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A New Discrete Mycorrhiza Optimization Nature-Inspired Algorithm

: This paper presents the discrete version of the Mycorrhiza Tree Optimization Algorithm (MTOA), using the Lotka–Volterra Discrete Equation System (LVDES) formed by the Predator–Prey, Cooperative and Competitive Models. The Discrete Mycorrhizal Optimization Algorithm (DMOA) is a stochastic metaheuristic that integrates randomness in its search processes. These algorithms are inspired by nature, speciﬁcally by the symbiosis between plant roots and a fungal network called the Mycorrhizal Network (MN). The communication in the network is performed using chemical signals of environmental conditions and hazards, the exchange of resources, such as Carbon Dioxide (CO 2 ) that plants perform through photosynthesis to the MN and to other seedlings or growing plants. The MN provides water (H 2 O) and nutrients to plants that may or may not be of the same species; therefore, the colonization of plants in arid lands would not have been possible without the MN. In this work, we performed a comparison with the CEC-2013 mathematical functions between MTOA and DMOA by conducting Hypothesis Tests to obtain the efﬁciency and performance of the algorithms, but in future research we will also propose optimization experiments in Neural Networks and Fuzzy Systems to verify with which methods these algorithms perform better.


Introduction
In the literature, multiple definitions of optimization can be found, and can be summarized as the search for the best probable solutions for a given problem. In [1], the desired concepts and objectives represent obtaining the best solution [2]. When integrating processes, optimization is an effective and powerful tool. The objective function being optimized quantifies the level of the solution being sought [3]. The constraints are actually the limitations of the system model in the search process. Consequently, in optimization we maximize or minimize the value of the objective function subject to constraints imposed on the decision variables [4].
The proposal presented in this work is a new optimization algorithm inspired by nature, where the roots of plants and the Mycorrhiza Network (MN) play the main role, i.e., the symbiosis that occurs between these two protagonists is modeled through LVDES.
The principal contribution of this research is to model the symbiosis between plant roots and the Mycorrhizal Fungal Network (MN) with the Lotka-Volterra Discrete Equation System (LVDES), unlike the previous research [5] for which we used the Lotka-Volterra Continuous Equation System (LVCES). We believe that this algorithm can provide more flexibility and operability in the handling of the parameters and that it adapts to iterative processes.
Many articles were developed using the Lotka-Volterra system of equations, and some investigations followed the direction of continuous equations such as the works of Atena Ghasemabadi and Mohammad Hossein Rahmani Doust in 2019, "Investigating the dynamics of Lotka-Volterra model with disease in the prey and predator species" [6]; James   lines between trees represent their linkage with Euclidean distances. The thickness of the line increases with the number of links between pairs of plants [16].
Thousands of mycorrhizae form in the subsoil of a forest with the roots of plants and different types of fungi [17]. Many plants share fungi whose mycelia connect the roots of different plants, as shown in Figure 3, resulting in the formation of mycorrhizal networks. Mycorrhizal networks of established plants can colonize other nearby plants and seedlings and create fungal pathways that allow plants to exchange carbon, nutrients or water [18][19][20][21][22][23]. In forest dynamics, mycorrhizal networks are recognized for their role in plant establishment, survival, growth and competitive ability [24][25][26][27][28][29]. The importance of mycorrhizal networks for seedling success in relation to other biotic or abiotic constraints is not well understood and may vary with growth stage, climatic and site conditions [30]. Mycorrhizal networks are considered to be the most important form of forest colonization; even when mycorrhizal networks are disrupted by soil disturbance (climatic factors such as snow, wind, ice, floods, fires, logging, animals, etc.), mycelial fragments retain inoculum potential, and the network can rapidly reform [31]. Fungal colonization of naturally regenerating seedlings occurs through mycorrhizal networks and through inoculum dispersed by wind, soil, or mammals, and the role of mycorrhizal networks decreases with an increasing severity of disturbance and loss of residual trees [32]. When disturbances are very severe, plants die, the forest floor is consumed and biomass is reduced. Recolonization requires other sources such as spores carried by air, soil, or mammals, and fungal hyphae found deeper in the soil [33,34].

