Collective dynamics of interacting particles

Flocking, swarming, aggregation there is a multitude of actual real-life phenomena that from the mathematical point of view can be interpreted as one of these concepts. The mathematical description of collective dynamics of self-propelled agents with nonlocal interaction originates from one of the basic equations of kinetic theory – Vlasov’s equation from 1938. Recently it was noted that such models provide a way to describe a wide range of phenomena that involve interacting agents with a tendency to aggregate their certain qualities. This approach proved to be useful and the language of aggregation now appears not only in the models of groups of animals but also in the description of seemingly unrelated phenomena such as the emergence of common languages in primitive societies, distribution of goods or reaching a consensus among individuals [3, 31, 33, 42]. The literature on such aggregation models is very rich thus we will only mention a few examples on some of the more popular branches of the research. Those branches include the analysis of time asymptotics (see e.g. [27]) and pattern formation (see e.g. [26, 41]) or analysis of the models with additional forces that simulate various natural factors (see e.g. [11, 20] deterministic forces or [15] stochastic forces). The other variations of the model include forcing particles to avoid collisions (see e.g. [13]) or to aggregate under the leadership of certain individuals (see e.g. [14]). A good example of a paper in which a well rounded analysis of a model that includes effects of attraction, repulsion and alignment is [7]. The general form of equations associated with aggregation models reads as follows: