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Periodic solution of a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control
ARTICLE IN PRESS Journal of Theoretical Biology 261 (2009) 23–32
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Periodic solution of a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control$ Zuxiong Li a,, Tieying Wang a,b, Lansun Chen a a b
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China College of Science, Dalian Nationalities University, Dalian 116605, PR China
a r t i c l e in fo
abstract
Article history: Received 17 January 2009 Received in revised form 17 July 2009 Accepted 17 July 2009 Available online 23 July 2009
In this paper, a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Feedback control Globally asymptotical stability Periodic solution Qualitative analysis Impulsive effect
1. Introduction The chemostat is a basic piece of laboratory apparatus, yet it has begun to occupy an increasingly central role in ecological studies. The ecological environment created by a chemostat is one of the few completely controlled experimental systems for testing microbial growth and competition. As a tool in biotechnology, the chemostat plays an important role in bio-processing. Chemostat has widely been used to study bacterial metabolism, population genetics, and plasmid stability, mostly because these are the simplest and most easily constructed types of continuous cultures. Many papers have investigated the mathematical models on the culture of the microorganisms. For example, mathematical (Armstrong and McGehee, 1980; Butler and Wolkowicz, 1985; Hsu, 1978; Hsu et al., 1977; Levins, 1979; Wolkowicz and Lu, 1992) and experimental (Grover, 1997; Hansen and Hubbell, 1980; Tilman, 1982) models exhibit the competitive exclusion principle only one species survives. Several modifications of the chemostat have been made to ensure the coexistence of species on a single nutrient (De Leenheer and Smith, 2003; Flegr, 1997; Grover, 1997; Tilman, 1982). However, there are a lot of factors such as temperature, dissolved oxygen content affecting the growth and reproduction of the microorganisms in the process of bio-reacts.
$
Supported by National Natural Science Foundation of China (no. 10771179).
Corresponding author. Tel.: +8615074417427.
E-mail addresses: [email protected] (Z. Li), [email protected] (T. Wang), [email protected] (L. Chen). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.07.016
In some cases, it is necessary to regulate the bio-react according to the concentration of microorganisms. The volume of the chemostat can be controlled either by using a pump (seen on the left of Fig. 1) or an overflow system (seen on the right of Fig. 1). In some cases, the high concentration of the microorganisms will often have caused a series of adverse effects, such as the production inhibition. To regulate the concentration of the microorganism, we modify overflow system of the chemostat into the right system of Fig. 2. When the concentration which can be measured by optoelectronic devices and other ways, reaches a certain threshold, it can be regulated through replenishing water to reduce the concentration of the microorganisms. Impulsive equation has become a widely concerned subject in recent years, particularly as applied to biological impulsive equation model more appropriate for the actual reality. Based on the principle of the right system of Fig. 2, we propose a mathematical model concerning the chemostat with Beddington– DeAnglis uptake function and impulsive state feedback control. As for some models with impulsive effect Bainov and Simeonov (1993) Meng et al. (2008, 2007), Meng and Chen (2008), Jiao et al. (2007), Jiao and Chen (2008), Shi and Chen (2007, 2008a, 2008b) have investigated and been well studied. As for some models with impulsive state feedback control, Zeng (2007), Jiang et al. (2007) and Tang and Chen (2004) discussed prey-predator models and obtained the complete expression of the periodic solution. However, so far, few papers have discussed the chemostat system using the impulsive differential equation with state feedback control.
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Feed Pump
Effluent Pump
Feed
Feed Pump Effluent
Feed
Level maintained by overflow of the effluent through a port on the side of the reactor
Level maintained by setting effluent pump to run faster than the feed pump
Effluent
Fig. 1. Two type of chemostat.
Feed Pump
Feed Pump Water
Feed
Feed
Control Value Level maintained by overflow of the effluent through a port on the side of the reactor
Level maintained by overflow of the effluent through a port on the side of the reactor
Effluent
Effluent Photo Cell Light Source
Fig. 2. The sketch map of chemostat and modified chemostat.
In this paper, we will prove that the model without impulsive effect has an globally asymptotically stable positive equilibrium under certain condition. we will also discuss the existence and stability of periodic solution of the chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control according to the existence criteria (Zeng, 2007) and the stability theorem (Simeonov and Bainov, 1988) of periodic solution of the general impulsive autonomous system. This paper is organized as follows. The model and some preliminary results are presented in the next section. The qualitative analysis of the system without impulsive effect is given in Section 3. In Section 4, the existence and stability of periodic solution of order one of differential equation with impulsive state feedback control are investigated. Numerical simulations are given in Section 5. Finally, some conclusion and biological discussions are provided in Section 6.
2. The model and preliminaries The general model of continuously culturing microorganism in a chemostat is described by the following differential equation (De Leenheer and Smith, 2003): 8 < S_ ¼ DðS SÞ 1f ðSÞx; in d ð2:1Þ : x_ ¼ f ðSÞx Dx:
In model (2.1), S denotes the concentration of substrate at time t and x denotes the concentration of the microorganism in the chemostat at time t; D and Sin 40 denote, respectively, the dilution rate of the chemostat and the concentration of the input substrate; d is the yield ; f is uptake function and we take f ðSÞ ¼ mm S=ðS þ Bx þ AÞ, that is, DeAngelis et al. (1975) and Beddington (1975) uptake function. Then (2.1) has the following form: 8 mm Sx > > ; > S_ ¼ DðSin SÞ < dðS þ Bx þ AÞ ð2:2Þ mm Sx > > > : x_ ¼ ðS þ Bx þ AÞ Dx: where mm 40 is called the maximal specific growth rate of the microorganisms, km 40 is the saturation constant, AZ0; BZ0 are constant. In some cases, if the concentration of the microorganisms is lower than a critical value, it is not necessary to adopt control measures. But the high concentration of the microorganisms will often have caused a series of adverse effects, such as the substrate inhibition, higher viscosity caused by the mass transfer efficiency, and so on. What is especially serious, many secondary metabolite’s production of microbial production are suppressed by highly concentrated glucose, the carbohydrate and the azotic compound
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degradation product. In order to overcome the above shortcomings, we often use fed-batch culture if the concentration of the substrate is low. In the process of fermentation, when the microorganism begins to consume the substrate, we take some measures to supplement certain material in the culture system so that the concentration of the culture medium maintains to retain a certain scope for a long time. To achieve increasing production volume, product concentration and product yield, we always maintain the growth of microorganisms in the scope. In this system, when the flow velocity of the culture medium is lower than the growth rate of microorganism, the concentration of microorganism increases, once the concentration reaching to a critical value which can be detected by optoelectronic devices, we should take measures to decrease the concentration of microorganism. Once the concentration is decreased, it takes a period of time to reach the critical value once more. Therefore, the feedback control can be carried out by the impulsive state control. In this paper, we consider that the substrate with the microorganism is discharged impulsively due to the added water and other regulator solution on the basis of continuous cultivation in the chemostat when x reaches the critical value (denoted by xm ). Therefore, by simplifying, (2.2) can be modified as follows by introducing the impulsive state feedback control: 8 9 mSx > > _ ¼1S > > S ; > > ðS þ Bx þ AÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x S þ Bx þ A ð2:3Þ ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40; 0 0 where 0obo1 is constant and is also the fraction of the concentration of microorganism that decreases due to the feedback control when the concentrate x reach h. By Observing, if A40; B ¼ 0, then system (2.3) reduces to a chemostat model with Monod growth rate and impulsive state feedback control 8 9 mSx > > > ;> > S_ ¼ 1 S > ðS þ AÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x SþA ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40; 0 0 if A ¼ 0; B40, the system (2.3) reduces to a chemostat model with ratio-dependent growth rate and impulsive state feedback control 8 9 mSx > > > ;> > S_ ¼ 1 S > ðS þ BxÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x S þ Bx ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40: 0
0
In this paper, we mainly discuss the existence and stability of periodic solution of system (2.3) by the existence criteria and stability criteria of the general impulsive autonomous system. Before introducing the existence criteria, we give some topological definitions about impulsive differential equations. Deftinition 2.1 (Laksmikantham et al., 1989). An triple ðX; p; Rþ Þ is said to be a semi-dynamical system if X is metric space, Rþ is the
25
set of all non-negative reals and p : X Rþ -X is a continuous function such that (i) pðx; 0Þ ¼ x for all x 2 X; (ii) pðpðx; tÞ; sÞ ¼ pðx; t þ sÞ for all x 2 X and t; s 2 Rþ . We denote some times a semi-dynamical system ðX; p; Rþ Þ by ðX; pÞ. For any x 2 X, the function px : Rþ -X defined by px ðtÞ ¼ pðx; tÞ is continuous and we call px the trajectory of x. The set C þ ðxÞ ¼ fpðx; tÞjt 2 Rþ g; is called the positive orbit of x. For any subset M of X, we let Mþ ðxÞ ¼ C þ ðxÞ \ M x
and
M ¼ GðxÞ \ M x;
and
Gðx; tÞ ¼ fyjpðy; tÞ ¼ xg;
where GðxÞ ¼ [fGðx; tÞjt 2 Rþ g
is the attainable set of x at t 2 Rþ . Finally we set MðxÞ ¼ Mþ ðxÞ [ M ðxÞ. Deftinition 2.2 (Laksmikantham et al., 1989). An impulsive semidynamical system ðX; p; M; IÞ consists of a semi-dynamical system ðX; pÞ 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 2 X is a limit point of MðxÞ, (ii) ½tjGðx; tÞ \ Ma is a closed subset of Rþ . We write N ¼ IðMÞ ¼ fy 2 Xjy ¼ IðxÞ; x 2 Mg and for any x 2 X, IðxÞ ¼ xþ . We call M the set of impulses, I the impulsive function. We define a function F : X-Rþ [ f1g as following: ( 1 if M þ ðxÞ ¼; FðxÞ ¼ 2M for 0otos and pðx; sÞ 2 M; s if pðx; tÞ= here we call s the time without impulse of x, that is to say s is the first time when pðx; 0Þ hits M. Deftinition 2.3 (Laksmikantham et al., 1989). Let ðX; p; M; IÞ be an impulsive semi-dynamical system and let x 2 X and x= 2M. The trajectory of x is a function p~ x defined on subset ½0; sÞ of Rþ (s may be 1) to X inductively as following:
p~ x ðtÞ ¼ pðxþnþ1 ; tÞ;
tn1 t tn ;
where fxn g is the sequence of impulse points of x, which satisfied
pðxþn1 ; Fðxþn1 ÞÞ ¼ xn . Ptn is the sequence of time of impulses þ relative to fxn g, tn ¼ n1 k¼0 Fðxk Þ. Deftinition 2.4 (Laksmikantham et al., 1989). A trajectory p~ x is said to be periodic of period t and order k if there exist positive integers mZ1 and kZ1 such that k is the smallest integer for Pmþk1 þ Fðxþ which xþ m ¼ xmþk and t ¼ i¼m i Þ. Theorem 2.1 (Brouwer’s fixed-point theorem, Griffel, 1981). Every continuous mapping of a closed bounded convex set in Rn into itself has a fixed point. In the following we give the existence criteria for an impulsive autonomous system with state-dependent proved by means of Theorem 2.1 (Zeng et al., 2006).
