# Nonlinear Modelling and Qualitative Analysis of a Real Chemostat with Pulse Feeding

Authors: Yuan Tian; Kaibiao Sun; Andrzej Kasperski; Lansun Chen
Journal: Discrete Dynamics in Nature and Society
 (2010)

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## Abstract

The control of substrate concentration in the bioreactor medium should be
due to the substrate inhibition phenomenon. Moreover, the oxygen demand
in a bioreactor should be lower than the dissolved oxygen content. The
biomass concentration is one of the most important factors which affect the
oxygen demand. In order to maintain the dissolved oxygen content in an
appropriate range, the biomass concentration should not exceed a critical
level. Based on the design ideas, a mathematical model of a chemostat with
Monod-type kinetics and impulsive state feedback control for microorganisms
of any biomass yield is proposed in this paper. By the existence criteria
of periodic solution of a general planar impulsive autonomous system, the
conditions for the existence of period-1 solution of the system are obtained.
The results simplify the choice of suitable operating conditions for continuous
culture systems. It also points out that the system is not chaotic according to
the analysis on the existence of period-2 solution. The results and numerical
simulations show that the chemostat system with state impulsive control
tends to a stable state or a period solution.


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## Body

## 1. Introduction

Bioreactor control is an active area of research on the continuous microorganism cultivation [1]. Modern control strategies require a mathematical model to check behavior of bioprocess and test its stability. Furthermore, they are necessary to optimize bioprocess and obtain maximal profit. According to different reactions and differential control technologies, many dynamic models concerning the culture of microorganism in the chemostat have been established [2–5]. However, there are a lot of factors affecting the growth and reproduction of the microorganisms in the process of bioreacts. For example, for some aerobic microbes, the dissolved oxygen content in the medium is a key factor to microbial growth. In order to maintain the dissolved oxygen content in an appropriate range, it is easy to prevent the process from the decrease of dissolved oxygen concentration (DOC) in the bioreactor medium below a low level by the monitoring of DOC oscillations. It is necessary because the low level of DOC decreases biomass yield and specific growth rate [6]. On the other hand, with the growth of the microorganisms, the effect of inhibition between the production and other negative effect will occur when the biomass concentration reaches a critical level. For the purpose of continuously culturing microorganisms and decreasing the inhibition effect, it is necessary to keep the biomass concentration lower than a critical level.Many biological phenomena involve thresholds, bursting rhythm models in, for example, medicine, biology, pharmacokinetics, and frequency modulated systems, that exhibit impulsive effects. Thus, impulsive differential equations appear as a natural description of the observed evolution phenomena resulting from several real-world problems [7]. Many papers have investigated the systems with sudden perturbations which are involving in impulsive differential equations. Authors in [8–11] introduced some impulsive differential equations in population dynamics and obtained some interesting results. The research on the chemostat model with impulsive perturbations was studied by Sun and Chen [4]. Tang and Chen [12] introduced a Lotka-Volterra model with state-dependent impulsion and analyzed the existence and stability of positive period-1 solution. Jiang et al. [13] and Smith [14] have studied the state-dependent models with impulsive state control, where the model has a first integral, and obtained the complete expression of the period of the periodic solution. Jiang et al. [15] and Zeng et al. [16] have also discussed the models concerning integrated pest management (IPM). In the bioprocess, Guo and Chen [17, 18], Sun et al. [19–21], and Tian et al. [22, 23] have studied state-dependent models with impulsive state control by applying the Poincaré principle and Poincaré-Bendixson theorem of the impulsive differential equation, respectively.This paper aims at proposing a mathematical model of a chemostat with variable yield and feedback control, described by the impulsive differential equation, and studying the dynamics of the bioreact. The rest of this paper is organized as follows. In Section2 we introduce a chemostat model with Monod's growth rate and impulsive state feedback control for microorganisms of any biomass yield. In Section 3, we obtain the conditions for the existence of positive period-1 solution by the Poincaré-Bendixson theorem. We also point out that the proposed system is not chaotic according to the analysis of period-2 solution. In Section 4, we give the numerical simulations to verify the theoretical results, such as the existence of period-1 solutions, obtained in this paper and discuss the biological essence. Finally in Section 5 we present the conclusions.


