Search efficiency of biased migration towards stationary or moving targets in heterogeneously structured environments

Abstract: Efficient search acts as a strong selective force in biological systems ranging from cellular populations to predator-prey systems. The search processes commonly involve finding a stationary or mobile target within a heterogeneously structured environment where obstacles limit migration. An open generic question is whether random or directionally biased motions or a combination of both provide an optimal search efficiency and how that depends on the motility and density of targets and obstacles. To address this question, we develop a simple model that involves a random walker searching for its targets in a heterogeneous medium of bond percolation square lattice and used mean first passage time (MFPT, $\langle T \rangle$) as an indication of average search time. Our analysis reveals a dual effect of directional bias on the minimum value of $\langle T \rangle$. For a homogeneous medium, directionality always decreases $\langle T \rangle$ and a pure directional migration (a ballistic motion) serves as the optimized strategy; while for a heterogeneous environment, we find that the optimized strategy involves a combination of directed and random migrations. The relative contribution of these modes is determined by the density of obstacles and motility of targets. Existence of randomness and motility of targets add to the efficiency of search. Our study reveals generic and simple rules that govern search efficiency. Our findings might find application in a number of areas including immunology, cell biology, and ecology.