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

Justin C. Tzou; Shuangquan Xie; Theodore Kolokolnikov

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arXiv · 1410.1391

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

Justin C. Tzou·Shuangquan Xie·Theodore Kolokolnikov

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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.

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Justin C. Tzou, Shuangquan Xie, Theodore Kolokolnikov. 2014-10-03. Drunken robber, tipsy cop: First passage times, mobile traps, and Hopf bifurcations. https://arxiv.org/abs/1410.1391

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