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To consider the following general autonomous impulsive differential equations: 8 9 dxðtÞ > > > > ¼ Pðx; yÞ; > = > > dt > > ðx; yÞ= 2M; > > > < dyðtÞ ; ¼ Q ðx; yÞ; > ð2:4Þ dt > ) > > > > > Dx ¼ I1 ðx; yÞ; ðx; yÞ 2 M: > > > : Dy ¼ I2 ðx; yÞ; Here ðx; yÞ 2 R2 , and P; Q ; I1 ; I2 are all functions mapping R2 into R, M R2 is the set of impulse, and we assume: (H2.1) Pðx; yÞ; Q ðx; yÞ are all continuous with respect to x; y in R2 . (H2.2) M R2 is a line, I1 ðx; yÞ and I2 ðx; yÞ are linear functions of x and y.
System (3.1) only has a boundary equilibrium (1,0) in region Rþ ¼ fðS; xÞjSZ0; xZ0g if 0om 1 þ A. The Jacobian matrix at equilibrium E ¼ ðS; xÞ is given by 0 B B JðS;xÞ ¼ B @
1
mxðBx þ AÞ
ðS þ Bx þ AÞ2 mðBx þ AÞ
ðS þ Bx þ AÞ2
1 mSðS þ AÞ ðS þ Bx þ AÞ2 C C C: mSðS þ AÞ A 1 2 ðS þ Bx þ AÞ
We now give some results about system (3.1). Theorem 3.1. If m41 þ A, then Bð1; 0Þ is a saddle point, the positive equilibrium AðS ; x Þ is globally asymptotically stable node and limt-1 SðtÞ ¼ S ; limt-1 xðtÞ ¼ x , where S ¼
For each point Sðx; yÞ 2 M, we define I : R2 -R2 :
AþB ; mþB1
x ¼
mA1 : mþB1
IðSÞ9Sþ ¼ ðxþ ; yþ Þ 2 R2 ; xþ ¼ x þ I1 ðx; yÞ;
yþ ¼ y þ I2 ðx; yÞ:
Obviously N ¼ IðMÞ is also a line of R2 or a subset of a line, and we assume that N \ M ¼. From Definition 2.2 we know (2.4) is an impulsive semi-dynamical system. The following theorem gives the conditions on which (2.4) has a periodic solution of order one defined by Definition 2.5. Theorem 2.2 (Zeng, 2007). If system (2.4) satisfies assumptions ðH2:1Þ and ðH2:2Þ, there exists a bounded closed simply connected region D which has following properties: (i) There is no singularity in it and the boundary @D of D satisfies ðD @DÞ \ M ¼. (ii) L1 ¼ D \ M cannot be tangent with trajectories of (2.4) except at end-points and IðL1 Þ D. (iii) Trajectories with initial point in @D L1 will enter into interior of D, then there must exist a periodic solution of system (2.4) of order one in region D. 3. Qualitative analysis of system (2.3) without impulsive effect In this section, we will study the qualitative characteristic of system (2.3) without the impulse effect. If no impulsive effect is introduced, then system (2.3) is 8 mSx > _ > < S ¼ 1 S ðS þ Bx þ AÞ; ð3:1Þ > mS > : x_ ¼ x 1 : S þ Bx þ A
Proof. Firstly, by 1 0 m 1 B Aþ1 C C; Jð1;0Þ ¼ B m A @ 1 0 Aþ1 we know that the eigenvalues are 1 and m=ðA þ 1Þ 1. If m41 þ A, then Bð1; 0Þ is a saddle point. We can calculate the Jacobian matrix at the positive equilibrium AðS ; x Þ as follow: 1 0 mx ðBx þ AÞ mS ðS þ AÞ 1 C B B ðS þ Bx þ AÞ2 ðS þ Bx þ AÞ2 C C JðS ;x Þ ¼ B C B mx ðBx þ AÞ mS ðS þ AÞ A @ 1 2 2 ðS þ Bx þ AÞ ðS þ Bx þ AÞ 0 B B ¼B @
1
mx ðBx þ AÞ
ðS þ Bx þ AÞ2 mx ðBx þ AÞ
ðS þ Bx þ AÞ2
1 S þ A ðS þ Bx þ AÞ C C C: Bx A S þ Bx þ A
Therefore, the eigenvalue equation is " # mx ðBx þ AÞ Bx Bx l2 þ 1 þ þ lþ 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ þ
mx ðBx þ AÞ ðS þ Bx þ AÞ2
¼ 0:
Obviously,
l1 l2 ¼
From 1S
mSx ¼ 0; ðS þ Bx þ AÞ
and mS 1 ¼ 0; x S þ Bx þ A we can obtain system (3.1) has a boundary equilibrium ð1; 0Þ and a positive equilibrium ðS ; x Þ if m41 þ A, where S ¼
AþB ; mþB1
x ¼
mA1 : mþB1
Bx mx ðBx þ AÞ þ 40; S þ Bx þ A ðS þ Bx þ AÞ2 "
l1 þ l2 ¼ 1 þ
mx ðBx þ AÞ ðS þ Bx þ AÞ
þ 2
# Bx o0; S þ Bx þ A
and because " #2 mx ðBx þ AÞ Bx Bx D¼ 1þ þ 4 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ #" #2 mx ðBx þ AÞ mx ðBx þ AÞ Bx 1 Z0; þ ðS þ Bx þ AÞ2 ðS þ Bx þ AÞ2 S þ Bx þ A
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then AðS ; x Þ is asymptotically stable node. Here l1 ; l2 are two roots of " # mx ðBx þ AÞ Bx Bx l2 þ 1 þ þ lþ 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ þ
mx ðBx þ AÞ ðS þ Bx þ AÞ2
¼ 0:
Next, we prove system (3.1) is uniformly bounded. Let the straight line S ¼ 1 and 0 intersects the straight line x ¼ 0 at the points Bð1; 0Þ; Oð0; 0Þ, respectively. Let the straight line x ¼ ðm A 1Þ=B intersects the straight line S ¼ 1 and 0 at the points Cð1; ðm A 1Þ=BÞ; Dð0; ðm A 1Þ=BÞ, respectively. By some calculations, we can obtain that dS=dtjODðS¼0Þ 40, dx=dtjx¼ðmA1Þ=B o0; dS=dtjODðS¼1Þ o0 and because dx=dtjx¼0 ¼ 0, the trajectory of system (3.1) cannot pass through the straight line x ¼ 0. Hence, system (3.1) is uniformly bounded. We denote the region O which is formed by straight lines BC ; CD; DO; OB, that is, the region O is composed of the &BCDOB. (See Fig. 3.) Finally, we prove the global stability of system (3.1). Firstly, we will give the following lemma. & Lemma 3.1. Suppose GðTÞ ¼ ðSðtÞ; xðtÞÞ is a periodic orbit with T of system (3.1), R is the set which consists of all the points in phase plane G.
Z
T
¼ 0
27
! BmxS mS 1 þ 1 dt; ðS þ Bx þ AÞ2 S þ Bx þ A S þ Bx þ A mxðBx þ AÞ
for SðtÞ is a function with period T, so we can calculate and obtain Z T Z T mS d lnðxðtÞÞ ¼ 0: 1 dt ¼ S þ Bx þ A 0 0
Therefore, N¼
Z
T
1 0
! BmxS dto0: ðS þ Bx þ AÞ2 S þ Bx þ A mxðBx þ AÞ
The proof of Lemma 3.1 is complete. & From the above discussions, we know that A is locally stable. According to Lemma 3.1, we can obtain that if there exists periodic solution ðSðtÞ; xðtÞÞ around AðS ; x Þ, then it is stable for any periodic solution. This is impossible. According to Poincare–Bendixson theorem, the limit set of all orbits must be equilibrium point A. This implies that AðS ; x Þ is globally asymptotically stable in O. This completes the proof of Theorem 3.1. & The vector graph of (3.1) can be seen in Fig. 4.
1
Denote Z T @P @Q N¼ ðSðtÞ; xðtÞÞ þ ðSðtÞ; xðtÞÞ dt; @S @x 0
0.8
where S_ ¼ PðSðtÞ; xðtÞÞ; x_ ¼ Q ðSðtÞ; xðtÞÞ. If the conditions of Theorem 3.1 hold, then we can obtain No0. x
0.6
Proof. By Z T @P @Q ðSðtÞ; xðtÞÞ þ ðSðtÞ; xðtÞÞ dt N¼ @S @x 0 Z
T
¼
1
ðS þ Bx þ AÞ2
þ
mSðS þ AÞ ðS þ Bx þ AÞ2
0.2
! 1 dt
0 0
0.2
0.4
0.6
0.8
1
S Fig. 4. Vector graph of system (3.1) for m41 þ A.
0.3 0.2 0.1 x
0
mxðBx þ AÞ
0.4
0 –0.1 –0.2 0.5
0.6
0.7
0.8
0.9
1
S Fig. 3. The region O is composed of the &BCDEB.
Fig. 5. Vector graph of (3.1) for mo1 þ A.
1.1
1.2
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By the Jacobian matrix at equilibrium (1,0), we easily obtain the following theorem: Theorem 3.2. If mo1 þ A, then the boundary equilibrium (1,0) is local asymptotical stability.
