Dynamics Analysis and Biomass Productivity Optimisation of a Microbial Cultivation Process through Substrate Regulation

Abstract

A microbial cultivation process model with variable biomass yield, control of substrate concentration, and biomass recycle is formulated, where the biochemical kinetics follows an extension of the Monod and Contois models. Control of substrate concentration allows for indirect monitoring of biomass and dissolved oxygen concentrations and consequently obtaining high yield and productivity of biomass. Dynamics analysis of the proposed model is carried out and the existence of order-1 periodic solution is deduced with a formulation of the period, which provides a theoretical possibility to convert the state-dependent control to a periodic one while keeping the dynamics unchanged. Moreover, the stability of the order-1 periodic solution is verified by a geometric method. The stability ensures a certain robustness of the adopted control; that is, even with an inaccurately detected substrate concentration or a deviation, the system will be always stable at the order-1 periodic solution under the control. The simulations are carried out to complement the theoretical results and optimisation of the biomass productivity is presented.

Kaibiao Sun 1 and Shan Liu 1 and Andrzej Kasperski 2 and Yuan Tian 3

Academic Editor:Carmen Coll

1, School of Control Science and Engineering, Dalian University of Technology, Dalian 116024, China
2, Faculty of Biological Sciences, Department of Biotechnology, University of Zielona Gora, Ulica Szafrana 1, 65-516 Zielona Gora, Poland
3, School of Information Engineering, Dalian University, Dalian 116622, China

Received 24 February 2016; Revised 1 April 2016; Accepted 13 April 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Microorganisms play an important role in nature and their activities have numerous industrial applications [1]. For that reason, bioreactor engineering, as a branch of chemical engineering and biotechnology, is an active area of research on microbial cultivation process, including development, control, and commercialization of new technology [2]. Reaching optimal results and attaining maximal profits require modern control strategies based on mathematical models or artificial intelligence methods. Many dynamic models concerning microbial cultivation processes, employing several types of reactions and control technologies, have been established [3-6]. In microbial cultivation process, there are a lot of factors affecting the growth and reproduction of the microorganisms. For example, for some aerobic microorganisms, the dissolved oxygen content in the bioreactor medium is a key factor to microbial growth. In order to maintain the dissolved oxygen content in an appropriate range, it is necessary to monitor and control the dissolved oxygen concentration (DOC) in the bioreactor medium since a low level of DOC decreases biomass yield and specific growth rate.

For a given microorganism, the oxygen demand is affected by several factors, for example, the biomass and substrate concentrations and cell metabolism state. The control of biomass and substrate concentrations is one of the most important ways to maintain an appropriate dissolved oxygen concentration. To obtain this goal, several approaches to bioprocess design have been presented in recent years. Especially attractive is the use of the self-cycling bioprocess. In such a bioprocess, impulsive controls are introduced once the monitored state reaches an established control level. In fact, many biological phenomena such as bursting rhythm models in biology and pharmacokinetics or frequency modulated systems exhibit impulsive effects, and periodic and state-dependent impulsive effects are two commonly occurring ones. In recent years, state-dependent impulsive control strategies have been used in the study on the interaction between wild and transgenic mosquito populations [7], predator-prey systems [8-11], management of pests and fisheries [12-14], pulse vaccination for human infectious diseases [15], and bioprocess models [16-20]. In bioprocess, self-cycling is a new approach to deal with important environmental problems [16], and for that reason it has become a subject of consideration presented in this work. In general, the self-cycling bioprocess is a computer-controlled system, which can be considered as a specific type of continuous bioprocess. The impulsive effect causes the removal of a certain fraction of the medium and the input of an equal volume of fresh medium [16, 21]. The aim of introduction of the impulsive effect is to provide among others appropriate oxygen conditions by limitation of the biomass concentration, which stems from the fact that insufficient dissolved oxygen concentration decreases the biomass yield, specific growth rate, and biomass productivity.

In bioprocess control, a key question is how to monitor reactant concentrations in a reliable and cost-effective manner [22]. However, in many industrial applications, not all of concentrations of the components (which are involved and critical for quality control) are available for online measurement. In this work, the substrate concentration is assumed to be measured online [16, 23, 24], and the biomass concentration is controlled indirectly by substrate concentration regulation. The microorganisms grow by assimilation of the substrate; for that reason the biomass concentration increases and substrate concentration decreases. Thus, in order to keep the biomass concentration lower than a critical level, the substrate concentration should be kept higher than a given level. This is also reasonable since a low level of the substrate concentration is disadvantageous to the transformation of microorganisms.

