Random searches with chemotaxis: Optimal strategy transitions induced by target scent

We investigate the optimal movement strategy in random searches, such as animal foraging, assisted by chemotaxis. To do that, we introduce a minimalist one-dimensional model of Lévy walkers in the presence of both a near and a far target. Each target produces a stationary concentration of scent molecules, exponentially decaying with distance. The decay or spread length of the scent depends on the diffusivity of its molecules. In the model, the scent concentration gradient is used by the searcher to increase its probability of choosing the near-target direction. Without scent, by employing very long steps in the far-target direction, finding a target is guaranteed, but at the cost of a long walk. Conversely, by employing very small steps, the searcher may get stuck in between targets for a long time. The optimal strategy corresponds to a certain superdiffusive balance between short and long steps. At intermediate scent spread length, however, the scent becomes so important that a discontinuous strategy transition occurs, over which the optimal step-length strategy becomes completely Brownian since longer steps would risk placing the searcher too far from where the scent is useful. At an even higher scent spread length, a second discontinuous transition occurs, now "backwards": the scent profile becomes so flat that, despite far reaching, it cannot be useful. Our results shine a light on how the degree and spatial arrangement of external information sampling changes optimal foraging strategies for searchers in scarce environments, which leads to ecological implications.