Discrete Mycorrhized Optimization Algorithm (DMOA)
Through photosynthesis, trees produce carbon-rich sugars (CO2) which they exchange with fungi that form a symbiotic association with plant roots. In the Mycorrhizae Network, exchange with plants consists mainly of water (H2O), nitrogen (N), phosphorus (P), sulfur (S) and other nutrients. Mycorrhizae have also been found to

Discrete Mycorrhized Optimization Algorithm (DMOA)
Through photosynthesis, trees produce carbon-rich sugars (CO 2 ) which they exchange with fungi that form a symbiotic association with plant roots. In the Mycorrhizae Network, exchange with plants consists mainly of water (H 2 O), nitrogen (N), phosphorus (P), sulfur (S) and other nutrients. Mycorrhizae have also been found to connect plants to each other. This algorithm, as mentioned above, is inspired by the symbiotic relationship between plant roots and the mycorrhizal fungal network. Research has shown that the following processes occur in these associations, which are the foundations on which this algorithm is based [16,35] In this research, the initial population for both × and y were different and were obtained by random solutions. With these populations, we found the best fitness solution. With the best solutions found, we used the Lotka-Volterra Cooperative Equation for parameters a and d; the result will influence one of the two LV Equations (Predator-Prey or Competitive), depending on the random diversity result. Diversity is controlled by the probability d [1,2] to choose one of the two Lotka-Volterra equations to try to simulate what happens in an ecosystem.
To overcome the stagnation of the algorithm at local minima, this is controlled by the epochs where the two populations are renewed after each cycle of 30 iterations.
In the Prey-Predator Model, trees are constantly subjected to abiotic stresses (external factors) that exert an influence on plants, which are as follows: water (water stress), salts (salt stress), temperature (heat stress), anoxia (lack of oxygen), oxidative stress, heavy metals, environmental pollutants and atmospheric pollutants. Both the trees and the mycorrhizal network, through chemical signals, are alerted to the danger of animals, insects or any other threat. The Cooperative Model, in which the resources of the entire ecosystem are shared as a way of survival of the entire habitat, the large trees supply the network and other plants with CO 2 because with their height they can perform photosynthesis, and the Mycorrhizae Network supplies the trees and other plants that are connected to the network, mainly with water and other nutrients such as phosphorus, nitrogen, potassium and zinc, among others. The Competitive Model simulates the colonization of the forest, the expansion of the Mycorrhizae network, and the inclusion of other plants in the ecosystem. This process of Defense, Cooperation and Competition ensures the survival of the entire habitat.  Objective min or max f(x), × = (x 1 , x 2 , ..., x d ) Define parameters (a, b, c, d, e, f, x, y) Initialize a population of n plants and mycorrhiza with random solutions Find the best solution fit in the initial population while (t < maxIter) for i = 1:n (for n plants and Micorrhiza population)

DMOA Pseudocode
Evaluate new solutions. If the new solutions are better, the best new solutions are updated. Find the current best fit solution. end while

DMOA Flowchart
The flowchart of the DMO algorithm is shown in Figure 4 where the biological operators of defense, resource exchange and colonization model the symbiosis between plants and MN. The flowchart illustrates how the information flows in the algorithm.

Lotka-Volterra Discrete Equation System
Discrete models governed by difference equations are more suitable than continuous models when reproductive generations last only one reproductive season (non-overlapping generations) [5,39]. The Discrete Lotka-Volterra models shown in Table 1 have many applications in applied sciences. These models were initially developed in mathematical biology, then their studies were extended to other areas [40][41][42][43].
The mathematical description of Discrete Equations (1) and (2) are part of the Lotka-Volterra System of Discrete Equations for the Predator-Prey-Defense Model, where the parameters a, b, d, and g are positive constants, and x i and y i represent the initial conditions of the population for the two species and are positive real numbers [37,63].

DMOA Flowchart
The flowchart of the DMO algorithm is shown in Figure 4 where the biological operators of defense, resource exchange and colonization model the symbiosis between plants and MN. The flowchart illustrates how the information flows in the algorithm.

Lotka-Volterra Discrete Equation System
Discrete models governed by difference equations are more suitable than continuous models when reproductive generations last only one reproductive season (non-overlapping generations) [5,39]. The Discrete Lotka-Volterra models shown in Table 1 have many applications in applied sciences. These models were initially developed in mathematical biology, then their studies were extended to other areas [40,41,42,43].

