{
  "id": "1410.1391",
  "version": "v1",
  "published": "2014-10-03T16:15:34.000Z",
  "updated": "2014-10-03T16:15:34.000Z",
  "title": "Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf bifurcations",
  "authors": [
    "Justin C. Tzou",
    "Shuangquan Xie",
    "Theodore Kolokolnikov"
  ],
  "comment": "14 pages, 5 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-03T16:15:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B20",
      "35C20",
      "35Q92",
      "35K57",
      "G.0"
    ],
    "keywords": [
      "first passage times",
      "hopf bifurcation",
      "mobile trap",
      "drunken robber",
      "tipsy cop"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.1391T"
    }
  }
}