
Citation: Yong Yao. Dynamics of a delay turbidostat system with contois growth rate[J]. Mathematical Biosciences and Engineering, 2019, 16(1): 56-77. doi: 10.3934/mbe.2019003
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The chemostat is a laboratory bio-reactor used for the continuous culture of microorganism (Figure 1). The apparatus is of both ecological and mathematical interest since it can be used to represent a lot of microorganism systems, such as the simple lake and the waste-water treatment, and plays an important role in micrology and population [8,9,30].
Recently, a large of research devoted to modifying the chemostat for higher economical values by controlling the dilution rate of the chemostat. The chemostat with the feedback control of the dilution rate is established, which is referred to as turbidostat by biologists [14] (Figure 1). In the turbidostat, the optical sensor measures the concentration of the microorganism and the signal feedback control the dilution rate. For example, Flegr [5] analyzed the coexistence of two species in the turbidostat by numerical analysis, and later Leenheer and Smith [13] also demonstrated results of Flegr by theoretical analysis. They showed that a turbidostat with monotone response functions permits a unique coexistence equilibrium and if it is locally asymptotically stable then it is globally attracting. Furthermore, they also obtained that coexistence is not achievable in the turbidostat with more than two competing organisms. Li [14] established a mathematical model of competition in a turbidostat for an inhibitory nutrient and achieved the stability of the equilibrium and the existence of limit cycles. Li and Chen [16] studied a mathematical model of the turbidostat with impulsive state feedback control and obtained the existence and the stability of periodic solutions of order one.
The uptake function in the model of microorganism continuous culture is the growth rate of the microorganism including many forms, such as, Monod [19], Moser [20] and Contois [4]. The most classical and common uptake function is Monod one, which is the function of concentration of microorganism and the constant half saturation term. However, Contois [4] presented experimental results to show that it is not necessarily a constant. Thus, the uptake function of Contois was introduced, which depends on the ratio substrate to microorganism. Jost [24] explored the relation of the Contois growth rate in microbiology and the ratio-dependent functional response in population ecology introduced by Arditi and Ginzburg [1]. As such, many researchers also call it ratio-dependent growth rate just as in population ecology [10,11,17,34]. Especially, Hu et al. [11] investigated the existence and the stability of the periodic solution of order one for the turbidostat system with ratio-dependent growth rate and impulsive state feedback control. The Contois model gave predictions that were in excellent agreement with experiment measurement. Especially, the Contois model was found to accurately describe the processing of industrial wastewaters. Nelson investigated a series of chemostat models with Contois growth rate. For example, Nelson and Sidhu [21] investigated the Hopf bifurcation and degenerate Hopf bifurcation of the chemostat model with Contois growth rate and the variable yield. Alqahtani et al. [22] studied the chemostat model with variable yield and contois growth kinetics with substrate inhibition and showed the Hopf bifurcation, degenerate Hopf bifurcation and Bogdanov-Takens bifurcation.
Researchers recognized that time delays are natural in the ecological systems [12,27,29]. Smith and Waltman [30] showed that there are two obvious sources of delays in the cultivation of microorganism, one of which is the possibility that the microorganism stores the nutrient. Caperon [3] first introduced the delay into the chemostat model by some experiments. Bush and Cook [2] corrected Caperon's model and established a chemostat model with delay in the growth rate of microorganism. The time delays yield some complex impacts to the chemostat model and biologists gave the reasonable explanations for the observations and experimental datas by investigating the chemostat model with delays [18,15,23,26,31]. For example, Li et al. [15] considered the global dynamics of the chemostat model with two populations of microorganisms competing for two perfectly complementary nutrients when distributed delays are involved. Ruan and Wolkowicz [26] studied the existence and the stability of the Hopf bifurcation of the chemostat model with a distributed delay and found that the periodic solutions become unstable if the dilution rate is increased. Wang and Wolkowicz [31] analyzed the local stability of the equilibria and the globally asymptotical stability of the single species survival equilibrium when n species compete in the chemostat with time delay for a single resource, hence the competitive exclusion principle holds. Yao et al. [32] discussed the delayed turbidostat model with Monod growth rate and obtained the phenomenon of oscillation. In the turbidostat, additional factor causing delay is due to the process that the sensor controls the dilution, for example, Tagashira and Hara [28] and Yuan et al. [33] considered the turbidostat models of the dilution rate with delayed feedback control.
Li and Xu [17] investigated the following chemostat system with Contois growth rate and time delay
{dS(t)dt=d(S0−S(t))−1α+βS(t)mS(t)x(t)ax(t)+S(t),dx(t)dt=x(t){mS(t−τ)ax(t−τ)+S(t−τ)−d}, | (1.1) |
where S(t) and x(t) present the concentration of substrate and the concentration of microorganism at time t , respectively, S0>0 stands for the input concentration of the nutrient, α+βS(t) (α>0,β>0) is variable yield, m>0 and a>0 are growth parameters of the microorganism, τ>0 is the time delay of the growth response of the microorganism, and d>0 presents the flow volume. They considered the local and global stability of the equilibria and Hopf bifurcation.
In this paper, motivated by the works of Bush and Cook ([2]), Contois ([4]) and Li and Xu ([17]), one focuses on the dynamics of a turbidostat model with Contois growth rate and time delay which follows as
{dS(t)dt=(d+kx(t))(S0−S(t))−1γmS(t)x(t)ax(t)+S(t),dx(t)dt=x(t){mS(t−τ)ax(t−τ)+S(t−τ)−d−kx(t)}, | (1.2) |
where S(t) , x(t) and parameters S0 , m , a , τ are defined as in (1.1), γ>0 is yield constant and d+kx(t) (d>0,k>0) presents the dilution rate of the turbidostat.
The aim of this paper is to investigate the qualitative properties of system (1.2) including the stability of the equilibria and the bifurcations. The outline of this paper is as follows. In Section 2, I analyze the existence and local stability of the equilibria, and the transcritical bifurcation at the boundary equilibrium and the Hopf bifurcation at the positive equilibrium. In Section 3 mostly focuses on the stability and type of the bifurcating periodic solutions induced by Hopf bifurcation to system (1.2). I in Section 4 further illustrate our main results by numerical simulations.
In this section, we investigate the existence and stability of the equilibria and the bifurcations at equilibria including the transcritical bifurcation and Hopf bifurcation of system (1.2).
