The aim of this work is to present a theoretical analysis and optimization of a biochemical reaction process by means of feedback control strategy. To begin with, a mathematical model of the biochemical reaction process with feedback control is formulated. Then, based on the formulated model, the analysis of system's dynamics is presented. The optimization of the bioprocess is carried out, in order to achieve maximal biomass productivity. It is shown that during the optimization, the bioprocess with impulse effects loses the possibility of synchronization and strives for a simple continuous bioprocess. The analytical results presented in the work are validated by numerical simulations for the Tessier kinetics model.
Chemical and bioprocess engineering play an important role in the production of many chemical products. In particular, bioreactor engineering as a branch of chemical engineering and biotechnology is an active area of research on bioprocesses, including among others development, control, and commercialization of new technology [
Since many biological phenomena such as bursting rhythm models in, for example, medicine, biology, pharmacokinetics, and frequency modulated systems exhibit impulsive effects [
The objective of this work is to illuminate theoretical and practical aspects of the nonlinear analysis of a universal mathematical model of the biochemical reaction process. The paper is organized as follows. In Section
The general model of continuously culturing microorganism in a chemostat is given by the following form of differential equations [
Crooke et al. [ (regularity) (monotonicity) (convexity)
Thus, in the study that the kinetic models only satisfy the regularity, monotonicity, and convexity are considered.
According to the Herbert's and Pirt's models [
Before presenting the main results, we recall the following definitions and lemmas first [
Lambert
In the followings, we simply denote
The property of Lambert
Assume that there exists a bounded closed region
Illustration of Bendixson field.
The
We focus our discussion on the case
The line
If one of the conditions holds (i)
Firstly, it is obvious that all trajectories of system (
In addition, there is no singularity in the region
Illustration of system (
Suppose the period of the period-1 solution is
By (
Let
Suppose that
Since
Next, we consider the following comparison system of system (
If
Suppose that
Suppose that
Suppose that
Let
Suppose that
Let
Since
From the expression of the isoclinal line
Consider the following comparison system of system (
Next, we will prove that
For the predator-prey model concerning IPM strategies, Tang and Cheke [
Suppose that
Let
The tr
ajectory of the solution
From the proof of Theorem
Here, we will take the Tessier kinetics model [
The simulations are carried out by changing one main parameter and fixing all other parameters. The following numerical simulations are given by the programs of Maple and Matlab softwares. The time interval is set two days, that is,
We assume in the following that
Illustration of vector graph of system (
The time series and phase portrait for
The time series and phase portrait when
Figure
The time series and phase portrait when
Figure
Dependence of the biomass and substrate concentration on
Figure
Dependence of the biomass and substrate concentration on
The numerical simulations are consistent with the theoretical results obtained and presented in Section
Next, we discuss aspects of the bioprocess optimization.
where
The aim of the optimization is to find the maximum of the objective function (
(
(
Let
Therefore, for given
In the paper the dynamic behavior of a universal biochemical reaction process with feedback control was analyzed, and it was shown that the stability of the bioprocess (i.e., the existence of the positive period-1 solution), depended on both the biomass yield and the microorganism growth rate. Furthermore, the conditions for the existence and stability of the system's period-1 solution were obtained (i.e., Theorems
After the analysis of the system stability, bioprocess optimization was covered in the work. In the first case, the optimization of a simple continuous bioprocess (i.e., bioprocess in which
This research is supported in part by National Natural Science Foundation of China (Grant no. 11101066) and the Fundamental Research Funds for the Central Universities.