First passage times, mobile traps, and Hopf bifurcations

Justin C. Tzou, Shuangquan Xie, Theodore Kolokolnikov

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


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.
Original languageEnglish
Article number062138
Number of pages10
JournalPhysical Review E - Statistical, Nonlinear, and Soft Matter Physics
Issue number6
Publication statusPublished - Dec 2014
Externally publishedYes


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