Impulsive State Feedback Control of Cheese Whey Fermentation for Single-Cell Protein Production

Abstract

The work is the analysis of a mathematical model of cheese whey fermentation for single-cell protein production with impulsive state feedback control. Through the analysis, the sufficient conditions of existence and stability of positive order-1 periodic solution are obtained. It is shown that the system either tends to a stable state or has a periodic solution, which depends on the feedback state, the control parameter of the dilution rate, and the initial concentrate of microorganism and substrate. For some special cases, it is also shown that the system may exist in order-2 periodic solution. Furthermore, our findings are confirmed by means of numerical simulations.

1. Introduction

Single-cell protein (SCP) is the protein extracted from cultivated microbial biomass. It can be used for protein supplementation of a staple diet by replacing costly conventional sources like soymeal and fishmeal to alleviate the problem of protein scarcity. Increasing demand for protein sources of high nutrition value have stimulated the application of single-cell proteins, yeast, bacteria, and algae in human foods or animal feeds [1]. The interest in agricultural and industrial wastes as substrates for the production of single-cell proteins has increased recently (see [2–5] and the references therein). For example, Kurbanoglu and Algur [6] studied single-cell protein production from ram horm hydrolysate by bacteria, and El-Saadany et al. [7] studied the production of single cell protein from agricultural wastes by fungi.

The production of microbial biomass is done mainly by a continuous fermentation system [8]. According to different reactions and differential control technologies, many dynamic models concerning the culture of microorganisms in the continuous bioreactor have been established [9–13]. However, there are a lot of factors affecting the growth and reproduction of the microorganisms in the process of bioreacts [14]. For example, for some aerobic microorganisms, the dissolved oxygen (DO) is a key factor to the growth of the aerobic microorganisms. In order to maintain the dissolved oxygen concentration in an appropriate range, it is necessary to regulate the microorganisms’ concentration to not exceed a set level [15].

As shown in Figure 1, if the microorganisms’ concentration reaches the preset value, the valve is opened by the photoelectric control devices, and a certain amount of regulator solution is input. Then, the microorganisms’ concentration is decreased and less than the preset value. At this time, the valve is closed by the control system until the next moment at which the microorganisms’ concentration reaches the preset value once again. Therefore, the microorganism concentration can be decreased through the control methods such as replenishing water or other regulator solution to reduce the concentration of the microorganism.

Figure 1 

Schematic diagram of microorganism culture system with feedback control.

In recent years, in order to obtain the optimal conditions for microorganism growth and decrease the inhibition of the microorganism concentration or other negative effects, many authors introduced the state-dependent impulsion into the microorganism culture process and analyzed the system's dynamic behavior [16–18]. However, so far, no papers have discussed the mathematical model of cheese whey fermentation for single-cell protein production with impulsive feedback control.

This paper aims to propose a mathematical model of cheese whey fermentation for single-cell protein production with feedback control. We will discuss the existence and stability of periodic solution of the system with impulsive feedback control by using the existence criteria [19] and the stability theorem [20]. The paper is organized as follows. The mathematical model and some preliminary results are introduced in Section 2. In Section 3, the qualitative analysis of the system without impulsive effect is given. In Section 4, the existence and stability of order-1 periodic solution of the system with impulsive state feedback control are investigated. Numerical simulations and some discussion are provided in Section 5.

2. Model Formulation

Ghaly et al. [8] proposed the following mathematical model of cheese whey fermentation for the production of single-cell protein: 𝑆^′(𝑡)=−𝜇𝑆𝑥𝛿₁𝐾_𝑠𝑆+𝑆−𝑚𝑥+𝐷_𝑖,𝑥−𝑆_′(𝑡)=𝜇𝑆𝑥𝐾_𝑠−𝐾+𝑆^𝑑+𝐷𝑥, (1) where 𝑥 denotes the concentration of the microorganism (g/L) and 𝑆 denotes the concentration of the limiting substrate (g/L) in the bioreactor at time 𝑡. 𝑆_𝑖 denotes the influent substrate concentration (g/L). 𝐷 is the dilution rate; 𝜇 is the maximum special growth rate; 𝐾_𝑠 is the saturation constant, equal to the substrate concentration at one-half the maximum special growth rate (g/L); 𝐾_𝑑 is the endogenous microbial decay coefficient (ℎ⁻¹); 𝛿₁ is the yield coefficient; 𝑚 is the maintenance coefficient; and 𝑚𝑥 denotes the substrate uptake rate for the microorganism maintenance.

In some cases, if the concentration of microorganism is lower than a critical value (a preset value given by experiments), it is not necessary to take the control measures. But if the concentration reaches the critical value, some negative effects such as airborne inhibition and production inhibition and so on may happen, and the control methods such as liquefying medium and replenishing water should be taken to decrease the concentration of the microorganism. Once the concentration is decreased, it also should take some time to reach the critical value. Therefore, the system (1) can be modified as follows by introducing the impulsive state feedback control: 𝑆^′(𝑡)=−𝜇𝑆𝑥𝛿₁𝐾_𝑠𝑆+𝑆−𝑚𝑥+𝐷_𝑖𝑥−𝑆_′(𝑡)=𝜇𝑆𝐾_𝑠𝐾+𝑆𝑥−_𝑑(𝑡)(𝑡)𝑆(0)+𝐷𝑥,𝑥<ℎ,Δ𝑆=−𝑏𝑆,Δ𝑥=−𝑏𝑥,𝑥=ℎ,=𝑆_𝑖,𝑥(0)=𝑥₀<ℎ, (2) where Δ𝑥=𝑥(𝑡⁺)−𝑥(𝑡), Δ𝑆=𝑆(𝑡⁺)−𝑆(𝑡), 0<𝑏<1, is the fraction of the concentration of microorganism and substrate that decreases due to the feedback control when the microorganism concentrate 𝑥 reaches the critical value ℎ.