## 2. Model Formulation

If the microorganisms' growth proceeds in accordance with Monod-type kinetics, that is, according to dependence(2.1)μ(S)=μmaxSKS+S,
which is commonly used to model a large variety of biochemical reaction [24, 25], then the deterministic model of microbial growth in the chemostat is of the form [26]:(2.2)dSdt=D(SF-S)-1Yx/SμmaxSKS+Sx,dxdt=μmaxSKS+Sx-Dx,S(0+)=S0,x(0+)=x0,
where x(t) denotes the biomass concentration, S(t) the substrate concentration, Yx/S the biomass yield, μmax the maximum specific growth rate, KS the saturation constant, SF the concentration of the feed substrate, D the dilution rate of the chemostat, t denotes time, and x0 and S0 denote the initial biomass concentration and substrate concentration in the bioreactor medium; all parameters are positive.Crooke et al. [27] showed that the biomass yield expression plays an important role for the generation of oscillatory behavior in continuous bioprocess models. Further, Crooke and Tanner [28] have proved that model (2.2) could not exhibit any periodic solution if the biomass yield in the model is constant. On the other hand, in reality, growth yields have not shown a constant pattern [19, 29, 30], so it is necessary to examine the bioprocess for the real biomass yield.Not all energy produced in catabolic processes is used for cellular material synthesis, part of it (i.e., so-called maintenance energy) is used for maintaining life functions, for that reason dependence of biomass yield onμ is conditioned physiologically. An effect of existence of maintenance energy is biomass yield dependant on growth rate. One of the most important models of quantification of maintenance energy in microorganism growth balance is Pirt's model [31, 32]. According to those models, for very low substrate concentration the amount of energy obtained from the substrate is not sufficient for maintenance energy. In this case, the energy obtained from the substrate is fully assigned for maintenance energy. For substrate concentration greater than a certain minimum substrate concentration, there occurs a positive rate of biomass growth. If the amount of energy assigned for maintenance energy makes a considerable part of energy produced in catabolic processes, then can be assumed an almost linear dependence Yx/S on specific growth rate. For high concentration of substrate, a high specific growth rate is obtained. In such conditions biomass yield achieves the maximum value (Yx/Smax, where Yx/Smax<1) and is practically constant, that is, it does not depend on the substrate concentration. When the amount of substrate essential to ensure maintenance energy is small, the described characteristics of cells can be approximated with sigmoid function, which has high flexibility in adapting to reality.In this work, the sigmoid function, that is,(2.3)Yx/S=(a+exp(-bS))-1,(seeFigure1)
which has high flexibility to fit any real biomass yield is used, where a=1/Yx/Smax, b is the cell sensitivity to the substrate under optimal growth conditions (optimal temperature, pH, DOC, and other); a>1 and b>0 are the biological constraints. For the selected known point (S,Yx/S), the coefficient b can be calculated as: (2.4)b=-ln(Yx/S-1-Yx/Smax-1)S.
Then model (2.2) has the following form:(2.5)dSdt=D(SF-S)-(a+exp(-bS))μmaxSKS+Sx,dxdt=μmaxSKS+Sx-Dx,S(0+)=S0,x(0+)=x0.
In particular, when substrate concentration (S) is high, Yx/S=1/a=Yx/Smax, and the biomass yield is constant. When the cell sensitivity to the substrate is very high (i.e., b>100), Yx/S is also practically constant for any substrate concentration different from zero. For example for Saccharomyces cerevisiae and glucose as the substrate, b≈200, what means that these microorganisms are very sensitive to glucose [30].According to the design ideas of the bioreactor, the biomass concentration should not exceed a critical level. When the biomass concentrationx(t) in the bioreactor reaches the set level xset (where 0<xset≤xcritical and xcritical is the critical level of biomass concentration in the bioreactor medium), then part of the medium containing biomass and substrate is discharged from the bioreactor, and the next portion of medium of a given substrate concentration is inputted impulsively. Therefore, system (2.2) can be modified as follows by introducing the impulsive state feedback control:(2.6)dSdt=D(SF-S)-(a+exp(-bS))μmaxSKS+Sx,dxdt=μmaxSKS+Sx-Dx,x<xset,ΔS=Wf1(SF-S)-Wf2S,Δx=-(Wf1+Wf2)x,x=xset,S(0+)=S0,x(0+)=x0,
where 0≤Wf1<1 is the part of substrate of a given concentration which is inputted into the bioreactor in each biomass oscillation cycle, and 0≤Wf2<1 is the part of “clear” medium which is inputted into the bioreactor in each biomass oscillation cycle. In addition, we make the following assumptions on D and Wf1: (1) D<μmax because all microorganisms are removed from the bioreactor when flow through the bioreactor is too fast, that is, when D≥μmax; (2) Wf1 is equal to D in value, that is, |Wf1|=|D|.We mainly discuss the dynamics, that is, existence of periodic solution of the model (2.6) in the region Ω={(S,x)∣S>0,x>0} according to the existence criteria of periodic solution of the general impulsive autonomous system in [16]. For convenience, we introduce a new notation W=Wf1+Wf2, which will be used in the following discussion.


