Phenology and physioloygy of cacao tree Theobroma cacao
The genus Theobroma originated millions of years ago in South America, to the east of the Andes. Theobroma has been divided into twenty-two species of which textit T. cacao is the most known. It was the Maya who provided tangible evidence of cocoa as a domesticated crop. Archaeological evidence in Costa Rica says cocoa was drunk by Maya traders. The first to drink chocolate was Christopher Columbus, who reached Nicaragua in 1502 searching for a road to the spices of the East. But it was Hernan Cortès, leader of an expedition in 1519 to the Aztec empire, who returned to Spain in 1528 bearing the Aztec recipe for cocktail (chocolate drink) with him. The drink was originally received unenthusiastically and it was not until it became a popular drink in the Spanish courts. Strong demand for chocolate has boosted cocoa cultivation worldwide. Amelonado cacao from Brazil was planted in Principe in 1822, Sao Tomé in 1830 and Fernando Po in 1854, then in Nigeria in 1874 and Ghana in 1879. In Cameroon, cocoa was introduced during the colonial period of 1925 to 1939.
Cacao (T. cacao) is a plant of the family of Sterculiaceae and originated from the wet tropical forests of Central America . The firsts exportations of cacao towards Europa are been done in 1 585. In 17e century, there are many plantations of cacao around the world and it was through Fernando Pô island (actually Malabo), Sao Tome and Principe, which cacao is introduced in Africa. Cacao will be introduced in Cameroon in 1892 by Germans (, , ).
Table 1.2 recapitulate the taxonomy of cacao tree. The growth of leaves and stems is done by successively thrusts separated by rest period. During these periods, the final buds take back their « dormancy » . Several factors influence the growth of leaves and stems in particular temperature, daylight, tenor of hydrate of carbon and regulators of growth.
Production needs several stage: flowering, fruition, harvesting and marketing of beans. Cacao tree is able to flower all the year. Cocoa is raised from seed. Seeds will germinate and produce good plants when taken from pods not more than 15 days under-ripe. A bud is cut from a tree and placed under a flap of bark on another tree. The budding patch is then bound with raﬃa and waxed tape of clear plastic to prevent moisture loss. When the bud is growing, the old tree above it is cut oﬀ. A strip of bark is removed from a branch and the area covered in sawdust and a polyethylene sheet. The area will produce roots and the branch can then be chopped oﬀ and planted. The flowering of cacao has the particularity to appear on branch and trunk. There exists a large flowers which succeed all the year. On mature cacao, flowering is cyclic so to speak that the high growing period alternate to the small growing period . The next period after flowering is fruiting. The number of fruits present on cacao tree depends on the genotype of cacao. There is function of fertility of land, availability of water and daylight. Maturity of fruits can be aﬀect by temperature: we remark that during the warm months, pods needs approximately 140 to 175 days for its total development whereas during the cold period, pods need approximately 167 to 207 days for its total development.
Seasonal variations of mirids populations
Mirids dynamics varied greatly during the year. Density of population is likely to be influenced by pods availability on the trees. Mirid population is low on cocoa during the period from February to March. From June to July, the populations start to grow more or less rapidly depending on external conditions like weather and fruits production on the trees. The peak of the population appears between September and November when the pods are almost mature (; ; ,). Between January and March, despite the low number of pods present on cacao tree, it is still possible to collect mirids principally on the greedy and branches. Even if the lack of resource (i.e pods) can explain the decrease in mirid population, other hypotheses have been formulated to explain the drastic decline that occurred between February and March. In Babin et al. 2010 , it is assumed that lower mirid populations observed in plantations during a certain period of the year (the period from November-December to June) is due to declining fertility of females and increased mortality of individuals 1.5. Thus, it seems that development parameters (longevity, fecundity) of mirids vary depending on season. The G8 and G9 generations obtained in June, July and August (during pods formation on the trees) were the most productive generations with high fecundity and female longevity. On the other hand, generations G11 and G12 obtained in November, December and January showed lowest fecundity and female longevity. In Kouame et al. 2014 , it is assumed that the lower mirid population is observed in plantations between February to June. This variation depends on temperature and pluviometry.
