Dynamics of Swarms and Opinions via Discrete Positive Systems

In the talk a general model for swarm formation of birds (or other agents) will be presented. Swarm formation means that birds approach asymptotically the same velocity, whereby distances among them do converge. The main result offers conditions on the local interaction of the birds for swarm formation to happen. Roughly speaking, the structure of interaction should not be "too loose" and the intensity of interaction should not decay "too fast". Furthermore, the various flight regimes, for example echelons, occurring in swarm formation, will be analyzed. What if the interaction of birds (or other agents) is too loose and/or decays too fast? In this case, a further result will describe how an ensemble of birds splits into sub-swarms. The same result also applies to opinion dynamics and the formation of a consensus or of a fragmentation among the opinions of a group of agents. For analyzing the convergence behavior, the field of positive dynamical systems in discrete time provides useful tools when dealing with infinite products of matrices or with the iteration of averaging maps.