## 3. The Existence of Positive Periodic Solution of System (2.6)

Before discussing the dynamics of system (2.6), we first consider the qualitative characteristic of system (2.5). Clearly, system (2.5) has a boundary equilibrium (SF,0) and a positive equilibrium (S⋆,x⋆) if KS<(μmax-D)SF/D where (3.1)S⋆=DKSμmax-D<SF,x⋆=(SF-S⋆)(a+exp(-bS⋆))-1>0.
The Jacobian of system (2.5) at (SF,0) is (3.2)J(SF,0)=[-D-(a+exp(-bSF))μmaxSFKS+SF0μmaxSFKS+SF-D],
then the equilibrium (SF,0) is stable if KS≥(μmax-D)SF/D. In this case, we can conclude that every solution of (2.5) tends to a stable equilibrium (SF,0) if KS≥(μmax-D)SF/D. Then microorganisms are not cultured successfully.For the caseKS<(μmax-D)SF/D, the equilibrium (SF,0) is a saddle point and S⋆>0, x⋆>0. Since (3.3)J(S⋆,x⋆)=[-D-μmaxΓ(S⋆)x⋆-D(SF-S⋆)x⋆x⋆μmaxKS(KS+S⋆)20],
where (3.4)Γ(S⋆)=D(SF-S⋆)μmaxS⋆x⋆KSKS+S⋆-bexp(-bS⋆)S⋆KS+S⋆.
The characteristic equation is (3.5)λ2+pλ+q=0,
where (3.6)p=D[(S⋆)2+KSSFS⋆(KS+S⋆)-bexp(-bS⋆)(SF-S⋆)(a+exp(-bS⋆))-1],q=μmaxKSD(SF-S⋆)(KS+S⋆)2>0.
Denote that(3.7)κ(b)=[b(SF-S⋆)S⋆(KS+S⋆)(S⋆)2+KSSF-1]exp(-bS⋆).
If a>κ(b), then p>0 and the equilibrium (S⋆,x⋆) is asymptotically stable; else if a<κ(b), then p<0 and (S⋆,x⋆) is unstable; else a=κ(b), then p=0 and (S⋆,x⋆) is a center. Since Ṡ<0 for S>SF, then any solution starting from the region {(S,x)∣S≥SF} will enter into the region {(S,x)∣S<SF} eventually, so in the following discussion we assume that S0=S(0)<SF.For the case where(S⋆,x⋆) is stable, all solutions of system (2.6) starting from the region {(S,x)∣0<S<SF,0<x<x⋆} will tend to the equilibrium (S⋆,x⋆) and no impulse will occur if xset>x⋆. So in this case, we mainly focus our attentions on the discussion of the following case.Assumption.
Consider thatS⋆>0, 0<xset<x⋆, and (S0,x0)∈Ω1={(S,x)∣0<x<xset,0<S<SF-(a+exp(-bSF))x}.For the case where(S⋆,x⋆) is unstable, it can be shown that there exists one limit cycle in Ω with an outer boundary OEFG of the Bendixson annular region (see Figure 2), where FG¯ is a segment on the line l1 passing the point (SF,0) with the slope -(a+e-bSF),(3.8)l1:x+Sa+exp(-bSF)-SFa+exp(-bSF)=0
for that the derivative of l1 along with system (2.5) is (3.9)dl1dt=dxdt|(2.5)+1a+exp(-bSF)dSdt|(2.5)=x(μmaxSKS+S-D)+[D(SF-S)-(a+exp(-bS))(μmaxS/(KS+S))x]a+exp(-bSF)=SF-Sa+exp(-bSF)μmaxSKS+S[1-a+exp(-bS)a+exp(-bSF)]<0.Figure 1
The biomass yield forYx/Smax=0.5 and the cell sensitivity to the substrate equals 1.5 (i.e., a=2, b=1.5).Figure 2
The case for(S⋆,x⋆) is unstable and system (2.5) has a limit cycle.All trajectories starting from the regionΩ tend to the limit cycle. In this case, we mainly focus our attentions on the discussion of the following case.Assumption.
Consider thatS⋆>0, x⋆>0 and (S0,x0)∈Ω2={(S,x)∣0<x<x⋆,0<S<SF-(a+exp(-bSF))x}.


