In this paper, we compute the non-detection probability of a randomly moving target by a stationary or moving searcher in a square search region. We find that when the searcher is stationary, the decay rate of the non-detection probability achieves the maximum value when the searcher is fixed at the center of the square search region; when both the searcher and the target diffuse with significant diffusion coefficients, the decay rate of the non-detection probability only depends on the sum of the diffusion coefficients of the target and searcher. When the searcher moves along prescribed deterministic tracks, our study shows that the fastest decay of the non-detection probability is achieved when the searcher scans horizontally and vertically.
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