In the following, we mainly discuss the existence of periodic solution of (2) by the existence criteria of the general impulsive autonomous system. Before starting our main results, we give some topological definitions about impulsive differential equations.

Definition 1 (Laksmikantham et al. [21]). A triple (𝑋,𝜋,𝑅₊) is said to be a semidynamical system if 𝑋 is metric space, 𝑅₊ is the set of all nonnegative reals, and 𝜋∶𝑋×𝑅₊→𝑋 is a continuous function such that(i)𝜋(𝑥,0)=𝑥 for all 𝑥∈𝑋;(ii)𝜋(𝜋(𝑥,𝑡),𝑠)=𝜋(𝑥,𝑡+𝑠) for all 𝑥∈𝑋 and 𝑡,𝑠∈𝑅₊.

We denote sometimes a semidynamical system (𝑋,𝜋,𝑅₊) by (𝑋,𝜋).

For any 𝑥∈𝑋, the function 𝜋_𝑥∶𝑅₊→𝑋 defined by 𝜋_𝑥(𝑡)=𝜋(𝑥,𝑡) is continuous, and we call 𝜋_𝑥 the trajectory of 𝑥. The set 𝐶⁺(𝑥)=𝜋(𝑥,𝑡)∣𝑡∈𝑅₊ (3) is called the positive orbit of 𝑥. For any subset 𝑀 of 𝑋, we let 𝑀⁺(𝑥)=𝐶⁺(𝑥)∩𝑀−𝑥,𝑀⁻=𝐺(𝑥)∩𝑀−𝑥, (4) where 𝐺(𝑥)=∪𝐺(𝑥,𝑡)∣𝑡∈𝑅₊,𝐺(𝑥,𝑡)=𝑦∣𝜋𝑦,𝑡=𝑥 (5) is the attainable set of 𝑥 at 𝑡∈𝑅₊. Finally, we set 𝑀(𝑥)=𝑀⁺(𝑥)∪𝑀⁻(𝑥).

Definition 2 (Laksmikantham et al. [21]). An impulsive semidynamical system (𝑋,𝜋;𝑀,𝐼) consists of a semidynamical system (𝑋,𝜋) together with a nonempty closed subset 𝑀 of 𝑋 and a continuous function 𝐼∶𝑀→𝑋 such that the following properties hold:(i)no point 𝑥∈𝑋 is a limit point of 𝑀(𝑥),(ii)[𝑡∣𝐺(𝑥,𝑡)∩𝑀≠∅] is a closed subset of 𝑅₊.

We write 𝑁=𝐼(𝑀)={𝑦∈𝑋∣𝑦=𝐼(𝑥),𝑥∈𝑀} and for any 𝑥∈𝑋, 𝐼(𝑥)=𝑥⁺. We call 𝑀 the impulsive set and 𝐼 the impulsive function.

We define a function Φ∶𝑋→𝑅₊∪{∞} as follows: Φ(𝑥)=∞if𝑀⁺(𝑥)()()=𝜙,𝑠if𝜋𝑥,𝑡∉𝑀for0<𝑡<𝑠,𝜋𝑥,𝑠∈𝑀; (6) here, we call 𝑠 the time without impulse of 𝑥, that is 𝑠 is the first time when 𝜋(𝑥,0) hits 𝑀.

Definition 3 (Laksmikantham et al. [21]). Let (𝑋,𝜋;𝑀,𝐼) be an impulsive semidynamical system, and let 𝑥∈𝑋 and 𝑥∉𝑀. The trajectory of 𝑥 is a function 𝜋_𝑥 defined on subset [0,𝑠) of 𝑅₊ (𝑠 may be ∞) to 𝑋 inductively as follows: 𝜋_𝑥(𝑡)=𝜋𝑥⁺_𝑛+1,𝑡,𝜏_𝑛−1≤𝑡≤𝜏_𝑛, (7) where {𝑥_𝑛} is the sequence of impulse points of 𝑥, which satisfied 𝜋(𝑥⁺_𝑛−1,Φ(𝑥⁺_𝑛−1))=𝑥_𝑛. 𝜏_𝑛 is the sequence of time of impulses relative to {𝑥_𝑛}, 𝜏_𝑛=∑^𝑛−1_𝑘=0Φ(𝑥⁺_𝑘).

Definition 4 (Laksmikantham et al. [21]). A trajectory 𝜋_𝑥 is said to be periodic of period 𝜏 and order 𝑘 if there exist positive integers 𝑚≥1 and 𝑘≥1 such that 𝑘 is the smallest integer for which 𝑥⁺_𝑚=𝑥⁺_𝑚+𝑘 and 𝜏=∑^{𝑚+𝑘−1}_𝑖=𝑚Φ(𝑥⁺_𝑖).

Theorem 5 (Brouwer's fixed-point theorem (Griffel [22])). Every continuous mapping of a closed bounded convex set in 𝑅^𝑛 into itself has a fixed point.

Zeng et al. [19] have proved the existence criteria of periodic solution of an impulsive autonomous system by means of Browers’ fixed-point theorem. For convenience of reading, we repeat the main results as follows.