It is convenient to introduce dimensionless variables. In particular, we set
S=˜SS0,x=˜xγS0,k=d˜kγS0,a=˜aγ,m=d˜m,t=˜td. |
System (1.2) becomes
{dS(t)dt=(1+kx(t))(1−S(t))−mS(t)x(t)ax(t)+S(t),dx(t)dt=x(t)(mS(t−τ)ax(t−τ)+S(t−τ)−1−kx(t)), | (2.1) |
where we still denote ˜S,˜x,˜k,˜a,˜m,˜t with S,x,k,a,m,t , respectively. First of all, we discuss the existence of the equilibria of system (2.1) and have the following result.
Lemma 2.1. System (2.1) always has a boundary equilibrium E0:(1,0) ; If m>1 , then system (2.1) has a unique positive equilibrium E1:(S∗,x∗) , where S∗=1+km+k when a=1 ,
S∗=−(2ak+a−k+m−1)+√(2ak+a−k+m−1)2+4ak(1−a)(1+k)2k(1−a), |
when a≠1 and x∗=1−S∗ .
Proof. From the right part of system (2.1), we can obtain the boundary equilibrium easily, which we denote as E0:(1,0) . In order to get the positive equilibrium, we need to discuss the roots of the equation mSa+(1−a)S−1−k(1−S)=0 on the interval (0,1) . We define
f(S):=(1−a)kS2+(2ak+a−k+m−1)S−a(1+k). | (2.2) |
Therefore, we just need to discuss the zeros of the function f(S) on the interval (0,1) , which need to be discussed in the following three cases.
(ⅰ). If a=1 , then we have f(S)=(k+m)S−1−K and f(S) has a zero S=1+km+k on the interval (0,1) when m>1 , and f(S) has no zero on the interval (0,1) when m≤1 . Thus, we obtain S∗=1+km+k when a=1 and m>1 .
(ⅱ). If 0<a<1 , then f(S) is a quadratic function of S . Since the coefficient of quadratic term k(1−a)>0 and f(0)=−a(1+k)<0 , f(S)=0 has two roots
S1:=−(2ak+a−k+m−1)−√(2ak+a−k+m−1)2+4ak(1−a)(1+k)2k(1−a),S2:=−(2ak+a−k+m−1)+√(2ak+a−k+m−1)2+4ak(1−a)(1+k)2k(1−a), | (2.3) |
and it is obvious that S1<0,S2>0 . By simplifying the inequality S2<1 , we can get that S2<1 if and only if m>1 . Thus, we have S∗=S2 when 0<a<1 and m>1 .
(ⅲ). If a>1 , then f(S) still is a quadratic function of S with the coefficient of quadratic term k(1−a)<0 and f(0)=−a(1+k)<0 . The discriminant of f(S) is
Δ:=a2+2(2km−k+m−1)a+(k−m+1)2. |
We can take Δ as the function of a . Therefore, the discriminant of Δ is ˜Δ:=km(m−1)(1+k) . For this case we need further discussion in the following three cases.
(a). If m=1 , then ˜Δ=0 and the double root of Δ=0 is −k . Therefore, Δ>0 holds when a>1 . We have f(S)=0 has two roots S1=a(1+k)k(a−1)>1 and S2=1 . Thus, f(S) has no zero on the interval (0,1) when a>1 and m=1 .
(b). If m<1 , then ˜Δ<0 , which implies Δ>0 . We have f(S) has two roots S1 and S2 presented in (2.3) and S2<S1 . By simplifying S2>1 , we can get that S2>1 if and only if m<1 . Thus, f(S) has no zero on the interval (0,1) when a>1 and m<1 .
(c). If m>1 , then ˜Δ>0 . Since 4km−2k+2m−2>0 and (k−m+1)2≥0 , the two roots of Δ=0 denoted by a1 and a2 satisfy a1<a2≤0 . Therefore, Δ>0 holds when a>1 and m>1 . We obtain f(S) has two roots presented in (2.3) and S1>S2 . Further, we can get 0<S2<1 if and only if m>1 and a>1 . Suppose that S1<1 , then we get m>1 , which is a contradiction. Thus, we have S∗=S2 when a>1 and m>1 .
This completes the proof.
From the above analysis we have the existence of equilibria of system (2.1). In the following, we consider qualitative properties of system (2.1) including stability of the equilibria and the bifurcations. We discuss system (2.1) in two cases: (ⅰ) τ=0 and (ⅱ) τ>0 .
Case (ⅰ): τ=0 , i.e. there is no time delay in system (2.1), then system (2.1) becomes
{dS(t)dt=(1+kx(t))(1−S(t))−mS(t)x(t)ax(t)+S(t),dx(t)dt=x(t)(mS(t)ax(t)+S(t)−1−kx(t)). | (2.4) |
We first investigate the stability of the equilibria E0 and E1 and have the following result.
Theorem 2.2. If m<1 , then E0 is a stable node; If m>1 , then E0 is a saddle and E1 is a stable node; If m=1 , then E0 is a saddle-node.
Proof. The Jacobian matrix at E0 is
J0:=(−1−m0m−1). |
The determinant, trace and discriminant are respectively
D0:=1−m,T0:=m−2,Δ0:=T20−4D0=m2>0. |
If m<1 , then D0>0 and T0<0 . Thus, E0 is a stable node. If m>1 , then D0<0 . Thus, E0 is a saddle. If m=1 , then D0=0 and T0=−1 . This is the degenerate case, which needs further discussion.
Translating the equilibrium E0 to the origin by ˜S=S−1 , system (2.4) becomes the following system
{dS(t)dt=−S(t)(1+kx(t))−(S(t)+1)x(t)ax(t)+S(t)+1,dx(t)dt=x(t)(S(t)+1ax(t)+S(t)+1−1−kx(t)), | (2.5) |
where we still use S to present ˜S . Using the linear transformation u=x , v=S+x and time-rescaling t1=−t to normalize the linear part of system (2.5), we can change system (2.5) into the following
{dudt=(a+k)u2+a(1−a)u3−au2v+⋯:=Φ(u,v),dvdt=v+kuv:=Ψ(u,v), | (2.6) |
where we still denote t1 as t . By the implicit function theorem, there is a unique function v=ϕ(u) such that Ψ(u,v)=0 . We can obtain v=ϕ(u)=0 by solving Ψ(u,v)=0 . Substituting v=0 into Φ(u,v)=0 , we get
Φ(u,v)=(a+k)u2+a(1−a)u3+⋯. |
Thus E0 is a saddle-node when m=1 by Theorem 7.1 in Zhang et al. ([35]). Moreover, the parabolic sector of the saddle-node lies on the right-hand plane in the (u,v) -coordinates and the two hyperbolic sectors lie on the left-hand plane since a+k>0 .