Based on above considerations, this work proposes a mathematical model of a microbial cultivation process with variable biomass yield and substrate regulation. The paper is organized as follows. In Section 2, a microbial cultivation process model of the considered bioprocess is formulated, the biochemical kinetics following an extension of the Monod and Contois models. In Section 3, the dynamic analysis of the proposed model is carried out, and the existence of order-1 periodic solution is deduced. The complete expression of the period of the order-1 periodic solution is also given. In addition, it is shown that there does not exist any order- [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) periodic solution. Next, a stability criterion for a general semicontinuous dynamic system is presented by a geometric method, and the stability of the order-1 periodic solution is verified. In Section 4, numerical simulations are carried out to complement the theoretical results and optimisation of the biomass productivity is presented. The final conclusions are presented in Section 5.

2. Model Formulation and Preliminaries

2.1. Model Formulation

A general model to describe the microbial growth on a limiting substrate is of the form [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote, respectively, the concentrations of biomass and substrate in the bioreactor medium at time [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ), [figure omitted; refer to PDF] denotes the substrate concentration in the input flow [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the dilution rate ( [figure omitted; refer to PDF] ) ( [figure omitted; refer to PDF] : chemostat; [figure omitted; refer to PDF] : batch process), [figure omitted; refer to PDF] is the biomass yield [figure omitted; refer to PDF] defined as the ratio of the biomass produced to the amount of substrate assimilated, and the function [figure omitted; refer to PDF] describes the biochemical kinetics, which is characterized by the cell concentration ( [figure omitted; refer to PDF] ) and the limiting substrate concentration [figure omitted; refer to PDF] .

The key to the proper functioning of a bioreactor lies mainly in understanding the growth rate of the biomass, which plays an important role in the intrinsic oscillation mechanisms [25]. Many kinetics models that have been proposed in the literature, including Tessier [26], Monod [27], Moser [28], Contois [29], and Andrews [30], address the kinetics of the growth of the cell mass in the bioreactor and most of these models report expressions for specific growth rate of cell mass. In this study, an extension of the Monod and Contois models is introduced; that is, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is a constant and [figure omitted; refer to PDF] is constant in the extended Contois model ( [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] is dimensionless biomass concentration.

The biomass yield expression plays an important role for the generation of oscillatory behaviour in continuous bioreactor models [25]. In many models of bioreactors, it is assumed that the biomass yield coefficient is a constant during the course of the reaction. However, several researchers also suggest that the experimentally found oscillatory behaviour of microbial population can be explained when the biomass yield coefficient is dependent on the substrate concentration [31, 32] and cannot when the biomass yield coefficient is constant. Thus a variable biomass yield is presented as [25, 33-35]: [figure omitted; refer to PDF]

To effectively utilize the limiting substrate and avoid unnecessary waste, a modification on the chemostat is made by changing the inflow and outflow from continuous to impulsive. The sketch map of the apparatus is illustrated in Figure 1. The apparatus includes an optical sensing device which continuously monitors the substrate concentration in the bioreactor medium and two switches which work in a synchronous way. When the substrate concentration is higher than a substrate low control level, the switches are closed. Once the substrate concentration [figure omitted; refer to PDF] in the bioreactor medium decreases to a control level [figure omitted; refer to PDF] (where [figure omitted; refer to PDF] ), then part of the medium containing biomass and substrate is discharged from the bioreactor to a membrane filter, and the next portion of medium of a given substrate concentration is inputted impulsively. Therefore, system (1) can be modified as follows by introducing the impulsive state feedback control: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is the concentration of the feed substrate which is inputted impulsively, [figure omitted; refer to PDF] is the proportion of medium which is removed from the bioreactor in each substrate oscillation cycle, [figure omitted; refer to PDF] represents the biomass filter efficiency constant, and [figure omitted; refer to PDF] is the substrate recycle part.

Figure 1: Schematic view of the impulsive bioprocess.

[figure omitted; refer to PDF]

2.2. Preliminaries

Let us consider a general model with impulsive effects: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are indefinitely differentiable with respect to [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are linearly dependent on [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constant.

