We investigate the general problem of autonomous random walkers whose sole source of energy are search targets that are themselves diffusing random walkers. We study how the energy accumulated by the searcher varies with the target density via numerical simulations and compare the results with an analytical model for fixed targets. We report that superdiffusion of either searcher or target confers substantial energetic advantages to the former. While superdiffusion may not play a crucial role for high target densities, in contrast it confers a vital advantage in the limit of low densities at the edge of extinction: diffusive searchers rapidly die but superdiffusive searchers can survive for long periods without entering into the extinction state. The validity and relevance of our findings in broader contexts are also discussed.

We analyze searchers looking for diffusive targets when the formers rely on the net energy gained from the encounters to maintain the process. The system properties are studied at very low target densities, for the searchers at the edge of extinction. We report that superdiffusion for both types of players confers a substantial increase in the searchers survival rate. A continuous phase transition is observed for any search strategy. From the critical exponents, we find that the problem belongs to the same universality class of directed percolation with absorbing walls. We finally discuss the implications of the random search process criticality to the endurance of searchers as a group and eventual connections with the preservation of biological species.

We investigate the problem of survival at the very low target-density limit and report on a second-order phase transition for one-dimensional random searches in which the energy cost of locomotion is a function of the distance traveled by the searcher. Surprisingly, from analytical calculations (also tested numerically) we find identical critical exponents for arbitrary energy cost functions. We conclude that there is a single universality class that describes this process.