Before revision:   When the bacterial populations are able to split into subpopulations to have two phenotypes with different average lag time , the fraction of the finite lag time population shows continuous transition when changing the severeness of antibiotics application . The result suggests the possibility of reducing tolerant bacteria by controlled usage of antibiotics application .
After revision:  When the bacterial populations are able to split into subpopulations to have two phenotypes with different average lag time , 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 .