Mean first passage time for diffuse and rest search in a confined spherical domain

Many search processes in nature exhibit saltatory behavior, alternating phases of diffusive movement with resting. The first passage time is an important measure to describe the efficiency of such processes in different domains. Here, we presented a theoretical formula for the mean first passage time to a small target located in the bulk of confined spherical domain, where the search combines phases of the standard diffusion and resting with the target detectable only during the diffusion phase. We used a two-state system to model the switching mechanism, which yields a system with two coupled differential equations. We solved the obtained system to obtain an analytical formula for computing the mean first passage time and analyzed its dependence on the transition rates between the two phases. Our results indicate that, the mean first passage time for this scenario was greater than for the pure diffusion. Further, it grew linearly with increasing the rate from diffusion to resting while decayed with increasing the rate from resting to diffusion. For comparison, we provided numerical simulation. There is good a agreement between the theoretical and the simulation results. Our model could be used to design and accelerate the target search like ecological, biochemical and biological processes.