Intermittent random walks for an optimal search strategy: one-dimensional case

Abstract

We study the search kinetics of an immobile target by a concentration of randomly moving searchers. The object of the study is to optimize the probability of detection within the constraints of our model. The target is hidden on a one-dimensional lattice in the sense that searchers have no a priori information about where it is, and may detect it only upon encounter. The searchers perform random walks in discrete time n = 0, 1, 2, . . . , N , where N is the maximal time the search process is allowed to run. With probability α the searchers step on a nearest-neighbour, and with probability (1−α) they leave the lattice and stay off until they land back on the lattice at a fixed distance L away from the departure point. The random walk is thus intermittent. We calculate the probability PN that the target remains undetected up to the maximal search time N , and seek to minimize this probability. We find that PN is a non-monotonic function of α, and show that there is an optimal choice αopt(N) of α well within the intermittent regime, 0 < αopt(N) < 1, whereby PN can be orders of magnitude smaller compared to the ‘pure’ random walk cases α = 0 and α = 1.

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