From the third equation of (2.3), we know that Sþ ¼ ð1 bÞS if x ¼ h and furthermore SG ¼ ð1 bÞSG SG . According to the value of SG and SH , we have the following cases: Case 1: SG ½ð1 bÞBh þ A=ðm 1Þ and ½ð1 bÞBh þ A=ðm 1Þ SH SHM (see Fig. 6), where SHM is the solution of the equation
The vector graph of (3.1) can be seen in Fig. 5. 1 SHM 4. Existence and stability of periodic solutions
that is,
4.1. Existence of periodic solution of order one
SHM ¼
mSHM ð1 bÞh ¼ 0; A þ Bð1 bÞh þ SHM
1 A ð1 bÞhðm þ BÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1 bÞhðm þ BÞ þ A 12 þ 4ðA þ Bð1 bÞhÞ 2
:
From discussions of Section 3, we can see that ðS ; x Þ is a stable node when m4A þ 1. For the initial points which satisfy xð0Þox and dS=dtjðS0 ;x0 Þ Z0 if hZx , then all the solutions of (2.3) tend to the equilibrium ðS ; x Þ and no impulsive will occur. It is not necessary to control by impulsive control. So we mainly pay attention to the case that the following assumption holds: ðHÞ hox ; x0 ox and Sð0Þ 1. We apply the existence criteria (Theorem 2.1 of Zeng, 2007) to prove periodic solution of system (2.3) of order one. We first need to construct a closed region such that all the solutions of (2.3) enter the closed region and retain there. Without loss of generality, we will illustrate the ideas by using Figs. 6–8. From Figs. 6–8, we can see that the line x ¼ h interacts the isoclinal line mS=ðS þ Bx þ AÞ 1 ¼ 0, that is dx=dt ¼ 0, at the point GððBh þ AÞ=ðm 1Þ; hÞ, and interacts dS=dt ¼ 1 S mSx=ðS þ Bx þ AÞ ¼ 0 at the points DðSD ; hÞ. Here SD is the root of the equation 1 SD
mSD h ¼ 0: SD þ Bh þ A
The impulsive set MDEC ; EC ¼ fðS; xÞjðBh þ AÞ=ðm 1Þ S 1g. The impulsive functions I1 and I2 map the impulsive set M as GH ¼ fðS; xÞj½ð1 bÞðBh þ AÞ=ðm 1Þ S 1 b; N ¼ IðMÞDGH, x ¼ ð1 bÞhg, here G ¼ JðSG ; ð1 bÞhÞ, H ¼ HðSH ; ð1 bÞhÞ, SG ¼ ð1 bÞðBh þ AÞ=ðm 1Þ; SH ¼ 1 b.
Fig. 6. Illustration of (2.3) for the case hox ; x0 oh; SG SF ; SF SH SHM .
Fig. 7. Illustration of (2.3) for the case hox ; x0 oh; SG oSH o½ð1 bÞBh þ A= ðm 1Þ.
Fig. 8. Illustration of (2.3) for the case hox ; x0 oh; ½ð1 bÞBh þ A= ðm 1Þo SG oSH oSHM .
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Case 2: SG oSH o½ð1 bÞBh þ A=ðm 1Þ (Fig. 7). Case 3: ½ð1 bÞBh þ A=ðm 1ÞoSG oSH oSHM (Fig. 8). Theorem 4.1. Suppose that m41 þ A; 0ohox ; x0 oh, S0 þ mS0 x0 = ðA þ Bx0 þ S0 Þ 1 and S0 o1, then system (2.3) has a periodic solution of order one. Proof. Under the condition of theorem, the trajectories of system (2.3) staring from the region shown in case 1 must interact with the segment EC . Next, we construct the closed region O1 . From the qualitative characteristic of system (3.1), we know that dS=dto0 for S ¼ 1, dx=dt ¼ 0 for x ¼ 0, dx=dto0 for x ¼ ð1 bÞh in line segment IF and dS=dt40 for S ¼ 0. The straight line x ¼ ð1 bÞh interacts S ¼ 0 and mS=ðA þ Bx þ SÞ 1 ¼ 0 at the points Ið0; ð1 bÞhÞ and Fð½ð1 bÞBh þ A=ðm 1Þ; ð1 bÞhÞ, respectively. Therefore, we have BC ; CE; EF ; FG; GI; IO, and OB. We can see that the trajectories of system (2.3) enter and retain the region O1 formed by BC ; CE; EF ; FG; GI; IO, and OB (Fig. 6). By Theorem 2.2, we can obtain that system (2.3) has a periodic solution of order one if the conditions of theorem and case 1 hold. Next, we will prove that system (2.3) has a periodic solution of order one if the conditions of theorem and case 2 hold. We now construct the closed region O2 . Similar to the discussions of case 1, we can easily obtain that the closed region BC ; CE; EF ; FH; HG; GI; IO and OB (see Fig. 7). By Theorem 2.2, we know that system (2.3) has a periodic solution of order one if the conditions of theorem and case 2 are satisfied. Finally, we prove that system (2.3) has a periodic solution of order one if the conditions of theorem and case 3 are satisfied. We construct the closed region consisted of ^ and IC (see in Fig. 8). Here HI ^ is a trajectory CE; EF ; FG; GH; HI of system (2.3) which passes through the point H and interacts the isocline equation dS=dt ¼ 0 at the point I. The vertical line passing through the point I interacts the line x ¼ h at the point C. By the qualitative characteristic of the system, we can know that all the solution enter the closed region and retain there. By Theorem 2.2, we can obtain that system (2.3) has an order one periodic solution if the conditions of theorem and case 3 are satisfied. To sum up, we have proved that system (2.3) have a periodic solution of order one under the condition of Theorem 4.1. By qualitative analysis of system (2.3), we can see that the trajectories of system (2.3) starting for the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1g will enter the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 Þ 1g after some time. Therefore, we have the following theorem: Theorem 4.2. Suppose that m41 þ A; 0ohox ; x0 oh, S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1 and S0 o1, then system (2.3) has a periodic solution of order one.
29
Lemma 4.1. The T-periodic solution S ¼ xðtÞ; x ¼ ZðtÞ of the system 8 9 dS > > > > > ¼ PðS; xÞ; = > > dt > > if fðS; xÞa0; > dx > > < ; ¼ Q ðS; xÞ; > ð4:1Þ dt > ) > > > > > DS ¼ aðS; xÞ; if fðS; xÞ ¼ 0; > > > : Dx ¼ bðS; xÞ; is orbitally asymptotically stable if the Floquet multiplier m2 satisfies the condition jm2 jo1, where Z T q Y @P @Q m2 ¼ Dk exp ð4:2Þ ðxðtÞ; ZðtÞÞ þ ðxðtÞ; ZðtÞÞ dt; @S @x 0 k¼1 with @b @f @b @f @f þ þ Qþ Dk ¼ Pþ @x @S @S @x @S
and P; Q ; @a=@S; @a=@x; @b=@S; @b=@x; @f=@S and @f=@x are calculated Þ; Zðtþ ÞÞ; Qþ ¼ Q ðxðtþ Þ; Zðtþ ÞÞ: at the point ðxðtk Þ; Zðtk ÞÞ; Pþ ¼ Pðxðtþ k k k k fðS; xÞ is a sufficiently smooth function with grad fðS; xÞa0, and tk ðk 2 NÞ is the time of the kth jump. The proof of this lemma is referred to Simeonov and Bainov (1988). In the following, we suppose this periodic solution of system (2.3) with period T passes through the points Eþ 1 ðð1 bÞ x0 ; ð1 bÞhÞ 2 GH and E1 ðx0 ; hÞ 2 EC (GH; EC seen Figs. 6–8). As the expression and the period of this solution are unknown, we discuss the stability of this positive periodic solution by Lemma 4.1. In our case, mSx mS ; ðS; xÞ ¼ x 1 ; PðS; xÞ ¼ 1 S A þ Bx þ S A þ Bx þ S
aðS; xÞ ¼ bS; fðS; xÞ ¼ x h;
bðS; xÞ ¼ bx; ðxðTÞ; ZðTÞÞ ¼ ðx0 ; hÞ;
ðxðT þ Þ; ZðT þ ÞÞ ¼ ðð1 bÞx0 ; ð1 bÞhÞ: Then @P mxðBx þ AÞ ; ¼ 1 @S ðS þ Bx þ AÞ2 @a ¼ b; @S
@a ¼ 0; @x
@Q mSðS þ AÞ 1; ¼ @x ðS þ Bx þ AÞ2
@b ¼ 0; @S
@b ¼ b; @x
From Theorems 4.1 and 4.2, it follows that Theorem 4.3. Suppose that m41 þ A; 0ohox ; x0 oh, and S0 o1, then system (2.3) has a periodic solution of order one.
4.2. Stability of periodic solution of order one In the following, we analysis the stability of periodic solution of order one in system (2.3). Firstly, we give one lemma to discuss the stability of this positive periodic solution of system (2.3).
@a @f @a @f @f þ @S @x @x @S @x ; @f @f þQ P @S @x
@b @f @b @f @f þ þ Qþ Dk ¼ Pþ @x @S @S @x @S
@f ¼ 0; @S
@a @f @a @f @f þ @S @x @x @S @x @f @f þQ P @S @x
Q ðxðT þ Þ; ZðT þ ÞÞð1 bÞ Q ðxðTÞ; ZðTÞÞ mð1 bÞx0 1 A þ Bð1 bÞZ0 þ ð1 bÞx0 : ¼ ð1 bÞ2 mx0 1 A þ BZ0 þ x0
¼
@f ¼ 1; @x
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Set GðtÞ ¼ @P=@SðxðtÞ; ZðtÞÞ þ @Q =@xðxðtÞ; ZðtÞÞ, then Z T @P @Q m2 ¼ D1 exp ðxðtÞ; ZðtÞÞ þ ðxðtÞ; ZðtÞÞ dt @S @x 0 mð1 bÞx0 Z T 1 A þ Bð1 bÞZ0 þ ð1 bÞx0 GðtÞ dt : ¼ ð1 bÞ2 exp mx0 0 1 A þ BZ0 þ x0
order one is orbitally asymptotically stable if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1: mx0 1 A þ BZ0 þ x0
Because ðSðtÞ; xðtÞÞ is periodic solution of system (2.3), we RT know that 0 GðtÞ dto0 by the proof of Lemma 3.1, that is, RT expð 0 GðtÞ dtÞo1. Obviously, jm2 jo1 if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1; mx0 1 A þ BZ0 þ x0
4.3. Periodic solution of order two
and conditions of Theorem 4.3 hold. Therefore, this periodic solution of system (2.3) is stable if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1; mx0 1 A þ BZ0 þ x0 and conditions of Theorem 4.3 hold. Theorem 4.4. System (2.3) with the conditions of Theorem 4.3 has an order one periodic solution. Therefore, the periodic solution of
Fig. 9. Existence of order two periodic solution.