Another hypothesis that can explain the drop of the mirid population on the plot is migration. In fact, cacao is cultivated in agroforestery systems in association with several other crops among which potential mirid host plants (Cola nitida, Ceiba pentandra). Mirids are originated from African forests therefore before the arrival of cocoa in Cameroon, those insects developed on other forest trees like D. dewevrei, Ceiba pentandra in Cameroon. So it could not be excluded that mirids take refuge in those trees when resources from cocoa trees become less suitable for the species development. An experience proved that S singularis is able to feed on D. dewevrei but its capacity to multiply on this tree remains to be confirm. Then, when development condition become more favourable on the cocoa trees, the mirids can re-infest the plantations.
Dispersal availability of S. singularis
Moving of S. singularis in the plot is an important factor using for evaluate the interaction between S. singularis and cacao. Eggs of S. singularis are immobile and wait for hatching to become nymphs. Nymphs of S. singularis are very mobile but only in the same tree. Nymphs do not have the ability to fly from one plant to another. They must complete their development and become adults for that; therefore, nymphs usually feed on the same cocoa tree view on the same tree. The dispersal potential of this species is entirely determined by the mirid adult flight capabilities. Concerning adults S. singularis, studies in Ghana and Nigeria, according to various methods have demonstrated that adult mirids can fly 24 hours after emergence, but the first long flight occurs only after 3 to 5 days. A fight of mirid can be started by coming of predator and in this case, flight is short and in zigzag or in spiral . He esteem that, on the average the distance travelled by adult of D. theobroma is 1.1 km for males and 2.3 km for females, with on the average speed of 3.4 meters per second. Youdeowei distinguish two types of flight of S. singularis: the trivial flight and the dispersion flight. A trivial flight only last for some seconds and have for main objective the feeding or researching of partner for mating. A dispersion flight can last more than thirty minutes and allow S. singularis to colonize a new habitats . In the same article, he observe that female of S. singularis are able to flight more than one hour without any interruption and males more than one hour and thirty minutes.
Interaction between mirids and cacao in agroforests.
Interaction between mirids and cacao is essentially due to the action of mirids on cacao. Mirids use cocoa for their feeding and their development. Mirids feed and lay on cacao tree: this action lead to damages observed on the tree: anecdotal damages on a pod and cumulated damages on the other parts of the tree.
Mirids tend to hide themselves to feed and move quickly. So,it is diﬃcult to establish a link between mirids population and damages observed in the yield because the treatment is based on the estimation of population of mirids and not on the damages observed in the yields. Moreover, several farmers cannot see the insect but they are able to recognize the damages on the pods. The bite on pod are check oﬀ on peduncle and according to the cortex size, damages on pods are remain anecdotal. In fact, the harvest remains interesting and the selling of beans continues. The most damages (damages which appears on the other parts of cacao tree) are cumulated over time and can lead to the destruction of the tree (, , , ).
Impact of the action of mirids on the development of cherelle: Cherelle withered
Mirid aggression lead to the death of several « cherelle ». Withered of « cherelle » is either due to mirid aggression, or internal factors of cocoa and environmental factors. Biting « cherelle » tend to be wither instead of to evolve as pods. In mean, 63% of biting « cherelle » whiter. In the plot, there has a small « cherelle » aggression. Mirid which prefer feed on pods are very low in the plot between February and June; this period correspond to the flowering and « cherelle » development. Mirid aggression on « cherelle », even if they are low, they are very harmful for the « cherelle » development.
Natural enemies and control S. singularis
Mirid S. singularis have many natural enemies: entomopathogenus, parasitoids, parasites and predators. Leiophron (Euphoros) sahlbergellae is the only parasite which lives in depend on the larvae of S. singularis. A parasitism rate of this specie is evaluated to less than 6 to 20% in Cameroon . Several species of toadstool are identified as harmful for the population of S. singularis: we can cite Hirsutella sp. and Beauveria sp.. More recently, a stock of Beauveria bassiana has been isolated in Cameroon . There is not several species of predators of mirids. We can cite species of spider and ants which are predators of D. theobroma. In consideration of the morphological similarity between S. singularis and D. theobroma, these species of predators of D. theobroma are probably predators of S. singularis.
