Chase and Escape from Sticks to Groups: Effects of Noise and Delay

Overview

The "chase and escape" problem is a classic topic in mathematical modeling. A simple example is the act of balancing a stick on one’s fingertips, which serves as an experimental paradigm for a one-on-one chase-and-escape dynamic. Recently, we have proposed an extended model in which one group chases another, termed "Group Chase and Escape." This extension connects the classical problem to contemporary studies of collective motion observed in animals, insects, vehicles, and more. In this talk, I will introduce our basic model and its surprisingly complex behaviors. Each chaser moves toward its nearest escapee, while each escapee moves away from its nearest chaser. Despite the absence of communication within each group, we observe emergent aggregate formations. I will discuss how these behaviors depend on key parameters, such as density. Additionally, I will present two variations of the model:

• The Effect of Noise: We introduce random fluctuations, where players make directional errors with some probability. Interestingly, a certain level of noise enhances the efficiency of catching.
• The Effect of Delay: We introduce reaction delays in chasers' movements, particularly making the delay distance-dependent. This modification leads to unexpectedly complex behaviors.

Furthermore, we examine the impact of probabilistically converting caught escapees into chasers, adding another layer of dynamical complexity to the system.

Ref: Group Chase and Escape, A. Kamimura and T. Ohira, Springer, 2019.