Mathematical modelling of soybean var. anjasmoro plant growth

Thorough understanding of interactions between all factors involved in soybean plant cultivation process is needed to increase the yield. This study is aimed to determine which interaction has the highest correlation according to the Pearson correlation coefficient, to define a certain model for said interaction, and to confirm the identicality of all samples that were observed using destructive sampling method. After being tested using Pearson correlation coefficient, the highest r value is 0,81 which is between plant height and number of leaves, proving they are highly related. Both parameters were simulated in several different ways until it was found that number of leaves variable is best described as function of plant height variable, the equation is y1 = 0,85x1 + c1 where y1 represents number of leaves and x1 represents plant height. Identicality between all plants is not confirmed, thus destructive sampling method is not recommended to be applied for similar studies.


Introduction
Soybean production in Indonesia still has not reached its full potential yet [1]. This is mostly due to the lack of detailed knowledge about the best cultivation method for soybean, espescially in the terms of fulfilling plant's needs of nutrient, water, sunlight, and other supporting components with the right amount and at the right time in order to produce the optimum yield [2]. To be able to meet the needs of plants in appropriate amount and time, an understanding of the relationship between plant growth supporting components and their effect on the plant's physical parameters is required, from which a certain model can be defined in order to understand those relationships more easily [3].
This study is conducted to define the interelationship between soybean plant growth supporting components such as water needs, total nitrogen, nitrogen fixing bacteria population, and soybean plant physical parameters which are plants height, and number of leaves [4]. Pearson correlation coefficient is used in this study due to its ability to show the coefficient of correlation between the two variables [5]. Furthermore, the plant growth pattern is also analyzed through plants height changes, and its interaction with the number of leaves changes using simulations with random numbers. The study was conducted by planting 39 soybean plants in a greenhouse using polybags by following an established method of soybean plant cultivation stated by Indonesian Legumes and Tuber Crop Research Institute [6]. The parameters being observed are divided into two categories. The first one is the destructive parameter [7], it includes total nitrogen content and nitrogen fixing bacteria population. To obtained the required data for this parameter, three soybean plants are chosen randomly every once in a week for 13 weeks to be destructed. The next one is non-destructive parameter, it includes plant height and number of leaves. the data for this parameter is obtained from all of the remaining plants every once in a week for 13 weeks.
All the average value of all parameters obtained every week for 13 weeks were then being tested using pearson correlation coefficient [8]. Plant height parameter continued to be simulated using logarithmic growth with limit function. Number of leaves parameter continued to be analyzed of its interaction with plant height, and also simulated as several different functions, which are as a function of plant height, as a function of time (week/s after planting), and using a data generated with random numbers.

Schematic of Components Relation
There are five main components in the soybean plant cultivation system considered in this study, which are environment [ ). Several of these components has an inevitable reciprocal relationship, for example the relationship between nitrogen content and nitrogen fixing bacteria population [9].
EV can not be controlled; EP in WN will depend to EV and NL; TP ini WN will depend to EV, NP and NL; PNB will depend to EV, EP and NS; NS will depend to PNB and NP; NP will depend to NS, PH, and NL; PH will depend to EV, WN, TN, and NL; while NL will also depend to EV, WN, TN, and PH.

Pearson Correlation Coefficient
The data obtained during soybean cultivation process are as follows:      From the figure 2 to 6, it can be seen that the data obtained is quite varied and not always forming a smooth line. The interactions between each of those parameters with all other parameters were then being tested using pearson correlation coefficient [8] and the results are as follows:

