendous political and economic pressures on farmers, rural population intensively abandons its farmland and moves to urban areas.

  Ministry of regional development and EU funds Ø Department for Islands Ø Island Council (consultative body to the Ministry) Ø Ministry of agriculture Ø Paying Agency for Agriculture, Fisheries and Rural Development Ø Advisory Office for Agriculture (Poljoprivredna savjetodavna služba) Ø Local agriculture cooperatives • since 1864 and first agricultural cooperative on Croatian islands established on Korčula • important local rural stakeholder during Yugoslavia (before 1991) • neglected after Croatian independence • recently reestablished in a modified form Ø LAGs (Leader Program) • focal point of comprehensive development with participation of local community • Škoji (Šolta, Hvar and Vis), LAG5 (Pelješac, islands Mljet, Korčula, Lastovo and municipality Dubravačko primorje) and LAG Brač (Brač) Despite significant infrastructural investments (public utilities, transportation network, electricity, telecommunications) depopulation trends continue. Its manifestation is size of school classes. Ø Relatively recent sources of data about agriculture in LA Dalmatian Islands: • 2015 -Paying Agency for Agriculture, Fisheries and Rural Development (PAAFRD) and National register for agricultural land use (ARKOD) • 2011 -Population Census • 2003 -Agricultural census (Croatian Bureau of Statistics [CBS])

Ø

  Due to karstic area specificities, although limited, agriculture represents very important aspect of life on islands. Together with non-agricultural activities agriculture on islands can assure decent life standard (National island development plan, 1997:22). Data from Census (2011) indicate certain level of specialization among islands: • Pastures -Brač which is famous for its lamb production • Olive groves: Brač, Hvar and Korčula • Vineyards: Pelješac and Vis • Brač: pigs and sheeps • Pelješac: goats and cattles Ø There were 5.831 family farms in LA Dalmatian Islands in 2015 • three quarters are governed by males and one quarter by females • 4818 members (beside holders of the farms, i.e. every farm has a holder, and some farms have additional members). • Thus, 10.649 persons are active in family farms. Ø Consequences of family farm dominance in numbers:

  Abdessater, C., Bitton-Lebeau, L., Bisiaux, L., Bourdin, M., Briand, P., Korsagni, C., Rochegonde, M. De, Gomes, M., Graziella-Cailleau, M., Ginter, Z., Le Priol, R., Litaudon, J., Lhuer, E., Normand, L, Parrot, I. Peyrin, S., Robinet, T. and Valentin, M. (2017): STRATEGIC ENVIRONMENTAL MANAGEMENT ANALYSIS OF HIGH NATURE VALUE FARMING IN THE DALMATIAN ISLANDS: Case studies from Mljet and Korčula Islands and Pelješac Peninsula in Croatia. Montpellier • Antić, LJ. (1995): The Economic Causes of Emigration from Croatia in the Period from the 1880's to the First World War, Povijesni prilozi, 14(14):291-300 • Antić-Brautović, J., Piteša-Orešković, A. and Gelo, J. (2008): Stanovništvo Pupnata na Korčuli (1870-1880): Početak demografske tranzicije, Anali Zavoda za povijesne znanosti Hrvatske akademije znanosti i umjetnosti u Dubrovniku, 47 (1):243-266 • Beug H.-J., 1967. On the Forest history of the Dalmatian coast. Review of Palaeobotany and Palynology, 2 , pp. 271-279. • Bašić, Ž (ur)(2007): Stoljeće Zadružnog saveza Dalmacije 1907.-2007.: prilozi iz prošlosti i sadašnjosti hrvatskojg i svjetskog zadrugarstva, Zadružni savez Dalmacije, Split • Defilippis, J. (1997): Dalmatinsko selo u promjenama, Avium, Split • Defilippis, J. (1969): Mogućnosti privredno-društvenog razvitka Dalmatinskih otoka, knjiga 5 (Poljoprivreda), Ekonomski institut Split, Split • Filipić, P. (ur.)(1985): Osnove dugoročnog razvoja Dalmacije, svodna studija, Zavod za ekonomska istraživanja, Ekonomski fakultet Split, Split • Goleš, I. (ur.)(2014): Zaboravljena Dalmacija na starim razglednicama, Artes Classicea: Galerija Naval Nugent, Split i Zagreb • Goleš, I. (ur.)(2015): Zaboravljena Dalmacija na starim razlgednicama, Pozdrav iz Dalmacije, Split • Jakšić, I. (2014): Brački vitalac, Ethnologica Dalmatica, 21 (1):125-137 • Jasprica, N. and Kovačić, S. (2014): Ugrožene, endemične i rijetke biljke hrvatske flore na Korčuli, Blatski ljetopis, pp. 73-96 • Jelavić, A. (1978): Dosadašnji razvoj, sadašnje stanje i mogućnosti razvoja dalmatinske industrije. Zbornik radova Ekonomskog fakulteta u Splitu, 3:135-164 • Krklec K., Ljubenkov I., Bensa A., 2011. Natural resources of Korčula island. Geoadria, 16 (1), pp. 3-25. • Krstulović, A. (2008): Vina Hrvatke, Profil, Zagreb • Klub jugoslavenskih pomoraca (1927): Ilustrovani pregled privrednih i kulturnih institucija u Kraljevini SHS, Zagreb • Mataga, Ž. (2005): Poljoprivredno zadrugarstvo u Hrvatskoj: razvoj i temeljni problemi, Sociologija sela, 43(1):17-42 • Maršić (Ed.) (1892): Dalmacija iz djela Austro-ugarska monarhija, Naklada međunarodne knjižare F. Maršića, Split • Nikolić, T., Topić, J. and Buković, N. (Eds.)(2009): Područja Hrvatske značajna za floru, Important Plant Area -Croatia, Working version 3.0, Zagreb• Republic of Croatia (1997): National island development program, available from 5. 11. 2012., dowloaded at: https://razvoj.gov.hr/o-ministarstvu/djelokrug-1939/regionalni-razvoj/otoci-i-priobalje/strateskidokumenti-i-propisi/142?trazi=1&=&page=1 (July 5, 2017) • Starc, N. and Stubbs, P. (2007): Islanders vs. the state?: Participation in Island Development Pgorammes in Croatia, 47th Congress of the European Regional Science Association -Local governance and sustainable development. Pariz : European Regional Science Association • Starc, N. (2001): Managing Island Development: the Croatian Case, Sociologija sela : časopis za istraživanje prostornoga i sociokulturnog razvoja. 39 (2001) , Vlahović, D. (2008): Zlatna zrnca Zlatnog rata, Turistička jednice Općine Bol.