0.55
x (t)
S (t)
0.6
0.5 0.45 0.4
0
5
10 t
15
20
0.45
0.45
0.4
0.4
0.35
0.35
x (t)
0.65
In this subsection, we discuss existence of periodic solution of system (2.3) of order two if periodic solution of order one is unstable. If ðS; xÞ is a periodic solution of (2.3), then ðS 0 ; x 0 Þ 2 NDGH and ðS 1 ; x 1 Þ 2 MDEC . We can obtain that S 0 S 1 because of Sþ ¼ ð1 bÞS by the third equation of system (2.3). Let ðS; xÞ is the arbitrary solution of system (2.3), the first interaction point of trajectory and the set Mðx ¼ hÞ is ðS1 ; hÞ and the corresponding interaction points are ðS2 ; hÞ; ðS3 ; hÞ; . . ., respectively. Therefore, under the effect of impulsive function I, the corresponding point þ þ after pulse are ðSþ 1 ; ð1 bÞhÞ; ðS2 ; ð1 bÞhÞ; ðS3 ; ð1 bÞhÞ; . . .. We know that dx=dt40 for S4½ð1 bÞBh þ A=ðm 1Þ and dx=dto0 for So½ð1 bÞBh þ A=ðm 1Þ by the qualitative analysis of system (2.3). So we consider the following cases: Case 1: It is easily obtained that the trajectory jumps to the point ðSþ 1 ; ð1 bÞhÞ from the point ðS1 ; hÞ if the periodic solution is in the region fðS; xÞjS4½ð1 bÞBh þ A=ðm 1Þg. Without loss of generality, we let S1 oS 1 , then Sþ 1 oS 0 . The trajectory from the point ðSþ 1 ; ð1 bÞhÞ will interact the set M at the point ðS2 ; hÞ and then jump to the point ðSþ 2 ; ð1 bÞhÞ; . . .. By the above discussion, there is one of the following sequences: Case (a) S1 S2 S3 S 1 and Case (b) S 1 ZS1 ZS2 ZS3 Z . From the sequences we can know that the trajectories tend to be periodic. Since the sequence is monotone, it is clearly that order two periodic solution does not exist in the case by Definition 2.3. Case 2. If the set N is in the region fðS; xÞjSo½ð1 bÞBh þ A=ðm 1Þg and system (2.3) has unstable periodic solution of order one, without loss of generality, let S1 4S0 (see Fig. 9), then we can obtain that S0 S2 oS1 or S2 S0 oS1. Therefore, there exists a periodic solution of order two if S0 ¼ S2 holds. If S0 oS2 oS1 holds, we can obtain 0oS0 oS2 o oS2k oS2kþ1 o oS3 oS1 o1, while if S2 oS0 oS1 holds, then we have 0o oS2k oS2ðk1Þ o oS2 oS0 oS1 oS3 o oS2kþ1 o o1. Similar to the above discussion and the proof of Proposition 3.2 of Jiang et al. (2007), we can obtain that there is no periodic solution of order k ðkZ3Þ in system (2.3) and the system is not chaotic.
0.3
0.3
0.25
0.25 0.2
0.2 0
5
10 t
15
20
0.4
0.45
0.5
0.55 S (t)
Fig. 10. Time series and portrait phase of system (2.3) when m ¼ 20; A ¼ 9; b ¼ 0:5; B ¼ 2; h ¼ 0:5; Sð0Þ ¼ 0:4, and xð0Þ ¼ 0:2.
0.6
0.65
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0.4
0.4
0.5
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0.35
0.4
x (t)
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0.3 0.25
0.3
0.25
0.2 0
5
10 t
15
20
0.3
0.2 0
5
10 t
15
20
0.3
0.45
0.45
0.75
0.44
0.44
0.7
0.43
0.43
0.65
0.42
0.42
0.5
x (t)
0.8
x (t)
S (t)
Fig. 11. Time series and portrait phase of system (2.3) when m ¼ 20; A ¼ 9; B ¼ 2; b ¼ 0:5; h ¼ 0:45ox ¼
0.55
0.41
5
10 t
15
0.4
0.39
0.39
10 21 ;
Sð0Þ ¼ 0:4, and xð0Þ ¼ 0:2.
0.38 0
20
0.6
0.41
0.4 0.38 0
0.5 S (t)
0.6
0.4
5
10 t
15
0.5 0.55 0.6 0.65 0.7 0.75 0.8
20
S (t)
Fig. 12. Time series and portrait phase of system (2.3) when m ¼ 20; b ¼ 0:15; A ¼ 9; B ¼ 2; h ¼ 0:45ox ; Sð0Þ ¼ 0:8, and xð0Þ ¼ 0:4.
0.552 x (t)
S (t)
0.55 0.548 0.546 0.544 0.542
0.4
0.4
0.398
0.398
0.396
0.396
x (t)
0.554
0.394
0.394
0.392
0.392
0.39 0
5
10
15
20
0.39 0
5
10
15
t
t
20
0.542 0.544 0.546 0.548 0.55 0.552 0.554 S (t)
Fig. 13. Time series and portrait phase of system (2.3) when m ¼ 20; b ¼ 0:02; A ¼ 9; B ¼ 2; h ¼ 0:4ox ; Sð0Þ ¼ 0:545, and xð0Þ ¼ 0:39.
5. Numerical simulation Now we consider the following example: 9 8 20Sx > > _ ¼1S > > S ; > = > 9 þ 2x þ S > > xoh; > > 20S > > < x_ ¼ x ; 1 ; > 9 þ 2x þ S ) > > > DS ¼ bS; > > > x ¼ h: > > > : Dx ¼ bx;
ð5:1Þ
10 , A ¼ 9; B ¼ 2; m ¼ 20; b ¼ 0:15; S0 ¼ 0:4; x0 ¼ 0:2; h ¼ 0:45ox ¼ 21 then system (2.3) has a periodic solution of order one, which verifies theoretical results in this paper. Fig. 12 shows that the trajectories of system (2.3) starting for the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1g will enter the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 Þ 1g, hence, system (2.3) has also periodic solution of order one. Fig. 13 gives the series times and phase portrait for S0 4½ð1 bÞBh þ A=ðm 1Þ and shows that the trajectory tends to a periodic trajectory.
6. Conclusion In numerical simulation, let A ¼ 9; m ¼ 20; B ¼ 2. By calculating, we know m41 þ A, holds. If h ¼ 0:54x ¼ ðm A 1Þ=ðm þ B 10 ; b ¼ 0:5; S0 ¼ 0:4; x0 ¼ 0:2ox , then the time series and 1Þ ¼ 21 phase portrait can be seen in Fig. 10. By analysis of Section 4, we know that no impulse occurs for system (2.3) if h4x holds. As shown in Fig. 10, numerical simulation also suggests that system (2.3) with the coefficients above admits no impulse to occur. By Theorem 4.1, we know that system (2.3) has a periodic solution of order one under conditions of Theorem 4.3. As shown in Fig. 11, If
In this paper, we built a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control. Firstly, we investigated qualitative characteristic of the system without impulsive effect, and obtained sufficient condition of globally asymptotically stable of the system. We obtained the system with impulsive state feedback control had periodic solution of order one, and sufficient conditions for existence and stability of periodic solution of order one. The case, in which it
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was possible that there was a periodic solution of order two, was also given in Section 4.3. The results show that the chemostat with Beddington–DeAnglis uptake function and impulsive state feedback control tends to a stable state or periodic, and the behavior of impulsive state feedback control on the concentrate of microorganism plays an important role on the periodic or stable state of system (2.3). When the concentrate of microorganism reaches an appropriate critical value, a state feedback measure for controlling concentrate of microorganism is taken. According to the analysis of Section 4, this measure is effective based on the fact that the system has stable periodic solution under some conditions. According to the theoretical results, the production of the microorganism will tend to a stable production level or the production will be periodic. The key to the production by applying the system with feedback control is to give the suitable feedback state (the value of h) and the control parameters (b) according to practice. It is seen from Figs. 11–13 that there are positive periodic trajectories under the impulsive state feedback control. Therefore, the periodic production of the microorganism can be achieved if the value of h and the suitable initial value of the substrate and the microorganism are taken. Indeed, we would like to point out that the model (2.3) must exist deficiency which we need further study.
Acknowledgments We would like to thank the anonymous referees and the editor for their careful reading of the original manuscript and their kind comments and valuable suggestions. References Armstrong, R.A., McGehee, R., 1980. Competitive exclusion. Am. Nat. 115, 151–170. Bainov, D.D., Simeonov, P.S., 1993. Impulsive Differential Equations: Periodic Solutions and Applications. Wiley, New York. Beddington, J.R., 1975. Mutual interference between parasites or predators and its effect on searching efficiency. J. Animal Ecol. 44, 331C40. Butler, G.J., Wolkowicz, G.S.K., 1985. A mathematical model of the chemostat with a general class of functions describing nutrient uptake. SIAM J. Appl. Math. 45, 138–151. De Leenheer, P., Smith, H.L., 2003. Feedback control for chemostat model. J. Math. Biol. 46, 48–70. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V., 1975. A model for trophic interaction. Ecology 56, 881C92.
Flegr, J., 1997. Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems. J. Theor. Biol. 188, 121–126. Griffel, D.H., 1981. Applied Functional Analysis. Ellis Horwood Ltd., Chichester, England. Grover, J.P., 1997. Resource Competition. Chapman & Hall, London. Hansen, S., Hubbell, S.P., 1980. Single-nutrient microbial competition: agreement between experimental and theoretical forecast outcomes. Science 207, 1491–1493. Hsu, S., 1978. Limiting behavior for competing species. SIAM J. Appl. Math. 34, 760–763. Hsu, S., Hubbell, S., Waltman, P., 1977. A mathematical theory of single-nutrient competition in continuous cultures of micro-organisms. SIAM J. Appl. Math. 32, 366–383. Jiang, G., Lu, Q., Qian, L., 2007. Complex dynamics of a Holling II prey– predator system with state feedback control. Chaos Soliton Fractal 31, 448–461. Jiao, J., Chen, L., 2008. Global attractivity of a stage-structure variable coefficients predator–prey system with time delay and impulsive perturbations on predators. Int. J. Biomath. 1, 197–208. Jiao, J., Meng, X., Chen, L., 2007. Stage-structured Holling mass defence predator–prey model with impulsive perturbations on predators. Applied Mathematics and Computation 189, 1448–1458. Laksmikantham, V., Bainov, D.D., Simeonov, P.S., 1989. Theory of Impulsive Differential Equations. World Scientific, Singapore. Levins, R., 1979. Coexistence in a variable environment. Am. Nat. 114, 765–783. Meng, X., Chen, L., 2008. Permanence and global stability in an impulsive Lotka–Volterra N-species competitive system with both discrete delays and continuous delays. Int. J. Biomath. 1, 179–196. Meng, X., Jiao, J., Chen, L., 2007. Two profitless delays for an SEIRS epidemic disease model with vertical transmission and pulse vaccination. Chaos, Solitons Fractals, doi: 10.1016/j.chaos.2007.09.096. Meng, X., Jiao, J., Chen, L., 2008. The dynamics of an age structured predator–prey model with disturbing pulse and time delayeds. Nonlinear Anal Real World Appl 9, 547–561. Shi, R., Chen, L., 2007. Stage-structured impulsive SI model for pest management. Discrete Dynamics in Nature and Society, doi: 10.1155/2007/97608. Shi, R., Chen, L., 2008a. Staged-structured Lotka–Volterra predator–prey models for pest management. Appl. Math. Comput, doi:10.1016/j.amc.2008. 04.032. Shi, R., Chen, L., 2008b. A predator–prey model with disease in the prey and two impulses for integrated pest management. Appl. Math. Modelling, doi:10.1016/ j.apm.2008.06.001. Simeonov, P.E., Bainov, D.D., 1988. Orbital stability of periodic solutions autonomous systems with impulse effect. Int. J. Syst. Sci. 19, 2562–2585. Tang, S., Chen, L., 2004. Modelling and analysis of integrated management strategy. Discrete contin. Dyn. Syst. Ser. B 4, 759–768. Tilman, D., 1982. Resource Competition and Community Structure. Princeton University Press, Princeton, NJ. Wolkowicz, G.S.K., Lu, Z., 1992. Global dynamics of a mathematical model of competition in the chemostat: general response function and differential death rates. SIAM J. Appl. Math. 52, 222–233. Zeng, G., 2007. Existence of periodic solution of order one of state-depended impulsive differential equations and its application in pest control. J. Biomath. 22, 652–660 (in China). Zeng, G., Chen, L., Sun, L., 2006. Existence of periodic solution of order one of planer impulsive autonomous system. J. Comput. Appl. Math. 186, 466–481.
