Mortality, Redundancy, and Diversity in Stochastic Search

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.

5 pages, 1 figure, 2-column revtex format; updated version: error corrected; general improvements; original figure deleted and 2 figures added

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