This project has received funding from the European Union Horizon 2020 research and innovations program under Grant Agreement No. 696391

Sub-periods of agricultural development:

Ø Associated labor organizations for cooperation (1967)(1968)(1969)(1970)(1971)(1972)(1973)(1974)(1975)(1976)(1977)(1978)(1979)(1980) Cooperatives transform into self-governed enterprises which are income oriented. Due to abolishment of the ban on purchase of mechanics, massive mechanization of small family farms take place in this period. Thus, small family farms become more productive.

Ø Economic and social crisis (1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990) • Introduction of new olive oil processing plants (1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990) • Population decreases, but at a lower rate, partly due to negative natural population growth rate, and partly due to emigration of youngs population from islands. Process of population aging continues. Emigrated population returns to islands in older age. • Since tourism development take place and services require a lot of labour force which is missing due to negative demographic trends, seasonal labour force is imported to island during summer period. • new hotels are built and tourist resorts are developed. There is increased demand for attractive land with the sea view. Price of real estates increase which increases political pressures to convert agricultural land to construction land. Price of real estates additionally limits young population from choosing islands for settling down. • Finally, inhabitants leave the island also due to physical isolation. Ferry connections are scarce out of the season.

With bad weather ferries remain in their port which additionally pronounces sense of isolation. New catamaran lines to islands are introduced only for tourism purpose during summer.

Actors: local and residential population and NGO-s.

Photo credit: Slađana Pavlinović, Supetar, island Brač transforming land use purpose from agriculture to touristic grassland decreases forest succession and forest fires continue due to missing of livestock to "clean" the terrain decrease of biodiversity Photo credit: Pero Poljanić (Pelješac peninsula -Župsko field-mosaic agriculture)


Josephine N A Tetteh

Sorin Olaru

César Parra-Rojas

Esteban A Hernandez-Vargas

César Parra

Lyapunov-based Switching to Mitigate Antimicrobial Resistance

Keywords: Bacterial resistance, switched positive systems, two-strain bacteria

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

INTRODUCTION

The alarming increase in morbidity and mortality rates attributed to drug-resistant infections has raised global concern in control and prevention of resistance related infections. Proposed treatment strategies to tackle this problem have been the subject of investigation of several mathematical modelling evaluations. Tackling resistance, however, will stem from understanding the mechanisms leading to resistance evolution at within-host level.

