  In this paper, a fractional Lotka-Volterra mathematical model for a
bioreactor is proposed and used to fit the data provided by a bioprocess known
as continuous fermentation of Zymomonas mobilis. The model contemplates a
time-delay $\tau$ due to the dead-time in obtaining the measurement of biomass
$x(t)$. A Hopf bifurcation analysis is performed to characterize the inherent
self oscillatory experimental bioprocess response. As consequence, stability
conditions for the equilibrium point together with conditions for limit cycles
using the delay $\tau$ as bifurcation parameter are obtained. Under the
assumptions that the use of observers, estimators or extra laboratory
measurements are avoided to prevent the rise of computational or monetary
costs, for the purpose of control, we will only consider the measurement of the
biomass. A simple controller that can be employed is the proportional action
controller $u(t)=k_px(t)$, which is shown to fail to stabilize the obtained
model under the proposed analysis. Another suitable choice is the use of a
delayed controller $u(t)=k_rx(t-h)$ which successfully stabilizes the model
even when it is unstable. Finally, the proposed theoretical results are
corroborated through numerical simulations.
