Cops and robbers...

Most (mobile) organisms are familiar with the challenge of pursuit-evasion; we ourselves encounter it at the young age in form of tag games such as Cops and robbers. In mathematics and computer science pursuit-evasion describes a class of problems where one group actively seeks to locate and capture members of another group within a defined environment. Benjamin Lindner and Davide Bernardi investigated mathematical models that describe the pursuit of a randomly moving object in one or two dimensions. If you want to learn more about (ii) analytically calculating the mean capture time, a crucial metric, (i) exploring how factors such as the target's diffusion coefficient and the number of pursuers influence the capture time, (iii) strategies that can optimize the pursuit process and (iv) potential generalizations and extensions of these models to enhance your understanding of pursuit dynamics in more complex scenarios, check out their Target Search Problems Chapter!

Abstract

In this chapter, we explore several mathematical models describing the pursuit of an object moving randomly in a one- or two-dimensional space. Our central focus lies in analytically calculating the mean capture time within these models, treating it as a mean first passage time. In certain special cases, we can derive exact expressions, which we subsequently utilize to develop approximations valid for the general scenarios. We examine how this statistical measure is influenced by both the diffusion coefficient of the target and the number of chasers involved. Additionally, we explore strategies for optimizing the pursuit from the hunters’ perspective. Looking ahead, we consider potential generalizations and extensions of the model.