  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.