Consider the following general autonomous impulsive differential equations: 𝑑𝑥(𝑡)(𝑡)𝑑𝑡=𝑃𝑥,𝑦𝑑𝑦𝑑𝑡=𝑄𝑥,𝑦𝑥,𝑦∉𝑀,Δ𝑥=𝐼₁𝑥,𝑦Δ𝑦=𝐼₂𝑥,𝑦𝑥,𝑦∈𝑀. (8) Here, (𝑥,𝑦)∈𝑅², 𝑃, 𝑄, 𝐼₁, and 𝐼₂ are all functions mapping 𝑅² into 𝑅, and 𝑀⊂𝑅² is the set of impulse, and we assume(H1)𝑃(𝑥,𝑦), 𝑄(𝑥,𝑦) are all continuous with respect to 𝑥, 𝑦 in 𝑅²;(H2)𝑀⊂𝑅² is a line, and 𝐼₁(𝑥,𝑦) and 𝐼₂(𝑥,𝑦) are linear functions of 𝑥 and 𝑦. For each point, 𝑆(𝑥,𝑦)∈𝑀, and we define that 𝐼∶𝑅²→𝑅²: 𝐼(𝑆)=𝑆⁺=𝑥⁺,𝑦⁺∈𝑅²,𝑥⁺=𝑥+𝐼₁𝑥,𝑦,𝑦⁺=𝑦+𝐼₂𝑥,𝑦. (9)

Obviously, 𝑁=𝐼(𝑀) is also a line of 𝑅² or a subset of a line, and we assume that 𝑁∩𝑀=𝜙. From Definition 2, we know that (8) is an impulsive semidynamical system. The following theorem gives the conditions on which (8) has an order-1 periodic solution defined by Definition 4.

Theorem 6 (see [19]). If system (8) satisfies assumptions (H1) and (H2), there exists a bounded closed simply connected region 𝐷 which has the following properties:(i)there is no singularity in it, and the boundary 𝜕𝐷 of 𝐷 satisfies (𝐷−𝜕𝐷)∩𝑀=∅;(ii)𝐿₁=𝐷∩𝑀 cannot be tangent with trajectories of (8) except at end points and 𝐼(𝐿₁)⊂𝐷;(iii)trajectories with initial point in 𝜕𝐷−𝐿₁ will enter into interior of 𝐷; then, there must exist an order-1 periodic solution of system (8) in region 𝐷.

3. Qualitative Analysis of System (1)

Before discussing the periodic solution of system (2), we should consider the qualitative characteristic of system (2) without impulsive effect, that is, consider the qualitative characteristic of system (1).

Lemma 7. The system (1) is uniformly bounded.

Proof. From the first equation of system (1), we have 𝑆^′(𝑡)=−𝜇𝑆𝑥𝐾_𝑠+𝑆𝛿₁−𝑚𝑥+𝐷𝑆_𝑖−𝑆. (10) The isocline 𝑆^′(𝑡)=0, that is, 𝑥=𝐷(𝑆_𝑖−𝑆)(𝐾_𝑠+𝑆)𝛿₁/((𝜇+𝑚𝛿₁)𝑆+𝑚𝐾_𝑠𝛿₁), which intersects the 𝑆-axis at points (−𝐾_𝑠,0) and (𝑆_𝑖,0), and 𝑥<𝐷(𝑆_𝑖−𝑆)(𝐾_𝑠+𝑆)𝛿₁/((𝜇+𝑚𝛿₁)𝑆+𝑚𝐾_𝑠𝛿₁), 𝑆^′(𝑡)>0. Furthermore, 𝑥→+∞, when 𝑆→−(𝑚𝛿₁𝐾_𝑠/(𝜇+𝑚𝛿₁)) from the right; therefore, there exists 𝑆=𝛿 and −𝐾_𝑠<𝛿<−(𝑚𝛿₁𝐾_𝑠/(𝜇+𝑚𝛿₁)) such that 𝑆^′(𝑡)>0. Let the straight line 𝑆=𝑆_𝑖, and 𝑆=𝛿 intersect 𝑥=0 at the points 𝐴(𝑆_𝑖,0), 𝐷(𝛿,0), respectively. Define a function 𝑉(𝑆,𝑥)=𝑆+(1/𝛿₁)𝑥−𝐾, where 𝐾>𝑆_𝑖. The function 𝑉(𝑆,𝑥) intersects the line 𝑆=𝑆_𝑖 and 𝑆=𝛿 at the points 𝐵 and 𝐶, respectively. Then, we have (𝑑𝑉/𝑑𝑡)|_𝐵𝐶=−(𝐷𝑥(𝑡)/𝛿₁)−(𝐾_𝑑𝑥(𝑡)/𝛿₁)−𝑚𝑥(𝑡)+𝐷(𝑆_𝑖−𝑆(𝑡)) = −(𝐾_𝑑𝑥(𝑡)/𝛿₁)−𝑚𝑥(𝑡)+𝐷(𝑆_𝑖−𝐾)<0, (𝑑𝑆/𝑑𝑡)|_𝐷𝐶>0, (𝑑𝑆/𝑑𝑡)|_𝐴𝐵=−(𝜇𝑆_𝑖/(𝐾_𝑠+𝑆_𝑖)𝛿₁)𝑥(𝑡)−𝑚𝑥(𝑡)<0. Hence, the system (1) is uniformly bounded. The proof is completed. We denote that the region Ω is formed by four points 𝐴, 𝐵, 𝐶, and 𝐷.

The equilibrium points of system (1) satisfies the following equation. −𝜇𝑆𝐾_𝑠+𝑆𝛿₁𝑥−𝑚𝑥+𝐷𝑆_𝑖−𝑆=0,𝜇𝑆𝐾_𝑠𝐾+𝑆𝑥−_𝑑+𝐷𝑥=0. (11) It can be seen that system (11) has a boundary equilibrium 𝑄(𝑆_𝑖,0) and a positive equilibrium 𝐻(𝑆^∗,𝑥^∗) if 𝑆_𝑖−(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷))>0, where 𝑆^∗=(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷)), 𝑥^∗=𝐷𝛿₁(𝑆_𝑖−𝑆^∗)/(𝐾_𝑑+𝐷+𝑚𝛿₁).

In the following, we will analyze the stability of equilibrium 𝑄(𝑆_𝑖,0) and 𝐻(𝑆^∗,𝑥^∗) of system (1). If 𝑆_𝑖−(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷))<0, the positive equilibrium point does not exist, and the boundary equilibrium is stable. That is, we have the following result.