### 3.1. Existence of Period-1 Solution forxset<x⋆

In order to apply the existence criteria of period-1 solution, that is, TheoremA.5, we need to construct a closed region ΩP such that all the solutions of system (2.6) enter into and retain it. The ideas will be illustrated as follows by using Figure 3.Illustration of system (2.6) (a) SC≥S⋆; (b) SC<S⋆.
(a)(b)As shown in Figure3, the line x=xset intersects the isoclinal line ẋ=0 at the point A(S⋆,xset) and intersects the line l1 at the point B(SB,xset). The line x=(1-W)xset intersects the line ẋ=0 at the point E(S⋆,(1-W)xset) and intersects the line l1 at point H(SH,(1-W)xset). Denote C=C(SC,(1-W)xset) and D=D(SD,(1-W)xset), where SC=(1-W)S⋆+DSF and SD=(1-W)SB+DSF. The impulsive set M⊆AB¯={(S,x)∣S⋆≤S≤SB,x=xset}, and N=I(M)⊆CD={(S,x)∣SC≤S≤SD,x=(1-W)xset}.Whether the equilibrium(S⋆,x⋆) is stable or not, the following theorem holds.Theorem 3.3.
System (2.6) has a period-1 solution under  Assumption 3.1.Proof.
Firstly, it can be easily shown that all trajectories of system (2.6) starting from the region Ω1 must intersect with the segment AB¯ and then jump to the segment CD¯. Next, we construct the closed region ΩP⊂Ω1 such that all the solutions of system (2.6) starting from Ω1 enter into and retain in ΩP. According to the values of SC and S⋆, we will discuss the following two cases.    Case 1.
Consider thatSC≥S⋆, that is, W≤(μmax-D)SF/KS;Case 2.
Consider thatSC<S⋆, that is, W>(μmax-D)SF/KS.
We first discuss Case1, where SC≥S⋆ and the illustration is shown in Figure 3(a).
By (3.8), we have SB=SF-(a+exp(-bSF))xset and SH=SF-(a+exp(-bSF))(1-W)xset. So we have(3.10)SD=(1-W)SB+DSF=(1-W)(SF-(a+exp(-bSF))xset)+DSF<SH.
From the qualitative characteristic of system (2.6), we know that (dl1/dt)∣(2.6)<0 and ẋ∣EH¯>0. Besides, Ṡ>0 for S=S⋆. Therefore, we have found a closed region ΩP, the boundary of which consists of AE¯,EH¯,HB¯, and BA¯.
For Case2, it follows from system (2.6) that x(t)=0 and S(t)=SF-(SF-S0)e-Dt for t∈(0,+∞). Especially, when S0=SC, FG⃗ is the semitrivial solution of system (2.6). Similar to the discussions of Case 1, we can obtain the closed region ΩP, the boundary of which consists of AE¯,EC¯,CF¯,FG¯,GB¯, and BA¯, which can be seen in Figure 3(b).
Summarizing Cases1 and 2 and following from Theorem A.5 that system (2.6) has a period-1 solution.