Denote [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] is referred to as the control set, which contains all the states at which the control strategy is taken on and [figure omitted; refer to PDF] and [figure omitted; refer to PDF] describe the effects of the control strategy. When the state variables [figure omitted; refer to PDF] arrive at the control set [figure omitted; refer to PDF] , the impulsive control measures are taken; then the state variables [figure omitted; refer to PDF] jump from [figure omitted; refer to PDF] to the set [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be any solution of system (5). Let [figure omitted; refer to PDF] , given the initial value [figure omitted; refer to PDF] . Then, the (forward) orbit is the set of all values that this trajectory obtains [figure omitted; refer to PDF] , also denoted as [figure omitted; refer to PDF] in short. Denote [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] with [figure omitted; refer to PDF] .

Let [figure omitted; refer to PDF] be an arbitrary set, let [figure omitted; refer to PDF] be an arbitrary point, and [figure omitted; refer to PDF] . Then the distance between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and the noncentral [figure omitted; refer to PDF] neighborhood of [figure omitted; refer to PDF] are denoted by [figure omitted; refer to PDF]

Definition 1 (periodic orbit [36, 37]).

An orbit [figure omitted; refer to PDF] of system (5) is said to be periodic if there exists a positive integer [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] . Denote [figure omitted; refer to PDF] . Then the orbit [figure omitted; refer to PDF] is said to be an order- [figure omitted; refer to PDF] periodic orbit.

Definition 2 ( [figure omitted; refer to PDF] -close [38]).

Two orbits [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -close if there is a reparameterization of time (a smooth, monotonic function) [figure omitted; refer to PDF] such that [figure omitted; refer to PDF]

Definition 3 (orbitally stable [36-38]).

An orbit [figure omitted; refer to PDF] is said to be orbitally stable if, for any [figure omitted; refer to PDF] , there is a neighborhood [figure omitted; refer to PDF] of [figure omitted; refer to PDF] so that, for all [figure omitted; refer to PDF] in [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are [figure omitted; refer to PDF] -close, as illustrated in Figure 2.

Figure 2: Orbital stability.

[figure omitted; refer to PDF]

Definition 4 (asymptotic orbital stability [36-38]).

An orbit [figure omitted; refer to PDF] is said to be asymptotically orbitally stable if it is orbitally stable and additionally [figure omitted; refer to PDF] may be chosen such that, for all [figure omitted; refer to PDF] , there exists a constant [figure omitted; refer to PDF] so that [figure omitted; refer to PDF] as [figure omitted; refer to PDF] , as illustrated in Figure 3.

Figure 3: Asymptotic orbital stability.

[figure omitted; refer to PDF]

Definition 5 (successor function [37]).

Suppose that the control set [figure omitted; refer to PDF] and the phase set [figure omitted; refer to PDF] are both lines, as illustrated in Figure 4. The coordinate on the phase set [figure omitted; refer to PDF] is defined as follows: the origin is defined as the intersection point between [figure omitted; refer to PDF] and the [figure omitted; refer to PDF] -axis, denoted as [figure omitted; refer to PDF] , and the coordinate of any point [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is defined as the distance between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] denote the intersection point between the trajectory starting from [figure omitted; refer to PDF] and the control set [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] denote the phase point of [figure omitted; refer to PDF] after control measures. Let [figure omitted; refer to PDF] denote the intersection point if the trajectory starting from [figure omitted; refer to PDF] intersects the phase set [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is called as the successor point of [figure omitted; refer to PDF] , and the mapping from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] constructs the successor function. If [figure omitted; refer to PDF] exists, the successor function is called type-II; that is, [figure omitted; refer to PDF] ; otherwise, it is called type-I; that is, [figure omitted; refer to PDF] . Moreover, the successor function is continuous on [figure omitted; refer to PDF] .

Figure 4: Illustration of successor function.

[figure omitted; refer to PDF]

Remark 6.

If there exists a point [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] or [figure omitted; refer to PDF] , then the orbit starting from [figure omitted; refer to PDF] forms a periodic orbit.

3. Main Results

It is visible that [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

3.1. Qualitative Analysis of the Bioprocess Model

By (4) it is easily obtained that [figure omitted; refer to PDF] Integrating both sides of (10) from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] yields [figure omitted; refer to PDF] Now it is assumed that [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be the first time for the trajectory to reach the control set [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] Denote [figure omitted; refer to PDF] . Due to the effects of the impulsive control, one has [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF]

Theorem 7.

System (4) admits a unique order-1 periodic solution.

Proof.