We continue to prove the properties of E1 .
The Jacobian matrix at E1 is
J1:=(−1−kx∗−ma(x∗)2(ax∗+S∗)2kx∗−m(S∗)2(ax∗+S∗)2ma(x∗)2(ax∗+S∗)2−1−2kx∗+m(S∗)2(ax∗+S∗)2). |
The determinant, trace and discriminant of J1 are respectively
D1:=(1+kx∗){maS∗x∗+kx∗(ax∗+S∗)2+max∗(ax∗+S∗)2}>0,T1:=−2−3kx∗+m(S∗)2(ax∗+S∗)2−ma(x∗)2(ax∗+S∗)2T1:=−{1+2k∗+maS∗x∗(ax∗+S∗)2+ma(x∗)2(ax∗+S∗)2}<0,Δ1:=T21−4D1={−kx∗+m(S∗)2(ax∗+S∗)2−ma(x∗)2(ax∗+S∗)2}2≥0. |
Thus, E1 is a stable node if it exists.
We complete the proof.
It is indicated in Lemma 2.1 that system (2.4) has either exact one equilibrium E0 when m≤1 or exact two equilibria E0 and E1 when m>1 . In the following we reduce the system to a 1-dimensional system on a center manifold and display the mechanism for E1 to arise.
Theorem 2.3. System (2.4) experiences a transcritical bifurcation at E0 when m=1 .
Proof. Let ε=m−1 and translate the equilibrium E0 to the origin by ˜S=S−1 , system (2.4) becomes the following system
{dSdt=−(1+kx)S−(1+ε)(S+1)ax+S+1x,dxdt=x{(1+ε)(S+1)ax+S+1−1−kx}, | (2.7) |
where we still denote ˜S as S .
Applying the transformation S=−u+v and x=u to diagonalize the linear part of system (2.7), we can change system (2.7) into the suspended system
{dudt=εu−(a+k)u2−aεu2+au2v+a(a−1)u3+⋯,dvdt=−v−kuv,dεdt=0. | (2.8) |
By the center manifold theory, system (2.8}) has a two-dimensional center manifold Wc:v=h(u,ε) near O , which is tangent to the plane v=0 at O in the (u,v,ε) -space. In order to obtain the second-order approximation of function h , we set
v=h(u,ε)=a1u2+b1uε+c1ε2+o(|u,ε|2). | (2.9) |
Substituting (2.9) into the equality ˙v=hu˙u and comparing the coefficient of u2 , ε2 and uε , we have a1=b1=c1=0 . Hence the center manifold is v=o(|u,ε|2) and the restricted system of (2.8}) on the center manifold (2.9) is
dudt=εu−(a+k+aε)u2+a(a−1)u3+⋯. | (2.10) |
The expression (2.10) shows that a transcritical bifurcation occurs at E0 as ε varies through the bifurcation value ε=0 ([6]). More concretely, when ε<0 , E0 is stable and the other equilibrium appears on the negative u -axis; when ε=0 , the two equilibria coincide at E0 , which is a saddle-node; when ε>0 , E0 remains an equilibrium but is unstable while a stable equilibrium E1 arises.
We complete the proof.
Case (ⅱ): τ>0 .
In this case we will investigate the effect of time delay on the system. The time delay can cause the loss of stability of E1 and can induce periodic solutions. Note that if m>1 , then system (2.1) has a unique positive equilibrium E1 . To further consider the local stability of E1 and the Hopf bifurcations induced by the delay, we set ˜x(t)=x(t)−x∗,˜S(t)=S(t)−S∗ and still denote ˜x,˜S as x,S . Then system (2.1) can be changed into the following system
{dS(t)dt=a11S(t)+a12x(t)+a13S2(t)+a14x2(t)+a15S(t)x(t)dS(t)dt=+a16x3(t)+a17S(t)x2(t)+a18S2(t)x(t)+a19S3(t)+⋯,dx(t)dt=b11S(t−τ)+b12x(t)+b13x(t−τ)+b14S2(t−τ)+b15x2(t)dx(t)dt=+b16x2(t−τ)+b17S(t−τ)x(t)+b18x(t)x(t−τ)dx(t)dt=+b19S(t−τ)x(t−τ)+b20x(t)x2(t−τ)+b21x3(t−τ)dx(t)dt=+b22x(t)x(t−τ)S(t−τ)+b23S(t−τ)x2(t−τ)dx(t)dt=+b24S2(t−τ)x(t)+b25x(t−τ)S2(t−τ)+b26S3(t−τ)+⋯, | (2.11) |
where
a11=−1−kx∗−ma(x∗)2(ax∗+S∗)2,a12=kx∗−m(S∗)2(ax∗+S∗)2,a13=ma(x∗)2(ax∗+S∗)3,a14=ma(S∗)2(ax∗+S∗)3,a15=−k−2maS∗x∗(ax∗+S∗)3,a16=−ma2(S∗)2(ax∗+S∗)4,a17=maS∗(2ax∗−S∗)(ax∗+S∗)4,a18=−max∗(ax∗−2S∗)(ax∗+S∗)4,a19=−ma(x∗)2(ax∗+S∗)4,b11=ma(x∗)2(ax∗+S∗)2,b12=−1−2kx∗+mS∗ax∗+S∗,b13=−maS∗x∗(ax∗+S∗)2,b14=−ma(x∗)2(ax∗+S∗)3,b15=−k,b16=ma2S∗x∗(ax∗+S∗)2,b17=max∗(ax∗+S∗)2,b18=−maS∗(ax∗+S∗)2,b19=−max∗(ax∗−S∗)(ax∗+S∗)3,b20=ma2S∗(ax∗+S∗)3,b21=−ma3S∗x∗(ax∗+S∗)4,b22=−ma(ax∗−S∗)(ax∗+S∗)3,b23=ma2x∗(ax∗−2S∗)(ax∗+S∗)4,b24=−max∗(ax∗+S∗)3,b25=max∗(2ax∗−S∗)(ax∗+S∗)4,b26=ma(x∗)2(ax∗+S∗)4. |
The following characteristic equation can be achieved from the linear system of system (2.11)
λ2−(a11+b12)λ+e−λτ(−b13λ+a11b13−a12b11)+a11b12=0. | (2.12) |
To investigate the stability of the equilibrium E1 and Hopf bifurcation of (2.1), we must study the distribution of the roots of (2.12).
