Dynamics of drug on-drug off models with mutations in morbidostat — Dedicated to the seventieth birthday of Professor Gail Wolkowicz

The morbidostat is a bacteria culture device that progressively increases antibiotic drug concentration. It is used to study the evolutionary pathway. In this article, we construct mathematical models for the morbidostat. First we consider the case of no mutations, we study limiting systems and obtain criteria for the large time behavior of the solutions. From the theoretical results and numerical simulations, we conclude that there are two competitive exclusion states of either wild type or mutant type as the threshold parameter $ U $ varies. There are three cases, wild type bacteria excludes all mutants; a mutant dominates in the competition; oscillation between the above two states. Next we study the systems of forward mutations and forward-backward mutations. Then we apply a result of pertubation for globally stable state.