Theorem 8. The equilibrium 𝑄(𝑆_𝑖,0) is locally asymptotically stable if 𝑅<1, where 𝑅=𝑆_𝑖(𝜇−(𝐾_𝑑+𝐷))/(𝐾_𝑑+𝐷)𝐾_𝑠. Otherwise, it is unstable.

Proof. The Jacobian matrix 𝐽_𝑄=𝐽(𝑆_𝑖,0) of system (1) at 𝐴 takes the form of −𝐷−𝜇𝑆_𝑖𝐾_𝑠+𝑆_𝑖𝛿₁0−𝑚𝜇𝑆_𝑖𝐾_𝑠+𝑆_𝑖−𝐾_𝑑+𝐷. (12)
It is easy to obtain that the eigenvalues for the matrix 𝐽_𝑄=𝐽(𝑆_𝑖,0) are 𝜆₁=−𝐷<0, 𝜆₂=(𝜇𝑆_𝑖/(𝐾_𝑠+𝑆_𝑖))−(𝐾_𝑑+𝐷). For 𝑅<1, then (𝜇𝑆_𝑖/(𝐾_𝑠+𝑆_𝑖))−(𝐾_𝑑+𝐷)<0, that is, 𝜆₂<0; hence, 𝑄(𝑆_𝑖,0) is locally asymptotically stable. Otherwise, 𝐴(𝑆_𝑖,0) is unstable. This completes the proof.

Theorem 9. The equilibrium point 𝑄(𝑆_𝑖,0) is globally asymptotically stable if 𝑅<1, where 𝑅=𝑆_𝑖(𝜇−(𝐾_𝑑+𝐷))/(𝐾_𝑑+𝐷)𝐾_𝑠.

Proof. Consider the Lyapunov function 𝑉(𝑡)=𝑥(𝑡); we can obtain the result immediately.

Theorem 10. If 𝑅>1, then the positive equilibrium point 𝐻 is locally asymptotically stable.

Proof. The Jacobian matrix 𝐽_𝐻=𝐽(𝑆^∗,𝑥^∗) of system (1) at 𝐻 takes the form of 𝐽_𝐻=−𝜇𝐾_𝑠𝑥^∗𝛿₁𝐾_𝑠+𝑆^∗2𝐾−𝐷−^𝑑+𝐷𝛿₁−𝑚𝜇𝐾_𝑠𝑥^∗𝐾_𝑠+𝑆^∗20. (13) The eigenvalue problem for the 𝐽_𝐻=𝐽(𝑆^∗,𝑥^∗) provides the following characteristic equation: 𝜆²+𝑄₁𝜆+𝑄₂=0, (14) where the coefficients 𝑄₁, 𝑄₂ are 𝑄₁=𝜇𝐾_𝑠𝑥^∗𝛿₁𝐾_𝑠+𝑆^∗2𝑄+𝐷,₂=𝜇𝐾_𝑠𝑥^∗𝐾_𝑠+𝑆^∗2𝐾_𝑑+𝐷𝛿₁+𝑚. (15) Note that 𝑄₁>0; 𝑄₂>0, then, we have that 𝐻 is locally asymptotically stable.

In the following, we will discuss the global stability of system (1). Firstly, we will give the following lemma.

Lemma 11. Suppose that Γ(𝑇)=(𝑆(𝑡),𝑥(𝑡)) is a periodic orbit with 𝑇 of system (1), and ℜ is the set which consists of all the points in Phase plane Γ.
Denote 𝑁=^𝑇₀𝜕𝑓₁(𝑆(𝑡)(𝑡+𝜕𝑆,𝑥))𝜕𝑓₂(𝑆(𝑡)(𝑡𝜕𝑥,𝑥))𝑑𝑡, (16) where 𝑆^′(𝑡)=𝑓₁(𝑆(𝑡),𝑥(𝑡)), 𝑥^′(𝑡)=𝑓₂(𝑆(𝑡),𝑥(𝑡)); then, we can obtain 𝑁<0.

Proof. 𝑁=^𝑇₀−𝜇𝐾_𝑠𝑥𝛿₁𝐾_𝑠+𝑆²−2𝐷−𝐾_𝑑+𝜇𝑆𝐾_𝑠=+𝑆𝑑𝑡_𝑇0−𝜇𝐾_𝑠𝑥𝛿₁𝐾_𝑠+𝑆²−𝐷+𝜇𝑆𝐾_𝑠+𝑆−𝐷−𝐾_𝑑𝑑𝑡, (17) for 𝑥(𝑡) is a period function with 𝑇, so ∫^𝑇₀((𝜇𝑆/𝐾_𝑠+𝑆)−𝐷−𝐾_𝑑)𝑑𝑡=∫^𝑇₀𝑑ln𝑥(𝑡)=0; hence, 𝑁=^𝑇₀−𝜇𝐾_𝑠𝑥𝛿₁𝐾_𝑠+𝑆²−𝐷𝑑𝑡. (18) It is evident that 𝑁<0; the proof is completed.

Theorem 12. If 𝑅>1, the positive equilibrium 𝐻 of system (1) is globally asymptotically stable.

Proof. From Theorem 10, we know that 𝐻 is locally stable. According to Lemma 11, we can obtain if there exists periodic solution (𝑆(𝑡),𝑥(𝑡)) around 𝐻(𝑆^∗,𝑥^∗); then, it is stable for any periodic solution. This is impossible. According to the Poincare-Bendixson theorem, limit set 𝜔 of all orbits must be equilibrium point 𝐻. This implies that 𝐻(𝑆^∗,𝑥^∗) is globally asymptotically stable in Ω. This completes the proof.