Lemma 2.4. If a11b12+a12b11−a11b13<0 , then ±iw0(w0>0) are the eigenvalues of (2.12) when τ=τj . The values of w0 and τj can be presented as follows
w0=√2a11b12+b213−(a11+b12)2+√{−2a11b12−b213+(a11+b12)2}2−4{a211b212−(a11b13−a12b11)2}2,τj=1w0{arccos((w20−a11b12)(a11b13−a12b11)−b13(a11+b12)w20(a11b13−a12b11)2+b213w20)+2jπ},j=0,1,2⋯. |
Proof. If λ=iw (w>0) is a root of (2.12), then we have
−w2−iw(a11+b12)+(coswτ−isinwτ)(−ib13w+a11b13−a12b11)+a11b12=0. |
Separating the real and imaginary parts, we obtain
(a11b13−a12b11)coswτ−b13wsinwτ=w2−a11b12,(a11b13−a12b11)sinwτ+b13wcoswτ=−w(a11+b12), | (2.13) |
which implies the following equation
w4+{−2a11b12+(a11+b12)2−b213}w2+a211b212−(a11b13−a12b11)2=0. |
Denote
h(w):=w4+(a211+b212−b213)w2+a211b212−(a11b13−a12b11)2. |
Since
a11b12+a11b13−a12b11=(1+kx∗){kx∗+maS∗x∗(ax∗+S∗)2}+m2aS∗(x∗)2(ax∗+S∗)(ax∗+S∗)4>0 |
and the condition a11b12+a12b11−a11b13<0 , we have the constant term of h(w) is negative. In addition, we can obtain the coefficient of quadratic term of h(w) is positive. In fact,
a211+b212−b213={1+kx∗+ma(x∗)2(ax∗+S∗)2}2+(1+2kx∗−mS∗ax∗+S∗)2−(maS∗x∗)2(ax∗+S∗)4a211+b212−b213={mS∗ax∗+S∗+ma(x∗)2(ax∗+S∗)2}2+(kx∗)2−(maS∗x∗)2(ax∗+S∗)4a211+b212−b213=m2{(S∗)2+a(x∗)2}2+2m2aS∗x∗{(S∗)2+a(x∗)2}(ax∗+S∗)4+(kx∗)2>0. |
Thus, h(w) has a unique positive real root
w0=√22{2a11b12+b213−(a11+b12)2w0+√(−2a11b12−b213+(a11+b12)2)2−4(a211b212−(a11b13−a12b11)2)}12. | (2.14) |
Substituting w0 into (2.13), we conclude that
![]() |
(2.15) |
Hence, (2.12) has a pair of purely imaginary roots ±iw0 as τ=τj, j=0,1,2⋯ .
This completes the proof.
Let λ(τ)=α(τ)+iβ(τ) denote the root of (2.12) near τ=τj satisfying α(τj)=0 and β(τj)=w0 . Then we have the following transversality condition.
Lemma 2.5. The following transversality condition holds
dRe(λ(τj))dτ>0,j=0,1,2⋯. |
Proof. Differentiating (2.12) with respect to τ yields that
(dλdτ)−1=(2λ−a11−b12)eλτ−b13λ(−b13λ+a11b13−a12b11)−τλ. |
Thus,
Re{(dλ(τj)dτ)−1}=Re{(2iw0−a11−b12)(cosw0τ+isinw0τ)−b13w20b13+iw0(a11b13−a12b11)}Re{(dλ(τj)dτ)−1}=h′(w0)2w30b213+2w0(a11b13−a12b11)2. |
Since h(w) is a quartic function of w and the leading coefficient of which is positive, furthermore, from the analysis of Lemma 2.4, we obtain that the four roots of equation h(w)=0 are a pair of conjugate complex roots, a negative real root and a positive real root w0 , respectively. Hence h(w) is monotonically increasing at w0 , i.e., h′(w0)>0 , which implies h′(w0)/{2w30b213+2w0(a11b13−a12b11)2}>0 . Thus, we have
sign{Re(dλ(τj)dτ)}=sign{Re(dλ(τj)dτ)−1}=1. |
The proof is completed.
Now we have the following result about the distribution of the roots of the exponential polynomial (2.12) by Corollary 2.4 in [25].
Lemma 2.6. Suppose that w0 and τj (j=0,1,2⋯) are defined by (2.14) and (2.15), respectively. We have the following results
(i ) If a11b12+a12b11−a11b13≥0 , then all the roots of (2.12) have negative real parts for all τ>0 .
(ii ) If a11b12+a12b11−a11b13<0 and τ=τj , then (2.12) has a pair of simple imaginary roots ±iw0 . Furthermore, if τ∈[0,τ0) , then all the roots of (2.12) have negative real parts; if τ∈(τj,τj+1),j=0,1,2⋯ , then (2.12) has at least one root with positive real parts.
From Lemma 2.4, Lemma 2.5, Lemma 2.6 and the Hopf bifurcation theorem, we have the following result of E1 .
Theorem 2.7. For system (2.1), we have
(i ) if a11b12+a12b11−a11b13≥0 , then the unique positive equilibrium E1 is asymptotically stable for all τ≥0 .
(ii ) if a11b12+a12b11−a11b13<0 , then the equilibrium E1 is asymptotically stable for τ∈[0,τ0) and unstable for τ>τ0 . Hopf bifurcation occurs when τ=τj,j=0,1,2⋯ .