### 3.2. Existence of Period-1 Solution forxset≥x⋆

For the case where(S⋆,x⋆) is unstable, let Γ be the limit cycle, x=xupper be the line tangent with the trajectory starting from the initial point K(SK,x⋆) and x=xmax(xmax>x⋆), and x=xmin(xmin<x⋆) be the lines tangent with Γ, where SK=SF-(a+exp(-bSF))x⋆. Then we have the following theorem.Theorem 3.4.
System (2.6) has no period-1 solution under Assumption 3.2 and the trajectory tends to the limit cycle if a<κ(b) and xset≥xupper, where κ(b) is determined by (3.7).Proof.
It can be easily shown that all trajectories starting from the regionΩ2 will not intersect the line x=xset and tend to the limit cycle eventually.For the case wherexmax<xset<xupper, we have the following theorem.Theorem 3.5.
There exists0<x̲<xmin and xmax<x¯<xupper such that, for xmax≤xset≤x¯, system (2.6) has a period-1 solution for W>1-x̲/xset under Assumption 3.2 if a<κ(b), where κ(b) is determined by (3.7).Proof.
As stated earlier, ifa<κ(b), then (S⋆,x⋆) is unstable and there exists one limit cycle Γ. Since Γ does not intersect the line x=0, then there exists 0<x̲<xmin such that the trajectory l starting from the initial point E'(S⋆,x̲) intersects the line S=S⋆ at the point E′′(S⋆,x¯), where xmax<x¯<xupper. For xmax≤xset≤x¯, let A' be the point of the trajectory l intersecting with the line x=xset and C' be the point of A' after impulsive function I. If W>1-x̲/xset, the similar to the discussion of Theorem 3.3, we can obtain the closed region ΩP, the boundary of which consists of A'E'̂,E'E¯,EC'¯,C'F¯,FG¯,GB¯, and BA'¯ for SC'<S⋆, which can be seen in Figure 4, where A′E′̂ is the part of the trajectory between A' and E'. Then by Theorem A.5 system (2.6) has a period-1 solution. Therefore, the trajectory starting from the region Ω2 either tends to the limit cycle or the period-1 solution.Figure 4
Illustration of system (2.6) for the case where xmax<xset<xupper.For the case wherex⋆≤xset≤xmax, we have the following theorem.Theorem 3.6.
System (2.6) has a period-1 solution under Assumption 3.2 if a<κ(b), x⋆≤xset≤xmax and W>1-x̲/xset, where κ(b) is determined by (3.7).Proof.
LetE′ and E′′ be the points of the limit cycle intersecting with the line S=S⋆. Let A′′ and A' be the points of the limit cycle intersecting with the line x=xset and C′′ be the point of A′′ after impulsive function I. It can be easily shown that all trajectories starting from the segment C′′D¯ will intersect the segment A′B¯. Since the equilibrium (S⋆,x⋆) is unstable, then all trajectories starting from the region Ω2 will intersect the segment A′′B¯ and jump to the segment C′′D¯ after at most one impulse. Similar to the discussion of Theorem 3.5, we can obtain the closed region ΩP, the boundary of which consists of A′E′̂,E′E¯,EC′′¯,C′′F¯,FG¯,GB¯, and BA'¯ for SC′′<S⋆, which can be seen in Figure 5, where A′E′̂ is the part of the limit cycle between A' and E'. Then by Theorem A.5 system (2.6) has a period-1 solution. Therefore, all trajectories starting from the region Ω2 tend to the period-1 solution.Figure 5
Illustration of system (2.6) for the case where x⋆≤xset≤xmax.


### 3.3. Existence of Period-2 Solution

In the last subsection, we have analyzed the existence of period-1 solution of system (2.6). Next, we will discuss existence of the period-2 solution.Suppose that(Ŝ,x̂) is a period-1 solution of system (2.6). Then we have (Ŝ0,x̂0)∈N⊆CD¯ and (Ŝ1,x̂1)∈M⊆AB¯, where Ŝ0=(1-W)Ŝ1+DSF. Let (S,x) be an arbitrary solution of system (2.6). Denote the first intersection point of the trajectory to the impulsive set M by (S1,xset), and the corresponding consecutive points are (S2,xset), (S3,xset),…, respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are (S1+,(1-W)xset), (S2+,(1-W)xset), (S3+,(1-W)xset), …. By the qualitative analysis of system (2.6), we know that ẋ>0 for S>S⋆ and ẋ<0 for S<S⋆. We will consider the following two cases according to the values Ŝ0 and S⋆.Case 1 (Ŝ0>S⋆).
In this case, the period-1 solution lies in the region{(S,x)∣S⋆<S<SF,(1-W)xset≤x≤xset}. The trajectory of the solution (S,x) jumps to the point (S1+,(1-W)xset) from the point (S1,xset). Without loss of generality, we suppose that S1>Ŝ1. By the dynamics of system (2.6), we have the following two sequences according to (S2,xset) lying on the right or left of (S1,xset), as shown in Figure 6:(a)
Ŝ1≤S1≤S2≤S3≤⋯or(b)
S1≥S2≥S3≥⋯≥Ŝ1.Figure 6
The trajectory of the solution(S,x) for the case where Ŝ0>S⋆.It is known from the sequence (b) that the trajectory tends to the period-1 solution. Since the sequences are monotone, it follows by DefinitionA.4 that period-2 solution does not exist in this case.Case 2 (Ŝ0≤S⋆).
Similarly, we assume thatS1>Ŝ1. By the dynamics of system (2.6), we have the following property: (S2k,xset)(k=1,2,…) lie on the left of (Ŝ1,xset) and (S2k-1,xset)(k=1,2,…) lie on the right of (Ŝ1,xset). For example, if S1<(S⋆-DSF)/(1-W) and (S3,xset) lies on the left or right of (S1,xset), the we have the following sequence as shown in Figure 7: (c)
S2≤S4≤S6≤⋯≤Ŝ1≤⋯≤S5≤S3≤S1.Figure 7
The trajectory of the solution(S,x) under sequence (c).From the above sequence, we know that it is possible that there exists period-2 solution. For example, whenS1=S3, the period-2 solution will occur.From the proof of Proposition3.3 in [5], it can also be obtained that there is no order k solution (k≥3) in system (2.6) and then the system is not chaotic.