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Journal of Theoretical Biology journal homepage: www.elsevier.com/locate/yjtbi
Periodic solution of a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control$ Zuxiong Li a,, Tieying Wang a,b, Lansun Chen a a b
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, PR China College of Science, Dalian Nationalities University, Dalian 116605, PR China
a r t i c l e in fo
abstract
Article history: Received 17 January 2009 Received in revised form 17 July 2009 Accepted 17 July 2009 Available online 23 July 2009
In this paper, a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control is considered. We obtain sufficient conditions of the global asymptotical stability of the system without impulsive state feedback control. We also obtain that the system with impulsive state feedback control has periodic solution of order one. Sufficient conditions for existence and stability of periodic solution of order one are given. In some cases, it is possible that the system exists periodic solution of order two. Our results show that the control measure is effective and reliable. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Feedback control Globally asymptotical stability Periodic solution Qualitative analysis Impulsive effect
1. Introduction The chemostat is a basic piece of laboratory apparatus, yet it has begun to occupy an increasingly central role in ecological studies. The ecological environment created by a chemostat is one of the few completely controlled experimental systems for testing microbial growth and competition. As a tool in biotechnology, the chemostat plays an important role in bio-processing. Chemostat has widely been used to study bacterial metabolism, population genetics, and plasmid stability, mostly because these are the simplest and most easily constructed types of continuous cultures. Many papers have investigated the mathematical models on the culture of the microorganisms. For example, mathematical (Armstrong and McGehee, 1980; Butler and Wolkowicz, 1985; Hsu, 1978; Hsu et al., 1977; Levins, 1979; Wolkowicz and Lu, 1992) and experimental (Grover, 1997; Hansen and Hubbell, 1980; Tilman, 1982) models exhibit the competitive exclusion principle only one species survives. Several modifications of the chemostat have been made to ensure the coexistence of species on a single nutrient (De Leenheer and Smith, 2003; Flegr, 1997; Grover, 1997; Tilman, 1982). However, there are a lot of factors such as temperature, dissolved oxygen content affecting the growth and reproduction of the microorganisms in the process of bio-reacts.
$
Supported by National Natural Science Foundation of China (no. 10771179).
Corresponding author. Tel.: +8615074417427.
E-mail addresses: [email protected] (Z. Li), [email protected] (T. Wang), [email protected] (L. Chen). 0022-5193/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jtbi.2009.07.016
In some cases, it is necessary to regulate the bio-react according to the concentration of microorganisms. The volume of the chemostat can be controlled either by using a pump (seen on the left of Fig. 1) or an overflow system (seen on the right of Fig. 1). In some cases, the high concentration of the microorganisms will often have caused a series of adverse effects, such as the production inhibition. To regulate the concentration of the microorganism, we modify overflow system of the chemostat into the right system of Fig. 2. When the concentration which can be measured by optoelectronic devices and other ways, reaches a certain threshold, it can be regulated through replenishing water to reduce the concentration of the microorganisms. Impulsive equation has become a widely concerned subject in recent years, particularly as applied to biological impulsive equation model more appropriate for the actual reality. Based on the principle of the right system of Fig. 2, we propose a mathematical model concerning the chemostat with Beddington– DeAnglis uptake function and impulsive state feedback control. As for some models with impulsive effect Bainov and Simeonov (1993) Meng et al. (2008, 2007), Meng and Chen (2008), Jiao et al. (2007), Jiao and Chen (2008), Shi and Chen (2007, 2008a, 2008b) have investigated and been well studied. As for some models with impulsive state feedback control, Zeng (2007), Jiang et al. (2007) and Tang and Chen (2004) discussed prey-predator models and obtained the complete expression of the periodic solution. However, so far, few papers have discussed the chemostat system using the impulsive differential equation with state feedback control.
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Feed Pump
Effluent Pump
Feed
Feed Pump Effluent
Feed
Level maintained by overflow of the effluent through a port on the side of the reactor
Level maintained by setting effluent pump to run faster than the feed pump
Effluent
Fig. 1. Two type of chemostat.
Feed Pump
Feed Pump Water
Feed
Feed
Control Value Level maintained by overflow of the effluent through a port on the side of the reactor
Level maintained by overflow of the effluent through a port on the side of the reactor
Effluent
Effluent Photo Cell Light Source
Fig. 2. The sketch map of chemostat and modified chemostat.
In this paper, we will prove that the model without impulsive effect has an globally asymptotically stable positive equilibrium under certain condition. we will also discuss the existence and stability of periodic solution of the chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control according to the existence criteria (Zeng, 2007) and the stability theorem (Simeonov and Bainov, 1988) of periodic solution of the general impulsive autonomous system. This paper is organized as follows. The model and some preliminary results are presented in the next section. The qualitative analysis of the system without impulsive effect is given in Section 3. In Section 4, the existence and stability of periodic solution of order one of differential equation with impulsive state feedback control are investigated. Numerical simulations are given in Section 5. Finally, some conclusion and biological discussions are provided in Section 6.
2. The model and preliminaries The general model of continuously culturing microorganism in a chemostat is described by the following differential equation (De Leenheer and Smith, 2003): 8 < S_ ¼ DðS SÞ 1f ðSÞx; in d ð2:1Þ : x_ ¼ f ðSÞx Dx:
In model (2.1), S denotes the concentration of substrate at time t and x denotes the concentration of the microorganism in the chemostat at time t; D and Sin 40 denote, respectively, the dilution rate of the chemostat and the concentration of the input substrate; d is the yield ; f is uptake function and we take f ðSÞ ¼ mm S=ðS þ Bx þ AÞ, that is, DeAngelis et al. (1975) and Beddington (1975) uptake function. Then (2.1) has the following form: 8 mm Sx > > ; > S_ ¼ DðSin SÞ < dðS þ Bx þ AÞ ð2:2Þ mm Sx > > > : x_ ¼ ðS þ Bx þ AÞ Dx: where mm 40 is called the maximal specific growth rate of the microorganisms, km 40 is the saturation constant, AZ0; BZ0 are constant. In some cases, if the concentration of the microorganisms is lower than a critical value, it is not necessary to adopt control measures. But the high concentration of the microorganisms will often have caused a series of adverse effects, such as the substrate inhibition, higher viscosity caused by the mass transfer efficiency, and so on. What is especially serious, many secondary metabolite’s production of microbial production are suppressed by highly concentrated glucose, the carbohydrate and the azotic compound
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degradation product. In order to overcome the above shortcomings, we often use fed-batch culture if the concentration of the substrate is low. In the process of fermentation, when the microorganism begins to consume the substrate, we take some measures to supplement certain material in the culture system so that the concentration of the culture medium maintains to retain a certain scope for a long time. To achieve increasing production volume, product concentration and product yield, we always maintain the growth of microorganisms in the scope. In this system, when the flow velocity of the culture medium is lower than the growth rate of microorganism, the concentration of microorganism increases, once the concentration reaching to a critical value which can be detected by optoelectronic devices, we should take measures to decrease the concentration of microorganism. Once the concentration is decreased, it takes a period of time to reach the critical value once more. Therefore, the feedback control can be carried out by the impulsive state control. In this paper, we consider that the substrate with the microorganism is discharged impulsively due to the added water and other regulator solution on the basis of continuous cultivation in the chemostat when x reaches the critical value (denoted by xm ). Therefore, by simplifying, (2.2) can be modified as follows by introducing the impulsive state feedback control: 8 9 mSx > > _ ¼1S > > S ; > > ðS þ Bx þ AÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x S þ Bx þ A ð2:3Þ ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40; 0 0 where 0obo1 is constant and is also the fraction of the concentration of microorganism that decreases due to the feedback control when the concentrate x reach h. By Observing, if A40; B ¼ 0, then system (2.3) reduces to a chemostat model with Monod growth rate and impulsive state feedback control 8 9 mSx > > > ;> > S_ ¼ 1 S > ðS þ AÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x SþA ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40; 0 0 if A ¼ 0; B40, the system (2.3) reduces to a chemostat model with ratio-dependent growth rate and impulsive state feedback control 8 9 mSx > > > ;> > S_ ¼ 1 S > ðS þ BxÞ = > > > xoh; > > > mS > > ; 1 ; > < x_ ¼ x S þ Bx ) > > D S ¼ bS; > > > x ¼ h; > > Dx ¼ bx; > > > > : Sð0Þ ¼ S 40; xð0Þ ¼ x 40: 0
0
In this paper, we mainly discuss the existence and stability of periodic solution of system (2.3) by the existence criteria and stability criteria of the general impulsive autonomous system. Before introducing the existence criteria, we give some topological definitions about impulsive differential equations. Deftinition 2.1 (Laksmikantham et al., 1989). An triple ðX; p; Rþ Þ is said to be a semi-dynamical system if X is metric space, Rþ is the
25
set of all non-negative reals and p : X Rþ -X is a continuous function such that (i) pðx; 0Þ ¼ x for all x 2 X; (ii) pðpðx; tÞ; sÞ ¼ pðx; t þ sÞ for all x 2 X and t; s 2 Rþ . We denote some times a semi-dynamical system ðX; p; Rþ Þ by ðX; pÞ. For any x 2 X, the function px : Rþ -X defined by px ðtÞ ¼ pðx; tÞ is continuous and we call px the trajectory of x. The set C þ ðxÞ ¼ fpðx; tÞjt 2 Rþ g; is called the positive orbit of x. For any subset M of X, we let Mþ ðxÞ ¼ C þ ðxÞ \ M x
and
M ¼ GðxÞ \ M x;
and
Gðx; tÞ ¼ fyjpðy; tÞ ¼ xg;
where GðxÞ ¼ [fGðx; tÞjt 2 Rþ g
is the attainable set of x at t 2 Rþ . Finally we set MðxÞ ¼ Mþ ðxÞ [ M ðxÞ. Deftinition 2.2 (Laksmikantham et al., 1989). An impulsive semidynamical system ðX; p; M; IÞ consists of a semi-dynamical system ðX; pÞ 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 2 X is a limit point of MðxÞ, (ii) ½tjGðx; tÞ \ Ma is a closed subset of Rþ . We write N ¼ IðMÞ ¼ fy 2 Xjy ¼ IðxÞ; x 2 Mg and for any x 2 X, IðxÞ ¼ xþ . We call M the set of impulses, I the impulsive function. We define a function F : X-Rþ [ f1g as following: ( 1 if M þ ðxÞ ¼; FðxÞ ¼ 2M for 0otos and pðx; sÞ 2 M; s if pðx; tÞ= here we call s the time without impulse of x, that is to say s is the first time when pðx; 0Þ hits M. Deftinition 2.3 (Laksmikantham et al., 1989). Let ðX; p; M; IÞ be an impulsive semi-dynamical system and let x 2 X and x= 2M. The trajectory of x is a function p~ x defined on subset ½0; sÞ of Rþ (s may be 1) to X inductively as following:
p~ x ðtÞ ¼ pðxþnþ1 ; tÞ;
tn1 t tn ;
where fxn g is the sequence of impulse points of x, which satisfied
pðxþn1 ; Fðxþn1 ÞÞ ¼ xn . Ptn is the sequence of time of impulses þ relative to fxn g, tn ¼ n1 k¼0 Fðxk Þ. Deftinition 2.4 (Laksmikantham et al., 1989). A trajectory p~ x is said to be periodic of period t and order k if there exist positive integers mZ1 and kZ1 such that k is the smallest integer for Pmþk1 þ Fðxþ which xþ m ¼ xmþk and t ¼ i¼m i Þ. Theorem 2.1 (Brouwer’s fixed-point theorem, Griffel, 1981). Every continuous mapping of a closed bounded convex set in Rn into itself has a fixed point. In the following we give the existence criteria for an impulsive autonomous system with state-dependent proved by means of Theorem 2.1 (Zeng et al., 2006).