In the literature, studies of infections have been conducted using the notion of switched positive dynamical systems [START_REF] Di Giamberardino | Hiv infection control: a constructive algorithm for a state-based switching control[END_REF][START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF]. A switched positive system generally refers to a hybrid dynamical system consisting of a family of continuous-time subsystems and a rule which brings about the switching between them. Interest in switched systems has increased recently due to their applications in diverse areas such as economics, engineering and biology to mention a few [START_REF] Liberzon | Switching in systems and control[END_REF]. The feedback stabilization of such systems has received a lot of attention and numerous tools have been developed to make such analysis. In particular, the use of control Lyapunov functions have gained wide popularity in studying the stabilization of both linear and nonlinear switched systems [START_REF] Zhao | Stability of switched positive linear systems with average dwell time switching[END_REF][START_REF] El-Farra | Output feedback control of switched nonlinear systems using multiple lyapunov functions[END_REF]. Studies show that for switched systems, control Lyapunov functions provide a robust feedback solution for achieving stability [START_REF] Blanchini | Switched positive linear systems[END_REF]. The notion of switched positive systems has also been applied in the context of infections (Di Giamber-This work was supported by the DAAD, the Alfons und Gertrud Kassel-Stiftung, and DFG (HE-7707/5-1). Corresponding authors: esteban@im.unam.mx, Sorin.Olaru@centralesupelec.fr ardino and [START_REF] Di Giamberardino | Hiv infection control: a constructive algorithm for a state-based switching control[END_REF][START_REF] Hernandez-Vargas | Discrete-time control for switched positive systems with application to mitigating viral escape[END_REF] and in epidemiological systems [START_REF] Rami | Stability criteria for sis epidemiological models under switching policies[END_REF].

In this work, we use a mathematical model which captures the mechanisms of resistance described in [START_REF] Hernandez-Vargas | Switching logistic to design cycling approaches against drug resistance[END_REF]. This model describes the interaction between genetic bacterial strains and helps tailor appropriate interventions. Employing notions in control engineering and positive switched systems, we develop a control technique based on Lyapunov functions to minimize the appearance of drug resistant bacteria within the host. The rest of this paper is organised as follows: description of the model in Section 2, analysis of the model without switching in Section 3, Lyapunov stability analysis in Section 4 and description and simulations of the switched system in Section 5. The paper ends with a conclusion in Section 6.

RESISTANCE MODELLING

This paper considers a particular case of the mathematical model proposed in [START_REF] Hernandez-Vargas | Switching logistic to design cycling approaches against drug resistance[END_REF] for describing the global dynamics of antibiotic resistance. The switched system is as follows

ẋi (t) = ρ i,σ(t) x i (t) 1 - x i (t) K -δx i (t) +µ j (M j,i x j (t) -M i,j x i (t)) (1)
defined for all t ≥ 0, and where x i : i = 1, 2, 3, ..., n with n represent different bacterial strains, µ is the mutation rate, δ is the bacterial clearance, ρ i is the proliferation rate of the strain i, M i,j is the mutation from strain i to strain j, K defines the maximum carrying capacity and σ(t) denotes the switching signal based on the treatment policy such that σ(t) = σ j for j ∈ {1, 2, • • • , N } with N representing the number of treatment policies. Further qualitative characterisation of the model with and without switching is considered in the subsequent sections.

MODEL ANALYSIS WITHOUT SWITCHING

Here we consider the model where there is no switch between therapies. With the use of therapy, it is possible for a strain susceptible to one drug to generate resistant strains which are susceptible to another drug by a phenomenon known as collateral sensitivity [START_REF] Pál | Collateral sensitivity of antibiotic-resistant microbes[END_REF]. Over time, the newly developed susceptible strains can also generate strains which are resistant to the current drug but susceptible to a different drug. This continuous occurrence of resistant and susceptible strains can eventually lead to the appearance of resistant strains susceptible to the initial drug used.

