title: First-passage times, mobile traps, and Hopf bifurcations.
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.
authors.affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada B3H 3J5.
authors.forename: Justin C
authors.initials: JC
authors.lastname: Tzou
authors.affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada B3H 3J5.
authors.forename: Shuangquan
authors.initials: S
authors.lastname: Xie
authors.affiliation: Department of Mathematics and Statistics, Dalhousie University, Halifax, Canada B3H 3J5.
authors.forename: Theodore
authors.initials: T
authors.lastname: Kolokolnikov