For any [figure omitted; refer to PDF] , by Definition 5, one has [figure omitted; refer to PDF] Define [figure omitted; refer to PDF] Denote [figure omitted; refer to PDF] . Then there is [figure omitted; refer to PDF] , which implies that the trajectory of (4) starting from [figure omitted; refer to PDF] first intersects the control set [figure omitted; refer to PDF] at [figure omitted; refer to PDF] and then jumps to [figure omitted; refer to PDF] again, which forms an order-1 periodic solution. In addition, [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . This means that [figure omitted; refer to PDF] if and only if [figure omitted; refer to PDF] ; that is, the order-1 periodic solution is unique.

Theorem 8.

System (4) does not admit any order- [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) periodic solution.

Proof.

Define [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In addition, there is [figure omitted; refer to PDF] . For any point [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] ; thus [figure omitted; refer to PDF] . Similarly, if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] ; thus [figure omitted; refer to PDF] . This implies that the order- [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) periodic solution does not exist.

Let [figure omitted; refer to PDF] be the order-1 periodic solution determined in Theorem 7; that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Theorem 9.

The period of the order-1 periodic solution is [figure omitted; refer to PDF] where [figure omitted; refer to PDF] .

Proof.

For the order-1 periodic solution [figure omitted; refer to PDF] , by (11), one has [figure omitted; refer to PDF] Meanwhile, by (4), one has [figure omitted; refer to PDF] Substituting (19) into (20) yields [figure omitted; refer to PDF] Integrating both sides of (21) from [figure omitted; refer to PDF] to [figure omitted; refer to PDF] yields [figure omitted; refer to PDF]

Besides the qualitative analysis of model (4) presented above, the stability of the controlled system is also an important issue to be considered. As far as the stability is concerned, most of the works in literature rely on the Analogue of Poincare Criterion [36]. Recently, a geometric method is used to verify the stability of the order-1 periodic solution for a general semicontinuous dynamic system [13]. For the convenience of readers, a detailed overview is presented.

3.2. Stability Criterion Representation

To test the stability of the order-1 periodic solution [figure omitted; refer to PDF] , the behaviour of the orbit of system (5) starting from [figure omitted; refer to PDF] for given small [figure omitted; refer to PDF] should be characterized. Let [figure omitted; refer to PDF] be a point on the order-1 periodic orbit with rectangular coordinates [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is the arc length between [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . To begin with, assume [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . For a given point [figure omitted; refer to PDF] , here assume that [figure omitted; refer to PDF] locates above [figure omitted; refer to PDF] , where the rectangular coordinates of [figure omitted; refer to PDF] are [figure omitted; refer to PDF] . The trajectory starting from [figure omitted; refer to PDF] will intersect the normal of [figure omitted; refer to PDF] and the intersection point is denoted by [figure omitted; refer to PDF] . When [figure omitted; refer to PDF] increases, this trajectory intersects the normal of [figure omitted; refer to PDF] and the control set [figure omitted; refer to PDF] at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively.

Define [figure omitted; refer to PDF] If [figure omitted; refer to PDF] , then the order-1 periodic solution [figure omitted; refer to PDF] is orbitally asymptotically stable.

Theorem 10 (stability criterion [13]).

The order-1 [figure omitted; refer to PDF] -periodic solution [figure omitted; refer to PDF] of system (5) is orbitally asymptotically stable if [figure omitted; refer to PDF] where [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the value of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] represents the value of [figure omitted; refer to PDF] at [figure omitted; refer to PDF] .

Proof.

As illustrated in Figure 5, the slope of the control set is [figure omitted; refer to PDF] (in case of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is equal to [figure omitted; refer to PDF] ). Due to the specific forms of the impulsive control, for any [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] are constants, one has [figure omitted; refer to PDF]

Denote [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] which implies that the phase set [figure omitted; refer to PDF] is also a straight line with slope [figure omitted; refer to PDF] , where [figure omitted; refer to PDF]

On the other hand, by the effect of the impulsive controls, one has [figure omitted; refer to PDF] Since [figure omitted; refer to PDF] , then [figure omitted; refer to PDF]

Let [figure omitted; refer to PDF] denote the angle between the normal at [figure omitted; refer to PDF] and the phase line and let [figure omitted; refer to PDF] denote the angle between the normal at [figure omitted; refer to PDF] and the impulse line. When [figure omitted; refer to PDF] is sufficiently close to [figure omitted; refer to PDF] , one has [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF]

Denote [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) as the angle between [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and the [figure omitted; refer to PDF] -axis and [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) as the angle between the tangent at [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) and the [figure omitted; refer to PDF] -axis. Then there are [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF]

Summing up (31)-(33) yields [figure omitted; refer to PDF]

To characterize the ratio [figure omitted; refer to PDF] , it needs to introduce the curvilinear coordinates [figure omitted; refer to PDF] ; the description is as follows: [figure omitted; refer to PDF] represents length of the arc starting from [figure omitted; refer to PDF] and its increasing direction is consistent with that of time [figure omitted; refer to PDF] ; [figure omitted; refer to PDF] represents the normal length and its positive direction is to the right side when traveling along the periodic orbit.