Our aim is to illustrate a hypothetical scenario of this phenomenon using a two-strain bacterial population, x 1 and x 2 , where each strain represents a family of strains sharing the same susceptibility to a given drug. Assuming that each mutation changes the susceptibility of the bacteria from one family to the other, the connection between the families can be written by the following symmetric matrix:

M = 0 1 1 0 .
Taking K = 1, the system of equations ( 1) can be rewritten according to the above assumptions as:

ẋ1 = [-δ + ρ 1 (1 -x 1 )] x 1 + µ(-x 1 + x 2 ) (2a) ẋ2 = [-δ + ρ 2 (1 -x 2 )] x 2 + µ(x 1 -x 2 ). ( 2b
)
All parameters and initial conditions are assumed to be non-negative so that x 1 (0) ≥ 0 and x 2 (0) ≥ 0.

Equilibrium points and stability

Equilibrium points The substitution method is used to find the equilibrium points of the system. Equating the right hand side of Eqns (2) to zero, an expression for x 2 in terms of x 1 is obtained from Eqn (2b) which is substituted into Eqn (2a) to obtain a quartic polynomial, H(x 1 ) described explicitly as :

H(x 1 ) = x 1 (-µρ 1 x 1 (δ + µ)) µ 2 + x 1 (ρ 1 (δ + µ -ρ 2 ) + ρ 2 (δ + µ) -δ(δ -2µ)) µ - x 1 ρ 2 ρ 1 (x 1 -1) (2x 1 (δ + µ) -µ) + x 1 (δ + µ) 2 + ρ 2 1 (x 1 -1) 2 x 1 µ 2 and x 2 (x 1 ) = δx 1 + µx 1 + ρ 1 x 2 1 -ρ 1 x 1 µ .
H(x 1 ) and x 2 (x 1 ) are plotted on the same axis. The intersection of H(x 1 ) with the abscissa gives the x 1components of the equilibrium points. The coordinates of these intersection points to the curve x 2 (x 1 ) give the corresponding x 2 -components and hence characterize the equilibrium points. This is illustrated in Fig. 1(a), which shows four equilibrium points: the trivial equilibrium point E 1 (0, 0), where both populations are zero, two others E 2 , E 3 and a coexistence equilibrium at E 4 (k 1 , k 2 ), where both populations are positive.

Depending on the parameter values, only two or all four equilibrium points are admissible: the origin (which always exists) and (under certain conditions) a coexistence equilibrium in the positive orthant. The points (E 2 and E 3 ) are not admissible because the values of x 1 and x 2 are negative at E 2 and E 3 respectively. Hence, the only biologically meaningful equilibrium points are the origin E 1 and the positive coexistence equilibrium at E 4 . A vector field plot of system (2) depicts these equilibrium points and gives a hint of the nature of their stability. This is shown in Stability The origin, E 1 (0, 0) is a common equilibrium point which always exists independent of the parameters thus, making it the point of interest from the stability point of view. Since E 1 (0, 0) is a common equilibrium point, once its stability is addressed, the stability of E 4 can be inferred. The Jacobian matrix for Eqn (2) is

J (x1,x2) = -δ -µ + ρ 1 -2ρ 1 x 1 µ µ -δ -µ + ρ 2 -2ρ 2 x 2 with eigenvalues λ 1,2 = 1 2 [-2δ -2µ + G(x 1 , x 2 )] ± 4µ 2 -G(x 1 , x 2 ) 2 (3) where G(x 1 , x 2 ) = 2ρ 1 x 1 + ρ 1 -2ρ 2 x 2 + ρ 2
At the origin, the trace and determinant of the Jacobian matrix are:

τ = -2(δ + µ) + ρ 1 + ρ 2 , ∆ = δ 2 + µ(-ρ 1 -ρ 2 ) + δ(2µ -ρ 1 -ρ 2 ) + ρ 1 ρ 2 .
Thus, for stability of E 1 (0, 0), the restriction of τ < 0 and ∆ > 0 is satisfied by the conditions:

C1 : µ > (δ -ρ 1 )(δ -ρ 2 ) -2δ + ρ 1 + ρ 2 = µ c and C2 : 2δ > ρ 1 + ρ 2
(4) where µ c is a critical mutation rate.

