Capture of a diffusive prey by multiple predators in confined space

The first passage search of a diffusing target (prey) by multiple searchers (predators) in confinement is an important problem in the stochastic process literature. While the analogous problem in open space has been studied in some details, a systematic study in confined space is still lacking. In this paper, we study the first passage times for this problem in 1,2 and $3-$dimensions. Due to confinement, the survival probability of the target takes a form $\sim e^{-t/\tau}$ at large times $t$. The characteristic capture timescale $\tau$ associated with the rare capture events are rather challenging to measure. We use a computational algorithm that allows us to estimate $\tau$ with high accuracy. We study in details the behavior of $\tau$ as a function of the system parameters, namely, the number of searchers $N$, the relative diffusivity $r$ of the target with respect to the searcher, and the system size. We find that $\tau$ deviates from the $\sim 1/N$ scaling seen in the case of a static target, and this deviation varies continuously with $r$ and the spatial dimensions.