  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.
