Macroscopic and microscopic self-organization by nonlocal anisotropic interactions

This paper is concerned with mathematical modeling of intelligent systems, such as human crowds and animal groups. In particular, the focus is on the emergence of different selforganized patterns from non-locality and anisotropy of the interactions among individuals. A mathematical technique by time-evolving measures is introduced to deal with both macroscopic and microscopic scales within a unified modeling framework. Then self-organization issues are investigated and numerically reproduced at the proper scale, according to the kind of agents under consideration. 1. Self-organization in many-particle systems One of the most outstanding expressions of intelligence in nonclassical physical systems, such as human crowds or animal groups, is their self-organization ability. Self-organization means that the individuals composing the system can give rise to complex patterns without using intercommunication as an essential mechanism. For instance, in normal conditions pedestrians are known to arrange in specific patterns, chiefly lanes, as demonstrated by many experimental investigations (Helbing et al. [13, 14, 17], Hoogendoorn et al. [21, 22]). Lane formation may be fostered by a suitable setup of the space, as reported by Helbing and his coauthors [13, 17]: a test performed in a tunnel connecting two subway stations in Budapest showed that a series of columns, placed in the middle of the walkway, induce pedestrians to organize in two oppositely walking lanes, preventing each of them to expand up to the full width of the corridor. More in general, lanes form also spontaneously, i.e., without the need for being triggered by environmental factors, provided the density of pedestrians is sufficiently large (Helbing et al. [14]). This is particularly evident if one considers the case of two groups of people, walking in opposite directions, which meet and cross (see also Hoogendoorn and Daamen [21]). Grouping and self-organization are well known and largely observed also in animals, see for example Krause and Ruxton [26]. These phenomena are in fact ubiquitous, ranging from bird flocks in the sky to migrating lobsters on the sea floor. Many papers on this subject (see, among others, Chaté et al. [4], Couzin et al. [6, 7], Cucker and Smale [8], Grégoire et al. [10], Gueron et al. [11], Huth and Wissel [25], Kunz and Hemelrijk [27], Li et al. [28], Vicsek et al. [35], Warburton and Lazarus [37]) proved by means of numerical simulations that few simple rules adopted by each animal can give rise to a complex organization of the whole group. Patterns commonly seen in nature are (see Heppner [19] for the nomenclature): globular clusters (e.g., starlings), extended and front clusters (e.g., wildebeests, antelopes, pigeons), lines (e.g., penguins, elephants, lobsters), Vees, Jays and echelons (e.g., geese). Motivations and convenience of these patterns are still under investigations, especially for Vees. It is commonly agreed that self-organization is the result of elementary actions that each subject performs to fulfill specific wills. Concerning pedestrians, the following basic guidelines can be identified: • The will to reach specific targets, e.g., an exit or a meeting point, which drives pedestrians along preferential paths, determined mainly by the geometry and the spatial arrangement of the walking area. Unlike animals, pedestrians experience strong interactions with the