4. Existence and Stability of Periodic Solution of System (2)

4.1. Existence of Order-1 Periodic Solution

From discussions of the qualitative characteristic of system (2) without the impulsive effects, we can see that if 𝑅>1, 𝐻(𝑆^∗,𝑥^∗) is globally stable. For the initial points which satisfy 𝑥(0)<𝑥^∗ and (𝑑𝑆/𝑑𝑡)|_{(𝑆0,𝑥0)}≥0, if ℎ>𝑥^∗, then all the solutions of system (2) will tend to the positive equilibrium (𝑆^∗,𝑥^∗) after at most finite times impulsive effects. Therefore, we mainly focus our attention on the cases ℎ<𝑥^∗, 𝑥(0)<𝑥^∗ and 𝑆(0)<𝑆_𝑖.

According to the existence criteria (Theorem 6), our ideas to prove the existence of periodic solution is to construct a closed region such that all the solutions of (2) enter the closed region and retain there. The ideas will be illustrated as follows using Figures 2 and 3.

Figure 2 

Illustration of (2) when 𝑅>1, ℎ<𝑥∗, 𝑥(0)≤ℎ, and 𝑆𝐷≥𝑆∗.

Figure 3 

Illustration of (2) when 𝑅>1, ℎ<𝑥∗, 𝑥(0)≤ℎ, and 𝑆𝐷<𝑆∗.

From Figure 2, we can see that the line 𝑥=ℎ interacts with the line 𝑆=𝑆^∗ at the point 𝐴(𝑆_𝐴,ℎ), where 𝑆_𝐴=𝑆^∗. The impulsive set 𝑀⊆𝐴𝐵, 𝐴𝐵={(𝑆,𝑥)∣𝑥=ℎ,𝑆^∗≤𝑆≤𝑆_𝑖}. The impulsive functions 𝐼₁ and 𝐼₂ map the impulsive set 𝑀 into 𝑁=𝐼(𝑀)⊆𝐶𝐷, 𝐶𝐷={(𝑆,𝑥)∣𝑥=(1−𝑏)ℎ,(1−𝑏)𝑆^∗≤𝑆≤(1−𝑏)𝑆_𝑖}, where 𝐶=(𝑆_𝐶,(1−𝑏)ℎ), 𝐷=(𝑆_𝐷,(1−𝑏)ℎ), 𝑆_𝐶=(1−𝑏)𝑆^∗, and 𝑆_𝐷=(1−𝑏)𝑆_𝑖. From the third equation of (2), we know that 𝑆^∗+=(1−𝑏)𝑆^∗ for 𝑥=ℎ, and furthermore, 𝑆_𝐶=(1−𝑏)𝑆_𝐴<𝑆_𝐴. Since the straight lines 𝐴𝐶 and 𝐵𝐷 satisfy 𝑥=ℎ𝑆/𝑆_𝐴 and 𝑥=ℎ𝑆/𝑆_𝑖, respectively, and both pass through the point (0,0), then it is clear that there are two cases of the point 𝐷, that is, (𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷))=𝑆^∗≤𝑆_𝐷 and 𝑆_𝐷<𝑆^∗=(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷)).

Theorem 13. Suppose that 𝑅>1, ℎ<𝑥^∗, 𝑥(0)≤ℎ, then system (2) has an order-1 periodic solution; furthermore, this order-1 periodic solution is unique.

Proof. Suppose that 𝑆_𝐷≥𝑆^∗=(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷)), then the trajectories of (2) starting from the region {𝑥(0)≤ℎ} must interact with the segment 𝐴𝐵. Next, we construct the closed region Ω₁. Let 𝐺 be the intersection point of the line 𝑆=𝑆^∗, and the trajectories of (2) starting from the point 𝐶, let 𝐺₁ be the intersection point of the line 𝑥=𝑥_𝐺, and 𝑆=𝑆_𝑖, let 𝐸 be the intersection point of the line 𝑆=𝑆^∗ and the segment 𝐶𝐷. From the qualitative characteristic of (1), it is easily obtained that (𝑑𝑆/𝑑𝑡)|_𝐵𝐺1<0, (𝑑𝑥/𝑑𝑡)|_𝐺𝐺1>0, (𝑑𝑥/𝑑𝑡)|_𝐶𝐸<0, (𝑑𝑆/𝑑𝑡)|_𝐴𝐸>0; thus, the closed region Ω₁ consists of 𝐴𝐸, 𝐸𝐶, 𝐶^⌢𝐺, 𝐺𝐺₁, 𝐺₁𝐵 and 𝐵𝐴 (see Figure 2), where the arc 𝐶^⌢𝐺 is the part of trajectory passing through 𝐶 and 𝐺. In addition, there is no singularity in the region Ω₁ and all the trajectories satisfying the conditions of the theorem enter the closed region Ω₁ and retain there; therefore, from Theorem 6, we know that system (2) has an order-1 periodic solution.
Suppose that 𝑆_𝐷<𝑆^∗=(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷)). Similar to the discussions of the case 𝑆_𝐷≥𝑆^∗=(𝐾_𝑑+𝐷)𝐾_𝑠/(𝜇−(𝐾_𝑑+𝐷)), let 𝐴₁ be the intersection point of the line 𝑥=ℎ and the trajectories of (2) starting from the point 𝐷; we can take the arc 𝐷^⌢𝐴₁ as a part of the boundaries of the closed region Ω₂. Then, we obtain the closed region Ω₂ which consists of 𝐷𝐶, 𝐶^⌢𝐺, 𝐺𝐺₁, 𝐺₁𝐵, 𝐵𝐴₁, and 𝐴^⌢₁𝐷 (see Figure 3); therefore, from Theorem 6, we know that system (2) has an order-1 periodic solution. To sum up, the system (2) has an order-1 periodic solution under the condition of Theorem 13. This completes the proof.

Remark 14. If system (2) has an order-1 periodic solution, then the order-1 periodic solution is unique.

We can prove the uniqueness of order-1 periodic solution in system (2) by using the method of successor functions which was introduced in [23], here omitted.