It follows that whenever both ρ 1 > δ and ρ 2 > δ, then µ c > 0 and -2δ+ρ 1 +ρ 2 > 0. The former is a contradiction to C2 and consequently E 1 is not stable. However, when both ρ 1 < δ and ρ 2 < δ, then -2δ+ρ 1 +ρ 2 < 0 and µ c < 0. Thus E 1 is stable for any value of µ > 0. Furthermore, if either ρ 1 > δ and ρ 2 ≤ δ or ρ 1 ≤ δ and ρ 2 > δ but their sum (ρ 1 + ρ 2 ) still satisfies C2, then (δ -ρ 1 )(δ -ρ 2 ) < 0 and -2δ + ρ 1 + ρ 2 < 0 so µ c > 0. The origin is not always stable in this case and there exist some values of µ which make E 1 stable. Proposition 1. The stability of E 4 is complementary to that of E 1 in that when E 1 is stable, E 4 does not exist but whenever E 4 exists, E 1 is unstable. In addition, the points E 2 and E 3 are saddles if they exist.

Note that the evolution of E 2 and E 3 as a function of changes in µ could lead to the escape of trajectories from the positive orthant and thus, a study of the positivity of the solutions is necessary. The feasible region of state space is required to be positive since the system is based on populations. The following subsection investigates the positivity of solutions by means of the invariance of the feasible region.

Set-theoretic characterisation of behaviour

The notion of positive systems relates to the positivity of solutions of the system. We aim to prove the positivity of the system (2) by means of the invariance of the statespace region of biological interest. Proposition 2. (Positivity of solutions). Solutions of the system with non-negative µ remain non-negative for any positive initial conditions in R 2 and for all time t > 0. 2

Proof : Let x i (t) > 0 ∀ t > 0 and non-negative values of µ.

From system (2) we have; ẋi

≥ x i (-δ -µ + ρ i (1 -x i )) with solution x i (t) ≥ (ρ i -δ) ρ i 1 + (ρi-δ) x0
-1 e -t(ρi-δ-µ)

(5) for all ρ i ≥ 0 , δ ≥ 0.

When

ρ i > δ, it follows (ρi-δ) ρi > 0.
When ρ i ≤ δ all components are converging monotonically and lim t→∞ x i (t) = 0. This proves that system (2) is non-negative ∀t > 0.

Let us now consider the region of invariance with respect to system (2) . Definition 1. [START_REF] Blanchini | Set invariance in control[END_REF]). A set X ∈ R n is positively invariant with respect to the system ẋ = f (x(t)) if ∀ x 0 ∈ X, the solution x(t, x 0 ) satisfies x(t, x 0 ) ∈ X ∀ t > 0.

Proposition 3. Given a set of positive parameters: ρ i , δ and µ, the set

Ω = (x 1 , x 2 ) ∈ R 2 | 0 ≤ x 1 ≤ k 1 , 0 ≤ x 2 ≤ k 2 (6
) is positively invariant with respect to system (2) above. 2

Proof : Consider the general dynamical system ẋ = F (x) (7) with F : R 2 → R 2 and let X be the set of all initial conditions in R 2 . Following Nagumo's theorem, assume that for each initial condition in X , there is a globally unique solution and let Ω ⊆ X be a closed and convex set. The boundary of the set Ω is denoted as ∂Ω and the tangent cone of Ω at x is denoted as T Ω(x) . Then, the set Ω is positively invariant for the system if and only if

F (x) ∈ T Ω(x) , ∀ x ∈ ∂Ω,
Thus, if for every x ∈ ∂Ω, the derivative ẋ(t) points inside the set Ω, then the trajectory x(t) remains in Ω [START_REF] Blanchini | Set invariance in control[END_REF].

In the particular case of Ω in Eqn (6) and ẋi in Eqn

(2) with

F (x) = f 1 (x) f 2 (x)
, ∂Ω is defined by the four line segments (see Fig. 1);

A = {(x 1 , x 2 ) | 0 < x 1 < k 1 , x 2 = 0} , B = {(x 1 , x 2 ) | x 1 = 0, 0 < x 2 < k 2 } , C = {(x 1 , x 2 ) | 0 < x 1 < k 1 , x 2 = k 2 } and D = {(x 1 , x 2 ) | x 1 = k 1 , 0 < x 2 < k 2 }.
To show that all points along ∂Ω point inwards, one has to use normals that point inside Ω by using the dot product. However, we recall Proposition 2, where it was shown that the system is positive and hence the points along the boundaries defined by A and B are proved to point inside Ω. Thus, it remains to show that points along C and D also point inside Ω.

