The topography of the environment alters the optimal search strategy for active particles

Significance Adopting the right search strategy is of critical importance in a broad range of natural and man-related activities. For example, when foraging in an environment with scarce resources, the search strategy used by animals to look for nourishment is a matter of life or death. It is generally accepted that, under most conditions, the optimal strategy alternates ballistic and diffusive steps. This, however, does not take into account that many search spaces feature complex topographies with boundaries, barriers, and obstacles. Here, we show that the presence of such complexity shifts the optimal strategy toward less ballistic and more diffusive searches. In environments with scarce resources, adopting the right search strategy can make the difference between succeeding and failing, even between life and death. At different scales, this applies to molecular encounters in the cell cytoplasm, to animals looking for food or mates in natural landscapes, to rescuers during search and rescue operations in disaster zones, and to genetic computer algorithms exploring parameter spaces. When looking for sparse targets in a homogeneous environment, a combination of ballistic and diffusive steps is considered optimal; in particular, more ballistic Lévy flights with exponent α≤1 are generally believed to optimize the search process. However, most search spaces present complex topographies. What is the best search strategy in these more realistic scenarios? Here, we show that the topography of the environment significantly alters the optimal search strategy toward less ballistic and more Brownian strategies. We consider an active particle performing a blind cruise search for nonregenerating sparse targets in a 2D space with steps drawn from a Lévy distribution with the exponent varying from α=1 to α=2 (Brownian). We show that, when boundaries, barriers, and obstacles are present, the optimal search strategy depends on the topography of the environment, with α assuming intermediate values in the whole range under consideration. We interpret these findings using simple scaling arguments and discuss their robustness to varying searcher’s size. Our results are relevant for search problems at different length scales from animal and human foraging to microswimmers’ taxis to biochemical rates of reaction.