Abstract
Evolutionary therapies regulate heterogeneous populations by dynamically altering selective pressures through sequential treatments in cancer and infectious diseases. The main result of this letter derives simplex-based invariant regions that bound the evolution of resistant populations under periodically switched therapies with mutation dynamics. For periodically switched systems, sufficient conditions are derived for the existence of such invariant regions. The simplex containment region is shown to remain robust under sufficiently small mutation-driven phenotype transitions. Analysis and simulations yield an explicit mutation threshold that separates regimes in which therapy cycling maintains containment from regimes in which mutation can enable evolutionary escape.