Adaptive Random Search on the simplex
The adaptive random search (ARS) algorithm consists in exploring a given bounded space, by alternating variance-selection and variance-exploitation phases [72, 148]. It will be used in Algorithm 5.1 to solve maximization Problem 5.2. We adapted this algorithm to the simplex An as follows. First, from a current point on the simplex, the displacement towards a new point of the simplex requires to randomly choose a direction ~d = (dk)k=1:::n such that Pn k=1 dk = 0, and jj~djj = 1. Then, if the length of the displacement, drawn from a normal distribution N(0; ), is too large and such that the new point falls the limit of the simplex, this point is discarded and another displacement is drawn randomly. The ARS algorithm, adapted to the n-simplex An, is described below. It aims at determining the optimal fallow distribution ARS(n) = ~ n; = ( 1 ; : : : ; n) that maximizes the profit R defined in equation (5.4) for a given number of fallows n.
Biology and cultivation of banana and plantain
Banana and plantain are subspecies of the genus Musa . Worldwide, there is no sharp distinction between « bananas » and « plantains ». Especially in the Americas and Europe, « banana » usually refers to soft, sweet, dessert bananas , particularly those of the Cavendish group, which are the main exports from bananagrowing countries . By contrast, Musa cultivars with firmer, starchier fruit, used for cooking are called « plantains », distinguishing them from dessert bananas. In some regions, many more kinds of banana are grown and eaten, so the binary distinction is not useful and is not made in local languages .
The term « banana » is also used as the common name for the plants that produce the fruit. It is a perennial herbaceous plant widely cultivated in the tropical and subtropical regions, and, as a non-seasonal crop, bananas are available fresh year-round. It is perennial because it produces succeeding generations of crops. The first cycle after planting is called the plant crop. The ratoon is the sucker (also called the follower) succeeding the harvested plant. The plant propagates itself by producing such suckers which are outgrowths of the vegetative buds set on the rhizome during leaf formation. During their initial development, the suckers share their parent rhizome . Hence, if the parent plant is infested, so are the suckers [24, 26]. The second cropping cycle after planting is called the first ratoon crop. The third cycle is the second ratoon crop, and so on. The growth cycle of banana consists of two phases. The vegetative phase (or ’shooting’) begins with the production of leaves by the planted tissue culture plant and ends when the inflorescence appearing at the top of the plant. During this phase, banana produces roots continuously. After it, the absorbed nutrients are essentially directed to the growing of the fruit bunch . The reproductive phase begins with the transition of the vegetative meristem into a floral shoot. The division of phases is arbitrary, and it takes normally about 7-8 months after planting before the inflorescence emerges at the top of the plant . The fruit filling period, that is,the time between flowering and harvest, completes the reproductive phase and the growth cycle. During the growth cycle, plants develop essentially three major components : an underground corm producing suckers and roots, a pseudo stem consisting of encircling leaf sheaths and carrying the leaves and an inflorescence, containing female flowers that develop into fruits. At the end of the growth cycle, the fruit bunch is harvested and after harvest, the aerial portions of a banana plant (leaves, pseudostem and fruit stalk) are normally cut down, or else they will die back naturally. The roots that are not involved in the growth of the sucker quickly lose their freshness by senescence .
The length of the growth cycle depends on the cultivar. The parent plant and the ratoon are in competition for resources and ratooning is generally followed only in those areas where there is an assured source of irrigation. During the vegetative phase, most of the resources are directed to the growing parent plant. During flowering, ratoon development increases. Hence ratoon management or de-suckering becomes very important. As a rule, a sucker is allowed only after the emergence of the inflorescence in the planted crop and the same package of practices are followed as that of the planted crop before allowing a sucker for the second ratoon. In commercial plantings, usually only one of the suckers is selected to grow out and regenerate the plant , in order to keep constant the number of plants by hectare. In East, West and Central Africa, where most of the world’s plantains are grown, very little attention has been given to them in terms of research. This was evidently because there were no major production problems in the context of limited input and small-scale subsistence farming systems, thus research was not considered a high priority. However, research is now critically important due to the serious threat of black Sigatoka , Xanthomonas wilt , and banana bunchy top virus (BBTV) , as well as rapid yield decline due to banana weevil, poor weed control, poor soil fertility and nematodes.
Biology of Radopholus similis
Radopholus similis is commonly known as the burrowing nematode and belongs to the nematodes phylum, Tylenchida order and Pratylenchidae family . It is with, among others, Radopholus kahikateae sp., Radopholus nelsonensis sp. nov., Radopholus nativus and R. vacuus, one of the many representatives of the genus Radopholus .
Radopholus similis has six life stages: egg, juvenile (4 stages), and adult. The sexual dimorphism is very pronounced in Radopholus similis: the male has a highly developed cephalic cap, thin labialrings, an aborted stylet, a reduced oesophagus and is probably incapable of feeding . On the other hand, females and juveniles have a large stylet with strong basal buttons and thick labial rings. The type specimen is 580 m long by 21 m in diameter . The size and especially the diameter of the adults can indeed vary from one individual to another. In general, males are slightly thinner than females and pregnant females are thicker than young adult females. Examining the offspring of an isolated female, adults can be ranged in size from 580 to 785 m and 22 to 26 m in diameter .
Table of contents :
II Mathematical Preliminaries
1 Mathematical preliminaries
1.1 Semi-discrete modelling and overview on impulsive differential equations
1.1.1 Floquet theory
1.1.2 Comparison principle
1.2 Singular perturbation theory for slow-fast dynamics
1.3 Random search algorithms
1.3.1 General random search and convergence
1.3.2 Adaptive Random Search on the simplex
III Literature Review
2 Biological background
2.1 Biology and cultivation of banana and plantain
2.2 Biology of Radopholus similis
3 Existing soilborne pest, plant epidemic and crop rotation mathematical models
3.1 Soilborne pest and plant epidemics mathematical models
3.1.1 The model of Gilligan and Kleczkowski 
3.1.2 The model of Madden and Van den Bosch 
3.1.3 The model of Mailleret et al. 
3.1.4 A cohort model adapted to Radopholus similis : the model of Tixier et al. .
3.2 Some crop rotation mathematical models
3.2.1 Taylor and Rodrìguez-Kábana’s model of rotation of peanuts and cotton to manage soilborne organisms 
3.2.2 The model of Van Den Berg and Rossing 
3.2.3 The model of Nilusmas et al. 
IV Results and discussion
4 Modelling and analysis of the dynamics of the banana burrowing nematode Radopholus similis in a multi-seasonal framework
4.1.1 Core model
4.1.2 Chemical control model
4.1.3 Fallow deployment model
4.1.4 Well-posedness of the problem
4.2 Analysis and results
4.2.1 Chemical control
4.2.2 Sufficient fallow deployment
5 Optimal fallow deployment for the sustainable management of Radopholus similis
5.0.1 Yield and profit
5.0.2 Parameter values
5.1.1 Location of the optimal solutions
5.1.2 Optimization algorithm
5.2 Numerical results
5.2.1 Small dimensions
5.2.2 High dimensions
5.2.3 Regulation of high dimension solutions
5.3 Discussion and future work
6 Toward a mixed control strategy
6.1 Optimization problem
6.2 Optimization with fixed-size chains and constant fallow
6.2.2 A case study
6.3 Optimization with fixed-size chains and varying fallows
6.3.2 A case study