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.

自然界中的许多搜索过程都表现出盐分行为，扩散运动与静止交替出现。第一次通过时间是描述不同领域中此类过程效率的重要指标。在这里，我们提出了一个理论公式，用于平均第一次通过时间到达位于密闭球形区域主体中的小目标，在该搜索中，将标准扩散阶段与只有在扩散阶段才可检测到的目标相结合。我们使用二态系统对切换机制进行建模，从而得出具有两个耦合微分方程的系统。我们解决了获得的系统，以获得用于计算平均首次通过时间的解析公式，并分析了其对两相之间过渡速率的依赖性。我们的结果表明，此方案的平均首次通过时间大于纯扩散的时间。此外，它随着从扩散到静止的速率的增加而线性增长，而随着从静止到扩散的速率的增加而衰减。为了进行比较，我们提供了数值模拟。理论和仿真结果之间有很好的一致性。我们的模型可用于设计和加速目标搜索，例如生态，生化和生物过程。