HYBRID MODELING PREY PREDATOR-GENETIC ALGORITHM OF MOLD GROWTH DURING CACAO BEANS STORAGE

The production and quality of cocoa beans remain key issues for the economy of Côte d'Ivoire. The fairly humid Ivorian climate promotes increased mold production, leading therefore to a considerable negative impact on beans quality. In this work, a prey predator model was proposed to simulate the molds growth on cocoa beans. Our model (MPM), based on that of Leslie Gower, was identified using a multiobjective genetic algorithm (MOGA). The developed model allows to stimulate the relationship between parameters (i.e.


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So mathematical approaches relating to molds growing on various supports (substrates) are proposed(E & F, 2002; Hukka & Viitanen, 1999;Kaleta & Górnicki, 2013). It is important to note that there are very few mathematical approaches modelling mold evolution in cocoa beans. Although, the literature about mold points out some typical models for some materials (Hukka & Viitanen, 1999;Ndiaye et al., 2002;Vereecken et al., 2015), and cereals (E & F, 2002; Kaleta & Górnicki, 2013). These so-called qualitative models set a critical threshold that depends on temperature and / or relative humidity above which a proven risk exists (Berger et al., 2018). For example, Jacobsen et al (E & F, 2002) proposed a linear model which takes account grain moisture (Aw) and temperature (T) as input parameters, in order to predict time elapsed before molds appearance during storage. Based on these same parameters, Ndiaye et al (Ndiaye et al., 2002) proposed a preventive approach (i.e. QualiGrain) of maximum storage time. Indeed, these models enable to predict the grains shelf life. However, they are static because they do not take into account the phenomenon temporal aspect. In addition, they are, a priori, based on predefined schedules with corrective interventions which are applied after damage observed (Fleurat-Lessard, 2017). To overcome these limits, several authors have turned to dynamic models defined by ordinary differential equations (ODE) (Berger et al., 2018;Hukka & Viitanen, 1999;Vereecken et al., 2015). For this purpose, approaches have been proposed in literature to predict risks of mold contamination as a function of relative humidity and temperature conditions (Barel, 1985). Among them, the VTT model has been proposed for wood and other building materials (Hukka & Viitanen, 1999). This model is one of the first in the construction industry, and has intensively been used. However, according to studies carried out by Berger et al (Berger et al., 2018), VTT model presents some limits because the determined parameters (e.g. k11, k12) are not included in the range of classes defined by this model. This work showed that mathematical formulation of mold growth VTT physical model is unreliable. Furthermore, these results showed that a small numerical value modification (about 1%) of VTT model parameter generates a prediction error of 94%. They therefore proposed a logistics model to improve the VTT one. However, even this last model (Berger et al) does not take into account mold growth variation under different humidity and temperature conditions. Most of these models focus only on mold population without considering the potential interactions that may exist in substrate environment in which they grow. In fact, mold evolution is strongly correlated to the substrates physicochemical parameters one, leading therefore to the proposition of a prey predator type differential equation model. This model considers each individual in a population and simulate its behavior and interaction with other individuals (Luckinbill, 1973). Prey-predator models enable to explain and provide solutions to the complexity of various ecosystem(Camara, s. d.). The first models were proposed by Lotka and Volterra in the 1920s, to explain the oscillatory phenomenon with phase shift of prey and predators abundance (Abid, 2016). These models have drawn considerable attention in applied mathematics and theoretical ecology. However, in (Abid, 2016)criticisms are made about the linear relationship between attack rates and predator size. Therefore, Leslie and Gower proposed a predation model improvement by integrating, as an hypothesis, a limit threshold representing resources maximum capacity for prey and predators (Leslie & Gower, 1960).Thus, this basic hypothesis (Du et al., 2019)confers to it the class of a realistic model and was therefore used for several areas (e.g. health, biological control, fisheries to regulate excessive fishing). These models (prey predator type) are still untapped for prediction of mold evolution in cocoa beans. Among all these models, Leslie-Gower one is suitable to investigate the case of interactions between cocoa beans and molds.
Unfortunately, in the case of cocoa beans storage, very few studies concerning modeling of mold growth were carried out. The purpose of this present study is to propose a growth model of molds evolution via the predator-prey model (Leslie Gower).

Materials and methods:-
In the purpose of modelling molds growth on cocoa beans during their storage, the Leslie-Gower model was used combined to a multi-objective genetic algorithm (MOGA). Indeed, the latter was used for Leslie Gower specific parameters identification.

