Mortality, Redundancy, and Diversity in Stochastic Search

TL;DR

This paper models a stochastic search process with mortality, redundancy, and diversity, revealing how these factors influence search time and efficiency, with applications to fertilization and chemotaxis.

Contribution

It introduces a one-dimensional stochastic search model incorporating mortality and diversity, analyzing their effects on search time and optimal search strategies.

Findings

Search time scales as τ_D/ln N when mortality is negligible.

High mortality leads to search time scaling as √(τ_D/μ), independent of N.

Optimal diffusivity exists for subpopulations with high mortality.

Abstract

We investigate a stochastic search process in one dimension under the competing roles of mortality, redundancy, and diversity of the searchers. This picture represents a toy model for the fertilization of an oocyte by sperm. A population of $N$ independent and mortal diffusing searchers all start at $x=L$ and attempt to reach the target at $x=0$. When mortality is irrelevant, the search time scales as $\tau_D/\ln N$ for $\ln N\gg 1$, where $\tau_D\sim L^2/D$ is the diffusive time scale. Conversely, when the mortality rate $\mu$ of the searchers is sufficiently large, the search time scales as $\sqrt{\tau_D/\mu}$, independent of $N$. When searchers have distinct and high mortalities, a subpopulation with a non-trivial optimal diffusivity are most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.