For C, x 2 is constant and x 1 changes from 0 to k 1 with the normal vector n C = (0, -1). For any point p ∈ [0, k 1 ], let X = p k 2 be the initial conditions. Then,

F (X) T .n C = F p k 2 T . 0 -1 = f 1 p k 2 f 2 p k 2 . 0 -1 = -f 2 p k 2 , ∀p.
It remains to show now that

F (X) T .n C = -f 2 p k 2 ≥ 0, ∀p ∈ [0, k 1 ]. (8) From Eqn (2b), ẋ2 = [-δ + ρ 2 (1 -x 2 )] x 2 + µ(-x 1 + x 2 ). (9) Since (k 1 , k 2 ) are roots of the system (2), ẋ2 = [-δ + ρ 2 (1 -k 2 )] k 2 + µ(-k 1 + k 2 ) = 0. (10) Thus, [-δ + ρ 2 (1 -k 2 ) -µ] k 2 + µk 1 = 0 [-δ + ρ 2 (1 -k 2 ) -µ] k 2 = -µk 1 (11) Now from Eqn (8), -f 2 p k 2 = -[ρ 2 k 2 (1 -k 2 ) -δk 2 -µk 2 + µp] = -[ρ 2 k 2 (1 -k 2 ) -δk 2 -µk 2 ] -µp = -(-µk 1 ) -µp from Eqn (11) = µ(k 1 -p) > 0 when p ∈ (0, k 1 )
The value of the dot product being positive, the vector fields point in the same direction as a normal field (i.e. inside the domain Ω).

For D, x 1 is constant and x 2 changes from 0 to k 2 with the normal vector n D = (-1, 0). Based on the same reasoning, similar results are obtained:

F (X) T .n D = f 1 k 1 p > 0, ∀p ∈ [0, k 2 ].
and the vector fields point inside the domain.

Eventually, since all points along ∂Ω point inwards, the set Ω is positively invariant with respect to system (2). Corollary 1. In the case when there is no mutation, that is when µ = 0, along the boundaries C and D,

f 2 p k 2 = 0 and f 1 k 1 p = 0
respectively. Therefore the set Ω is positively invariant with respect to system (2).

The invariant region Ω bounded by the line segments A, B, C, D is illustrated in Fig. 1.

LYAPUNOV STABILITY ANALYSIS

We have established that all solutions of the system with positive initial conditions are positive and remain in the positively invariant region Ω for all t > 0. We have also shown that the system converges to either the origin or the coexistence equilibrium point. However, the convergence of interest is the one at the origin which indicates the eradication of both strains of bacteria.

The stability of the origin is analysed in this section using Lyapunov's theory on stability of dynamical systems. The following lemma from [START_REF] Khalil | Nonlinear systems[END_REF] states Lyapunov's stability theorem for a general system of ordinary differential equations.

Theorem 1. (Lyapunov theorem stability [START_REF] Khalil | Nonlinear systems[END_REF]). Let x be an equilibrium point for the system (7) and D ⊂ R n be a domain containing x. Let V : D → R be a continuously differentiable function such that

V (x) = 0 and V (x) > 0 in D \ {x} (12) V (x) ≤ 0 in D (13) Then, x is stable. Moreover, if V (x) < 0 in D \ {x} (14)
then x is asymptotically stable. A continuously differentiable function V (x) satisfying these conditions is called a Lyapunov function.

Using a Lyapunov function, we can determine the Lyapunov stability of the equilibrium point, E 1 for system (2). Theorem 2. The system (2) under the conditions (4) has x = 0 as stable equilibrium with a domain of attraction

Ω D = {(x 1 , x 2 ) ∈ Ω | (x 1 , x 2 ) = (k 1 , k 2 )}. 2
Proof: Starting from the positive invariance of Ω ⊂ R 2 , we construct a candidate Lyapunov function in the form

V (x) = (x 1 -x1 ) 2 + (x 2 -x2 ) 2 . (15) Over the domain Ω D , V (x) is continuously differentiable, V (x) = 0 and V (x) > 0 for all x = x. Thus V (x) is a valid Lyapunov candidate. The derivative, V (x) of the Lyapunov function V (x) is calculated along the trajecto- ries as V (x) = 2(x 1 -x1 ) ẋ1 + 2(x 2 -x2 ) ẋ2 (16)
Clearly, V (x) = 0 for x = (0, 0) and x = (k 1 , k 2 ). Next, we consider the Lyapunov stability of the origin and develop Eqn (15) for x ∈ Ω D and the equilibrium point x = (0, 0)

