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

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 a the searchers step on a nearest-neighbour, and with probability (1-alpha) 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 P-N that the target remains undetected up to the maximal search time N, and seek to minimize this probability. We find that P-N is a non-monotonic function of alpha, and show that there is an optimal choice alpha(opt)(N) of a well within the intermittent regime, 0 < alpha(opt)(N) < 1, whereby P-N can be orders of magnitude smaller compared to the 'pure' random walk cases alpha = 0 and alpha = 1.