The relationship between the rectangular coordinates [figure omitted; refer to PDF] and the curvilinear coordinates [figure omitted; refer to PDF] on point [figure omitted; refer to PDF] is [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] [figure omitted; refer to PDF] and [figure omitted; refer to PDF] denote the values of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] at the order-1 periodic orbit [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . It can be inferred that [figure omitted; refer to PDF]

Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are periodic functions of [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] in (36) is also a periodic coefficient nonlinear equation with [figure omitted; refer to PDF] as the variable and [figure omitted; refer to PDF] as the undetermined function. By (35) it can be obtained that [figure omitted; refer to PDF] is a solution of (36) corresponding to the order-1 periodic orbit. Since [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are continuously differentiable, [figure omitted; refer to PDF] also has continuous partial derivatives with respect to [figure omitted; refer to PDF] , and (36) can be rewritten as [figure omitted; refer to PDF]

In order to calculate [figure omitted; refer to PDF] , it is noted that [figure omitted; refer to PDF] , which implies that [figure omitted; refer to PDF] ; that is, [figure omitted; refer to PDF] . Thus, [figure omitted; refer to PDF] Therefore, [figure omitted; refer to PDF] where [figure omitted; refer to PDF] represents the curvature of the orthogonal trajectory of (5) at point [figure omitted; refer to PDF] . The first-order approximation of (37) is [figure omitted; refer to PDF] , with the solution [figure omitted; refer to PDF]

Note that [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] which results in the fact that [figure omitted; refer to PDF] Combined with (34), this yields the conclusion.

Figure 5: Illustration of the disturbance nearby the periodic orbit of model (5).

[figure omitted; refer to PDF]

Theorem 11.

The order-1 periodic solution [figure omitted; refer to PDF] , [figure omitted; refer to PDF] determined in Theorem 7 is orbitally asymptotically stable.

Proof.

Denote [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is defined by (9) and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] are defined by (16). In system (4), there are [figure omitted; refer to PDF] where [figure omitted; refer to PDF] Then [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] Hence, the left side of (24) is [figure omitted; refer to PDF] Therefore, the order-1 periodic solution [figure omitted; refer to PDF] is orbitally asymptotically stable.

4. Verifications and Biomass Productivity Optimisation

The microbial cultivation process for pulse dosage supply of substrates and removal of bioreactor medium has been analyzed theoretically. In order to complement the theoretical results, simulations of system (4)'s dynamics are presented firstly. The model parameters used in simulations are as follows: [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . The low substrate control level is a decision parameter, which should be set according to an actual demand. Firstly, in order to verify the theoretical results, [figure omitted; refer to PDF] is set to [figure omitted; refer to PDF] . Next, an optimal [figure omitted; refer to PDF] is determined by a biomass productivity minimization problem.

4.1. Verification of the Main Results

The change of substrate concentration ( [figure omitted; refer to PDF] ), biomass concentration [figure omitted; refer to PDF] , and phase diagram [figure omitted; refer to PDF] starting from [figure omitted; refer to PDF] are shown in Figure 6. It can be observed that the trajectories tend to be periodic, and this phenomenon is not dependent on the model parameters. Furthermore, from Figure 6(b), it can be concluded that the biomass concentration is controlled below a certain level through substrate regulation. Figure 7 illustrates the order-1 periodic solution with period [figure omitted; refer to PDF] . The phase diagrams for different initial biomass concentrations on [figure omitted; refer to PDF] are illustrated in Figure 8, from which the stability of the order-1 periodic solution can be observed.

