Research article Special Issues

Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders

  • In this paper, a delayed fractional Lotka-Volterra food chain chemostat model with incommensurate orders is proposed, and the effect on system stability and bifurcation of this model are discussed. First, for the system with no controller, the stability and Hopf bifurcation with respect to time delay are investigated. Taking the time delay as the bifurcation parameter, the relevant characteristic equations are analyzed, and the conditions for Hopf bifurcation are proposed. The results show that the controller can fundamentally affect the stability of the system, and that they both have an important impact on the generation of bifurcation at the same time. Finally, numerical simulation is carried out to support the theoretical data.

    Citation: Xiaomeng Ma, Zhanbing Bai, Sujing Sun. Stability and bifurcation control for a fractional-order chemostat model with time delays and incommensurate orders[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 437-455. doi: 10.3934/mbe.2023020

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  • In this paper, a delayed fractional Lotka-Volterra food chain chemostat model with incommensurate orders is proposed, and the effect on system stability and bifurcation of this model are discussed. First, for the system with no controller, the stability and Hopf bifurcation with respect to time delay are investigated. Taking the time delay as the bifurcation parameter, the relevant characteristic equations are analyzed, and the conditions for Hopf bifurcation are proposed. The results show that the controller can fundamentally affect the stability of the system, and that they both have an important impact on the generation of bifurcation at the same time. Finally, numerical simulation is carried out to support the theoretical data.



    In the last 100 years, with the continuous development of mathematical biology, food chain models [1] have received much attention from scientists. In this field, mathematical models are established by scientific methods and reasonable assumptions, then, the specific problems are explained, predicted and controlled. In the 1920s, the Lotka-Volterra model explained the fluctuations in the number of fish and shark populations. The chemostat is a simple and well-adopted laboratory apparatus used to culture microorganisms. It can be used to investigate microbial growth and it has the advantage that the parameters are easily measurable. A sterile growth medium enters the chemostat at a constant rate; the volume within the chemostat is preserved by allowing excess medium to flow out through a siphon. We inoculate this chemostat with a heterotrophic bacterium that finds, in the medium, all of the necessary nutrients but one. This last nutrient is the limiting substrate. Based on previous works, the Lotka-Volterra food chain chemostat model has attracted a lot of attention. It has long been recognized that there is a time lag in the growth response of the population to environmental change. The growth of predators is based on the number of prey over a period of time. Therefore, it is feasible to add distributed delay to the food chain chemostat model. In [2], a model of the chemostat involving two species of microorganisms competing for two perfectly complementary, growth-limiting nutrients is considered.

    With the increasing complexity of the actual biological mathematical system and the increasing requirements for control, the control technology of integer-order calculus theory has been unable to achieve satisfactory results. However, using the model of an integer-order system can-not well describe some systems' dynamic processes; fractional calculus not only provides a new mathematical tool for the model of biological mathematics, but it also can solve some problems in real life; fractional calculus theory [3] can solve this problem. The order of integrals and differentials in fractional calculus can be changed at will. To a certain extent, it better expands the description ability of integral calculus. According to the historical information of the system, it is necessary and urgent to analyze the impact of time delay on the dynamics of a biological model in order to accurately describe the dynamics of the food chain model in theory and practice. In [4], a new integrated pest management predator-prey model is presented, and the existence and stability of the order-1 periodic orbit of the proposed model is discussed. Whether in population dynamics or epidemic dynamics [5,6,7], the relevant research of mathematical biology has been ongoing. In the field, fractional calculus has attracted the attention of engineers and scientists. It has been successfully applied in various fields, such as medicine, industry, finance, physics, security communication, system biology [8,9,10] and so on. With the rapid development of fractional calculus, the Hopf bifurcation of fractional-order models [11,12] has attracted more and more attention. In [13], the author mainly uses fractional-order differential equations to describe the dynamic behavior of the chemostat system. The integer-order chemostat model in the form of the ordinary differential equation was extended to the fractional-order differential equations. The stability and bifurcation analyses of the fractional-order chemostat model have been investigated using the Adams-type predictor-corrector method. In [14], the fractional-order form of a three dimensional chemostat model with variable yields is introduced. The stability analysis of this fractional system is discussed in detail. In order to study the dynamic behaviors of the mentioned fractional system, the well known the non-standard finite difference scheme was implemented.