V (x) = 2x 1 ẋ1 + 2x 2 ẋ2 = 2x 1 [(-δ + ρ 1 (1 -x 1 ))x 1 + µ(-x 1 + x 2 )] + 2x 2 [(-δ + ρ 2 (1 -x 2 ))x 2 + µ(x 1 -x 2 )] = 2x 2 1 (ρ 1 -δ) + 2x 2 2 (ρ 2 -δ) -2(ρ 1 x 3 1 + ρ 2 x 3 2 ) -2µ(x 1 -x 2 ) 2
Let µ be expressed in terms of the critical mutation rate µ c from Eqn (4) as µ = µ c + . Substituting this into V (x) yields:

V (x) = 2x 2 1 (ρ 1 -δ) + 2x 2 2 (ρ 2 -δ) -2 ρ 1 x 3 1 + ρ 2 x 3 2 -2(x 1 -x 2 ) 2 (δ -ρ 1 )(δ -ρ 2 ) -2δ + ρ 1 + ρ 2 + = -2ρ 1 x 3 1 -2ρ 2 x 3 2 -2 x 2 1 -2 x 2 2 + 4 x 1 x 2 + 2(δ -ρ 1 ) 2 x 2 1 + 2(δ -ρ 2 ) 2 x 2 2 + 4(δ -ρ 1 )(δ -ρ 2 )x 1 x 2 -2δ + ρ 1 + ρ 2 = -2ρ 1 x 3 1 -2ρ 2 x 3 2 -2 (x 1 -x 2 ) 2 + 2((δ -ρ 1 )x 1 + (δ -ρ 2 )x 2 ) 2 -2δ + ρ 1 + ρ 2 When C1 is satisfied then 2((δ-ρ1)x1+(δ-ρ2)x2) 2 -2δ+ρ1+ρ2 < 0. When C2 is satisfied then > 0 and -2 (x 1 -x 2 ) 2 < 0.
Therefore satisfying both conditions (4) ensures that V (x) < 0, ∀ x ∈ Ω.

When any of the conditions (4) are not satisfied, there exists at least one combination of (x 1 , x 2 ) ∈ Ω D which makes the expression (16) non-negative, thus invalidating the stability of the origin over this domain.

SWITCHING TO SCHEDULE THERAPIES

Model description with therapies

By considering the switched system (2), we introduce here a mechanism for improving the stabilisation and the rate of convergence of the system (2) when more than one treatment policy is employed. Definition 2. A switching signal is defined as the consecutive choice between therapies at the rate T and is represented by a piecewise constant function σ(t) : R → {σ 1 , σ 2 , . . . , σ N } such that σ(t) is constant for all switch times t k ∈ [kT, (k + 1)T ) and for k = 0, 1, 2, . . . The objective next is to employ switching for the gradual eradication of resistant strains in a system on the long term based on a feedback decision. We denote a periodic switching as the switching between therapies at regular intervals.

Switching based on control Lyapunov function

The use of the Lyapunov function serves as a tool for improving the convergence. Based on the Lyapunov function argument, we can have a switching strategy which ensures the stability of the origin under a given choice of the parameters. Proposition 4. The system (2) has the origin as a stable equilibrium if ∀x ∈ Ω D there exists at least one therapy σ i such that V (x, σ i ) < 0. 2 Proof. Let the Lyapunov function defined as in Eqn (15) be with respect to the dynamics of system (2) changing V (x) into V (x, σ). If for every x ∈ Ω D there is at least one therapy σ i that yields V (x, σ i ) < 0 at any switching time, we can construct a switching policy by choosing the therapy such that σ(t) = arg min σi V (x, σ i ).

This constructive solution proves the existence of a switching policy σ(t), such that V (x, σ(t)) < 0, ∀x ∈ Ω D .

Hence, the system has a stable equilibrium at the origin if there exists a ρ i,σ which makes V (x, σ) < 0. Thus, given two configurations both with the common equilibrium point (0, 0), a policy based on the Lyapunov function derivative can be used to stabilize the system provided that at least one of the configurations guarantees V (x(t), σ(t)) < 0. In addition, switching based on the Lyapunov function derivative can lead to faster convergence than periodic switching. In the following, some scenarios of these conditions are discussed.

Discussion on switching therapies

If the switching system (2) has a common Lyapunov function, then the system is asymptotically stable at the origin for any switching signal σ(t). To illustrate this, we consider three different scenarios with different proliferation rates and under two treatment policies (see Table 1). For all simulations, an initial condition vector x = [10 3 , 10] is chosen and the following parameter values are used: δ = 0.25, K = 10 5 , T = 10 days and µ = 10 -4 .

