Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf bifurcations

TL;DR

This paper investigates how mobile traps affect mean first passage times in a one-dimensional domain, revealing that slow movement can hinder capture efficiency while fast movement improves it, and links these findings to Hopf bifurcations in a related model.

Contribution

It provides explicit thresholds for trap speed regimes and uncovers a novel connection between trap dynamics and Hopf bifurcations in the Gray-Scott model.

Findings

Stationary traps outperform slow-moving traps in capture times.

Fast-moving traps outperform stationary traps beyond a certain speed threshold.

A Hopf bifurcation corresponds to the transition point between trap regimes.

Abstract

For a random walk on a confined one-dimensional domain, we consider mean first passage times (MFPT) in the presence of a mobile trap. The question we address is whether a mobile trap can improve capture times over a stationary trap. We consider two scenarios: a randomly moving trap and an oscillating trap. In both cases, we find that a stationary trap actually performs better (in terms of reducing expected capture time) than a very slowly moving trap; however, a trap moving sufficiently fast performs better than a stationary trap. We explicitly compute the thresholds that separate the two regimes. In addition, we find a surprising relation between the oscillating trap problem and a moving-sink problem that describes reduced dynamics of a single spike in a certain regime of the Gray-Scott model. Namely, the above-mentioned threshold corresponds precisely to a Hopf bifurcation that induces oscillatory motion in the location of the spike. We use this correspondence to prove the uniqueness of the Hopf bifurcation.