The model of hierarchical growth processess by differential equations – the analysis of the sales of the video game machines assuming two stage growth processes

In this study, we introduce the differential equation model which can explain some hierarchical growth processes. This model is based on the assumptions that some growth processes can be divided into plural stages, and that in a given stage, increasing speed of population (human or otherwise) depends on the size of population in the previous stage. For example, urbanization process can be divided into residential land development stage and building up stage, and the expanding speed of the built-up area depends on the residential development area. In accordance with these assumptions, we formulate the simultaneous differential equations, then calculate their general solutions. We consider that this model can describe the sales of the video game machines, because the process of buying them has the following two stages: first a person wants to buy the machine, then he actually buys it. In addition, the person who gets the game machine will buy some video game software. Deriving a new model with these thoughts, we can confirm that the model can explain the sales of video game machines well especially shortly after the machine's being put on the market. This consequence indicates that the hierarchical model is appropriate for the explanation of the growth tendency of the beginning period. Furthermore, the application of this model to explain the sudden growth in a growth process is proposed in this paper. When a price reduction campaign is launched, the sales of the machines shows a tendency of sudden increase. The model can explain this change by adding only two parameters to the original model. Our model has such simple structure that it will be able to represent some underlying mechanisms of various growth processes.