In this work, we study the problem of maximizing microbial production within a continuous stirred-tank reactor with n >= 1 species and one substrate. The interaction between species is described by the chemostat system including a mutation factor. The optimization problem falls into an optimal control problem of Lagrange type in which the control parameter is the dilution rate of the reactor. Thanks to the Pontryagin Maximum Principle, we obtain necessary conditions on optimal controls that involve a singular arc. These computations are highlighted thanks to numerical simulations via a direct method. We also study the related optimization problem at steady-state which provides an insight into optimal solutions of the optimal control problem in terms of turnpike phenomenon.