    The time delay is inevitable in most practical dynamical networks, including biological models, neural networks [15] and evolutionary dynamics. For delayed fractional-order systems, the bifurcation problem has attracted more and more attention. Fractional calculus can be used as a mathematical analysis tool to study arbitrary order integrals and derivatives. It can describe many systems in the real world. However, due to these remarkable results, the influence of a time delay on bifurcation is ignored. The chemostat is a simple and well-adopted laboratory apparatus used to culture microorganisms. It can be used to investigate microbial growth and it has the advantage that the parameters are easily measurable. A sterile growth medium enters the chemostat at a constant rate; the volume within the chemostat is preserved by allowing excess medium to flow out through a siphon. We inoculate this chemostat with a heterotrophic bacterium that finds, in the medium, all of the necessary nutrients but one. This last nutrient is the limiting substrate.

    In [16], we study the following n-dimensional linear fractional differential system with multiple time delays

    {dq1x1(t)dq1t=a11x1(tτ11)+a12x2(tτ12)++a1nxn(tτ1n),dq2x2(t)dq2t=a21x1(tτ21)+a22x2(tτ22)++a2nxn(tτ2n),dqnxn(t)dqnt=an1x1(tτn1)+an2x2(tτn2)++annxn(tτnn). (1.1)

    In [17], the authors considered the chaotic control of integer-orders and fractional-orders of a chaotic Burke-Shaw system by using time delayed feedback control

    {Dαx(t)=S(x(t)+y(t)),Dαy(t)=y(t)Sx(t)z(t)+K[y(t)y(tτ)],Dαz(t)=V+Sx(t)y(t).

    They investigated the control of a chaotic Burke-Shaw system using the Pyragas method. This system is derived from a Lorenz system which has several applications in physics and engineering. The linear stability and the existence of Hopf bifurcation of this system were investigated.

    In [18], the authors promote and consider a Lotka-Volterra food chain chemostat model that incorporates both distributed delay and stochastic perturbations. In this paper, our main work is to consider a fractional-order model with time delays

    {dS(t)dt=d(aS)m1SXε,dX(t)dt=m1SXdXm2XYη,dY(t)dt=m2XYdY. (1.2)

    Next, in order to better study the control of System (1.2), an extended feedback controller can be added; the controller is represented as follows

    μ(t)=h[X(t)X(tν)];

    clearly, if h=0 or ν=0, it is obtained that the controller is meaningless. In this case, it will not change the final result of the equilibrium point of System (1.2).

    In this paper, the controller is added to an incommensurate order a delayed fractional-order model, shown as the following system

    {Dα1tS(t)=d(aS)m1SXε,Dα2tX(t)=m1SXdXm2XYη+μ(t),Dα3tY(t)=m2X(tτ)Y(tτ)dY, (1.3)

    where αi(0,1](i=1,2,3), and the initial values are as follows:

    S(t)=ϕ1(t), X(t)=ϕ2(t), Y(t)=ϕ3(t),ϕ1(t)0,ϕ2(t)0,ϕ3(t)0, t[max(τ,ν),0];

    the model is based on the Caputo derivative. The meanings of various parameters in the system are shown in the following figure.