## 3.1. Existence of Period-1 Solution forxset<x⋆

In order to apply the existence criteria of period-1 solution, that is, TheoremA.5, we need to construct a closed region ΩP such that all the solutions of system (2.6) enter into and retain it. The ideas will be illustrated as follows by using Figure 3.Illustration of system (2.6) (a) SC≥S⋆; (b) SC<S⋆.
(a)(b)As shown in Figure3, the line x=xset intersects the isoclinal line ẋ=0 at the point A(S⋆,xset) and intersects the line l1 at the point B(SB,xset). The line x=(1-W)xset intersects the line ẋ=0 at the point E(S⋆,(1-W)xset) and intersects the line l1 at point H(SH,(1-W)xset). Denote C=C(SC,(1-W)xset) and D=D(SD,(1-W)xset), where SC=(1-W)S⋆+DSF and SD=(1-W)SB+DSF. The impulsive set M⊆AB¯={(S,x)∣S⋆≤S≤SB,x=xset}, and N=I(M)⊆CD={(S,x)∣SC≤S≤SD,x=(1-W)xset}.Whether the equilibrium(S⋆,x⋆) is stable or not, the following theorem holds.Theorem 3.3.
System (2.6) has a period-1 solution under  Assumption 3.1.Proof.
Firstly, it can be easily shown that all trajectories of system (2.6) starting from the region Ω1 must intersect with the segment AB¯ and then jump to the segment CD¯. Next, we construct the closed region ΩP⊂Ω1 such that all the solutions of system (2.6) starting from Ω1 enter into and retain in ΩP. According to the values of SC and S⋆, we will discuss the following two cases.    Case 1.
Consider thatSC≥S⋆, that is, W≤(μmax-D)SF/KS;Case 2.
Consider thatSC<S⋆, that is, W>(μmax-D)SF/KS.
We first discuss Case1, where SC≥S⋆ and the illustration is shown in Figure 3(a).
By (3.8), we have SB=SF-(a+exp(-bSF))xset and SH=SF-(a+exp(-bSF))(1-W)xset. So we have(3.10)SD=(1-W)SB+DSF=(1-W)(SF-(a+exp(-bSF))xset)+DSF<SH.
From the qualitative characteristic of system (2.6), we know that (dl1/dt)∣(2.6)<0 and ẋ∣EH¯>0. Besides, Ṡ>0 for S=S⋆. Therefore, we have found a closed region ΩP, the boundary of which consists of AE¯,EH¯,HB¯, and BA¯.
For Case2, it follows from system (2.6) that x(t)=0 and S(t)=SF-(SF-S0)e-Dt for t∈(0,+∞). Especially, when S0=SC, FG⃗ is the semitrivial solution of system (2.6). Similar to the discussions of Case 1, we can obtain the closed region ΩP, the boundary of which consists of AE¯,EC¯,CF¯,FG¯,GB¯, and BA¯, which can be seen in Figure 3(b).
Summarizing Cases1 and 2 and following from Theorem A.5 that system (2.6) has a period-1 solution.


