Collective behaviors emerging from chases and escapes.

"Chases and Escapes" is a classical mathematical problem. Recently, we proposed a simple extension, called "Group Chase and Escape," where one group chases another. This extension bridges the traditional problem with the current interest in studying collective motion among animals, insects, and cars. In this presentation, I will introduce our fundamental model and explore its intricate emergent behaviors. In our model, each chaser approaches the nearest escapee, while each escapee moves away from its closest chaser. Interestingly, despite the absence of communication within each group, we observe the formation of aggregate patterns. Furthermore, the effectiveness of capture varies as we adjust the ratio of chasers to escapees, which can be attributed to a group effect. I will delve into how these behaviors manifest in relation to various parameters, such as densities. Moreover, we have explored different expansions of this basic model. First, we introduced fluctuations, where players now make errors in their step directions with a certain probability. We found that a moderate level of fluctuations improves the efficiency of catching. Second, we incorporated a delay in the chasers' reactions to catch their targets. This distance-dependent reaction delay can lead to highly complex behaviors. Additionally, I will provide an overview of other groups' extensions of the model and the latest developments in this field. [ABSTRACT FROM AUTHOR]