    Table 1.  Definitions of parameters.
    Parameter Description
    S(t) The concentration of a nutrient at time t
    X(t) The concentration of the prey at time t
    Y(t) The concentration of the predator at time t
    m1 The per capita growth rate of the prey
    m2 The per capita growth rate of the predator
    a The concentration of the growth limiting nutrient in the feed vessel
    d The dilution rate
    ε The yield constant for prey growth on a nutrient
    η The yield constant for predator growth on the prey
    h The negative feedback gain
    ν The feedback control delay

     | Show Table
    DownLoad: CSV

    Motivated by the works mentioned above, this paper introduces a controller to a delayed fractional Lotka-Volterra food chain chemostat model with incommensurate orders. The architecture of our current paper is as follows. In the second section, some preliminary preparations are made. In the third section, some properties of the system, bifurcation control strategy and the stability of the system under the influence of the controller are studied. In the fourth section, the numerical simulation is described according to the theoretical knowledge of the previous sections. Finally, the corresponding conclusions are given.

    In this paper, all results are based on the Caputo derivative definition. The definition of the Caputo derivative can form the initial conditions of the fractional equation expressed in the form of integer derivative. With this advantage, some practical problems can be better solved. For the convenience of the reader, we present some necessary fractional definitions. The definitions can be found in recent literature.

    Definition 2.1. ([19]) The Caputo fractional derivative of order α of a function f(t) is defined as

    Dαtf(t)=1Γ(nα)tt0f(n)(τ)(tτ)α+1ndτ,

    where n is the positive integer and n1<α<n.

    Definition 2.2. ([20]) Let f(x) denote a function which vanishes for negative values of x. Its Laplace's transform Lα{f(x)} of order α (or α-th fractional Laplace transform) is defined by the following expression when it is finite:

    Lα{f(x)}:=:Fα(s)=0Eα(sαxα)f(x)(dx)α,:=limMM0Eα(sαxα)f(x)(dx)α,

    where sC and Eα(u) is the Mittag-Leffler function uk(kα)!. The following operational formulae can be easily obtained:

    Lα{xαf(x)}=DαsL{f(x)},Lα{f(ax)}s=(1/a)αLα{f(x)}x/a,Lα{f(xb)}=Eα(sαbα)Lα{f(x)},Lα{Eα(cαxα)f(x)}s=Lα{f(x)}s+c,Lα{xαf(x)}=DαsLα{f(x)},Lα{x0f(u)(du)α}=Γ1(1+α)sαLα{f(x)}.

    Furthermore, using the properties of the Mittag-Leffler function and integration by parts, we find that

    Lα{f(α)(x)}=sαLα{f(x)}Γ(1+α)f(0).

    According to the relevant conclusions of [15], the system (1.1) is extended to a more general linear system, System (2.1)

    {dq1x1(t)dq1t=a11x1(tτ11)+b11x1(t)+a12x2(tτ12)+b12x2(t)+a1nxn(tτ1n)+b1nxn(t),dq2x2(t)dq2t=a21x1(tτ21)+b21x1(t)+a22x2(tτ22)+b22x2(t)+a2nxn(tτ2n)+b2nxn(t),dqnxn(t)dqnt=an1x1(tτn1)+bn1x1(t)+an2x2(tτn2)+bn2x2(t)+annxn(tτnn)+bnnxn(t); (2.1)

    the characteristic matrix of System (2.1) is as follows:

    Δ(s)=(sq1a11esτ11b11a12esτ12b12a1nesτ1nb1na21esτ21b21sq2a22esτ22b22a2nesτ2nb2nan1esτn1bn1an2esτn2bn2sqnannesτnnbnn)
    =(sq1000sq2000sqn)(a11esτ11a12esτ12a1nesτ1na21esτ21a22esτ22a2nesτ2nan1esτn1an2esτn2annesτnn)(b11b12b1nb21b22b2nbn1bn2bnn);

    then the following conclusion can be established.

    Theorem 2.1. If all roots of the characteristic equation det(Δ(s))=0 have negative real parts, then the zero solution of System (2.1) is locally asymptotically stable.

