### Abstract

We study the mean capture time of an unbiased random walker by multiple absorbing mobile traps in bounded domains of one and two spatial dimensions. In one dimension, we consider multiple traps undergoing prescribed oscillatory motion on an interval with reflecting or absorbing boundary conditions. We develop trap cooperation strategies which optimize the mean capture time.

We find that as the frequency of oscillation passes through certain fixed values, the optimal trap strategy alternates between oscillating exactly in phase and exactly out of phase with neighboring traps. We also demonstrate a scenario in which the optimal configuration is neither in phase nor antiphase. In two dimensions, we consider two small traps rotating with the same angular velocity

ω inside a unit disk and characterize the optimal positions (radii of rotation and relative phase) of the two traps as a function of ω and trap radius ε 1. We identify several distinguished regimes in ω where the optimal configuration can be distinctly characterized. In particular, in the ω ∼ O(1) regime, the optimal configuration jumps from one in which two traps rotate antipodal and along the

same radius to one where the two traps rotate on the same side of the disk but at different radii. In addition, we demonstrate an algebraic approach to obtaining optimal configurations of N rotating traps as ω → ∞.

We find that as the frequency of oscillation passes through certain fixed values, the optimal trap strategy alternates between oscillating exactly in phase and exactly out of phase with neighboring traps. We also demonstrate a scenario in which the optimal configuration is neither in phase nor antiphase. In two dimensions, we consider two small traps rotating with the same angular velocity

ω inside a unit disk and characterize the optimal positions (radii of rotation and relative phase) of the two traps as a function of ω and trap radius ε 1. We identify several distinguished regimes in ω where the optimal configuration can be distinctly characterized. In particular, in the ω ∼ O(1) regime, the optimal configuration jumps from one in which two traps rotate antipodal and along the

same radius to one where the two traps rotate on the same side of the disk but at different radii. In addition, we demonstrate an algebraic approach to obtaining optimal configurations of N rotating traps as ω → ∞.

Original language | English |
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Pages (from-to) | 920-947 |

Number of pages | 28 |

Journal | Multiscale Modeling and Simulation |

Volume | 15 |

Issue number | 2 |

DOIs | |

Publication status | Published - 2017 |

Externally published | Yes |

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### Keywords

- mean first passage time
- multiple mobile traps
- trap cooperation
- asymptotic analysis
- optimization strategy

### Cite this

Lindsay, A. E., Tzou, J. C., & Kolokolnikov, T. (2017). Optimization of first passage times by multiple cooperating mobile traps.

*Multiscale Modeling and Simulation*,*15*(2), 920-947. https://doi.org/10.1137/16M1060169