Pursuit problem with a stochastic prey that sees its chasers

Abstract

A recent stochastic pursuit model describes a pack of chasers (hounds) that actively move toward a target (hare) that undergoes pure Brownian diffusion (Bernardi and Lindner 2022 Phys. Rev. Lett. 128 040601). Here, this model is extended by introducing a deterministic ‘escape term’, which depends on the hounds’ positions. In other words, the hare can ‘see’ the approaching hounds and run away from them, in addition to the ‘blind’ random diffusion. In the case of a single chaser, the mean capture time (CT) can still be computed analytically. At weak noise, the qualitative behavior of the system depends on whether the hare’s maximum running drift speed is above or below a critical value (the pursuers’ speed), but not on the target’s viewing range, whereas the capture statistics at strong noise is similar to those of the original model without escape term. When multiple hounds are present, the behavior of the system is surprisingly similar to the original model with purely diffusing target, because the escape terms tend to compensate each other if the prey is encircled. At weak noise levels and ‘supracritical’ maximum escape speed, the hare can slip through the chaser pack and lead to a very strong increase of the mean CT with respect to the blind case. This large difference is due to rare events, which are enhanced when the symmetry in the initial conditions is disrupted by some randomness. Comparing the median of the CT probability density (which reflects the typical CT) with the mean CT makes clear the contribution of rare events with exceptionally long CTs.