    According to the solution method for the equilibrium point and the relevant definitions of the stability of the equilibrium point, we calculated that the positive equilibrium E(S,I,Y) of System (1.3) is shown as

    S=am2εm1+m2ε,X=dm2,Y=(am1εm1+m2εdm2)η,

    if am1m2εd(m1+m2ε)>0; then, the system (1.3) has a positive equilibrium E(S,I,Y).

    The linearized system of System (1.3) at the positive equilibrium E(S,I,Y) is

    {Dα1tS(t)=(dm1Xε)S(t)m1SεX(t),Dα2tX(t)=(m1X)S(t)+(m1Sdm2Yη+h)X(t)hX(tv)m2XηY(t),Dα3tY(t)=m2YX(tτ)+m2XY(tτ)dY(t). (2.2)

    By a Laplace transform, we have

    {sα1L[S(t)]sα11ϕ1(0)=(dm1Xε)L[S(t)]m1SεL[X(t)],sα2L[X(t)]sα21ϕ2(0)=(m1X)L[S(t)]+(m1Sdm2Yη+h)L[X(t)]hesν(L[X(t)]+0νestϕ2(t)dt)m2XηL[Y(t)],sα3L[Y(t)]sα31ϕ3(0)=m2Yesτ(L[X(t)]+0τestϕ2(t)dt)+m2Xesτ(L[Y(t)]+0τestϕ3(t)dt)dL[Y(t)], (2.3)

    where L[F(t)] represents the Laplace transform of F(t). Let Δ(s) represents the characteristic matrix of System (2.3); then, System (2.3) can be rewritten as follows:

    Δ(s)(L[S(t)]L[X(t)]L[Y(t)])=(b1(s)b2(s)b3(s)) (2.4)
    {b1(s)=sα11ϕ1(0),b2(s)=sα21ϕ2(0)hesv0vestϕ2(t)dt,b3(s)=sα31ϕ3(0)+m2Yesτ0τestϕ2(t)dt+m2Xesτ0τestϕ3(t)dt, (2.5)

    and

    Δ(s)=(sα1a11a120a21sα2a22h+hesνa230a32esτsα3a33esτ+d), (2.6)

    where

    (a11a12a13a21a22a23a31a32a33)=(dm1Xεm1Sε0m1Xm1Sdm2Yηm2Xη0m2Ym2X). (2.7)

    In this subsection, the stability and bifurcation of the positive equilibrium E are discussed with μ(t)0.

    Theorem 3.1. Suppose that μ(t)=0,τ=0 and αi(0,1](i=1,2,3), and the following condition holds:

    (T1):am1m2εd(m1+m2ε)>0.

    Then, the positive equilibrium E of System (1.3) is locally asymptotically stable.

    Proof. When μ(t)=0,

    Δ(s)=(sα1a11a120a21sα2a22a230a32esτsα3a33esτ+d); (3.1)

    the characteristic polynomial is written as

    a1(s)+a2(s)esτ=0, (3.2)

    where

    a1(s)=sα1+α2+α3+dsα1+α2a22sα1+α3a11sα2+α3a22dsα1a11dsα2+(a11a22+a12a21)sα3+a11a22d+a12a21d,a2(s)=a33sα1+α2+(a22a33+a23a32)sα1+a11a33sα2a11a22a33a11a23a32a12a21a33; (3.3)

    assuming that τ=0 and α1=α2=α3=α, (3.2) becomes

    λ3+H1λ2+H2λ+H3=0, (3.4)

    where

    λ=sα,H1=da22a11a33,H2=a22a33+a23a32+a11a33+a11a22+a12a21a22da11d,H3=a11a22d+a12a21da11a22a33a11a23a32a12a21a33. (3.5)

    When τ=0 and α1=α2=α3=α, (3.4) can be written as

    (λz1)(λz2)(λz3)=0,

    where z1,z2 and z3 are the roots of (3.4). If (T1) is satisfied, all roots of (3.2) have negative real parts, i.e., all eigenvalues λi of the Jacobian matrix evaluated at the equilibrium points satisfy |arg(λi)|>απ2(i=1,2,3); then, the positive equilibrium E is locally asymptotically stable.

