  Biological events are often initiated when a random "searcher" finds a
"target," which is called a first passage time (FPT). In some biological
systems involving multiple searchers, an important timescale is the time it
takes the slowest searcher(s) to find a target. For example, of the hundreds of
thousands of primordial follicles in a woman's ovarian reserve, it is the
slowest to leave that trigger the onset of menopause. Such slowest FPTs may
also contribute to the reliability of cell signaling pathways and influence the
ability of a cell to locate an external stimulus. In this paper, we use extreme
value theory and asymptotic analysis to obtain rigorous approximations to the
full probability distribution and moments of slowest FPTs. Though the results
are proven in the limit of many searchers, numerical simulations reveal that
the approximations are accurate for any number of searchers in typical
scenarios of interest. We apply these general mathematical results to models of
ovarian aging and menopause timing, which reveals the role of slowest FPTs for
understanding redundancy in biological systems. We also apply the theory to
several popular models of stochastic search, including search by diffusive,
subdiffusive, and mortal searchers.
