Dynamic redundancy and mortality in stochastic search

TL;DR

The paper introduces Dynamic Redundancy and Mortality (DRM), a framework for first-passage times with a fluctuating number of searchers due to recruitment at rate $\lambda$ and death at rate $\mu$. It derives an exact expression for the DRM survival probability $S_{\lambda,\mu}(t)$ in terms of the single-mortal survival $S_{0,\mu}(t)$, enabling precise MFPT analysis and revealing connections to stochastic resetting, including a universal lower bound $\mathbb{E}[T_{\lambda,\mu}] \ge (1-p_\mu)/(\lambda p_\mu)$. In the 1D Brownian case, the authors obtain explicit survival forms and rigorous upper bounds, showing $\mathbb{E}[T_{r,r}] \sim 1/(r p_r)$ as $r\to\infty$, and demonstrate parameter regimes where DRM outpaces resetting, especially when redundancy dominates mortality (large $\lambda/\mu=\alpha>1$). The results provide a rigorous foundation for first-passage problems with fluctuating searcher populations and have broad implications for physical, biological, and algorithmic search strategies.

Abstract

Search processes are a fundamental part of natural and artificial systems. In such settings, the number of searchers is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search efficiency remains unexplored. Here we present a general framework for stochastic search in which independent agents progressively join and leave the process, a mechanism we term \emph{dynamic redundancy and mortality} (DRM). Under minimal assumptions on the underlying search dynamics, this framework yields exact first-passage time statistics. It further reveals surprising connections to stochastic resetting, including a regime in which the resetting mean first-passage time emerges as a universal lower bound for DRM, as well as regimes in which DRM search is faster. We illustrate our results through a detailed analysis of one-dimensional Brownian DRM search. Altogether, this work provides a rigorous foundation for studying first-passage processes with a fluctuating number of searchers, with direct relevance across physical, biological, and algorithmic systems.

Figures (5)

• Figure 1: A schematic for the stochastic search process where searchers are recruited to the search at rate $\lambda$ (dynamic redundancy) and abandon the search process at rate $\mu$ (mortality). The central focus of this Letter is to characterize the time taken for the target to be found for the first time under dynamic redundancy and mortality (DRM).
• Figure 2: Balanced ($\lambda\!=\!\mu\!\equiv\!r$) DRM MFPT $\mathbb{E}[T_{r,r}]$ for Brownian motion on $\mathbb{R}$ with $x_0\!=\!1$, and $D\!=\!1$. The solid teal curve indicates the solution computed from Eq. \ref{['mfpt']} via quadrature (see details in Ref. SI); the solid orange curve indicates the exact solution in resetting theory in Eq. \ref{['low']}; the solid purple curve indicates an algebraically computed upper bound in Eq. \ref{['low']}; the dashed pink line indicates an upper bound computed in Ref. SI. The inset shows ratios of each bound to the numerical DRM MFPT curve.
• Figure 3: DRM MFPT for Brownian motion on $\mathbb{R}$ with $\alpha\!>\! 0$ where $\mu\!=\! r$ and $\lambda\!=\!\alpha r$. (a) The solid pink curve indicates the balanced DRM MFPT and the dashed curve denotes the stochastic resetting MFPT. (b) The no mortality result is an asymptotic result for the frequent recruitment limit recently derived in Ref. tung2025first and is extrapolated here for finite but large recruitment rates (solid black line). In both (a) and (b), $x_0\!=\!1$, $D\!=\!1$, and the target is placed at the origin.
• Figure S1: DRM position density on $\mathbb{R}$ as in Eqs. \ref{['eq:rho_time']} and \ref{['rhostar']} for different times $t$ in (a) a mortality-dominant system: $\mu\!=\!2$, $\lambda\!=\!1$, (b) a balanced system: $\mu\!=\!\lambda\!=\!1$, and (c) a recruitment-dominant system: $\mu\!=\!1$, $\lambda\!=\!2$. Here $x_0\!=\!1$ (dashed black line) and $D\!=\!2$.
• Figure S2: (a) The DRM MFPT for Brownian motion on $\mathbb{R}$ where $\mu\!\equiv\! r$ and $\lambda\!\equiv\!\alpha r$. The dashed black line indicates the stochastic resetting MFPT with rate $r$. (b) In reference to (a), $r^*$ denotes the value of $r$ at which the redundant-dominant DRM MFPT equals the stochastic resetting MFPT. DRM therefore outpaces stochastic resetting for values of $r>r^*$. (c) A phase diagram for when the DRM MFPT outpaces the optimal resetting MFPT (above) and vice versa (below). Throughout, $x_0\!=\!1$ and $D\!=\!1$.