In this section, we consider the direction and stability of the bifurcating periodic solutions of system (2.1) induced by the Hopf bifurcation by using the normal form theory and the center manifold theorem by Hassard et al. ([7]). We compute (see Appendix for details of the computation)
g(z,ˉz)=g20z22+g11zˉz+g02ˉz22+g21z2ˉz2+⋯, |
where the first four coefficients g20 , g11 , g02 and g21 that we need for determining the properties of the Hopf bifurcation are of the following forms
g20=2τjˉD{a14q21+a15q1+a13+ˉq∗1(b15q21+(b18q21+b17q1)e−iw0τj+(b16q21g21=+b19q1+b14)e−2iw0τj)},g11=τjˉD{a15q1+a15ˉq1+2a13+2a14ˉq1q1+ˉq∗1(2b14+b19q1+b19ˉq1+2b15ˉq1q1g21=+2b16ˉq1q1+(b17ˉq1+b18q1ˉq1)e−iw0τj+(b17q1+b18ˉq1q1)eiw0τj)},g02=2τjˉD{a13+a14ˉq21+a15ˉq1+ˉq∗1(b15ˉq21+(b16ˉq21+b19ˉq1+b14)e2iw0τj+(b18ˉq21g21=+b17ˉq1)eiw0τj)},g21=τjˉD{3a16q21ˉq1+a17q21+2a17q1ˉq1+2a18q1+a18ˉq1+3a19+(12a15g21=+a14ˉq1)W(2)20(0)+(a15+2a14q1)W(2)11(0)+(a15q1+2a13)W(1)11(0)+(12a15ˉq1g21=+a13)W(1)20(0)+ˉq∗1(2b24q1+(b22q1+b22ˉq1)q1+(3b26+2b23ˉq1q1+3b21ˉq1q21g21=+b23q21+b25ˉq1+2b25q1)e−iw0τj+(b24ˉq1+b20q21ˉq1+b22q1ˉq1)e−2iw0τjg21=+2b20ˉq1q21+((b19q1+2b14)e−iw0τj+b17q1)W(1)11(−1)+((12b19ˉq1+b14)eiw0τjg21=+12b17ˉq1)W(1)20(−1)+((b18q1+b17)e−iw0τj+2b15q1)W(2)11(0)+(12(b18ˉq1g21=+b17)eiw0τj+b15ˉq1)W(2)20(0)+((2b16q1+b19)e−iw0τj+b18q1)W(2)11(−1)g21=+((b16ˉq1+12b19)eiw0τj+12b18ˉq1)W(2)20(−1))}, |
in which the terms W(1)11(0) , W(2)11(0) , W(1)11(−1) , W(2)11(−1) , W(1)20(0) , W(2)20(0) , W(1)20(−1) , W(2)20(−1) , q1 , q∗1 and ˉD are calculated in Appendix.
Now using the four coefficients we obtain the values of the parameters μ2 , β2 and T2
c1(0)=i2w0τj(g11g20−2|g11|2−13|g02|2)+g212,μ2=−Re(c1(0))Re(λ′(τj)),β2=2Re(c1(0)),T2=−Im(c1(0))+μ2Im(λ′(τj))w0τj. |
Thus, using the quantities above, the properties of the Hopf bifurcation are determined by the following theorem.
Theorem 3.1. The properties of the Hopf bifurcation are determined by the parameters μ2 , β2 and T2 , where μ2 determines the direction of the Hopf bifurcation: if μ2>0 (<0) , then the Hopf bifurcation is supercritical (subcritical); β2 determines the stability of the bifurcating periodic solutions: if β2<0 (>0) , the bifurcating periodic solutions are stable (unstable); and T2 determines the period of the bifurcating periodic solutions: if T2>0 (<0) , the period increases (decreases).
We discussed a turbidostat system with Contois growth rate and time delay in this paper. We investigated the qualitative properties of system (1.2) including the existence and the stability of the equilibria and the bifurcations. In the case of no time delay, the boundary equilibrium E0 is globally asymptotically stable if it is the unique equilibrium, and the positive equilibrium E1 arising from a transcritical bifurcation is globally asymptotically stable if it exists. In the case of system with delay, the stability of E1 is changed and the Hopf bifurcation occurs by choosing the delay as the bifurcation parameter. Furthermore, the stability and direction of the bifurcating periodic solutions are discussed by using the normal form and center manifold theorem. More concretely, when the delay is greater than the critical value τ0 , the positive equilibrium E1 loses its stability and the stable periodic solution appears. The qualitative analysis and theoretical results show that the delay can change the topological structures of the system and produces more complicated dynamic behaviors.
We next offer an example to illustrate the feasibility of our results. When system (2.1) without delay, i.e. τ=0 , we set a=1 and k=0.6 , the system has the unique equilibrium E0 , which is a stable node as m=0.5 ; the system has a saddle E0 and a stable node E1=(0.76,0.24) as m=1.5 ; the system has the unique equilibrium E0 , which is a saddle-node as m=1 (Figure 2). However, when m=1.5 , a=1 and k=0.6 , by choosing the time delay τ as the bifurcation parameter, we obtained the critical value of bifurcation τ0≈7 . Thus, the positive equilibrium E1 is asymptotically stable when τ=6<τ0 , which is supported by Figure 3. The positive equilibrium E1 is unstable and a stable bifurcating periodic solution occurs from E1 when τ=8>τ0 , which can be seen clearly in Figure 4. According to (4.10) we can compute
g20≈−4.25−9.16i,g11≈0.21+1.3i,g02≈11.79−0.01i,g21≈−47.16−5.86i. |
Further, we can get
c1(0)≈−21.71−12.63i. |
Then, in accordance with (4.22), we can obtain
μ2≈2131.70>0,β2≈−43.42<0,T2≈38.7>0. |
Therefore, when τ=8 , μ2>0 and β2<0 , then the Hopf bifurcation for system (2.1) is supercritical and the stable bifurcating periodic solutions occur from the positive equilibrium E1 . From Figure 3 and Figure 4, we can find that the dynamics of system (2.1) change when τ locates near τ0 . The bifurcation diagram of x−τ is presented in Figure 5, from which we find that even for parameter values not chosen, the stable periodic solutions occur in a large region of time delay.
Remark 1. This is a remark about Figure 2. In fact, considering the biological background, the set Ω={(S,x):0≤S≤1,x≥0} is positively invariant with respect to system (2.4). Further, E0 is globally asymptotically stable with respect to Ω if m<1 ; E0 is globally attractive with respect to Ω if m=1 ; and E1 is globally asymptotically stable with respect to Ω if m>1 . These results can be proved easily by Liapunov-LaSalle invariant principle and Poincare-Bendixson Theorem respectively.
The author is very grateful to the anonymous referees for their careful reading and valuable suggestions, which have notably improved the quality of this paper.
All authors declare no conflicts of interest in this paper.
We obtained in Section 2 that system (2.1) undergoes the Hopf bifurcation at the positive equilibrium E1 when τ=τj . In the following the properties of the Hopf bifurcation are determined. We first consider system (2.1) by the transformation ˜y1(t)=S(τt), ˜y2(t)=x(τt), τ=τj+μ and still denote ˜y1(t),˜y2(t) as y1(t),y2(t) . Then system (2.1) is equivalent to a functional differential equation defined in C=C([−1,0],R2)
˙y(t)=Lμ(yt)+h(μ,yt), | (4.1) |
where y(t)=(y1(t),y2(t))T∈R2 , and Lμ:C→R2 and h:R×C→R2 are given respectively by
Lμφ=(τj+μ)(a11a120b12)(φ1(0)φ2(0))+(τj+μ)(00b11b13)(φ1(−1)φ2(−1)) |
and
h(μ,φ)=(τj+μ)(h1h2), |
where
h1=a13φ21(0)+a14φ22(0)+a15φ1(0)φ2(0)+a16φ32(0)+a17φ1(0)φ22(0)h1=+a18φ21(0)φ2(0)+a19φ31(0)+⋯,h2=b14φ21(−1)+b15φ22(0)+b16φ22(−1)+b17φ1(−1)φ2(0)+b18φ2(0)φ2(−1)h1=+b19φ1(−1)φ2(−1)+b20φ22(−1)φ2(0)+b21φ32(−1)h1=+b22φ2(0)φ2(−1)φ1(−1)+b23φ1(−1)φ22(−1)+b24φ21(−1)φ2(0)h1=+b25φ21(−1)φ2(−1)+b26φ31(−1)+⋯, |
and φ=(φ1,φ2)T∈C.