## 3.2. Existence of Period-1 Solution forxset≥x⋆

For the case where(S⋆,x⋆) is unstable, let Γ be the limit cycle, x=xupper be the line tangent with the trajectory starting from the initial point K(SK,x⋆) and x=xmax(xmax>x⋆), and x=xmin(xmin<x⋆) be the lines tangent with Γ, where SK=SF-(a+exp(-bSF))x⋆. Then we have the following theorem.Theorem 3.4.
System (2.6) has no period-1 solution under Assumption 3.2 and the trajectory tends to the limit cycle if a<κ(b) and xset≥xupper, where κ(b) is determined by (3.7).Proof.
It can be easily shown that all trajectories starting from the regionΩ2 will not intersect the line x=xset and tend to the limit cycle eventually.For the case wherexmax<xset<xupper, we have the following theorem.Theorem 3.5.
There exists0<x̲<xmin and xmax<x¯<xupper such that, for xmax≤xset≤x¯, system (2.6) has a period-1 solution for W>1-x̲/xset under Assumption 3.2 if a<κ(b), where κ(b) is determined by (3.7).Proof.
As stated earlier, ifa<κ(b), then (S⋆,x⋆) is unstable and there exists one limit cycle Γ. Since Γ does not intersect the line x=0, then there exists 0<x̲<xmin such that the trajectory l starting from the initial point E'(S⋆,x̲) intersects the line S=S⋆ at the point E′′(S⋆,x¯), where xmax<x¯<xupper. For xmax≤xset≤x¯, let A' be the point of the trajectory l intersecting with the line x=xset and C' be the point of A' after impulsive function I. If W>1-x̲/xset, the similar to the discussion of Theorem 3.3, we can obtain the closed region ΩP, the boundary of which consists of A'E'̂,E'E¯,EC'¯,C'F¯,FG¯,GB¯, and BA'¯ for SC'<S⋆, which can be seen in Figure 4, where A′E′̂ is the part of the trajectory between A' and E'. Then by Theorem A.5 system (2.6) has a period-1 solution. Therefore, the trajectory starting from the region Ω2 either tends to the limit cycle or the period-1 solution.Figure 4
Illustration of system (2.6) for the case where xmax<xset<xupper.For the case wherex⋆≤xset≤xmax, we have the following theorem.Theorem 3.6.
System (2.6) has a period-1 solution under Assumption 3.2 if a<κ(b), x⋆≤xset≤xmax and W>1-x̲/xset, where κ(b) is determined by (3.7).Proof.
LetE′ and E′′ be the points of the limit cycle intersecting with the line S=S⋆. Let A′′ and A' be the points of the limit cycle intersecting with the line x=xset and C′′ be the point of A′′ after impulsive function I. It can be easily shown that all trajectories starting from the segment C′′D¯ will intersect the segment A′B¯. Since the equilibrium (S⋆,x⋆) is unstable, then all trajectories starting from the region Ω2 will intersect the segment A′′B¯ and jump to the segment C′′D¯ after at most one impulse. Similar to the discussion of Theorem 3.5, we can obtain the closed region ΩP, the boundary of which consists of A′E′̂,E′E¯,EC′′¯,C′′F¯,FG¯,GB¯, and BA'¯ for SC′′<S⋆, which can be seen in Figure 5, where A′E′̂ is the part of the limit cycle between A' and E'. Then by Theorem A.5 system (2.6) has a period-1 solution. Therefore, all trajectories starting from the region Ω2 tend to the period-1 solution.Figure 5
Illustration of system (2.6) for the case where x⋆≤xset≤xmax.


## 3.3. Existence of Period-2 Solution

In the last subsection, we have analyzed the existence of period-1 solution of system (2.6). Next, we will discuss existence of the period-2 solution.Suppose that(Ŝ,x̂) is a period-1 solution of system (2.6). Then we have (Ŝ0,x̂0)∈N⊆CD¯ and (Ŝ1,x̂1)∈M⊆AB¯, where Ŝ0=(1-W)Ŝ1+DSF. Let (S,x) be an arbitrary solution of system (2.6). Denote the first intersection point of the trajectory to the impulsive set M by (S1,xset), and the corresponding consecutive points are (S2,xset), (S3,xset),…, respectively. Consequently, under the effect of impulsive function I, the corresponding points after pulse are (S1+,(1-W)xset), (S2+,(1-W)xset), (S3+,(1-W)xset), …. By the qualitative analysis of system (2.6), we know that ẋ>0 for S>S⋆ and ẋ<0 for S<S⋆. We will consider the following two cases according to the values Ŝ0 and S⋆.Case 1 (Ŝ0>S⋆).
In this case, the period-1 solution lies in the region{(S,x)∣S⋆<S<SF,(1-W)xset≤x≤xset}. The trajectory of the solution (S,x) jumps to the point (S1+,(1-W)xset) from the point (S1,xset). Without loss of generality, we suppose that S1>Ŝ1. By the dynamics of system (2.6), we have the following two sequences according to (S2,xset) lying on the right or left of (S1,xset), as shown in Figure 6:(a)
Ŝ1≤S1≤S2≤S3≤⋯or(b)
S1≥S2≥S3≥⋯≥Ŝ1.Figure 6
The trajectory of the solution(S,x) for the case where Ŝ0>S⋆.It is known from the sequence (b) that the trajectory tends to the period-1 solution. Since the sequences are monotone, it follows by DefinitionA.4 that period-2 solution does not exist in this case.Case 2 (Ŝ0≤S⋆).
Similarly, we assume thatS1>Ŝ1. By the dynamics of system (2.6), we have the following property: (S2k,xset)(k=1,2,…) lie on the left of (Ŝ1,xset) and (S2k-1,xset)(k=1,2,…) lie on the right of (Ŝ1,xset). For example, if S1<(S⋆-DSF)/(1-W) and (S3,xset) lies on the left or right of (S1,xset), the we have the following sequence as shown in Figure 7: (c)
S2≤S4≤S6≤⋯≤Ŝ1≤⋯≤S5≤S3≤S1.Figure 7
The trajectory of the solution(S,x) under sequence (c).From the above sequence, we know that it is possible that there exists period-2 solution. For example, whenS1=S3, the period-2 solution will occur.From the proof of Proposition3.3 in [5], it can also be obtained that there is no order k solution (k≥3) in system (2.6) and then the system is not chaotic.


