Mathematical models of swarming and social aggregation, invited lecture

Abstract | I survey some of the problems (both mathematical and biological) connected with aggre-gation of social organisms and indicate some mathe-matical and modelling challenges. I describe recent work with collegues on swarming behaviour. Exam-ples discussed include (1) a model for locust migra-tion swarms, (2) the eect of non-local interactions on swarm shape and dynamics, and (3) an individual-based model for the spacing of neighbors in a group. I. Background and previous work Many chemical and physical systems are characterized by formation of patterns, clusters and aggregates, or phenomena such as wave and pulse propagation. In bi-ology, swarming and social aggregation form a rich and diverse collection of such phenomena. The size scale of groups ranges from microscopic cellular populations to herds, flocks, schools, and swarms of macroscopic, and sometimes enormous size. In some cases, notably swarms of locusts, these aggregates have serious im-pact on ecology and human activity. These examples, and many others, have motivated studies in which both biological aspects as well as theoretical modeling and mathematical aspects of the phenomena have been investigated. Early works in the 1950’s were largely descriptive biological investi-gations of sh schools [3] or bird flocks [4]. Some theoretical concepts for the formation of herds and groups were discussed in a classic paper by Hamilton [9]. See also [22] for simulations and a model for ani-mal group structure. The classic book by Okubo(1980) [16] (recently modernized [18]) presented some aspects of the initial mathematical concepts applied to study-ing swarms. Since that time, the literature has grown substantially, with many signicant contributions ad-dressing such problems from the perspective of biology, mathematical modeling, or physics. Notable amon