    When τ=0, for all αi(0,1] (i=1,2,3), let sα1=λ1, sα2=λ2 and sα3=λ3. Without loss of generality, (3.2) can be written as

    (λ1z1)(λ2z2)(λ3z3)=0. (3.6)

    According to relevant calculations, the characteristic equation satisfies the Hurwitz criterion, and if (T1) is satisfied, all roots of (3.6) have negative real parts, i.e., all eigenvalues λi of the Jacobian matrix evaluated at the equilibrium points satisfy |arg(λi)|>απ2(i=1,2,3); then, the positive equilibrium E is locally asymptotically stable. This completes the proof.

    Theorem 3.2. Assume μ(t)=0,τ>0 and the condition (T1) holds. If the following condition holds:

    (T3):ϖ1ϑ1+ϖ2ϑ2>0, where

    ϖ1=(Re[a1(iφ0)]+Re[a2(iφ0)])cosφ0τ0(Im[a1(iφ0)]+Im[a2(iφ0)])sinφ0τ0,ϖ2=(Re[a1(iφ0)]+Re[a2(iφ0)])sinφ0τ0+(Im[a1(iφ0)]+Im[a2(iφ0)])cosφ0τ0,ϑ1=φ0Im[a2(iφ0)],   ϑ2=φ0Re[a2(iφ0)];

    then

    1) The positive equilibrium of System (1.3) is locally asymptotically stable for τ[0,τ0).

    2) There exists a constant τ0>0 such that System (1.3) has a Hopf bifurcation.

    Proof. Let s=iφ=φeπi2 and substituting it into (3.2), it can be obtained that

    (cosφτisinφτ)=a1(iφ)a2(iφ),a2(iφ)0; (3.7)

    separating the real and imaginary parts, one gets

    {cosφτ=Re[a1(iφ)]Re[a2(iφ)]+Im[a1(iφ)]Im[a2(iφ)]Re2[a2(iφ)]+Im2[a2(iφ)],sinφτ=Im[a1(iφ)]Re[a2(iφ)]Re[a1(iφ)]Im[a2(iφ)]Re2[a2(iφ)]+Im2[a2(iφ)],

    where

    Re[a1(iφ)]=φα1+α2+α3cos(α1+α2+α3)π2+dφα1+α2cos(α1+α2)π2a22φα1+α3cos(α1+α3)π2a11φα2+α3cos(α2+α3)π2a22dφα1cosα1π2a11dφα2cosα2π2+(a11a22+a12a21)φα3cosα3π2+a11a22d+a12a21d,Im[a1(iφ)]=φα1+α2+α3sin(α1+α2+α3)π2+dφα1+α2sin(α1+α2)π2a22φα1+α3sin(α1+α3)π2a11φα2+α3sin(α2+α3)π2a22dφα1sinα1π2a11dφα2sinα2π2+(a11a22+a12a21)φα3sinα3π2,Re[a2(iφ)]=a33φα1+α2cos(α1+α2)π2+(a22a33+a23a32)φα1cosα1π2+a11a33φα2cosα2π2a12a21a33a11a22a33a11a23a32,Im[a2(iφ)]=a33φα1+α2sin(α1+α2)π2+(a22a33+a23a32)φα1sinα1π2+a11a33φα2sinα2π2.

