Two dimensional extension of the optimal velocity model

将最优速度（OV）模型推广到二维模型，并对模型的性质进行了解析和数值研究。我们阐明了交通流的性质与颗粒流、行人流和集体生物运动的性质之间的关系。第一年，我们研究了一个多变量的一维OV模型。我们发现该模型的性质可以在单变量OV模型的框架下得到解释，并将OV模型直接推广到二维模型。在该模型中，有序状态和无序（拥挤）状态以相对低的密度存在。111该模型还可以再现混合流和逆向流的车道形成。我们还分析了日本高速公路上的真实的交通数据。第二年，我们对二维OV模型进行了改进，使之适用于非常高密度的情况。为了考察模型的多样性，我们观察了马拉松运动员的运动，发现了车道形成和密度波动的存在。观测结果与二维OV模式的模拟结果一致。我们对二维模型进行了分析，得到了均匀流的稳定性条件，并将二维模型应用于生物群体运动，特别是群体的形成。我们证明了群是从均匀态自发形成的，这种现象可以用OV模型来解释。我们还证明了所谓的同步流可以用OV模型来再现。

An optimal velocity (OV) model is extended to a 2-dimensional model and properties of the model are investigated analytically and numerically. We clarify a relation among the property of traffic flow and those of granular flow, pedstrian flow, and collective biological motion.In the first year, a 1-dimensional OV model with many variables is investigated. We found that the property of the model can be interpreted within a framework of the OV model with single variable.We extended the OV model straightforwardly to a 2-dimensional model. In the model an ordered state and a disorderd (congested) state exit at relatively low densities. The model can also reproduce lane formation in mixed flow and counter flowWe have also analyzed real traffic data in Japanese highwaysIn the second year, we modified the 2-dimensional OV model and made the model applicable to the case at very high density. In order to investigate the varidity of the model, we have observed the motion of marathon runners and found the existence of lane formation and density fluctuation. The observations are consistent with the result of simulation in the 2-dimensional OV model. We have analyzed the model and found the stability condition of uniform flowThe 2-dimensional model have been applied to the collective biological motion, especially the formation of groups. We have shown that groups is formed from a homogeneous state spontaneously and this phenomenon can be explained by the OV modelWe have also shown that so-called syncronized flow can be reproduced by the OV model