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 alpha 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 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 alpha, and show that there is an optimal choice alphaopt(N) of alpha well within the intermittent regime, 0 < alphaopt(N) < 1, whereby PN can be orders of magnitude smaller compared to the 'pure' random walk cases alpha = 0 and alpha = 1.