    Then, it can be obtained that

    Ψ12(φ)+Ψ22(φ)=1, (3.8)

    where

    {Ψ1(φ)=cosφτ=Re[a1(iφ)]Re[a2(iφ)]+Im[a1(iφ)]Im[a2(iφ)]Re2[a2(iφ)]+Im2[a2(iφ)],Ψ2(φ)=sinφτ=Im[a1(iφ)]Re[a2(iφ)]Re[a1(iφ)]Im[a2(iφ)]Re2[a2(iφ)]+Im2[a2(iφ)];

    suppose that (3.8) has at least a positive real root φ0. Then, it is obtained that

    {τ(k)0=1φ0[arccosΨ1(φ0)+2kπ],k=0,1,2τ(k)0=1φ0[arcsinΨ2(φ0)+2kπ],k=0,1,2 (3.9)

    here, the bifurcation point is defined as τ0=min{τ(k)0}. Then, we get the positive equilibrium of system (1.3) is locally asymptotically stable.

    Next, differentiating (3.2) with respect to τ, we obtain

    (a1(s)+a2(s)τa2(s)esτ)dsdτsa2(s)esτ=0;

    then,

    (dsdτ)1=a1(s)esτ+a2(s)sa2(s)τs; (3.10)

    substituting s=iφ0 and τ=τ0 into (3.10), it gives

    (dsdτ)1|s=iφ0=ϖ1+iϖ2ϑ1+iϑ2τ0iφ0;

    thus, one obtains

    Re[(dsdτ)1]=ϖ1ϑ1+ϖ2ϑ2ϑ12+ϑ22.

    If ϖ1ϑ1+ϖ2ϑ2>0, we have that Re[(dsdτ)1]τ=τ0>0. According to Theorem 2.1, this section demonstrates completion.

    In this subsection, consider the case with the controller below. First, under the influence of μ(t), the bifurcation and stability of E with respect to time delay are studied. Given the controller, the characteristic equation is written as follows

    Λ1(s)+Λ2(s)esτ=0, (3.11)

    where

    Λ1(s)=a1(s)+hesvsα1+α3+hdesvsα1a11hesvsα3a11hdesvhsα1+α3dhsα1+a11hsα3+a1hd,Λ2(s)=a2(s)ha33esvsα1+a11a33hesv+ha33sα1a11a33h. (3.12)

    Theorem 3.3. Assume τ>0 and the condition (T1) hold. If the following condition holds:

    (T4):κ1ι1+κ2ι2>0, where

    κ1=(Re[Λ1(iφ1)]+Re[Λ2(iφ1)])cosφ1τ1(Im[Λ1(iφ1)]+Im[Λ2(iφ1)])sinφ1τ1,κ2=(Re[Λ1(iφ1)]+Re[Λ2(iφ1)])sinφ1τ1+(Im[Λ1(iφ1)]+Im[Λ2(iφ1)])cosφ1τ1,ι1=φ1Im[Λ2(iφ1)],   ι2=φ1Re[Λ2(iφ1)];

    then

    1) The positive equilibrium of System (1.3) is locally asymptotically stable for τ[0,τ01).

    2) There exists a constant τ01>0 such that System (1.3) has a Hopf bifurcation.

    Proof. Let s=iφ=φeπi2 and substituting it into (3.2), it can be obtained that

    (cosφτisinφτ)=Λ1(iφ)Λ2(iφ),Λ2(iφ)0; (3.13)

    separating the real and imaginary parts, one gets

    {cosφτ=Re[Λ1(iφ)]Re[Λ2(iφ)]+Im[Λ1(iφ)]Im[Λ2(iφ)]Re2[Λ2(iφ)]+Im2[Λ2(iφ)],sinφτ=Im[Λ1(iφ)]Re[Λ2(iφ)]Re[Λ1(iφ)]Im[Λ2(iφ)]Re2[Λ2(iφ)]+Im2[Λ2(iφ)],