Model Discrete Equations
Predator-Prey (Defense) and x i and y i are the populations for each of the species and are positive real numbers. Each of the parameters of the aforementioned equations are described below [9,36].
The main differences among the three biological operators are as follows: 1.
The six equations are different and each pair of equations corresponding to each biological operator model the conditions under which said operators work. 2.
The Cooperative model infers randomly in one of the two biological operators of either Defense and Competitive, to try to simulate what can happen in a living and changing ecosystem over time.
The inspiration of the MTOA Algorithm is the symbiosis between the Trees and the Mycorrhizal Network and uses the System of Continuous Equations of Lotka-Volterra. The algorithm, DMOA is inspired by the symbiosis between the Plants and the Mycorrhizal Network and tries to simulate in part what happens over time in this ecosystem. There is no modification of the algorithm, with the difference being the use of the Lotka-Volterra System of Discrete Equations. In the MTOA algorithm, the best fitness values are decided by taking the lowest result of the three biological operators. In the DMOA algorithm, the result of the cooperative operator is randomly inferred through the parameter x i (grow rates of populations x at time t) in one of the two biological operators of Predator-Prey or Competitive, resulting in the best fitness; we used this parameter because in the experimentation it provided us with better results for the desired purpose. Table 2 lists the parameters of population, growth rate, dimensions, epochs, iterations, etc., that were used in the experiments with DMOA and MTOA algorithms.  The parameters a and d result from the populations for x and y, which change in each iteration followed by the smallest values being selected. These results are assigned to the population growth rates x i and y i of the discrete Lotka-Volterra equations, while the other parameters such as population size, dimensions size, number of epochs, and iteration size are positive integers that are initially defined and usually have the following values: 20, (30,50,50,100,200), 20, and 30, respectively, however there is flexibility to play with other data to achieve better results. Table 3 shows the 36 mathematical functions of the CEC-2013 competition that we used to carry out the experiments. Table 1 shows the name of the functions, their range and their nature (Unimodal or Multimodal) and Figure 5 shows the corresponding graphs of a sample of the functions for illustration purpose.

Results of the MTOA Algorithm Experiments with CEC-2013
Tables 4-6 show the results of the experiments of the 36 mathematical functions of the CEC-2013. The tables show the means and standard deviations that were obtained for 30, 50, and 100 dimensions and for 30, 50, 100, and 500 iterations, respectively.

Results of the DMOA Algorithm Experiments with CEC-2013
Tables 7-9 show the results of the experiments of the 36 mathematical functions of the CEC-2013 of the mean and standard deviation that were carried out for 30, 50 and 100 dimensions and for 30, 50, 100, and 500 iterations, respectively. Figure 9 show the convergence of the following three mathematical functions: Sphere, Ackley, and Rosenbrock for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms. Figure 10 show the convergence of the following three mathematical functions: Ackley, Rosenbrock, and Rastrigin for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms. Figure 11 show the convergence of the following three mathematical functions: Rosenbrock, Rastrigin, and Griewank for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms.

Results of the DMOA Algorithm Experiments with CEC-2013
Tables 7-9 show the results of the experiments of the 36 mathematical functions of the CEC-2013 of the mean and standard deviation that were carried out for 30, 50 and 100 dimensions and for 30, 50, 100, and 500 iterations, respectively. Figure 9 show the convergence of the following three mathematical functions: Sphere, Ackley, and Rosenbrock for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms. Figure 10 show the convergence of the following three mathematical functions: Ackley, Rosenbrock, and Rastrigin for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms. Figure 11 show the convergence of the following three mathematical functions: Rosenbrock, Rastrigin, and Griewank for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms.

Results of the DMOA Algorithm Experiments with CEC-2013
Tables 7-9 show the results of the experiments of the 36 mathematical functions of the CEC-2013 of the mean and standard deviation that were carried out for 30, 50 and 100 dimensions and for 30, 50, 100, and 500 iterations, respectively. Figure 9 show the convergence of the following three mathematical functions: Sphere, Ackley, and Rosenbrock for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms.         Figure 10 show the convergence of the following three mathematical functions: Ackley, Rosenbrock, and Rastrigin for 30, 50, and 100 dimensions for the DMOA and MTOA algorithms.