4.2. Stability of Order-1 Periodic Solution

In the following, we will analyze the stability of order-1 periodic solution in system (2). Firstly, we give one lemma to discuss the stability of this periodic solution of system (2).

Lemma 15. The 𝑇-periodic solution 𝑆=𝜉(𝑡),𝑥=𝜂(𝑡) of the following system 𝑑𝑆(),𝑑𝑡=𝑃𝑆,𝑥𝑑𝑥(),()(),()()𝑑𝑡=𝑄𝑆,𝑥𝑖𝑓𝜙𝑆,𝑥≠0,Δ𝑆=𝛼𝑆,𝑥Δ𝑥=𝛽𝑆,𝑥𝑖𝑓𝜙𝑆,𝑥=0, (19) is orbitally asymptotically stable if the Floquet multiplier 𝜇₂ satisfies the condition |𝜇₂|<1, where 𝜇₂=Π^𝑞_𝑘=1Δ_𝑘exp^𝑇₀𝜕𝑃𝜉(𝑡)(𝑡)+𝜕𝑆,𝜂𝜕𝑄𝜉(𝑡)(𝑡),𝜕𝑥,𝜂𝑑𝑡 (20) with Δ_𝑘=𝑃₊𝜕𝛽𝜕𝑥𝜕𝜙−𝜕𝑆𝜕𝛽𝜕𝑆𝜕𝜙+𝜕𝑥𝜕𝜙𝜕𝑆+𝑄₊𝜕𝛼𝜕𝑆𝜕𝜙−𝜕𝑥𝜕𝛼𝜕𝑥𝜕𝜙+𝜕𝑆𝜕𝜙×𝑃𝜕𝑥𝜕𝜙𝜕𝑆+𝑄𝜕𝜙𝜕𝑥⁻¹, (21) and 𝑃, 𝑄, 𝜕𝛼/𝜕𝑆, 𝜕𝛼/𝜕𝑥, 𝜕𝛽/𝜕𝑆, 𝜕𝛽/𝜕𝑥, 𝜕𝜙/𝜕𝑆 and 𝜕𝜙/𝜕𝑥 are calculated at the point (𝜉(𝜏_𝑘),𝜂(𝜏_𝑘)), 𝑃₊=𝑃(𝜉(𝜏⁺_𝑘),𝜂(𝜏⁺_𝑘)), 𝑄₊=𝑄(𝜉(𝜏⁺_𝑘),𝜂(𝜏⁺_𝑘)). 𝜙(𝑆,𝑥) is a sufficiently smooth function with grad𝜙(𝑆,𝑥)≠0, and 𝜏_𝑘(𝑘∈𝑁) is the time of the 𝑘th jump.

The proof of this lemma referres to Simeonov and Bainov [20].

In the following, we suppose that this periodic solution of system (2) with period 𝑇 passes through the points 𝐷⁺₁((1−𝑏)𝜁₀,(1−𝑏)ℎ)∈𝐶𝐷 and 𝐷₁(𝜁₀,ℎ)∈𝐴𝐵 (see Figures 2 and 3). As the expression and the period of this solution are unknown, we discuss the stability of this periodic solution by Lemma 15. In our case, 𝑃(𝑆,𝑥)=−𝜇𝑆𝑥𝛿₁𝐾_𝑠𝑆+𝑆−𝑚𝑥+𝐷_𝑖,𝑄()=−𝑆𝑆,𝑥𝜇𝑆𝐾_𝑠𝐾+𝑆𝑥−_𝑑𝛼()()𝜙()𝜉(𝑇)(𝑇)=𝜁+𝐷𝑥,𝑆,𝑥=−𝑏𝑆,𝛽𝑆,𝑥=−𝑏𝑥,𝑆,𝑥=𝑥−ℎ,,𝜂₀,𝜉𝑇,ℎ₊,𝜂𝑇⁺=(1−𝑏)𝜁₀,(1−𝑏)ℎ. (22)

Then, 𝜕𝑃𝜕𝑆=−𝜇𝐾_𝑠𝑥𝛿₁𝐾_𝑠+𝑆²−𝐷,𝜕𝑄=𝜕𝑥𝜇𝑆𝐾_𝑠−𝐾+𝑆^𝑑+𝐷,𝜕𝛼𝜕𝑆=−𝑏,𝜕𝛼𝜕𝑥=0,𝜕𝛽𝜕𝑆=0,𝜕𝛽𝜕𝑥=−𝑏,𝜕𝜙𝜕𝑆=0,𝜕𝜙Δ𝜕𝑥=1,_𝑘=𝑃₊𝜕𝛽𝜕𝑥𝜕𝜙−𝜕𝑆𝜕𝛽𝜕𝑆𝜕𝜙+𝜕𝑥𝜕𝜙𝜕𝑆+𝑄₊𝜕𝛼𝜕𝑆𝜕𝜙−𝜕𝑥𝜕𝛼𝜕𝑥𝜕𝜙+𝜕𝑆𝜕𝜙×𝑃𝜕𝑥𝜕𝜙𝜕𝑆+𝑄𝜕𝜙𝜕𝑥⁻¹=𝑄₊𝜉𝑇⁺,𝜂𝑇⁺(1−𝑏)𝑄𝜉(𝑇)(𝑇)=(),𝜂1−𝑏²𝜇(1−𝑏)𝜁₀𝐾_𝑠+(1−𝑏)𝜁₀−𝐾_𝑑×+𝐷𝜇𝜁₀𝐾_𝑠+𝜁₀−𝐾_𝑑+𝐷⁻¹. (23)