    where

    Re[Λ1(iφ)]=Re[a1(iφ)]+hφ(α1+α3)(cosφvcos(α1+α3)π2+sinφvsin(α1+α3)π2cos(α1+α3)π2)+dhφα1(cosφvcosα1π2+sinφvsinα1π2cosα1π2)a11hφα3(cosφvcosα3π2+sinφvsinα3π2cosα3π2)a11hdcosφv+a11hd,Im[Λ1(iφ)]=Im[a1(iφ)]+hφ(α1+α3)(cosφvsin(α1+α3)π2+sinφvcos(α1+α3)π2sin(α1+α3)π2)+dhφα1(cosφvsinα1π2+sinφvcosα1π2sinα1π2)a11hφα3(cosφvsinα3π2+sinφvcosα3π2sinα3π2)a11hdsinφv,Re[Λ2(iφ)]=Re[a2(iφ)]a33hφα1cosφvcosα1π2a33hφα1sinφvsinα1π2+a11a33hcosφv+a33hφα1cosα1π2a11a33h,Im[Λ2(iφ)]=Im[a2(iφ)]a33hφα1cosφvsinα1π2+a33hφα1sinφvcosα1π2+a11a33hsinφv+a33hφα1sinα1π2.

    Then, it can be obtained that

    j12(φ)+j22(φ)=1, (3.14)

    where

    {j1(φ)=cosφτ=Re[Λ1(iφ)]Re[Λ2(iφ)]+Im[Λ1(iφ)]Im[Λ2(iφ)]Re2[Λ2(iφ)]+Im2[Λ2(iφ)],j2(φ)=sinφτ=Im[Λ1(iφ)]Re[Λ2(iφ)]Re[Λ1(iφ)]Im[Λ2(iφ)]Re2[Λ2(iφ)]+Im2[Λ2(iφ)];

    suppose that (3.14) has at least a positive real root φ1. Then, it is obtained that

    {τ(k)1=1φ1[arccosj1(φ1)+2kπ],k=0,1,2τ(k)1=1φ1[arcsinj2(φ1)+2kπ],k=0,1,2 (3.15)

    the bifurcation point is defined as τ1=min{τ(k)1}.

    Next, differentiating (3.11) with respect to τ gives

    (Λ1(s)+Λ2(s)τΛ2(s)esτ)dsdτsΛ2(s)esτ=0;

    then,

    (dsdτ)1=Λ1(s)esτ+Λ2(s)sΛ2(s)τs; (3.16)

    substituting s=iφ1 and τ=τ1 into (3.16), it gives

    (dsdτ)1|s=iφ1=κ1+iκ2ι1+iι2τ1iφ1;

    thus, one obtains

    Re[(dsdτ)1]=κ1ι1+κ2ι2ι12+ι22.

    If κ1ι1+κ2ι2>0, we have that Re[(dsdτ)1]τ=τ1>0. According to Theorem 2.1, this section demonstrates completion.

    Theorem 3.4 mainly provides a method to find a given μ(t) bifurcation point. However, the most important point should be to find the appropriate control parameters for the time delay. And the characteristic equation is written as follows

    W1(s)+W2(s)esυ=0, (3.17)

    where

    W1(s)=a1(s)+A2(s)esτW2(s),W2(s)=h[(a11a33a33sα1)esτ+dsα1+sα1+α3a11da11sα3]. (3.18)

    Theorem 3.4. Assume the condition (T1) holds. If the following condition holds:

    (T5):P1Q1+P2Q2>0, where

    P1=(Re[W1(iφ2)]+Re[W2(iφ2)])cosφ2υ0(Im[W1(iφ2)]+Im[W2(iφ2)])sinφ2υ0,P2=(Re[W1(iφ2)]+Re[W2(iφ2)])sinφ2υ0+(Im[W1(iφ2)]+Im[W2(iφ2)])cosφ2υ0,Q1=φ2Im[W2(iφ2)],   Q2=φ2Re[W2(iφ2)];

    then

    (1) The positive equilibrium of System (1.3) is locally asymptotically stable for υ[0,υ0).

    (2) There exists a constant υ=υ0>0 such that System (1.3) has a Hopf bifurcation.

