  Prolonged lag time can be induced by starvation contributing to the
antibiotic tolerance of bacteria. We analyze the optimal lag time to survive
and grow the iterative and stochastic application of antibiotics. A simple
model shows that the optimal lag time exhibits a discontinuous transition when
the severeness of the antibiotic is increased. This suggests the possibility of
reducing tolerant bacteria by controlled usage of antibiotics application. When
the bacterial populations are able to have two phenotypes with different lag
times, the fraction of the second phenotype that has different lag time shows a
continuous transition. We then present a generic framework to investigate the
optimal lag time distribution for total population fitness for a given
distribution of the antibiotic application duration. The obtained optimal
distributions have multiple peaks for a wide range of the antibiotic
application duration distributions, including the case where the latter is
monotonically decreasing. The analysis supports the advantage in evolving
multiple, possibly discrete phenotypes in lag time for bacterial long-term
fitness.
