Animals forage, immune cells patrol the body for pathogens, and biological molecules search for receptors on cell surfaces. Algorithms trawl for optimal solutions to thorny problems. In natural and artificial systems alike, the act of searching is ubiquitous, says SFI Complexity Postdoctoral Fellow Aanjaneya Kumar. It’s also complicated: Searchers may die off or disappear, and new searchers can join the hunt. However, most rigorous investigations of searching assume that there’s only a fixed number of agents, notes Kumar. The influence of a changing cast of searchers has until now remained largely unexplored.
In a recent paper in Physical Review Letters, Kumar and Samantha Linn, a mathematician at Imperial College London, describe how a mix of agents influences the search process. A system can gain and lose agents during a random search, and that “dynamic redundancy and mortality” can shape the system’s efficiency. That efficiency is often evaluated by how long it takes an agent (or agents) to find the target for the first time. This is called the “first-passage time.” An example might be how long it takes an animal to find a meal after it begins searching.
After building a general framework, Kumar and Linn explore the example of a particle bouncing around in one dimension. Their work shows that its first-passage time may be different than the case of multiple particles.
The new approach, says Kumar, more accurately represents real-world situations than those that only focus on a fixed number of searchers. He points to biological phenomena as an example. “Molecules can be degraded by the environment, but they can also be synthesized,” he says. “Similarly, T cells are not fixed. They keep joining the surge [of an immune response], they keep dying, they drop out.”
Or it could get even more complex, says Linn. “Perhaps there are many targets in the system, each with specific downstream behavior induced when a searcher detects the target.”
The work began when Linn was visiting SFI in 2025, and the two began talking about the “method of images,” a clever shortcut in physics often used in first-passage calculations. But their conversations led to an interesting problem: The shortcut proved useless when they tried to apply it to systems where searchers came and went. They quickly recognized that the same limitation arose in every search situation, natural or artificial. Most approaches used in first-passage problems assumed a single searcher.
“When we try to use conventional tools to solve this problem where the number of searchers is fluctuating, the tools can fail,” says Kumar.
Once they recognized the problem, their framework fell into place quickly. The PRL paper represents a first attempt at building a theory for these processes, Kumar says. It also connects this new avenue of investigation to previous work. It shows, for example, that when searchers join and leave the search at equal rates, the problem begins to look like “stochastic resetting,” a well-known process in which a searcher randomly returns to its starting point. But that connection disappears as the rate of searcher turnover changes.
Kumar sees the new work as introducing an untapped resource in studying search behaviors. In future work, he says, researchers could apply it to a spectrum of relevant applications, from biology to computer search algorithms. And there’s room to expand the theory: He and Linn assumed each agent worked independently, but in the real world, searchers can collaborate and interact to achieve a common goal.
“There are a bunch of interesting questions to answer before we move on to interactions,” he says, “but I think that’s where the next conceptual advance is going to happen.”
Read the paper "Dynamic Redundancy and Mortality in Stochastic Search" in Physical Review Letters (August 25, 2026). DOI: 10.1103/s79q-l1dq