Mortality, Redundancy, and Diversity in Stochastic Search

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 nontrivial optimal diffusivity is most likely to reach the target. We also discuss the effect of chemotaxis on the search time and its fluctuations.

Figure 1

Contour plot of $⟨T_N⟩/{\tau }_D$ from Eq. (12) versus $M$ and $\ln N$.

Figure 2

(a) Nonmonotonic dependence of $⟨T_N⟩$ from Eq. (12) on $M$ for $N=10$. The large-$M$ asymptotic $⟨T_N⟩/{\tau }_D\simeq 1/M$ [Eq. (14)] is shown dashed. (b) $⟨T_N⟩/{\tau }_D$ versus $\ln N$ for $M=2$. The dashed and dotted lines are the respective asymptotics (9) and (10).