MONDAY, January 20, 2003

3:00 PM

CAB 657

Dr. Chad Topaz

Department of Mathematics
Duke Unviversity


A simple kinematic model for two-dimensional swarms


A biological swarm is a group of organisms undergoing large-scale coordinated movement. Typically, this movement is not due to centralized control, but rather to social interactions with other organisms which occur on a length scale smaller than that of the global swarm formation. Biological swarms, which occur in ants, locusts, fish, birds, and other organisms, are often observed to have sharp boundaries and a roughly spatially-constant population density.  

In this talk, I will discuss preliminary results for a simple continuum model for swarms in two dimensions. The population density rho satisfies an advection equation. The velocity depends nonlocally on rho by means of a convolution with a spatially decaying kernel K, which describes the social interaction between organisms. Using the Hodge decomposition theorem, the velocity field may be decomposed into a divergence-free component and a gradient component. This framework provides a convenient way to characterize the two-dimensional dynamics. The divergence-free component is responsible for the rotational motion of the population, while the gradient component controls its expansion or contraction. When the velocity is divergence free, the model has constant-density solutions of compact support including vortex states similar to those observed in biological swarms.