# Self-Organization When Pedestrians Move in Opposite Directions. Multi-Lane Circular Track Model
## 1 Introduction  
Spontaneous organization in systems composed of several units is known as self-organization. Self-organization is pervasive and is observed even when the units act responding to local stimuli, without considering the rest of the group. Researchers are interested in understanding how local interactions among units and with the environment leads to self-organization. Some books and review articles on self-organization in biological systems include [1; 2; 3; 4]. Self-organization is observed when humans are walking in crowded environments. As a first example, consider two adjacent rooms connected by a door that is initially closed. Assume a large number of individuals is in each room and they all want to go to the other room. Suddenly, the door is opened. Under these circumstances, the spontaneous formation of an alternating flow is frequently observed. By alternating flow, we mean that the individuals take turns. Individuals from only one of the rooms cross the door for a period of time and then there is a sudden switch, individuals from the other room are the only ones crossing the door after the switch. This switching, or turn taking, persists while the environment remains crowded [5]. A second example of self-organization due to walking individuals interacting among themselves and with the environment is the formation of trails. More precisely, assume individuals need to cross, possibly in different directions, the same mildly dense grass field on a regular basis. Individuals will try to walk along the areas where the grass is shorter and less dense, even if the path they take as a result is not the shortest. Stepping on the grass is how the individuals affect the environment. Grass has a harder time growing in the areas heavily transited. As a result, dirt path trails form [6]. A third example of self-organization is the different pattern formations observed when the crowd is a collection of several smaller groups. For example, groups of friends or families. We refer the reader to [7] for more details. In this and the previous two examples, the coordinated behaviors observed are not planned by the crowd as a whole or by any member or group of members within the crowd. Instead, they emerge spontaneously, as a consequence of the individuals acting in response to local stimuli and motivated by their own goals. The study of the dynamics of crowds is of interest for several reasons. It is a source of examples of self-organization. It may lead to strategies to increase the safety in crowded areas such as bridges [8] or stadiums. It can provide guidelines in the design of movie theaters, shopping malls, or other similar types of heavily transited buildings, where optimal crowd flow is desired because of economical and safety reasons. Accordingly, the study of dynamics of crowds, both theoretically (early works include [9; 10], more recent work includes [11; 12; 13; 14; 15; 16], see also [1; 17] for reviews) and experimentally [18; 19; 20], is a very active area of research. Microscopic mathematical models to study dynamics of crowds are those that keep track of each individual. Cellular automata models [18; 21; 22; 23], lattice gas automata models [24; 25], algorithms [26], and large systems of odes, are all examples of microscopic models. When large systems of odes are used, the mass times acceleration of each individual is set equal to the sum of generalized or social forces the individual feels [27; 28; 29; 30; 31]. These social forces are not real forces. They model the responses of the individuals to the environment and the presence and actions of the other individuals [32]. Some of these models are known as self-propelled particles models [33] and others, as individual based models [32; 34]. Mesoscopic, kinetic or Boltzman-type models to study the dynamics of crowds, are integro-partial differential equations that describe the evolution of probability densities of the position and velocities of the individuals [35; 36].