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To consider the following general autonomous impulsive differential equations: 8 9 dxðtÞ > > > > ¼ Pðx; yÞ; > = > > dt > > ðx; yÞ= 2M; > > > < dyðtÞ ; ¼ Q ðx; yÞ; > ð2:4Þ dt > ) > > > > > Dx ¼ I1 ðx; yÞ; ðx; yÞ 2 M: > > > : Dy ¼ I2 ðx; yÞ; Here ðx; yÞ 2 R2 , and P; Q ; I1 ; I2 are all functions mapping R2 into R, M R2 is the set of impulse, and we assume: (H2.1) Pðx; yÞ; Q ðx; yÞ are all continuous with respect to x; y in R2 . (H2.2) M R2 is a line, I1 ðx; yÞ and I2 ðx; yÞ are linear functions of x and y.
System (3.1) only has a boundary equilibrium (1,0) in region Rþ ¼ fðS; xÞjSZ0; xZ0g if 0om 1 þ A. The Jacobian matrix at equilibrium E ¼ ðS; xÞ is given by 0 B B JðS;xÞ ¼ B @
1
mxðBx þ AÞ
ðS þ Bx þ AÞ2 mðBx þ AÞ
ðS þ Bx þ AÞ2
1 mSðS þ AÞ ðS þ Bx þ AÞ2 C C C: mSðS þ AÞ A 1 2 ðS þ Bx þ AÞ
We now give some results about system (3.1). Theorem 3.1. If m41 þ A, then Bð1; 0Þ is a saddle point, the positive equilibrium AðS ; x Þ is globally asymptotically stable node and limt-1 SðtÞ ¼ S ; limt-1 xðtÞ ¼ x , where S ¼
For each point Sðx; yÞ 2 M, we define I : R2 -R2 :
AþB ; mþB1
x ¼
mA1 : mþB1
IðSÞ9Sþ ¼ ðxþ ; yþ Þ 2 R2 ; xþ ¼ x þ I1 ðx; yÞ;
yþ ¼ y þ I2 ðx; yÞ:
Obviously N ¼ IðMÞ is also a line of R2 or a subset of a line, and we assume that N \ M ¼. From Definition 2.2 we know (2.4) is an impulsive semi-dynamical system. The following theorem gives the conditions on which (2.4) has a periodic solution of order one defined by Definition 2.5. Theorem 2.2 (Zeng, 2007). If system (2.4) satisfies assumptions ðH2:1Þ and ðH2:2Þ, there exists a bounded closed simply connected region D which has following properties: (i) There is no singularity in it and the boundary @D of D satisfies ðD @DÞ \ M ¼. (ii) L1 ¼ D \ M cannot be tangent with trajectories of (2.4) except at end-points and IðL1 Þ D. (iii) Trajectories with initial point in @D L1 will enter into interior of D, then there must exist a periodic solution of system (2.4) of order one in region D. 3. Qualitative analysis of system (2.3) without impulsive effect In this section, we will study the qualitative characteristic of system (2.3) without the impulse effect. If no impulsive effect is introduced, then system (2.3) is 8 mSx > _ > < S ¼ 1 S ðS þ Bx þ AÞ; ð3:1Þ > mS > : x_ ¼ x 1 : S þ Bx þ A
Proof. Firstly, by 1 0 m 1 B Aþ1 C C; Jð1;0Þ ¼ B m A @ 1 0 Aþ1 we know that the eigenvalues are 1 and m=ðA þ 1Þ 1. If m41 þ A, then Bð1; 0Þ is a saddle point. We can calculate the Jacobian matrix at the positive equilibrium AðS ; x Þ as follow: 1 0 mx ðBx þ AÞ mS ðS þ AÞ 1 C B B ðS þ Bx þ AÞ2 ðS þ Bx þ AÞ2 C C JðS ;x Þ ¼ B C B mx ðBx þ AÞ mS ðS þ AÞ A @ 1 2 2 ðS þ Bx þ AÞ ðS þ Bx þ AÞ 0 B B ¼B @
1
mx ðBx þ AÞ
ðS þ Bx þ AÞ2 mx ðBx þ AÞ
ðS þ Bx þ AÞ2
1 S þ A ðS þ Bx þ AÞ C C C: Bx A S þ Bx þ A
Therefore, the eigenvalue equation is " # mx ðBx þ AÞ Bx Bx l2 þ 1 þ þ lþ 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ þ
mx ðBx þ AÞ ðS þ Bx þ AÞ2
¼ 0:
Obviously,
l1 l2 ¼
From 1S
mSx ¼ 0; ðS þ Bx þ AÞ
and mS 1 ¼ 0; x S þ Bx þ A we can obtain system (3.1) has a boundary equilibrium ð1; 0Þ and a positive equilibrium ðS ; x Þ if m41 þ A, where S ¼
AþB ; mþB1
x ¼
mA1 : mþB1
Bx mx ðBx þ AÞ þ 40; S þ Bx þ A ðS þ Bx þ AÞ2 "
l1 þ l2 ¼ 1 þ
mx ðBx þ AÞ ðS þ Bx þ AÞ
þ 2
# Bx o0; S þ Bx þ A
and because " #2 mx ðBx þ AÞ Bx Bx D¼ 1þ þ 4 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ #" #2 mx ðBx þ AÞ mx ðBx þ AÞ Bx 1 Z0; þ ðS þ Bx þ AÞ2 ðS þ Bx þ AÞ2 S þ Bx þ A
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then AðS ; x Þ is asymptotically stable node. Here l1 ; l2 are two roots of " # mx ðBx þ AÞ Bx Bx l2 þ 1 þ þ lþ 2 S þ Bx þ A S þ Bx þ A ðS þ Bx þ AÞ þ
mx ðBx þ AÞ ðS þ Bx þ AÞ2
¼ 0:
Next, we prove system (3.1) is uniformly bounded. Let the straight line S ¼ 1 and 0 intersects the straight line x ¼ 0 at the points Bð1; 0Þ; Oð0; 0Þ, respectively. Let the straight line x ¼ ðm A 1Þ=B intersects the straight line S ¼ 1 and 0 at the points Cð1; ðm A 1Þ=BÞ; Dð0; ðm A 1Þ=BÞ, respectively. By some calculations, we can obtain that dS=dtjODðS¼0Þ 40, dx=dtjx¼ðmA1Þ=B o0; dS=dtjODðS¼1Þ o0 and because dx=dtjx¼0 ¼ 0, the trajectory of system (3.1) cannot pass through the straight line x ¼ 0. Hence, system (3.1) is uniformly bounded. We denote the region O which is formed by straight lines BC ; CD; DO; OB, that is, the region O is composed of the &BCDOB. (See Fig. 3.) Finally, we prove the global stability of system (3.1). Firstly, we will give the following lemma. & Lemma 3.1. Suppose GðTÞ ¼ ðSðtÞ; xðtÞÞ is a periodic orbit with T of system (3.1), R is the set which consists of all the points in phase plane G.
Z
T
¼ 0
27
! BmxS mS 1 þ 1 dt; ðS þ Bx þ AÞ2 S þ Bx þ A S þ Bx þ A mxðBx þ AÞ
for SðtÞ is a function with period T, so we can calculate and obtain Z T Z T mS d lnðxðtÞÞ ¼ 0: 1 dt ¼ S þ Bx þ A 0 0
Therefore, N¼
Z
T
1 0
! BmxS dto0: ðS þ Bx þ AÞ2 S þ Bx þ A mxðBx þ AÞ
The proof of Lemma 3.1 is complete. & From the above discussions, we know that A is locally stable. According to Lemma 3.1, we can obtain that if there exists periodic solution ðSðtÞ; xðtÞÞ around AðS ; x Þ, then it is stable for any periodic solution. This is impossible. According to Poincare–Bendixson theorem, the limit set of all orbits must be equilibrium point A. This implies that AðS ; x Þ is globally asymptotically stable in O. This completes the proof of Theorem 3.1. & The vector graph of (3.1) can be seen in Fig. 4.