Leslie Gower Model:
In our approach, we considered Leslie Gower (Eq. 1) prey predator model as the one that simultaneously obtains the prey and predators density over time by taking into account the available resource relative to each population. In this model, the predation term can be written in different ways depending on the response functional type. There are three types according to the problem to be solved (Abid, 2016). On this subject, in (Luo et al., 2020) the authors state that the functional response is a determining element of predation. Thus, whether it is linear (Holling type I), nonlinear (Holling type II) or sigmoid (Holling type III), it helps to better understand the relationship between prey and predator. In the case of our study, we opted for the Holling type I functional response which results in (Abid, 258 2016). And this means that the variation in predation evolves proportionally to encounter rate between predator and prey. This model looks like this: (1 ) (1 ) Where U (t) represents prey population and V (t) the predator one at time t. μ ≥ 0, λ≥0, ρ≥0 and θ> 0 are parameters representing the interaction between these two species. μ denotes the prey population growth rate in the absence of predator. λ represents predator attack rate, θ represents the conversion efficiency (percentage of consumed biomass that is converted into predator biomass), ρ the predator mortality rate and μ the growth rate of the prey population in the absence of predator. In our approach, 5 parameters were identified by optimizing the objective or fitness function that has been used to evaluate individuals in a well-defined research space. It is also necessary to assign data to the GA operators, namely selection, crossover and mutation. To accomplish this, each parameter to be identified must be encoded by a gene using either binary or real representation. In this present work, real representation was chosen to avoid difficulties that can be raised on the binary encoding and decoding of chromosome. It is important to note that AGs can operate with one or more objective functions. This last case concerns multi-objective optimization. This optimization uses the Pareto dominance notion to classify solutions and define selection strategies (reproduction or survival). They keep the best solutions in the population over generations or save them in an archive. From the compromises of objective functions, an equilibrium called Pareto optimum is reached. According to (Murugan et  The principle of MOGAs is more specific at the assessment stage preceding the selection one. Indeed, this step is subdivided into two operations, namely the solution quality assessment in terms of convergence and diversity. A multi-objective optimization problem is as follows ( Proposed Approach: Our approach to predict mold evolution encompasses the growth and predation of mold on cocoa beans surface sugars. This predation is illustrated in Fig. 1;   Figure 1:-Prey Predator diagram (molds -sugar).

Problem Formulation:
Leslie Gower model aims modelling prey predator interactions according to verhust law. Thus, this technique use is justified in our case, because molds have an impact similar to that of a predator on cocoa beans. In fact, to feed themselves, molds proceed by absorption of surrounding organic matter (carbon source); hence their so-called heterotrophic way of life (« L'univers des champignons », 2018). To obtain the herein prey predator model, sugar content was indexed as prey and molds as the predator. According to literature, water content promotes molds development. However, sugars which coating the tegument, constitute one of the carbon sources and therefore food for molds. Thus, Leslie Gower model enables to determine molds evolution according to sugar amount observed regularly. In our work we assume that: (i) mold development depends mainly on sugar coated on beans; (ii) all beans have the same sugar content after drying.

Mathematical Model:
Using the Leslie Gower model as defined previously (section 3), it is necessary to determine molds amount as well as sugar content over time. Then formulation (1) after correspondence of the different terms, becomes formula (3). All paragraphs must be indented.

Multi-objective Function:
In this section, the optimized bi-objective function Jr (r = molds and sugar) is defined. Indeed, Jr represents the sum of the squares of the differences between the observed sugar / mold values and those calculated by Eq.(1) or (2), as follows : Then, a genetic algorithm is applied in order to find the parameters best values that minimize these functions concomitantly. The optimisation using of the genetic algorithm was repeated 100 times with the operators values defined in Table 2. The experimental data were obtained by measurements of biomass (molds) and sugar amount on cocoa beans tegument stored in jute bags for 24 weeks. The different bags were placed on pallets (1 m x 70 cm), in a store at room temperature (20-36 ° C).
Each week, three bags were taken randomly considering beans heterogeneity. The parameter value at a given time is then the mean one.

Mold determination:
Total fungal population count was carried out by surface inoculation on PDA medium (Potato dextrose agar) with chloramphenicol. This is a method that allows uniform molds growth because of equal access to oxygen (Hocking, 261 1991

Sugar content determination:
Sugar content determination was carried out by using Dubois sulfuric phenol method (Bourely, 1980).

Results and Discussion:-
The different simulations were carried out using Matlab R2013a. The computer used characteristics were: 64bitoperating system computer, 8 GB of RAM, Processor Intel Core i7.
The parameters 1 r , u S  , λ , 2 r , α of the approach presented in section 4 were determined by minimizing the multiobjective function Jr using experimental data. The results of these simulations are presented by Fig.2, represent ing the Pareto front. It appears that the values of Jsugar are lower when Jbiomass ones are higher. However, the best value of Jr function two components is shown in red in this figure. It coordinates are (0.083; 0.547) and represents the best compromise between Jbiomass and Jsugar.
This minimum value corresponds to the individual presented in Table 1.  The analysis of Fig. 3, shows that the biomass increases gradually from week 0 (0.30 log CFU / g) to week 15 (4.11 log CFU / g). Then it stabilizes until the end of the experiment (24 weeks). On the other hand, the sugar content decreases steadily until week 15 to stabilize (Fig 4).
Furthermore, it is clear from the two figures that the calculated values fit correctly the experimental ones with a correlation coefficient R = 0.9963 and R = 0.8382, respectively for the biomass and the sugar content. This result shows a good accuracy of Leslie Gower model for modeling mold growth on cocoa beans during 24 weeks storage. These observations are in agreement with those reported in Ref. 16 for molds on different supports.
However, there is a slight discrepancy between experimental and observed data in particular, with regard to the sugar content, between the 7th and 16th weeks. This fact could be due to phenomena (impact of relative humidity, temperature ...) not considered in our model.

Conclusion:-
In this paper, a mathematical model of mold evolution in cocoa beans based on prey predator models of the Leslie Gower type was proposed. A multi-objective genetic algorithm was also used for its parameters identification. The results obtained show a good accuracy between experimental and the proposed model calculated values, indicating that Leslie Gower models are suitable for modeling molds growth during cocoa beans storage.