Optimizing the success of random searches

We address the general question of what is the best statistical strategy to adapt in order to search efficiently for randomly located objects (target sites). It is often assumed in foraging theory that the flight lengths of a forager have a characteristic scale: from this assumption Gaussian, Rayleigh and other classical distributions with well defined variances have arisen. However such theories can't explain the long tailed power-law distributions of flight lengths or flight times that are observed experimentally.Here we study how the search efficiency depends on the probability distribution of flight lengths taken by a forager that can detect target sites only in its limited vicinity. We show that when target sites are sparse and can be visited any number of times, an inverse square power-law distribution of flight lengths, corresponding to Levy flight motion, is an optimal strategy. We test the theory by analyzing experimental foraging data on selected insect, mammal and bird species and find that they are consistent with the predicted inverse square power-law distributions.
In the second part we will discuss a more recent work which shows that the distribution of target sites plays a crucial role in the random search problem. The main result is,  in the case of a single target site in a bounded domain or regular patterns of targets; in contrast to repeated statements in the literature, persistent random walk can minimize the search time and in that sense perform better than any Levy walk.

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