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Efficient search acts as a strong selective force in biological systemsranging from cellular populations to predator-prey systems. The searchprocesses commonly involve finding a stationary or mobile target within aheterogeneously structured environment where obstacles limit migration. An opengeneric question is whether random or directionally biased motions or acombination of both provide an optimal search efficiency and how that dependson the motility and density of targets and obstacles. To address this question,we develop a simple model that involves a random walker searching for itstargets in a heterogeneous medium of bond percolation square lattice and usedmean first passage time (MFPT, $\langle T \rangle$) as an indication of averagesearch time. Our analysis reveals a dual effect of directional bias on theminimum value of $\langle T \rangle$. For a homogeneous medium, directionalityalways decreases $\langle T \rangle$ and a pure directional migration (aballistic motion) serves as the optimized strategy; while for a heterogeneousenvironment, we find that the optimized strategy involves a combination ofdirected and random migrations. The relative contribution of these modes isdetermined by the density of obstacles and motility of targets. Existence ofrandomness and motility of targets add to the efficiency of search. Our studyreveals generic and simple rules that govern search efficiency. Our findingsmight find application in a number of areas including immunology, cell biology,and ecology.

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