MATHEMATICAL MODELLING OF MOTORCYCLE SALE COMPETITION IN INDONESIA

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Motorbike is a means of transportation that is widely used in Indonesia and other developing countries. In this paper, we apply the Lotka-Volterra competition mathematical model to describe the dynamics of motorcycle sale competition between two motorcycle producers in Indonesia. We also estimate the parameter values of the model using the particle swarm optimization method. We found that the mathematical model quite well to explain the annual motorcycle sales of both companies. The results of the model analysis show that the competition between the two motorcycle producers is pure competition, with the first producer dominates market share of motorcycle selling in Indonesia.


INTRODUCTION
The population increase causes an increase in the need for transportation facilities. People need transportation to support their mobility. Motorbikes are a transportation mode that is widely used in Indonesia and other developing countries. The use of motorbikes in Indonesia is common because of their relatively low price. The motorcycle also has the advantage of being able to maneuver between traffic jams. The use of motorbikes also provides efficiency in travel costs. The use of motorbikes can also increase the population's mobility in terms of flexibility and shorten travel time. Furthermore, motorbikes can foster economic activity in the form of goods or goods delivery services. In the last ten years, online transportation services using motorbikes have even begun to develop in Indonesia.
Indonesia is the country with the most purchases of motorbikes in Southeast Asia. Also, Indonesia is the third-largest motor vehicle user in the world after India  The potential for motorcycle sales in Indonesia is large enough, so it encourages motorcycle manufacturers to produce motorcycles that attract consumer interest. Sales competition illustrates consumer trends in choosing motorbikes over a long period. Accurate predictions can help companies identify potential growth or potential decline in demand for a motorbike brand.
Moreover, a company can also develop strategies to maintain and increase the market share of motorbike demand in the years to come. Accurate forecasts related to potential motorcycle sales are also a reference for investors in identifying investment targets.
The Lotka-Volterra type competition mathematical model has been used to describe the competition between species. In 1934, Gause applied the Lotka-Volterra competition model to explain competition between two Protozoa species, namely Paramecium caudatum and Paramecium aurelia species [4]. Recently, Novoa-Munoz et al. used the Lotka-Volterra competition model to explain the competition dynamics between two lizards species, namely Liolaemus cyanogaster and Liolaemus tenuis species [5]. The Lotka-Volterra type competition model was also used to explain competition between companies in the struggle for share, including competition between the Korean stock markets [6], the retail industry in Taiwan [7], and commercial banks and rural banks competition in Indonesia [8].
In this paper, we apply the Lotka-Volterra type competition mathematical model to explain the competitive dynamics of motorcycle sales from two motorcycle producers in Indonesia.
We also estimate the parameters of the model using the particle swarm optimization method.
Moreover, we perform a stability analysis of the model by changing the estimation results' parameter values and finding the model's equilibria. We also perform a stability analysis of each equilibrium model, and we simulate the model to study the future competitive conditions of the two motor producers.

MATHEMATICAL MODEL OF MOTORCYCLE SALES COMPETITION
In this section, we present an application of the Lotka-Volterra type competition mathematical model to describe the dynamics of motorcycle sales competition in Indonesia. We examine the dynamics of competition between two motorcycle producers (companies) in Indonesia. Here, the first and second motorcycle producers are the market-leader producer and the market-leader main competitor. Suppose y 1 and y 2 represent the number of annual motorcycle sales of the first producer and the second producer in Indonesia. The mathematical model of the two motorbike producers' competition is constructed based on the following assumptions: (1) There are only two motorcycle producers (manufacturers) considered in the model.
(2) Competition between the two motorcycle manufacturers and other manufacturers was ignored because the two companies' motorcycle sales were much higher than other motorbikes' sales.
(3) In the absence of competition, the two motorcycle producers' sales grow following the logistic growth model. (6) The decline rate of motorcycle sales due to the competition is proportional to the two motorcycle producers' motorcycle sales.
Based on these assumptions, the competitive dynamics of the two motor manufacturers can be represented in the following differential equation system: Here, parameters a 1 , a 2 are the growth rates of the first and second producer. Parameter K represents the maximum number of annual motor sales from both producers. Parameters b 1 , and b 2 denote the decline rates in the first producer and the second producer's annual sales of motorcycles, respectively. We assume that a 1 , a 2 , K > 0 and b 1 , b 2 ≥ 0.

PARAMETER ESTIMATION ON THE MATHEMATICAL MODEL OF MOTORCYCLE SELL-ING COMPETITION
In this section, we estimate the parameter values in the mathematical model of motorbike sales competition in Indonesia, as presented in equations (1) -(2). Because the analytic solution of the differential equation system in equations (1) -(2) is unknown, then we could apply heuristic methods such as genetic algorithms and particle swarm optimization method to determine the parameter values of the competition mathematical model [9,10]. Here we use the particle swarm optimization method as a parameter estimation method since the particle swarm optimization method is more robust than the genetic algorithm [10]. We use annual motorcycle The parameter value is selected in such a way that the parameter produces the smallest error value. In this article, we use MAPE (Mean Absolute Percentage Error) as an objective function.
The MAPE (E) value for this problem is given by Here, n is the number of data, y 1,i , y 2,i are the annual motorcycle sales data from the first and second producers in the year i , where and i = 0, 1, 2, . . . , n. Besides, y * 1,i , y * 2,i are the predicted annual motorcycle sales quantities of the first and second producers in the year i.
The particle swarm optimization method was carried out as many as 50 trials with 300 iterations for each trial. We obtained the best objective function value (the smallest MAPE value) of 0.086222 or 8.622%. The best parameter values from the parameter estimation process are presented in Table 1.  By substituting the parameter values in Table 1 into equations (1) -(2), the competitive mathematical model of motorcycle sales in Indonesia, between two producers, can be described in the following form.
By using the eigenvalues method, the equilibria P 1 , P 2 , P 3 are unstable, and the coexistence equilibrium P 4 is locally asymptotically stable. The coexistence equilibrium P 4 is asymptotically stable so that this equilibrium may occur in real situations. From a practical point of view, the motorcycle sales competition from the two producers is a pure competition with the domination of the first producer as the market-leader producer.
Next, we present numerical simulations to predict annual motorcycle sales and the dynamics of competition between the first and second companies for the future. We also perform numerical simulations by using parameter values in Table 1

CONCLUSION
In this paper, the mathematical model of the Lotka-Volterra competition has been successfully applied to explain the dynamics of the motorcycle sales competition between two manufacturers in Indonesia. The Lotka-Volterra competition model adequately describes the annual motorcycle sales figures of both manufacturers. The results of the model analysis show that the competition between the two motorbike manufacturers is pure competition, with the first manufacturer controlling the motorcycle market in Indonesia.

This research was funded by the Ministry of Research and Technology/National Research and
Innovation Agency, Republic of Indonesia through "Penelitian Dasar" project scheme.

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.