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The dynamical behaviour for a generic two-dimensional model of a continuous bioreactor is studied in this article. The state variables for the bio-reacting system are restricted to concentrations of substrate and biomass, where the specific growth rate is a smooth function of the substrate concentration, which can be a kinetic function, monotone or non-monotone (as Monod, Haldane, Teissier, etc.). The effect of input (dilution rate) on multiplicity and bifurcation of equilibrium is shown in open-loop configuration. The absence of limit cycles on open-loop configuration and through state-feedback on the dilution rate are demonstrated. The aim of analysing oscillations under state feedback control is related to the possible improvement of reactor yields under this operation regime. An example for a cell-producing bioreactor illustrates the analytical results.