1
Denote Z T @P @Q N¼ ðSðtÞ; xðtÞÞ þ ðSðtÞ; xðtÞÞ dt; @S @x 0
0.8
where S_ ¼ PðSðtÞ; xðtÞÞ; x_ ¼ Q ðSðtÞ; xðtÞÞ. If the conditions of Theorem 3.1 hold, then we can obtain No0. x
0.6
Proof. By Z T @P @Q ðSðtÞ; xðtÞÞ þ ðSðtÞ; xðtÞÞ dt N¼ @S @x 0 Z
T
¼
1
ðS þ Bx þ AÞ2
þ
mSðS þ AÞ ðS þ Bx þ AÞ2
0.2
! 1 dt
0 0
0.2
0.4
0.6
0.8
1
S Fig. 4. Vector graph of system (3.1) for m41 þ A.
0.3 0.2 0.1 x
0
mxðBx þ AÞ
0.4
0 –0.1 –0.2 0.5
0.6
0.7
0.8
0.9
1
S Fig. 3. The region O is composed of the &BCDEB.
Fig. 5. Vector graph of (3.1) for mo1 þ A.
1.1
1.2
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Z. Li et al. / Journal of Theoretical Biology 261 (2009) 23–32
By the Jacobian matrix at equilibrium (1,0), we easily obtain the following theorem: Theorem 3.2. If mo1 þ A, then the boundary equilibrium (1,0) is local asymptotical stability.
From the third equation of (2.3), we know that Sþ ¼ ð1 bÞS if x ¼ h and furthermore SG ¼ ð1 bÞSG SG . According to the value of SG and SH , we have the following cases: Case 1: SG ½ð1 bÞBh þ A=ðm 1Þ and ½ð1 bÞBh þ A=ðm 1Þ SH SHM (see Fig. 6), where SHM is the solution of the equation
The vector graph of (3.1) can be seen in Fig. 5. 1 SHM 4. Existence and stability of periodic solutions
that is,
4.1. Existence of periodic solution of order one
SHM ¼
mSHM ð1 bÞh ¼ 0; A þ Bð1 bÞh þ SHM
1 A ð1 bÞhðm þ BÞ þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ð1 bÞhðm þ BÞ þ A 12 þ 4ðA þ Bð1 bÞhÞ 2
:
From discussions of Section 3, we can see that ðS ; x Þ is a stable node when m4A þ 1. For the initial points which satisfy xð0Þox and dS=dtjðS0 ;x0 Þ Z0 if hZx , then all the solutions of (2.3) tend to the equilibrium ðS ; x Þ and no impulsive will occur. It is not necessary to control by impulsive control. So we mainly pay attention to the case that the following assumption holds: ðHÞ hox ; x0 ox and Sð0Þ 1. We apply the existence criteria (Theorem 2.1 of Zeng, 2007) to prove periodic solution of system (2.3) of order one. We first need to construct a closed region such that all the solutions of (2.3) enter the closed region and retain there. Without loss of generality, we will illustrate the ideas by using Figs. 6–8. From Figs. 6–8, we can see that the line x ¼ h interacts the isoclinal line mS=ðS þ Bx þ AÞ 1 ¼ 0, that is dx=dt ¼ 0, at the point GððBh þ AÞ=ðm 1Þ; hÞ, and interacts dS=dt ¼ 1 S mSx=ðS þ Bx þ AÞ ¼ 0 at the points DðSD ; hÞ. Here SD is the root of the equation 1 SD
mSD h ¼ 0: SD þ Bh þ A
The impulsive set MDEC ; EC ¼ fðS; xÞjðBh þ AÞ=ðm 1Þ S 1g. The impulsive functions I1 and I2 map the impulsive set M as GH ¼ fðS; xÞj½ð1 bÞðBh þ AÞ=ðm 1Þ S 1 b; N ¼ IðMÞDGH, x ¼ ð1 bÞhg, here G ¼ JðSG ; ð1 bÞhÞ, H ¼ HðSH ; ð1 bÞhÞ, SG ¼ ð1 bÞðBh þ AÞ=ðm 1Þ; SH ¼ 1 b.
Fig. 6. Illustration of (2.3) for the case hox ; x0 oh; SG SF ; SF SH SHM .
Fig. 7. Illustration of (2.3) for the case hox ; x0 oh; SG oSH o½ð1 bÞBh þ A= ðm 1Þ.
Fig. 8. Illustration of (2.3) for the case hox ; x0 oh; ½ð1 bÞBh þ A= ðm 1Þo SG oSH oSHM .
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Case 2: SG oSH o½ð1 bÞBh þ A=ðm 1Þ (Fig. 7). Case 3: ½ð1 bÞBh þ A=ðm 1ÞoSG oSH oSHM (Fig. 8). Theorem 4.1. Suppose that m41 þ A; 0ohox ; x0 oh, S0 þ mS0 x0 = ðA þ Bx0 þ S0 Þ 1 and S0 o1, then system (2.3) has a periodic solution of order one. Proof. Under the condition of theorem, the trajectories of system (2.3) staring from the region shown in case 1 must interact with the segment EC . Next, we construct the closed region O1 . From the qualitative characteristic of system (3.1), we know that dS=dto0 for S ¼ 1, dx=dt ¼ 0 for x ¼ 0, dx=dto0 for x ¼ ð1 bÞh in line segment IF and dS=dt40 for S ¼ 0. The straight line x ¼ ð1 bÞh interacts S ¼ 0 and mS=ðA þ Bx þ SÞ 1 ¼ 0 at the points Ið0; ð1 bÞhÞ and Fð½ð1 bÞBh þ A=ðm 1Þ; ð1 bÞhÞ, respectively. Therefore, we have BC ; CE; EF ; FG; GI; IO, and OB. We can see that the trajectories of system (2.3) enter and retain the region O1 formed by BC ; CE; EF ; FG; GI; IO, and OB (Fig. 6). By Theorem 2.2, we can obtain that system (2.3) has a periodic solution of order one if the conditions of theorem and case 1 hold. Next, we will prove that system (2.3) has a periodic solution of order one if the conditions of theorem and case 2 hold. We now construct the closed region O2 . Similar to the discussions of case 1, we can easily obtain that the closed region BC ; CE; EF ; FH; HG; GI; IO and OB (see Fig. 7). By Theorem 2.2, we know that system (2.3) has a periodic solution of order one if the conditions of theorem and case 2 are satisfied. Finally, we prove that system (2.3) has a periodic solution of order one if the conditions of theorem and case 3 are satisfied. We construct the closed region consisted of ^ and IC (see in Fig. 8). Here HI ^ is a trajectory CE; EF ; FG; GH; HI of system (2.3) which passes through the point H and interacts the isocline equation dS=dt ¼ 0 at the point I. The vertical line passing through the point I interacts the line x ¼ h at the point C. By the qualitative characteristic of the system, we can know that all the solution enter the closed region and retain there. By Theorem 2.2, we can obtain that system (2.3) has an order one periodic solution if the conditions of theorem and case 3 are satisfied. To sum up, we have proved that system (2.3) have a periodic solution of order one under the condition of Theorem 4.1. By qualitative analysis of system (2.3), we can see that the trajectories of system (2.3) starting for the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1g will enter the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 Þ 1g after some time. Therefore, we have the following theorem: Theorem 4.2. Suppose that m41 þ A; 0ohox ; x0 oh, S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1 and S0 o1, then system (2.3) has a periodic solution of order one.
29
Lemma 4.1. The T-periodic solution S ¼ xðtÞ; x ¼ ZðtÞ of the system 8 9 dS > > > > > ¼ PðS; xÞ; = > > dt > > if fðS; xÞa0; > dx > > < ; ¼ Q ðS; xÞ; > ð4:1Þ dt > ) > > > > > DS ¼ aðS; xÞ; if fðS; xÞ ¼ 0; > > > : Dx ¼ bðS; xÞ; is orbitally asymptotically stable if the Floquet multiplier m2 satisfies the condition jm2 jo1, where Z T q Y @P @Q m2 ¼ Dk exp ð4:2Þ ðxðtÞ; ZðtÞÞ þ ðxðtÞ; ZðtÞÞ dt; @S @x 0 k¼1 with @b @f @b @f @f þ þ Qþ Dk ¼ Pþ @x @S @S @x @S
and P; Q ; @a=@S; @a=@x; @b=@S; @b=@x; @f=@S and @f=@x are calculated Þ; Zðtþ ÞÞ; Qþ ¼ Q ðxðtþ Þ; Zðtþ ÞÞ: at the point ðxðtk Þ; Zðtk ÞÞ; Pþ ¼ Pðxðtþ k k k k fðS; xÞ is a sufficiently smooth function with grad fðS; xÞa0, and tk ðk 2 NÞ is the time of the kth jump. The proof of this lemma is referred to Simeonov and Bainov (1988). In the following, we suppose this periodic solution of system (2.3) with period T passes through the points Eþ 1 ðð1 bÞ x0 ; ð1 bÞhÞ 2 GH and E1 ðx0 ; hÞ 2 EC (GH; EC seen Figs. 6–8). As the expression and the period of this solution are unknown, we discuss the stability of this positive periodic solution by Lemma 4.1. In our case, mSx mS ; ðS; xÞ ¼ x 1 ; PðS; xÞ ¼ 1 S A þ Bx þ S A þ Bx þ S
aðS; xÞ ¼ bS; fðS; xÞ ¼ x h;
bðS; xÞ ¼ bx; ðxðTÞ; ZðTÞÞ ¼ ðx0 ; hÞ;
ðxðT þ Þ; ZðT þ ÞÞ ¼ ðð1 bÞx0 ; ð1 bÞhÞ: Then @P mxðBx þ AÞ ; ¼ 1 @S ðS þ Bx þ AÞ2 @a ¼ b; @S
@a ¼ 0; @x
@Q mSðS þ AÞ 1; ¼ @x ðS þ Bx þ AÞ2
@b ¼ 0; @S
@b ¼ b; @x
From Theorems 4.1 and 4.2, it follows that Theorem 4.3. Suppose that m41 þ A; 0ohox ; x0 oh, and S0 o1, then system (2.3) has a periodic solution of order one.
4.2. Stability of periodic solution of order one In the following, we analysis the stability of periodic solution of order one in system (2.3). Firstly, we give one lemma to discuss the stability of this positive periodic solution of system (2.3).
