Fractional Lotka-Volterra model with time-delay and delayed controller for a bioreactor

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.