Bioreactors that employ a synthetic microbial community hold potential to overcome limitations of those based on a single species, which embrace a higher level of complexity due to the inter-species interactions. In this work, a number of generic system structures involving two cross-feeding species and various types of inhibition have been studied, together with two three-species cases where a third species is introduced to fulfil a specific function. These cases are represented by mathematical models and inspected through bifurcation analysis and numerical simulation to reveal how the system structure and parametrization affect stability and productivity of the bioreactor. The results show that inhibitions generally lead to reduction in both productivity and stability, and that the presence of a negative feedback loop and a positive feedback loop may give rise to oscillation and bi-stability, respectively, depending on the strength of the inhibitions involved. The intended gains by the introduction of a third species may be achieved when its negative side-effect is sufficiently moderate, and at the cost of reduced stability. As observed in several cases, the changes in stability and productivity do not always follow the same trend, implying trade-off between the two objectives in the engineering of such bioreactors.