Hypothesis Tests
All hypothesis tests were performed for two independent samples with 30 experiments each, where µ 1 , µ 2 , σ 1 , and σ 2 were known [64][65][66][67]. Table 10 shows the null and alternative hypothesis, while Equation (7) shows the Z test statistics formula and Table 11 shows the hypothesis test parameters. Table 12 shows 108 hypothesis tests (z) that were performed on the MTOA and DMOA algorithms for 30, 50, and 100 dimensions and 30, 500, and 100 iterations, respectively. In most of the tests, in 92% of cases there was significant evidence that the MTOA algorithm is favorable over the DMOA algorithm, and the DMOA algorithm only performed better in eight tests (7.4%), in two functions for 30 × 30, in 2 for 50 × 50, and in four functions for 100 × 100.  Table 11. Hypothesis Test Parameters.
x 1 = Mean of sample 1 x 1 = Mean of sample 2 σ 1 = Standard Deviation of sample 1 σ 2 = Standard Deviation of sample 2 n 1 = Number of sample data 1 n 2 = Number of sample data 2

Compared with Others Methods
In Tables 13 and 14, a comparative study of the DMOA Algorithm with other methods of the mean and Standard Deviation is shown, including MTOA (Mycorrhiza Tree Optimization Algorithm) [14], Self-Defense (A New Bio-inspired Optimization Algorithm Based on the Self-defense Mechanism of Plants in Nature) [68], FA (FireFly Algorithm) [69], GSA (Gravitational Search Algorithm) [69], CS (Cuckoo Search) [69], GA (Genetic Algorithm) [69], DE (Differential Evolution) [70], HS (Harmony Search) [69], and GA (Genetic Algorithm) [69] for 30 and 50 dimensions. In Figures 12 and 13, we observe the behavior of the Standard Deviation of the DMOA Algorithm and the aforementioned methods for 30 and 50 dimensions and realize that in both behaviors the Mean and Standard Deviation of the DMOA were superior in most cases compared to the other methods.

Programming Environment
The language used in the programming of the DMOA algorithm in MATLAB  R2019b and

Programming Environment
The language used in the programming of the DMOA algorithm in MATLAB  R2019b and

Programming Environment
The language used in the programming of the DMOA algorithm in MATLAB R2019b and the equipment where the programming and experiments were carried out is a Desktop Computer Intel Core i5 4460S 2.90 GHz, RAM DDR3 16 Gb, Intel HD Graphics 4600, and Operating System Windows 10 Professional.

Conclusions
The MTOA and DMOA algorithms have the same inspiration in nature, derived from the symbiosis between the roots of the plants and the Mycorrhiza Network (MN). The main difference between both algorithms is the use of the Lotka-Volterra equations system; in the MTOA algorithm, we previously used continuous LV equations and in the DMOA algorithm we used the discrete LV equations that according to the literature have better handling, flexibility and results in the algorithms in the use of time.
In this research, 36 mathematical functions from CEC-2013 were used. All 36 experiments of the MTOA and DMOA algorithms were performed in 30, 50, and 100 dimensions, and for 30, 50, 100, and 500 iterations. Additionally, 108 hypothesis tests were performed for two samples and there was significant evidence that the DMOA algorithm was better in only eight tests, as shown in Table 9. The research will be furthered as experiments will be carried out in the future with Type-1 and Interval Type-2 Fuzzy Logic Systems (T1FLS-IT2FLS) in the adaptation and optimization of parameters, as well as in the optimization of neural networks and it is possible that each of these algorithms could be better suited to different fields, recalling the No Free Lunch Theorem (NFLT) of Optimization principle that "a general-purpose, universal optimization strategy is impossible" [71,72].
A comparative analysis of the behavior of the Mean and the Standard Deviation for 30 and 50 dimensions was carried out with the methods of MTOA (Mycorrhiza Tree Optimization Algorithm) [14], Self-Defense (A New Bio-inspired Optimization Algorithm Based on the Self-defense Mechanism of Plants in Nature) [68], FA (FireFly Algorithm) [69], GSA (Gravitational Search Algorithm) [69], CS (Cuckoo Search) [69], GA (Genetic Algorithm) [69], DE (Differential Evolution) [70], HS (Harmony Search) [70], and GA (Genetic Algorithm) [69] for five mathematical functions of Sphere, Ackley, Rosenbrock, Rastrigin and Griewank. The results we obtained revealed that the Mean and the Deviation of the DMOA standard were lower than in the other six methods and was only surpassed by the MTOA in the five functions, the Self-Defense in two functions (F1 and F3) and the GSA in two functions (F1 and F2). We can conclude that the DMOA is a competitive algorithm in relation to the other compared methods.