Set 𝑁(𝑡)=(𝜕𝑃/𝜕𝑆)(𝜉(𝑡),𝜂(𝑡))+(𝜕𝑄/𝜕𝑥)(𝜉(𝑡),𝜂(𝑡)); then; 𝜇₂=Δ₁exp^𝑇₀𝜕𝑃𝜉(𝑡)(𝑡)+𝜕𝑆,𝜂𝜕𝑄𝜉(𝑡)(𝑡)=()𝜕𝑥,𝜂𝑑𝑡1−𝑏²𝜇(1−𝑏)𝜁₀𝐾_𝑠+(1−𝑏)𝜁₀−𝐾_𝑑×+𝐷𝜇𝜁₀𝐾_𝑠+𝜁₀−𝐾_𝑑+𝐷⁻¹exp^𝑇₀𝑁(𝑡)𝑑𝑡. (24)

Because (𝑆(𝑡),𝑥(𝑡)) is a periodic solution of system (2), we know that ∫^𝑇₀𝑁(𝑡)𝑑𝑡<0, by the proof of Lemma 11, that is, exp[∫^𝑇₀𝑁(𝑡)𝑑𝑡]<1. Obviously, |𝜇₂|<1 if |(1−𝑏)²[(𝜇(1−𝑏)𝜁₀/(𝐾_𝑠+(1−𝑏)𝜁₀))−(𝐾_𝑑+𝐷)]/[(𝜇𝜁₀/(𝐾_𝑠+𝜁₀))−(𝐾_𝑑+𝐷)]|<1. Therefore, we have the following theorem.

Theorem 16. If 𝑅>1, ℎ<𝑥^∗, 𝑥(0)≤ℎ and |(1−𝑏)²[(𝜇(1−𝑏)𝜁₀/(𝐾_𝑠+(1−𝑏)𝜁₀))−(𝐾_𝑑+𝐷)]/[(𝜇𝜁₀/(𝐾_𝑠+𝜁₀))−(𝐾_𝑑+𝐷)]|<1, then the order-1 periodic solution of system (2) is orbitally asymptotically stable.

4.3. Order-2 Periodic Solution

From Theorem 13, we know that system (2) has an order-1 periodic solution. In this subsection, we will discuss the existence of order-2 periodic solution.

Suppose that (𝑆,𝑥) is a periodic solution of system (2), then (𝑆₀,(1−𝑏)ℎ)∈𝑁⊆𝐶𝐷 and (𝑆₁,ℎ)∈𝑀⊆𝐴𝐵. It is easy to obtain that 𝑆₀<𝑆₁ for 𝑆⁺=(1−𝑏)𝑆 by the third equation of system (2). Let (𝑆,𝑥) be the arbitrary solution of system (2) starting from the point (𝑆⁺₀,(1−𝑏)ℎ), and let the first interaction point of trajectory and the set 𝑀(𝑥=ℎ) be (𝑆₁,ℎ) and let the corresponding interaction points of every impulse be (𝑆₂,ℎ),(𝑆₃,ℎ),…,  respectively. Therefore, under the effect of impulsive function 𝐼, the corresponding points after impulse are (𝑆⁺₁,(1−𝑏)ℎ),(𝑆⁺₂,(1−𝑏)ℎ),…,  respectively.

By the qualitative analysis of system (2), if there exists the 𝑖th impulse effect such that 𝑆⁺_𝑗≥𝑆^∗ for all 𝑗>𝑖≥1, then the sequence 𝑆⁺_𝑗 will be monotone, that is, 𝑆⁺_𝑗≥𝑆⁺_𝑗+1≥𝑆⁺_𝑗+2≥⋯≥𝑆^∗ or 𝑆^∗≤𝑆⁺_𝑗≤𝑆⁺_𝑗+1≤𝑆⁺_𝑗+2≤⋯. In this case, system (2) has no order 𝑘(𝑘≥2) periodic solution. Therefore, if system (2) has order 𝑘(𝑘≥2) periodic solution, there must exist a 𝑆⁺_𝑗 such that 𝑆⁺_𝑗<𝑆^∗. Besides, when 𝑏>1−(𝑆^∗/𝑆_𝑖), we can know that 𝑆⁺_𝑗<𝑆^∗ for 𝑗=1,2,…. So, in the following, we mainly discuss the existence of order-2 periodic solution for 𝑏>1−(𝑆^∗/𝑆_𝑖).

If 𝑆⁺₀=𝑆⁺₁, then system (2) has an order-1 periodic solution.

If 𝑆⁺₀≠𝑆⁺₁, then 𝑆⁺₀<𝑆⁺₁ or 𝑆⁺₀>𝑆⁺₁. Suppose that 𝑆⁺₀<𝑆⁺₁, for 𝑏>1−(𝑆^∗/𝑆_𝑖) and the qualitative properties of system (2), we know that 𝑆⁺₂<𝑆⁺₁ and 𝑆⁺₂≤𝑆⁺₀<𝑆⁺₁ or 𝑆⁺₀≤𝑆⁺₂<𝑆⁺₁. Therefore, there exists an order-2 periodic solution if 𝑆⁺₂=𝑆⁺₀ holds.