Table 1. Proliferation rates for bacterial strains under therapy combinations Case Therapy

x 1 x 2 E 1 1 1 ρ 1,1 = 0.2 ρ 2,1 = 0.1 Stable 2 ρ 1,1 = 0.1 ρ 2,1 = 0.2 Stable 2 1 ρ 1,1 = 0.5 ρ 2,1 = 0.1 Unstable 2 ρ 1,1 = 0.1 ρ 2,1 = 0.2 Stable 3 1 ρ 1,1 = 0.35 ρ 2,1 = 0.1 Unstable 2 ρ 1,1 = 0.1 ρ 2,1 = 0.35 Unstable
In case 1, both subsystems are asymptotically stable at the origin. Therefore, any switching policy leads to eradication of both strains. In case 2, the first subsystem is unstable at the origin but the second is stable. Switching periodically between therapies will not stabilize the entire system unconditionally. In cases 1 and 2, switching based on the Lyapunov function argument not only ensures convergence to the origin but also achieves convergence at a faster rate compared to switching periodically between policies. We can see in Fig. 2(a) and 2(b) switching results for cases 1 and 2 respectively. For case 3, the two subsystems are not stable at the origin and thus, using the policies independently does not lead to eradication. However, switching between the two policies periodically leads to a convergence at the origin. Moreover, the Lyapunov switching converges to such a periodic stabilizing sequence implicitly and additionally ensures a better convergence of the closed-loop system (see Fig. 2(c)).

For this last case, subsystems a priori unstable at the origin can be brought to convergence under the Lyapunov switching argument based on feedback. Moreover, an analysis of the Lyapunov function derivative shows that a sliding mode can occur at the intersection of the two configuration as shown in Fig. 3(b). Thus, the feedback switching can be seen as a reaching law to the sliding surface. Once the sliding mode is reached, the control can be completed with a sliding law defined based on periodic switching. x(0) Fig. 3. A sliding mode for case 3 occurs at the intersection of V (x 1 , x 2 ) for therapy 1 and 2 when the minimum of V (x 1 , x 2 ) for each pair (x 1 , x 2 ) with respect to both therapies is plotted. The right hand side shows the values of V (x 1 , x 2 ) under therapy 1 and the left hand side shows the values of V (x 1 , x 2 ) under therapy 2. t r denotes the time to reach the sliding mode.

CONCLUSION

Switching therapies can be considered as a mechanism for improving the stabilisation and rate of convergence of a positive system when more than one treatment policy is used. For the particular case of antibiotic resistance infections, switching can be employed for the gradual eradication of resistant strains on the long term based on a feedback decision. Our numerical simulations suggest that switching between therapies based on the switching rule from the Lyapunov function leads to a better convergence at the origin than an open-loop periodic switching.

A key biological aspect of the proposed control-theoretical approach is that in a first stage one needs to identify the distribution of the bacterial population. Based on this initial information, a switching trajectory (see Fig. 3(a)) can be computed to indirectly define a sliding mode (see Fig. 3(b)). Once on the sliding mode, a periodic switching between therapies can be employed to eradicate the bacterial colony. In conclusion, this approach can be used as an instrument in the future to guide therapies to tackle bacterial resistance with a minimal collection of information.

Fig. 1

 1 (b). Fig.1(c) illustrates the dependence of E 4 on parameter values.

Fig. 1 .

 1 Fig. 1. Graphical view of equilibrium points for system (2). (a) Equilibrium points as a result of substitution method. Red lines: trace of intersection points to x 2 (x 1 ) curve. (b) Vector fields around the four equilibrium points; two with non-negative coordinates (E 1 and E 4 ) and two with negative components (E 2 and E 3 ). (c) Dependence of E 4 on parameters. Red dots represent the value of E 4 for µ = 0 and blue dots for µ > 0. The green lines show the translation of equilibrium point (k 1 , k 2 ) from µ = 0 to µ > 0. If ρ 1 = ρ 2 , E 4 is unchanged for both µ = 0 and µ > 0.

Fig. 2 .

 2 Fig. 2. Illustration of therapy switching for all three cases. (a) Case 1; (b) Case 2; (c) Case 3. Solid lines: periodic switching. Dashed lines: Lyapunov based switching. Blue lines: strain x 1 . Red lines: strain x 2 .