Continuum models of cohesive stochastic swarms
by Professor Barry Hughes
Abstract: Mathematical models of swarms of moving agents with non-local interactions have many applications and have been the subject of considerable recent interest. For modest numbers of agents, cellular automata or related algorithms can be used to study such systems, but in the present work, instead of considering discrete agents, I discuss a class of one-dimensional continuum models, in which the agents possess a density rho(x,t) at location x at time t. The agents are subject to a stochastic motility mechanism and to a global cohesive inter-agent force. The motility mechanisms covered include classical diffusion, nonlinear diffusion (which may be used to model, in a phenomenological way, volume exclusion or other short-range local interactions), and a family of linear redistribution operators related to fractional diffusion equations.
A variety of exact analytic results are discussed, including equilibrium solutions and criteria for unimodality of equilibrium distributions, full time-dependent solutions, and transitions between asymptotic collapse and asymptotic escape. Of particular interest is the combination of a double-well potential and weak linear diffusion, which raises intriguing mathematical problems.
The talk, which is an updated version of a lecture I gave at the 2012 ANZIAM meeting, includes a discussion of work done on my recent study leave at the University of Graz (Austria) in collaboration with Professor Klemens Fellner.