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phenomena, such as consensus decision and eective leadership, arise in the decision-
making process in animal groups so that the group as a whole can benet of the
nal decision.
Group composition is also a dynamical process. For instance, in African elephant
societies [Couzin (2006)] groups merge or split as the elephants move through the
environment. In general, animals such as dolphins, chimpanzees and elephants take
advantages from living in ssion-fusion societies at dierent timescales; for instance,
evading a predator requires continuous vigilance and rapid reactions, while locating
suitable habitat requires much longer timescale.
All these examples can be explained by an unifying law, that of self-organization.
The phenomena observed, in fact, arise from the local interactions between the
units and not from a hierarchical organization of the system. In a certain sense, the
individual is submerged by the group, but at the same time the individual variety,
the information owned by the single, and the individual itself can be of fundamental
importance for the group.
Besides the animal world, self-organization has been observed in many physical
systems, with the emergence of complex patterns such as, for example, sand dunes
in the desert or autowaves in the BZ reaction. Nevertheless, self-organization in
animal world has a peculiar feature: in biological systems complexity also arises at
the unit level. This is pointed out in [Sumpter (2006)], which individuates several
principles underlying self-organized collective behavior in biological systems such
as integrity and variability (each animal in the group is an individual dierent
from the others), positive feedback (observed for instance when an ant nds a food
source and its trail is followed by other ants), response threshold (animals change
their behavior when the stimulus reaches a threshold), leadership (often assumed
by individuals possessing a particular information), redundancy (animal groups, for
instance, insect societies, are often formed by a vast number of replaceable units),
selshness (the cost/benet ratio of forming a group should be advantageous for
each individual) and so on.
However, many scientists have shown that the essential features of collective
motion patterns of animal groups can be captured by minimal models based on the
assumption that an individual can be considered a self-propelled particle. In these
models local interaction rules are able to explain the coordinated behavior of the
group as well as the variety of shapes and motions observed and the eective lead-
ership by a subset of informed individuals. Early studies date to last decade. One
of the most cited works on the subject is the discrete-time model discussed in [Vic-
sek et al. (1995); Czirok et al. (1997); Czirok et al. (1999)]. This was inspired by a
former model previously introduced in [Reynolds (1987)] with the aim of visualizing
realistic ocks and schools for the animation industry. Other early approaches are
based on hydrodynamic principles [Toner and Tu (1995, 1998)], on continuous-time
models with all-to-all and attractive long-range interactions [Mikhailov and Zanette
(1999)], and on reaction-diusion equations [Schenk et al. (1998)].