## 4. Simulations and Discussion

We have analyzed theoretically the feedback control for microorganisms of any biomass yield and Monod-type kinetics. The results are new and significant, which not only provide the possibility of a check of system dynamic property, that is, the existence of period-1 solution for different microorganisms and several parameters, but also provide a possibility of making simulation of real process according to the mathematical models determined in the article. To verify the received results, the numerical simulations of system (2.6) are shown. Because of the large practical importance, the case where (S⋆,x⋆) is stable is presented and discussed. Moreover, galactose as the substrate and the microorganisms with a=2 (i.e., Yx/Smax=1/2=0.5[g/g]), b=1.5, μmax=0.3[1/h], and KS=2[g/l] are used for the demonstration of system behavior. In order to ensure the existence of positive equilibrium, we set SF=6[g/l], and D=0.1[1/h]. Then we have (μmax-D)SF/KS=0.6, S⋆=1[g/l] and κ(b)≃0.16. Because κ(b)<a, (S⋆,x⋆)=(1,2.25) is a stable focus. Next, we check and show the influence of xset and Wf2 changes on the existence of period-1 solution.Firstly, we setxset=2[g/l]. The time series and phase diagram for system (2.6) with Wf2=0.1 is presented in Figure 8, from which we can see that the trajectory tends to be periodic.The time series and phase diagram for system (2.6) starting from initial point (S0,x0)=(1.2,1.2) with a=2, b=1.5, xset=2[g/l], and Wf2=0.1.
(a)(b)(c)Secondly, we change the value ofWf2. The phase diagram for system (2.6) with Wf2=0.2,0.4,0.6 is displayed in Figure 9. From Figure 9, we can see that the existence of period-1 solution does not depend on the value of Wf in this case, but the value of Wf2 affects the position and tendency of the period-1 solution.The phase diagram for system (2.6) starting from initial point (S0,x0)=(1.2,1.2) with a=2, b=1.5, xset=2[g/l], and Wf2=0.2,0.4,0.6.
(a)(b)(c)Thirdly, we setxset=2.5[g/l]. The time series and phase diagram is presented in Figure 10. It can be easily seen from Figure 10 that no impulse occurs when xset>x⋆ and the trajectory tends to the stable node (1.5,1.2).The time series and phase diagram for system (2.6) starting from initial point (S0,x0)=(1.2,1.2) with a=2, b=1.5, and xset=2.5[g/l].
(a)(b)(c)Therefore, the numerical simulations are consistent with the theorems obtained and presented in Section3. A potential application area of the chemostat with feedback control is the commercial and industrial biomass production. In such a chemostat, the microorganisms always keep the suitable growth rate and the biomass concentration should be controlled to a given set level for which the dissolved oxygen concentration is considered as optimal. Therefore, in order to eliminate the negative effects such as the insufficiency of dissolved oxygen and decrease of the inhibition effect, one can use the chemostat models with impulsive state feedback control.


## 5. Conclusions

The article established the mathematical model of a chemostat with variable biomass yield and feedback control in maintaining the biomass concentration in a desired range. The flexible sigmoid function was proposed in describing the dependence of the biomass yield on the substrate concentration. It was shown that the stability of the bioprocess (i.e., the existence of the positive period-1 solution) depends on the biomass yield and the microorganisms growth rate. It was also shown that for the system there may exist a period-2 solution, but not having period-k (k≥3), then it is not chaos. The existence of period-1 solution indicates that, in the biomass production, a stable output of the biomass can be achieved. The key to the production is to given a suitable feedback state (i.e., xset) and the control parameter (i.e., D and Wf2) according to the practice. In addition, the appropriate initial biomass and substrate concentration should also be considered such that the culture achieves the periodic state as soon as possible. The results also provide a possibility of making simulation of real process according to the mathematical models and the parameters determined in this paper.



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