By the Riesz representation theorem, there exists a matrix whose components are bounded variation functions η(θ,μ) for θ∈[−1,0] such that
Lμφ=∫0−1dη(θ,μ)φ(θ)forφ∈C. |
Indeed, we may choose
η(θ,μ)=(τj+μ)(a11a120b12)δ(θ)−(τj+μ)(00b11b13)δ(θ+1), |
where δ is the Dirac delta function.
For φ∈C1([−1,0],R2) , we further define the operators A and B as
A(μ)φ={dφ(θ)dθ,θ∈[−1,0),∫0−1dη(θ,μ)φ(θ),θ=0,B(μ)φ={0,θ∈[−1,0),h(μ,φ),θ=0. |
Then system (4.1) is equivalent to
˙yt=A(μ)yt+B(μ)yt, | (4.2) |
where yt(θ)=y(t+θ) for θ∈[−1,0] .
For ψ∈C1([0,1],(R2)∗) , we define A∗ of A as
A∗ψ={−dψ(s)ds,s∈(0,1],∫0−1dηT(t,0)ψ(−t),s=0. |
In order to normalize the eigenvectors of A and A∗ , we define a bilinear inner product
⟨ψ(s),φ(θ)⟩=ˉψT(0)φ(0)−∫0−1∫θξ=0ˉψT(ξ−θ)dη(θ)φ(ξ)dξ, | (4.3) |
where η(θ)=η(θ,0) .
From the above discussion we assume q(θ) and q∗(s) are eigenvectors of A and A∗ corresponding to iw0τj and −iw0τj . Suppose that q(θ)=(1,q1)Teiw0τjθ is the eigenvector of A(0) corresponding to iw0τj , then Aq(0)=iw0τjq(0) . On the basis of the definitions of A(0) , Lμφ and η(θ,μ) , the following conclusion can be achieved
τj(a11−iw0a12b11e−iw0τjb12−b13e−iw0τj−iw0)(1q1)=(00). |
We can get
q1=iw0−a11a12. | (4.4) |
Similarly, by the definition of A∗ , we have
τj(a11+iw0b11eiw0τja12b12+b13eiw0τj+iw0)(1q∗1)=(00) |
and
q∗1=iw0−a11b11eiw0τj. | (4.5) |
We can then obtain the value of D by ⟨q∗,q⟩=1 . Further it holds from (4.3) that
⟨q∗(s),q(θ)⟩=ˉD(1,ˉq∗1)(1,q1)T⟨q∗(S),q(θ)⟩=−∫0−1∫θξ=0ˉD(1,ˉq∗1)e−iw0τj(ξ−θ)dη(θ)(1,q1)Teiξw0τjdξ⟨q∗(S),q(θ)⟩=ˉD{1+ˉq∗1q1−∫0−1(1,ˉq∗1)θeiθw0τjdη(θ)(1,q1)T}⟨q∗(S),q(θ)⟩=ˉD{1+ˉq∗1q1+τje−iw0τjˉq∗1(b11+b13q1)}. |
Hence
D=11+q∗1ˉq1+τjeiw0τjq∗1(b11+b13ˉq1). | (4.6) |
In the following part, we compute the coordinates describing center manifold C0 at μ=0 by using the theory in [7]. Define
z(t)=⟨q∗(s),yt(θ)⟩,W(t,θ)=yt(θ)−2Re{z(t)q(θ)}. | (4.7) |
Then on the center manifold C0 , we have
W(t,θ)=W(z(t),ˉz(t),θ)=W20(θ)z22+W11(θ)zˉz+W02(θ)ˉz22+⋯, |
where z and ˉz are local coordinates for center manifold C0 in the direction of q∗ and ˉq∗ . It is easy to see that W is real if yt is real. Thereby, we next only consider real solutions of (4.2). If yt∈C0 is the solution of (4.2), since μ=0 , which implies that
˙z(t)=<q∗,˙yt>=<q∗,A˙yt+B˙yt>=<A∗q∗,˙yt>+<q∗,B˙yt>˙z(t)=iw0τjz+ˉq∗(0)h(0,W(z,ˉz,0)+2Re{zq(0)})˙z(t)def=iw0τjz+ˉq∗(0)h0(z,ˉz)=iw0τjz+g(z,ˉz), |
where
g(z,ˉz)=ˉq∗(0)h0(z,ˉz)=g20z22+g11zˉz+g02ˉz22+g21z2ˉz2+⋯. | (4.8) |
In the following we need to compute the coefficients g20 , g11 , g02 and g21 . Note that yt(θ)=(y1t(θ),y2t(θ))T=W(t,θ)+zq(θ)+¯zq(θ) and q(θ)=(1,q1)Teiw0τjθ . It then follows that
{y1t(0)=z+ˉz+W(1)20(0)z22+W(1)11(0)zˉz+W(1)02(0)ˉz22y1t(0)=+O(|(z,ˉz)|3),y2t(0)=q1z+¯q1ˉz+W(2)20(0)z22+W(2)11(0)zˉz+W(2)02(0)ˉz22y1t(0)=+O(|(z,ˉz)|3),y1t(−1)=ze−iw0τj+ˉzeiw0τj+W(1)20(−1)z22+W(1)11(−1)zˉzy1t(−1)=+W(1)02(−1)ˉz22+O(|(z,ˉz)|3),y2t(−1)=q1ze−iw0τj+ˉq1ˉzeiw0τj+W(2)20(−1)z22+W(2)11(−1)zˉzy1t(−1)=+W(2)02(−1)ˉz22+O(|(z,ˉz)|3). | (4.9) |
It follows together with the definition h(μ,φ) that
