Scaling law in target-hunting processes

Abstract

We study the hunting process for a target, in which the hunter tracks the goal by smelling odors it emits. The odor intensity is supposed to decrease with the distance it diffuses. The Monte Carlo experiment is carried out on a 2-dimensional square lattice. Having no idea of the location of the target, the hunter determines its moves only by random attempts in each direction. By sorting the searching time in each simulation and introducing a variable $x$ to reflect the sequence of searching time, we obtain a curve with a wide plateau, indicating a most probable time of successfully finding out the target. The simulations reveal a scaling law for the searching time versus the distance to the position of the target. The scaling exponent depends on the sensitivity of the hunter. Our model may be a prototype in studying such the searching processes as various foods-foraging behavior of the wild animals.