DEPARTMENT OF MATHEMATICAL
& STATISTICAL SCIENCES

UNIVERSITY OF
ALBERTA

**MATHEMATICAL BIOLOGY SEMINAR**

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.