Figure 6: Verification of Theorem 7. The change of substrate concentration ( [figure omitted; refer to PDF] ), biomass concentration [figure omitted; refer to PDF] , and phase diagrams with [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 7: Verification of Theorem 7. The change of substrate concentration ( [figure omitted; refer to PDF] ), biomass concentration [figure omitted; refer to PDF] , and phase diagrams with [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 8: Verification of Theorem 7. The phase diagrams for different initial biomass concentrations on [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

4.2. Biomass Productivity Optimisation

Besides analyzing the dynamics of the proposed bioprocess model, another important aspect is to find the optimal substrate control level such that the biomass productivity is maximized. In the proposed bioprocess, the biomass productivity [figure omitted; refer to PDF] is formulated as follows: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] is determined by (16) and [figure omitted; refer to PDF] is determined by (18). For the given model parameters, the dependence of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] and the substrate control level [figure omitted; refer to PDF] is shown in Figure 9. The maximum biomass productivity occurs for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Note that, for the maximum biomass productivity, the bioreactor medium removal part is also kept at the minimum value, which increases the frequency of the operation. For bigger [figure omitted; refer to PDF] , the dependence of [figure omitted; refer to PDF] on the substrate control level [figure omitted; refer to PDF] is shown in Figure 10, and, for different substrate control level [figure omitted; refer to PDF] , the dependence of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] is illustrated in Figure 11.

Figure 9: The dependence of [figure omitted; refer to PDF] on the control parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 10: The dependence of [figure omitted; refer to PDF] on the substrate control level [figure omitted; refer to PDF] for different [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 11: The dependence of [figure omitted; refer to PDF] on [figure omitted; refer to PDF] for different substrate control level [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

The model parameters also have a certain impact on the biomass productivity. For given [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , the biomass productivity achieves its maximum at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , as illustrated in Figure 12. This is easily understood since in case of full filter efficiency the recycling operation is unnecessary. The recycling takes effect only for low filter efficiency. The dependence of [figure omitted; refer to PDF] on the model parameters [figure omitted; refer to PDF] for different [figure omitted; refer to PDF] is given in Figure 13, and the dependence of [figure omitted; refer to PDF] on the model parameter [figure omitted; refer to PDF] for different [figure omitted; refer to PDF] is given in Figure 14. It can be concluded that the optimal substrate recycling proportion [figure omitted; refer to PDF] is dependent on [figure omitted; refer to PDF] ; that is, when [figure omitted; refer to PDF] is small, for example, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] should be set equal to [figure omitted; refer to PDF] ; when [figure omitted; refer to PDF] , [figure omitted; refer to PDF] should be set equal to [figure omitted; refer to PDF] ; while, for bigger [figure omitted; refer to PDF] , for example, [figure omitted; refer to PDF] , [figure omitted; refer to PDF] should be set equal to zero, which means that the substrate recycling operation is unnecessary. But it should be pointed that the recycling operation is indeed beneficial in saving the substrate loss.

Figure 12: The dependence of [figure omitted; refer to PDF] on the model parameters [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for the given [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 13: The dependence of [figure omitted; refer to PDF] on the model parameters [figure omitted; refer to PDF] for the given [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

Figure 14: The dependence of [figure omitted; refer to PDF] on the model parameters [figure omitted; refer to PDF] for the given [figure omitted; refer to PDF] , [figure omitted; refer to PDF] .

[figure omitted; refer to PDF]

5. Conclusions

In the paper the dynamic behaviour of a microbial cultivation process model with variable biomass yield and substrate regulation was analyzed. For the proposed bioprocess model, it was shown that (i) the existence of the order-1 periodic solution does not depend on the biomass yield and the kinetic model (Theorem 7), but the bioprocess behaviour (i.e., the order-1 periodic solution) is dependent on the biomass yield and the kinetic model (see (11)-(18)); (ii) the system is not chaotic due to nonexistence of the order- [figure omitted; refer to PDF] periodic solution (Theorem 8) and orbit asymptotic stability of the order-1 periodic solution (Theorem 11); (iii) the stability of the order-1 periodic solution is independent from the biomass yield and the kinetic model. The analytical results offer the possibility of establishing general and more systematic operation and control strategies based on the counteraction of the mechanisms underlying the adverse effects of bioreactor dynamics. In addition, microorganisms in the considered bioprocess always kept a suitable growth rate through substrate regulation. The biomass concentration could also be controlled to a certain level, which should ensure an appropriate dissolved oxygen concentration. Moreover, through biomass productivity optimisation, the optimal substrate control level and bioreactor medium removal proportion were obtained. The study showed how the biomass productivity depended on the bioprocess parameters (i.e., the substrate recycling proportion [figure omitted; refer to PDF] and the biomass filter efficiency [figure omitted; refer to PDF] ).

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (nos. 11401068, 61473327) and the Liaoning Province Natural Science Foundation of China (no. 2014020133).

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