g(z,ˉz)=ˉq∗(0)h0(z,ˉz)g(z,ˉz)=τjˉD{a13φ21(0)+a14φ22(0)+a15φ1(0)φ2(0)+a16φ32(0)+a17φ1(0)φ22(0)g(z,ˉz)=+a18φ21(0)φ2(0)+a19φ31(0)+⋯+ˉq∗1(b14φ21(−1)+b15φ22(0)g(z,ˉz)=+b16φ22(−1)+b17φ1(−1)φ2(0)+b18φ2(0)φ2(−1)+b19φ1(−1)φ2(−1)g(z,ˉz)=+b20φ22(−1)φ2(0)+b21φ32(−1)+b22φ2(0)φ2(−1)φ1(−1)g(z,ˉz)=+b23φ1(−1)φ22(−1)+b24φ21(−1)φ2(0)+b25φ21(−1)φ2(−1)g(z,ˉz)=+b26φ31(−1)+⋯)}. |
By substituting (4.9) into the above equation and comparing the coefficients with (4.8) we have
{g20=2τjˉD{a14q21+a15q1+a13+ˉq∗1(b15q21+(b18q21+b17q1)e−iw0τjg21=+(b16q21+b19q1+b14)e−2iw0τj)},g11=τjˉD{a15q1+a15ˉq1+2a13+2a14ˉq1q1+ˉq∗1(2b14+b19q1+b19ˉq1g21=+2b15ˉq1q1+2b16ˉq1q1+(b17ˉq1+b18q1ˉq1)e−iw0τjg21=+(b17q1+b18ˉq1q1)eiw0τj)},g02=2τjˉD{a13+a14ˉq21+a15ˉq1+ˉq∗1(b15ˉq21+(b16ˉq21+b19ˉq1g21=+b14)e2iw0τj+(b18ˉq21+b17ˉq1)eiw0τj)},g21=τjˉD{3a16q21ˉq1+a17q21+2a17q1ˉq1+2a18q1+a18ˉq1+3a19g21=+(12a15+a14ˉq1)W(2)20(0)+(a15+2a14q1)W(2)11(0)g21=+(a15q1+2a13)W(1)11(0)+(12a15ˉq1+a13)W(1)20(0)g21=+ˉq∗1(2b24q1+(b22q1+b22ˉq1)q1+(3b26+2b23ˉq1q1+3b21ˉq1q21g21=+b23q21+b25ˉq1+2b25q1)e−iw0τj+(b24ˉq1+b20q21ˉq1g21=+b22q1ˉq1)e−2iw0τj+2b20ˉq1q21+((b19q1+2b14)e−iw0τjg21=+b17q1)W(1)11(−1)+((12b19ˉq1+b14)eiw0τj+12b17ˉq1)W(1)20(−1)g21=+((b18q1+b17)e−iw0τj+2b15q1)W(2)11(0)+(12(b18ˉq1+b17)eiw0τjg21=+b15ˉq1)W(2)20(0)+((2b16q1+b19)e−iw0τj+b18q1)W(2)11(−1)g21=+((b16ˉq1+12b19)eiw0τj+12b18ˉq1)W(2)20(−1))}. | (4.10) |
In order to determine g21 we need to compute W20(θ) and W11(θ) . In the light of (4.2) and (4.7), we obtain
˙W=˙yt−˙zq−˙ˉzˉq˙W={AW−2Re{ˉq∗(0)h0(z,ˉz)q(θ)},θ∈[−1,0),AW−2Re{ˉq∗(0)h0(z,ˉz)q(θ)}+h0(z,ˉz),θ=0,˙Wdef=AW+H(z,ˉz,θ), | (4.11) |
where
H(z,ˉz,θ)=H20(θ)z22+H11(θ)zˉz+H02(θ)ˉz22+H20(θ)z36+⋯. | (4.12) |
Expanding the previous series and comparing the coefficients, we obtain that
{(A−2iw0τjI)W20(θ)=−H20(θ),AW11(θ)=−H11(θ),⋯. | (4.13) |
Case 1: We first consider the case θ∈[−1,0) , it follows from (4.11) that
H(z,ˉz,θ)=−ˉq∗(0)h0(z,ˉz)q(θ)−q∗(0)ˉh0(z,ˉz)ˉq(θ)H(z,ˉz,θ)=−g(z,ˉz)q(θ)−ˉg(z,ˉz)ˉq(θ). | (4.14) |
Comparing the coefficients with (4.8), we obtain
{H20(θ)=−g20q(θ)−ˉg02ˉq(θ),H11(θ)=−g11q(θ)−ˉg11ˉq(θ). | (4.15) |
It follows from (4.13), (4.15) and the definition of A that
{˙W20(θ)=2iw0τjW20(θ)+g20q(θ)+ˉg02ˉq(θ),˙W11(θ)=g11q(θ)+ˉg11ˉq(θ). | (4.16) |
Note that q(θ)=(1,q1)Teiw0τjθ , then we have
{W20(θ)=ig20w0τjq(0)eiw0τjθ+iˉg023w0τjˉq(0)e−iw0τjθ+e2iw0τjθE,W11(θ)=−ig11w0τjq(0)eiw0τjθ+iˉg11w0τjˉq(0)e−iw0τjθ+F, | (4.17) |
where E=(E(1),E(2))T∈R2 and F=(F(1),F(2))T∈R2 are all constant vectors. In what follows, we will seek appropriate E and F .
Case 2: We now consider the case θ=0 . From (4.11), we have
H(z,ˉz,θ)=−2Re{ˉq∗(0)h0(z,ˉz)q(θ)}+h0(z,ˉz). |
It then follows from H(z,ˉz,θ) and (4.12) that
{H20(0)=−g20q(0)−ˉg02ˉq(0)+τj(d1d2),H11(0)=−g11q(0)−ˉg11ˉq(0)+τj(d3d4), | (4.18) |
where
d1=2(a13+a14q21+a15q1),d2=2{b15q21+(b14+b16q21+b19q1)e−2iw0τj+(b17q1+b18q21)e−iw0τj},d3=2a13+2a14q1ˉq1+a15(q1+ˉq1),d4=2b14+2q1ˉq1(a15+b16)+b19(q1+ˉq1)+b17(q1eiw0τj+ˉq1e−iw0τj)d4=+b18q1ˉq1(eiw0τj+e−iw0τj). |
From (4.13) and the definition of A , we obtain
{∫0−1dη(θ)W20(θ)=2iw0τjW20(0)−H20(0),∫0−1dη(θ)W11(θ)=−H11(0), | (4.19) |
where η(θ)=η(0,θ) .
