Exploring a Terrain with Random Direction Changes

Explore a mathematical model of an explorer moving through an inhomogeneous environment in this lecture by Prof. Jan Wehr from the University of Arizona. Delve into a system of stochastic delay equations approximating the explorer's movement, adapting speed to terrain changes with a delay. Examine the limit case where direction changes occur rapidly, and investigate the intriguing possibility of negative delays. Discover how this model applies to light-sensitive mini-robots reacting to local light intensity changes. Analyze the qualitative behavioral shifts occurring at specific delay values and extend findings to swarm behavior through mean field arguments. Learn about the aggregation-deaggregation transition at a critical delay parameter. Gain insights into the mathematical analysis of this problem as a particular case of singular limits in stochastic differential equation systems.