The Mathematics of Emergence∗

A common situation occurring in a number of disciplines is that in which a number of autonomous agents reach a consensus without a central direction. An example of this is the emergence of a common belief in a price system when activity takes place in a given market. Another example is the emergence of common languages in primitive societies, or the dawn of vowel systems. Yet a third example is the way in which populations of animals move together (referred as “schooling”, “flocking”, or “herding” depending on the considered animals). As a motivating example in this introduction we consider a population, say of birds, whose members are moving in IE = IR3. This situation has been recently studied in [6] and in what follows we freely draw from this paper. It has been observed that under some initial conditions, for example on the positions and velocities of the birds, the state of the flock converges to one in which all birds fly with the same velocity. A way to justify this observation is to postulate a model for the evolution of the flock and exhibit conditions on the initial state under which a convergence as above is established. In case these conditions are not satisfied, dispersion of the flock may occur.