Predicting search time when hunting for multiple moving targets: A recursive harmonic law

We investigate searching for multiple mobile objects on networks and introduce the concept of mean random search time (MRST) to quantify the expected time a searcher takes to capture moving targets specified in advance. We consider this quantity averaged over all initial conditions for a searcher and multiple targets called the global MRST. We find that the growth of global MRST follows a recursive harmonic law with respect to that of stalking the individuals. In particular, when the diffusive laws of moving targets are identical, the global MRST shows a logarithmic increase with the number of moving targets. Moreover, utilizing the recursive harmonic law, we can accurately predict the expected successive time interval for capturing a new moving target. The recursive harmonic law unveils the underlying mechanism governing the search time when hunting for multiple moving targets on networks.