Theory of monotone dynamical systems
Monotone dynamical systems is systems which preserve some partial order on the state space. Monotonicity has been shown to constrain system behaviour in various ways, for example ruling out attracting nontrivial periodic orbits, under fairly general assumptions. When the system is strongly monotone, behaviour is constrained further: for almost all initial conditions bounded solutions con-verge to the set of equilibria. Sometimes such generic convergence claims can be strengthened: for instance, convergence of every bounded orbit can be obtained in a variety of special cases. Mathematical models of biosystems, in general, and in population studies, in particular, are often represented by continuous dynamical systems. The qualitative analysis of such systems regarding the long term behaviour of their solutions is one of the main mathematics involvements is such studies. The aim of our study concerns the properties of global nature like basins of attraction and global asymptotic stability of equilibria. The theory of monotone dynamical systems oﬀers alternative monotonicity-based approach which is to a large extend independent of the dimensionality of the system. There exists two types of monotone systems: Cooperative and competitive systems.
We denote by F : X ! Rn a C1 vector field generating the (local) flow = f tg, in X. Thus the solution to the initial value problem u = F (u), u(0) = x is the curve t ! t x, defined for t in some open interval Ix = ( x; x), x 0 x +1. We call F (or ) cooperative if @Fi=@xj 0 f or i 6= j.
For some results we need the additional assumption that F is irreducible, i.e., the Jacobian matrices DF (x) are irreducible. When this holds and F is cooperative then D t(x) > 0 for t > 0, and p-convexity then implies that is strongly monotone : t(x) < t(y) if x < y and t > 0.
Table of contents :
1 Literature review and Mathematical tools
1.1 The cacao tree: Theobroma cacao
1.1.1 Origin and importance
1.1.2 Characteristics of agroforests in Cameroon
1.1.3 Phenology and physioloygy of cacao tree Theobroma cacao
1.1.4 Diseases and bugs of T. cacao
1.2 Cacao mirids
1.2.1 The cacao mirid: Sahlbergella singularis
188.8.131.52 Presentation of Sahlbergella singularis
1.2.2 Life cycle of S.singularis
1.2.3 Biology of S. singularis
184.108.40.206 Development of S. singularis
220.127.116.11 Mating, reproduction and egg-laying
18.104.22.168 Feeding behavior of S. singularis
1.2.4 Ecology of S.singularis
22.214.171.124 Estimated mirid populations in plots
126.96.36.199 Seasonal variations of mirids populations
188.8.131.52 Dispersal availability of S. singularis
1.2.5 Interaction between mirids and cacao in agroforests.
1.2.6 Damage caused by mirids
184.108.40.206 Effects of the action of mirids on the growth of cacao tree
220.127.116.11 Impact of the action of mirids on the development of cherelle: Cherelle withered .
1.2.7 Natural enemies and control S. singularis
1.2.8 How to control S. singularis?
1.4 Dynamical systems
1.4.1 Theory of delays differential equations (DDE)
18.104.22.168 Useful results about delays differential equations
1.4.2 Theory of monotone dynamical systems
22.214.171.124 Cooperative systems
126.96.36.199 Cooperative systems with concave nonlinearities
188.8.131.52 Cooperative delayed systems
1.4.3 Theory of piecewise dynamical systems
2 Mathematical modelling of the time evolution of Sahlbergella singularis: Application to control’s improvement
2.1 The ODE model with constant parameters
2.1.1 Formulation of the model
2.1.2 Study of the ODEs model
2.1.3 Sensitivity analysis
2.1.4 Mirid system with periodic coefficients
2.2 A model with delays
2.2.1 Sensitivity analysis
2.2.2 Numerical simulation
2.3 Application to control strategies
2.3.1 Chemical control
2.3.2 Semio-chemical control
2.3.3 Comparative study between chemical control and semio-chemical control
3 Miridae control using sex-pheromones, trapping. Modeling, analysis and simulations.
3.1 A sex-structured model of mirid population
3.2 Control using mating disruption and trapping
3.2.1 Case with Male abundance: M > F + Fp
3.2.2 Case with male scarcity: M < F + Fp
3.2.3 Study of the bifurcation for the threshold F p and F p .
3.2.4 Long term behaviour of system (3.2) when Fp > 0
3.2.5 Control strategy related to the level of infestation of Mirids
3.2.6 Applications – Numerical simulations
3.3 About mating disruption strategy when the pods carrying capacity is periodic
3.3.1 Periodic case – Simulations
.1 Equilibria of Linear and Non-linear Systems
.2 Stability of solutions and bifurcations
.2.1 Basic offspring number
.3 Irreducible Cooperative Systems
.4 Uniform persistence theory