Optimal random searches of revisitable targets: Crossover from superdiffusive to ballistic random walks

One of the most important aspects in the general search problem of finding randomly located target sites concerns how to characterize the role played by the non-revisit ability delay time T during which a previously found target becomes unavailable to the searcher. By using an appropriate parameterization of the number of random walk steps undertaken between successive targets, we show that for the case of sparse randomly distributed sites the optimal search strategy shifts from a superdiffusive to a ballistic strategy consisting of essentially rectilinear motion between the targets, as T increases from tau --> 0 to tau --> infinity, respectively. The crossover between these limiting regimes occurs as a function of tau. These conclusions are shown to hold even if dissipative phenomena are considered in the searching dynamics. We discuss the results in the context of their application to animal foraging.