Plant Height
Living creature's growth, including soybean plant growth, as can be seen through its height growth, commonly can be justified using the function of logarithmic growth with limit (equation 1) [10], Thus, it will follow the sigmoid curve as seen in figure 7 below:  However, the plant height changes data obtained from the field observation in this study as seen at Figure 2, is scaterred randomly and does not form a smooth sigmoid curve. The comparison between the original data obtained and simulated data using the equation of logarithmic growth with limit is as seen at figure 8.
This happened due to the usage of destructive method for the data sampling, in which the plant is being destructed routinely during the cultivation process, and all the plants involved in the study is assumed to be identical, therefore the remaning plants are assumed to be able to represent the already destructed plants until the end of the cultivation period. In fact, according to the previous research reported [3], each individual of plant, although it comes from the same variety and is claimed to have high similarity, has their each own way to expressed their genetic information, and it will very much depend on the conditions they are exposed into.
Hence, to mimic the process of data sampling using destructive method, we simulate the plant height data using equation (1). The two variables are Hmax that represents the maximum height possibly reached by the soybean plant, and r that represents the growth rate of soybean plants, both are modified using random numbers to picture the difference of genetic expression of each individual plant. As for H0, it is equated with the value of H0 from the field observation, which is 11.95. The sigmoid curves generated from equation (1) should be able to cover the maximum and minimum value of the original data, and other curves that lie between those two. This process resulted on figure 9 below. The data forming the chart at figure 9 is then eliminated one by one for each week to demonstrate the process of destructive sampling process as follows: The yellow box represents the data absent due to destructive sampling activity. The average value is then calculated from the remaining data, and then it is plotted into a graph and is compared with the original data obtained by field observation to see the similarity.

Number of Leaves
According to pearson correlation coefficient calculation, the r value of plant height interaction to number of leaves is 0.81, wich is a strong possitive correlation, it means that high score at plant height variable will go with high score of number of leaves [5]. Hence we proposed number of leaves variables as the function of plant height generated from one of simulated data in the previous section, using the following equation: With y1 represents the number of leaves and x1 represents the plant height. Next we also proposed number of leaves as the function of time, in this case is weeks after planting, using the following equation: With y1 represents the number of leaves and x2 represents the weeks after planting Last we proposed number of leaves data generated using random numbers. The result of these three simulations are then compared to the the field observation data of number of leaves and its correlation with the plant height are also tested using pearson correlation coefficient. The result is shown in the following figure  Figure 10. Simulation of soybean plant number of leaves and plant height as several different functions.

Result and Discussion
From the pearson correlation coefficient (table 1), it is known that the highest r value 0.81 is obtained from the interaction between plant height variable and number of leaves variable. This phenomena corresponds to the previous research reported that mentions that interrelationship between plants height and number of leaves is possitive, both has possitive correlation with relative maturity of the plant [4]. The next highest r value is 0.72 and is obtained from the interaction between total nitrogen content and water needs, which also corresponds with a previous research that mention the possitive correlation between precipitation and net nitrogen availability in a cultivation system [11]. The original data of plant height obtained from field observation of this study has a relatively high similarity with the simulated data of plant height generated from logarithmic growth with limit function with the destructive data sampling method being considered. The result as seen at the following figure:  It proves that the plants used in the study are not identical, thus assuming that the remaining plants could represent the already destructed plants in the practice of destructive method of data sampling is not reccomended.
The highest pearson coefficient correlation is obtained from the interactions between plant height and number of leaves simulation data generated from the equation (2) with the r value of 0.93. Hence it confirms that number of leaves can be well described as the function of plant height. But this equation has a limitation, that is can not quite describe the declining in the number of leaves towards the end of the cultivation process that is caused by the soybean plant sheds its leaves as a signal telling that it is ready to be harvested.

Conclusion
According to peason correlation coefficient, interactions between components involved in this study with the highest correlations are interactions between plant height and number of leaves with r value of 0.81, and interaction between total nitrogen rate and transpiration rate with an r value of 0.72.
Data sampling using destructive method with a small number of sample will lead into the collection of less representative data, because the plants are not identical, each plants has their own way of expressing genetic information depends to the condition they are exposed into.
The number of leaves variable is the function of plant height variable, the equation is y1 = 0,85x1 + c1 where y1 represents number of leaves and x1 represents plant height. The disadvantage of this equation is its inability to describe the declining in the number of leaves towards the end of the cultivation period.