Substituting (4.17), (4.18) into (4.19) and noting that
(iw0τjI−∫0−1eiw0τjθdη(θ))q(0)=0, |
and
(−iw0τjI−∫0−1e−iw0τjθdη(θ))ˉq(0)=0. |
We conclude that
(2iw0τjI−∫0−1e2iw0τjθdη(θ))E1=τj(d1d2), |
that is
(2iw0−a11−a12−b11e−2iw0τj2iw0−b12−b13e−2iw0τj)E1=(d1d2). |
Thus
E1=(2iw0−a11−a12−b11e−2iw0τj2iw0−b12−b13e−2iw0τj)−1(d1d2). | (4.20) |
Similarly, we have
(a11a12b11b12+b13)E2=−(d3d4), |
which enables us to assert that
E2=−(a11a12b11b12+b13)−1(d3d4). | (4.21) |
Still now we can determine W20(θ) and W11(θ) from (4.17). Therefore, all gij in (4.10) can be determined. Furthermore, we can compute the following values
{c1(0)=i2w0τj(g11g20−2|g11|2−13|g02|2)+g212,μ2=−Re(c1(0))Re(λ′(τj)),β2=2Re(c1(0)),T2=−Im(c1(0))+μ2Im(λ′(τj))w0τj. | (4.22) |
[1] | R. Arditi and L.R. Ginzburg, Coupling in predator-prey dynamics: ratio-dependence, J. Theor. Biol., 139 (1989), 311–332. |
[2] | A.W. Bush and A.E. Cook, The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J. Theoret. Biol., 63 (1975), 385–396. |
[3] | J. Caperon, Time lag in population growth response of isochrysis galbana to a variable nitrate environment, Ecology, 50 (1969), 188–192. |
[4] | D.E. Contois, Kinetics of bacterial growth: relationship between population density and specific growth rate of continuous cultures, J. Gen. Microbiol., 21 (1959), 40–50. |
[5] | J. Flegr, Two distinct types of natural selection in turbidostat-like and chemostat-like ecosystems, J. Theor. Biol., 188 (1997), 121–126. |
[6] | J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. |
[7] | B.D. Hassard, N.D. Kazarinoff and Y.H. Wan, Theory and applications of Hopf bifurcation, Cambridge University Press, Cambridge, 1981. |
[8] | D. Herbert, R. Elsworth and B.C. Telling, The continuous culture of bacteria; a theoretical and experimental study, J. Gen. Microbiol., 14 (1956), 601–622. |
[9] | S.B. Hsu, S. Hubbell and P. Waltman, A mathematical theory for single-nutrient competition in continuous culture of microorganism, SIAM J. Appl. Math., 32 (1997), 366–383. |
[10] | Z.X. Hu, G.K. Gao and W.B. Ma, Dynamics of a three-species ratio-dependent diffusive model, Nonlinear Anal. Real, 11 (2010), 2106–2114. |
[11] | X.Y. Hu, Z.X. Li and X.G. Xiang, Feedback control for a turbidostat model with ratio-dependent growth rate, J. Appl. Math. Inform., 31 (2013), 385–398. |
[12] | Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, New York, 1993. |
[13] | P.D. Leenheer and H. Smith, Feedback control for the chemostat, J. Math. Biol., 46 (2003), 48–70. |
[14] | B.T. Li, Competition in a turbidostat for an inhibitory nutrient, J. Biol. Dynam., 2 (2008), 208–220. |
[15] | B.T. Li, G.S.K. Wolkowicz and Y. Kuang, Global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay, SIAM J. Appl. Math., 60 (2000), 2058–2086. |
[16] | Z.X. Li and L.S. Chen, Periodic solution of a turbidostat model with impulsive state feedback control, Nonlinear Dynam., 58 (2009), 525–538. |
[17] | Z. Li and R. Xu, Stability analysis of a ratio-dependent chemostat model with time delay and variable yield, Int. J. Biomath., 3 (2010), 243–253. |
[18] | N. MacDonald, Time lag in simple chemostat models, Biot. echnol. Bioeng., 18 (1976), 805–812. |
[19] | J. Monod, The growth of bacterial culture, Annu. Rev. Microbiol., 3 (1949), 371–394. |
[20] | H. Moser, The dynamics of bacterial populations maintained in the chemostat, Cold Spring Harb. Sym., 22 (1957), 121. |
[21] | M.I. Nelson and H.S. Sidhu, Reducing the emission of pollutants in food processing wastewaters, Chem. Eng. Process., 46 (2007), 429–436. |
[22] | R.T. Alqahtani, M.I. Nelson and A.L. Worthy, Analysis of a chemostat model with variable yield coefficient and substrate inhibition: contois growth kinetics, Chem. Eng. Process., 202 (2015), 332–344. |
[23] | Z.C. Jiang andW.B. Ma, Delayed feedback control and bifurcation analysis in a chaotic chemostat system, Int. J. Bifurcat. Chaos, 25 (2015), 1550087. |
[24] | C. Jost, Predator-prey theory: hidden twins in ecology and microbiology, Oikos, 90 (2000), 202–208. |
[25] | S.G. Ruan and J.J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynam. Cont. Discrete Impul. syst. Ser. A., 10 (2003), 863–874. |
[26] | S.G. Ruan and G.S.K. Wolkowicz, Bifurcation analysis of a chemostat model with a distributed delay, J. Math. Anal. Appl., 204 (1996), 786–812. |
[27] | H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springe, New York, 2010. |
[28] | O. Tagashira and T. Hara, Delayed feedback control for a chemostat model, Math. Biosci., 201 (2006), 101–112. |
[29] | Y. Tian, Y. Bai and P. Yu, Impact of delay on HIV-1 dynamics of fighting a virus with another virus, Math. Biosci. Eng., 11 (2014), 1181–1198. |
[30] | L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, UK, 1995. |
[31] | L.Wang and G.S.K.Wolkowicz, A delayed chemostat model with general nonmonotone response functions and differential removal rates, J. Math. Anal. Appl., 321 (2006), 452–468. |
[32] | Y. Yao, Z.X. Li and Z.J. Liu, Hopf bifurcation analysis of a turbidostat model with discrete delay, Appl. Math. Comput., 262 (2015), 267–281. |
[33] | S.L. Yuan, P. Li and Y.L. Song, Delay induced oscillations in a turbidostat with feedback control, J. Math. Chem., 49 (2011), 1646–1666. |
[34] | Z. Zhao and X.Y. Song, Bifurcation and complexity in a ratio-dependent predator-prey chemostat with pulsed input, Appl. Mat. Ser. B, 22 (2007), 379–387. |
[35] | Z.F. Zhang, T.R. Ding, W.Z. Huang and Z.X. Dong, Qualitative Theory of Differential Equations, American Mathematical Society, Providence, RI., 1992. |
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