Continuous cultures of budding yeast are known to exhibit autonomous oscillations that adversely affect bioreactor stability and productivity. We demonstrate that this phenomenon can be modeled by coupling the population balance equation (PBE) for the cell mass distribution to the mass balance of the rate limiting substrate. An efficient and robust numerical solution procedure using orthogonal collocation on finite elements is developed to approximate the PBE model by a coupled set of nonlinear ordinary differential equations (ODEs). A controller design model is obtained by linearizing and temporally discretizing the ODEs derived from spatial discretization of the PBE model. The resulting linear state-space model is used to develop model predictive control (MPC) strategies that regulate the discretized cell number distribution by manipulating the dilution rate and the feed substrate concentration. Two choices of the controlled output Vector are considered: (i) the entire discretized distribution; and (ii) a subset of the discretized distribution. The ability of the MPC controllers to stabilize steady-state and periodic solutions is evaluated via simulation. We show that superior closed-loop performance is obtained when a subset of the distribution is employed as controlled outputs.