Imagine a swarm (or a few) agents (we will call them swimmers) moving in a region. To better explore the terrain they keep changing the direction of motion randomly. They also shine light on each other and this influences the speed of their motion: the more light a swimmer senses, the slower it moves, But to change the speed in response to light intensity may take time---this is called sensory delay. Perhaps surprisingly, in actual experiments the sensory delay can be made negative as well as positive---I will explain how. These experiments show that the collective behavior of the swimmers vary with the value of the sensory delay, changing qualitatively at a certain critical value. Above this value the swimmers tend to aggregate close to each other, and below it they deaggregate, i.e. stay away from one another. The phenomenon can be explained by studying the stochastic differential equations (SDE) of motion of the swimmers in what is technically known as the homogenization limit. But. so far, this has been carried out only in the simplest case when the random direction changes are uncorrelated in time. In the parlance of the SDE theory, these direction changes are driven by a white noise. The colored (i.e. time-correlated) noise case is open and very interesting. Experimental results are available and their quantitative explanation will constitute a great project for an ambitious graduate student.