If 𝑆⁺₂<𝑆⁺₀<𝑆⁺₁ holds, we can obtain(i)(1−𝑏)𝑆^∗<⋯<𝑆⁺_2𝑘<⋯<𝑆⁺₄<𝑆⁺₂<𝑆⁺₀<𝑆⁺₁<𝑆⁺₃<𝑆⁺₅<⋯<𝑆⁺_2𝑘+1<⋯<𝑆^∗.  While if 𝑆⁺₀<𝑆⁺₂<𝑆⁺₁ holds, we have (ii)(1−𝑏)𝑆^∗≤𝑆⁺₀<𝑆⁺₂<𝑆⁺₄<⋯<𝑆⁺_2𝑘<⋯<𝑆⁺_2𝑘+1<⋯<𝑆⁺₅<𝑆⁺₃<𝑆⁺₁<𝑆^∗.  The same discussion for the case 𝑆⁺₀>𝑆⁺₁, that is, we have 𝑆⁺₁<𝑆⁺₂<𝑆⁺₀ or 𝑆⁺₁<𝑆⁺₀<𝑆⁺₂. If 𝑆⁺₁<𝑆⁺₀<𝑆⁺₂ holds, we have(iii)(1−𝑏)𝑆^∗<⋯<𝑆⁺_2𝑘+1<⋯<𝑆⁺₅<𝑆⁺₃<𝑆⁺₁<𝑆⁺₀<𝑆⁺₂<𝑆⁺₄<⋯<𝑆⁺_2𝑘<⋯<𝑆^∗.  While if 𝑆⁺₁<𝑆⁺₂<𝑆⁺₀ holds, we have(iv)(1−𝑏)𝑆^∗≤𝑆⁺₁<𝑆⁺₃<𝑆⁺₅<⋯<𝑆⁺_2𝑘+1<⋯<𝑆⁺_2𝑘<⋯<𝑆⁺₄<𝑆⁺₂<𝑆⁺₀<𝑆^∗.

If the case of (ii) and (iv) holds, system (2) has an order-1 periodic solution. It follows from (i) (or (iii)) that lim_𝑘→∞𝑆⁺_2𝑘=𝑆₀^+∗, lim_𝑘→∞𝑆⁺_2𝑘+1=𝑆₁^+∗, which implies that system (2) exists with an order-2 periodic solution. Similar to the previous discussion and the proof of Proposition 3.2 in [24], we can obtain that there is no order 𝑘(𝑘≥3) periodic solution in system (2) for 𝑏>1−(𝑆^∗/𝑆_𝑖).

Remark 17. Since the exact expression of the solution to system (2) is unknown, it is difficult to give the sufficient conditions to guarantee that the system (2) has an order-2 periodic solution or not.

5. Numerical Analysis and Discussion

In this paper, a mathematical model of cheese whey fermentation for single-cell protein production with impulsive state feedback control is proposed. We investigated the qualitative characteristic of the system without impulsive effect and show that the system is globally asymptotical stability. It is shown that the system with impulsive state feedback control has an order-1 periodic solution, and sufficient conditions for existence and stability of order-1 periodic solution are also obtained. The case, in which it is possible that there is an order-2 periodic solution, is also observed; furthermore, it is also pointed out that the system has no order 𝑘(𝑘≥3) periodic solution for 𝑏>1−(𝑆^∗/𝑆_𝑖). These results show that the cheese whey fermentation system for single-cell protein production with impulsive state feedback control either tends to a stable state or has a periodic solution.

To verify the theoretical results obtained in this paper, we give the numerical simulations of system (2). Let 𝛿₁=0.6, 𝐾_𝑠=2, 𝑆_𝑖=4.7, 𝐷=0.4, 𝜇=3.8 and 𝐾_𝑑=0.7; then, we have 𝑆^∗=0.8148, 𝑥^∗=0.6135. If 𝑏=0.8, ℎ=0.78>𝑆^∗, 𝑆₀=0.5, and 𝑥₀=0.25, then the system (2) tends to the positive equilibrium point, which can be seen in Figure 4. Figure 4 shows that no impulse occurs when ℎ>𝑆^∗. Figure 5 gives the time series and phase portrait when 𝑏=0.34, ℎ=0.25<𝑆^∗, 𝑆₀=2.8, and 𝑥₀=0.05 and show the trajectory tends to be periodic. Figure 6 is the time series and phase portrait when 𝑏=0.8, ℎ=0.25<𝑆^∗, 𝑆₀=0.15, and 𝑥₀=0.08. Figures 5 and 6 show different positions of the periodic solution under different parameter value 𝑏 and different initial values. Furthermore, the phase portrait of Figure 5 indicates that the microorganism concentration always keeps increasing, but Figure 6 indicates that the microorganism concentration firstly decreases and then begins to increase. Therefore, a different control parameter 𝑏 can result in different bioprocesses and different efficiencies of microorganism culture. Which will give a conclusion theoretically to the researchers in the field of single-cell protein production. Researchers should give suitable control parameter 𝑏 and appropriate initial concentration in order to obtain a steady and optimal production.

(a)

(b)

(c)

Figure 4 

Time series and phase portrait of system (2) with 𝛿1=0.6, 𝐾𝑠=2, 𝑆𝑖=4.7, 𝐷=0.4, 𝜇=3.8, 𝐾𝑑=0.7, 𝑏=0.8, 𝑆0=0.5, 𝑥0=0.25, and ℎ=0.78>𝑆∗.

(a)

(b)

(c)

Figure 5 

Time series and phase portrait of system (2) with 𝛿1=0.6, 𝐾𝑠=2, 𝑆𝑖=4.7, 𝐷=0.4, 𝜇=3.8, 𝐾𝑑=0.7, 𝑏=0.34, 𝑆0=2.8, 𝑥0=0.05, ℎ=0.25<𝑆∗.

(a)

(b)

(c)

Figure 6 

Time series and phase portrait of system (2) with 𝛿1=0.6, 𝐾𝑠=2, 𝑆𝑖=4.7, 𝐷=0.4, 𝜇=3.8, 𝐾𝑑=0.7, 𝑏=0.8, 𝑆0=0.15, 𝑥0=0.08, ℎ=0.25<𝑆∗.

In conclusion, we give a mathematical conclusion to the researchers in the field as follows: researchers of cheese whey fermentation for single-cell protein production should give suitable feedback state (the value of ℎ), control parameters (𝑏), and appropriate initial concentration of microorganism and substrate in order to obtain a steady and optimal production.

Acknowledgments

This paper supported by National Natural Science Foundation of China (11171284, 31272653), Fujian Provincial Natural Science Foundation of China (2012J01012), the Fujian Provincial Education Foundation (JA12198), and the Scientific Research Foundation of Jimei University of China (ZC2011003).