@a @f @a @f @f þ @S @x @x @S @x ; @f @f þQ P @S @x
@b @f @b @f @f þ þ Qþ Dk ¼ Pþ @x @S @S @x @S
@f ¼ 0; @S
@a @f @a @f @f þ @S @x @x @S @x @f @f þQ P @S @x
Q ðxðT þ Þ; ZðT þ ÞÞð1 bÞ Q ðxðTÞ; ZðTÞÞ mð1 bÞx0 1 A þ Bð1 bÞZ0 þ ð1 bÞx0 : ¼ ð1 bÞ2 mx0 1 A þ BZ0 þ x0
¼
@f ¼ 1; @x
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Set GðtÞ ¼ @P=@SðxðtÞ; ZðtÞÞ þ @Q =@xðxðtÞ; ZðtÞÞ, then Z T @P @Q m2 ¼ D1 exp ðxðtÞ; ZðtÞÞ þ ðxðtÞ; ZðtÞÞ dt @S @x 0 mð1 bÞx0 Z T 1 A þ Bð1 bÞZ0 þ ð1 bÞx0 GðtÞ dt : ¼ ð1 bÞ2 exp mx0 0 1 A þ BZ0 þ x0
order one is orbitally asymptotically stable if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1: mx0 1 A þ BZ0 þ x0
Because ðSðtÞ; xðtÞÞ is periodic solution of system (2.3), we RT know that 0 GðtÞ dto0 by the proof of Lemma 3.1, that is, RT expð 0 GðtÞ dtÞo1. Obviously, jm2 jo1 if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1; mx0 1 A þ BZ0 þ x0
4.3. Periodic solution of order two
and conditions of Theorem 4.3 hold. Therefore, this periodic solution of system (2.3) is stable if mð1 bÞx0 1 ð1 bÞ2 A þ Bð1 bÞZ0 þ ð1 bÞx0 j 1; mx0 1 A þ BZ0 þ x0 and conditions of Theorem 4.3 hold. Theorem 4.4. System (2.3) with the conditions of Theorem 4.3 has an order one periodic solution. Therefore, the periodic solution of
Fig. 9. Existence of order two periodic solution.
0.55
x (t)
S (t)
0.6
0.5 0.45 0.4
0
5
10 t
15
20
0.45
0.45
0.4
0.4
0.35
0.35
x (t)
0.65
In this subsection, we discuss existence of periodic solution of system (2.3) of order two if periodic solution of order one is unstable. If ðS; xÞ is a periodic solution of (2.3), then ðS 0 ; x 0 Þ 2 NDGH and ðS 1 ; x 1 Þ 2 MDEC . We can obtain that S 0 S 1 because of Sþ ¼ ð1 bÞS by the third equation of system (2.3). Let ðS; xÞ is the arbitrary solution of system (2.3), the first interaction point of trajectory and the set Mðx ¼ hÞ is ðS1 ; hÞ and the corresponding interaction points are ðS2 ; hÞ; ðS3 ; hÞ; . . ., respectively. Therefore, under the effect of impulsive function I, the corresponding point þ þ after pulse are ðSþ 1 ; ð1 bÞhÞ; ðS2 ; ð1 bÞhÞ; ðS3 ; ð1 bÞhÞ; . . .. We know that dx=dt40 for S4½ð1 bÞBh þ A=ðm 1Þ and dx=dto0 for So½ð1 bÞBh þ A=ðm 1Þ by the qualitative analysis of system (2.3). So we consider the following cases: Case 1: It is easily obtained that the trajectory jumps to the point ðSþ 1 ; ð1 bÞhÞ from the point ðS1 ; hÞ if the periodic solution is in the region fðS; xÞjS4½ð1 bÞBh þ A=ðm 1Þg. Without loss of generality, we let S1 oS 1 , then Sþ 1 oS 0 . The trajectory from the point ðSþ 1 ; ð1 bÞhÞ will interact the set M at the point ðS2 ; hÞ and then jump to the point ðSþ 2 ; ð1 bÞhÞ; . . .. By the above discussion, there is one of the following sequences: Case (a) S1 S2 S3 S 1 and Case (b) S 1 ZS1 ZS2 ZS3 Z . From the sequences we can know that the trajectories tend to be periodic. Since the sequence is monotone, it is clearly that order two periodic solution does not exist in the case by Definition 2.3. Case 2. If the set N is in the region fðS; xÞjSo½ð1 bÞBh þ A=ðm 1Þg and system (2.3) has unstable periodic solution of order one, without loss of generality, let S1 4S0 (see Fig. 9), then we can obtain that S0 S2 oS1 or S2 S0 oS1. Therefore, there exists a periodic solution of order two if S0 ¼ S2 holds. If S0 oS2 oS1 holds, we can obtain 0oS0 oS2 o oS2k oS2kþ1 o oS3 oS1 o1, while if S2 oS0 oS1 holds, then we have 0o oS2k oS2ðk1Þ o oS2 oS0 oS1 oS3 o oS2kþ1 o o1. Similar to the above discussion and the proof of Proposition 3.2 of Jiang et al. (2007), we can obtain that there is no periodic solution of order k ðkZ3Þ in system (2.3) and the system is not chaotic.
0.3
0.3
0.25
0.25 0.2
0.2 0
5
10 t
15
20
0.4
0.45
0.5
0.55 S (t)
Fig. 10. Time series and portrait phase of system (2.3) when m ¼ 20; A ¼ 9; b ¼ 0:5; B ¼ 2; h ¼ 0:5; Sð0Þ ¼ 0:4, and xð0Þ ¼ 0:2.
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0.45
0.4
0.4
0.5
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0.35
0.4
x (t)
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S (t)
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0.3 0.25
0.3
0.25
0.2 0
5
10 t
15
20
0.3
0.2 0
5
10 t
15
20
0.3
0.45
0.45
0.75
0.44
0.44
0.7
0.43
0.43
0.65
0.42
0.42
0.5
x (t)
0.8
x (t)
S (t)
Fig. 11. Time series and portrait phase of system (2.3) when m ¼ 20; A ¼ 9; B ¼ 2; b ¼ 0:5; h ¼ 0:45ox ¼
0.55
0.41
5
10 t
15
0.4
0.39
0.39
10 21 ;
Sð0Þ ¼ 0:4, and xð0Þ ¼ 0:2.
0.38 0
20
0.6
0.41
0.4 0.38 0
0.5 S (t)
0.6
0.4
5
10 t
15
0.5 0.55 0.6 0.65 0.7 0.75 0.8
20
S (t)
Fig. 12. Time series and portrait phase of system (2.3) when m ¼ 20; b ¼ 0:15; A ¼ 9; B ¼ 2; h ¼ 0:45ox ; Sð0Þ ¼ 0:8, and xð0Þ ¼ 0:4.
0.552 x (t)
S (t)
0.55 0.548 0.546 0.544 0.542
0.4
0.4
0.398
0.398
0.396
0.396
x (t)
0.554
0.394
0.394
0.392
0.392
0.39 0
5
10
15
20
0.39 0
5
10
15
t
t
20
0.542 0.544 0.546 0.548 0.55 0.552 0.554 S (t)
Fig. 13. Time series and portrait phase of system (2.3) when m ¼ 20; b ¼ 0:02; A ¼ 9; B ¼ 2; h ¼ 0:4ox ; Sð0Þ ¼ 0:545, and xð0Þ ¼ 0:39.
5. Numerical simulation Now we consider the following example: 9 8 20Sx > > _ ¼1S > > S ; > = > 9 þ 2x þ S > > xoh; > > 20S > > < x_ ¼ x ; 1 ; > 9 þ 2x þ S ) > > > DS ¼ bS; > > > x ¼ h: > > > : Dx ¼ bx;
ð5:1Þ
10 , A ¼ 9; B ¼ 2; m ¼ 20; b ¼ 0:15; S0 ¼ 0:4; x0 ¼ 0:2; h ¼ 0:45ox ¼ 21 then system (2.3) has a periodic solution of order one, which verifies theoretical results in this paper. Fig. 12 shows that the trajectories of system (2.3) starting for the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 ÞZ1g will enter the region fðS0 ; x0 Þjx0 ohox ; S0 þ mS0 x0 =ðA þ Bx0 þ S0 Þ 1g, hence, system (2.3) has also periodic solution of order one. Fig. 13 gives the series times and phase portrait for S0 4½ð1 bÞBh þ A=ðm 1Þ and shows that the trajectory tends to a periodic trajectory.
6. Conclusion In numerical simulation, let A ¼ 9; m ¼ 20; B ¼ 2. By calculating, we know m41 þ A, holds. If h ¼ 0:54x ¼ ðm A 1Þ=ðm þ B 10 ; b ¼ 0:5; S0 ¼ 0:4; x0 ¼ 0:2ox , then the time series and 1Þ ¼ 21 phase portrait can be seen in Fig. 10. By analysis of Section 4, we know that no impulse occurs for system (2.3) if h4x holds. As shown in Fig. 10, numerical simulation also suggests that system (2.3) with the coefficients above admits no impulse to occur. By Theorem 4.1, we know that system (2.3) has a periodic solution of order one under conditions of Theorem 4.3. As shown in Fig. 11, If
In this paper, we built a chemostat model with Beddington–DeAnglis uptake function and impulsive state feedback control. Firstly, we investigated qualitative characteristic of the system without impulsive effect, and obtained sufficient condition of globally asymptotically stable of the system. We obtained the system with impulsive state feedback control had periodic solution of order one, and sufficient conditions for existence and stability of periodic solution of order one. The case, in which it
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was possible that there was a periodic solution of order two, was also given in Section 4.3. The results show that the chemostat with Beddington–DeAnglis uptake function and impulsive state feedback control tends to a stable state or periodic, and the behavior of impulsive state feedback control on the concentrate of microorganism plays an important role on the periodic or stable state of system (2.3). When the concentrate of microorganism reaches an appropriate critical value, a state feedback measure for controlling concentrate of microorganism is taken. According to the analysis of Section 4, this measure is effective based on the fact that the system has stable periodic solution under some conditions. According to the theoretical results, the production of the microorganism will tend to a stable production level or the production will be periodic. The key to the production by applying the system with feedback control is to give the suitable feedback state (the value of h) and the control parameters (b) according to practice. It is seen from Figs. 11–13 that there are positive periodic trajectories under the impulsive state feedback control. Therefore, the periodic production of the microorganism can be achieved if the value of h and the suitable initial value of the substrate and the microorganism are taken. Indeed, we would like to point out that the model (2.3) must exist deficiency which we need further study.
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