    Proof. Let s=iφ=φeπi2 and substituting it into (3.17), it can be obtained that

    (cosφυisinφυ)=W1(iφ)W2(iφ),W2(iφ)0; (3.19)

    separating the real and imaginary parts, one gets

    {cosφυ=Re[W1(iφ)]Re[W2(iφ)]+Im[W1(iφ)]Im[W2(iφ)]Re2[W2(iφ)]+Im2[W2(iφ)],sinφυ=Im[W1(iφ)]Re[W2(iφ)]Re[W1(iφ)]Im[W2(iφ)]Re2[W2(iφ)]+Im2[W2(iφ)],

    where

    Re[W1(iφ)]=Re[a1(iφ)]+Re[a2(iφ)]cosφτ+Im[a2(iφ)]sinφτRe[W2(iφ)],Im[W1(iφ)]=Im[a1(iφ)]+Im[a2(iφ)]cosφτRe[a2(iφ)]sinφτIm[W2(iφ)],Re[W2(iφ)]=ha11a33cosφvha33φα1(cosφvcosα1π2+sinα1π2sinφv)+hdφα1cosα1π2,+hφα1+α3cos(α1+α3)π2a11dha11hφα3cosα3π2Im[W2(iφ)]=ha11a33sinφvha33φα1(cosφvsinα1π2+sinα1π2cosφv)+hdφα1sinα1π2+hφα1+α3sin(α1+α3)π2a11dha11hφα3sinα3π2.

    Then, it can be obtained that

    r12(φ)+r22(φ)=1, (3.20)

    where

    {r1(φ)=cosφυ=Re[W1(iφ)]Re[W2(iφ)]+Im[W1(iφ)]Im[W2(iφ)]Re2[W2(iφ)]+Im2[W2(iφ)],r2(φ)=sinφυ=Im[W1(iφ)]Re[W2(iφ)]Re[W1(iφ)]Im[W2(iφ)]Re2[W2(iφ)]+Im2[W2(iφ)];

    suppose that (3.20) has at least a positive real root φ2. Then, it is obtained that

    {υ(k)0=1φ2[arccosr1(φ1)+2kπ],k=0,1,2υ(k)0=1φ2[arcsinr2(φ1)+2kπ],k=0,1,2 (3.21)

    the bifurcation point is defined as υ0=min{υ(k)0}.

    Next, differentiating (3.17) with respect to υ gives

    (W1(s)+W2(s)τW2(s)esυ)dsdυsW2(s)esυ=0;

    then, we have

    (dsdυ)1=W1(s)esυ+W2(s)sW2(s)υs; (3.22)

    substituting s=iφ2 and υ=υ0 into (3.22), it gives

    (dsdυ)1|s=iφ2=P1+iP2Q1+iQ2υ0iφ2;

    thus, one obtains

    Re[(dsdυ)1]=P1Q1+P2Q2Q12+Q22.

    If P1Q1+P2Q2>0, we have that Re[(dsdυ)1]υ=υ0>0. According to Theorem 2.1, this section demonstrates completion.

    In this section, some concrete examples are given to illustrate the theory presented in the previous section. First, the system unaffected by the controller is considered; the system is shown as

    Example 4.1.

    {Dα1tS(t)=720(32S)15SX7,S(0)=0.7,Dα2tX(t)=32SX720X75XY8,X(0)=0.1,Dα3tY(t)=3X(tτ)Y(tτ)720Y,Y(0)=0.08. (4.1)

    According to the above system, the equilibrium point can be calculated as E=(0.8750,0.1167,0.1027), and all of the conditions in Theorem 3.1 can be satisfied. Thus, if τ=0, the positive equilibrium of System (4.1) is locally asymptotically stable, in which α1=0.98, α2=0.96 and α3=0.97, as shown in Figure 1.

    Figure 1.  Local asymptotic stability of the positive equilibrium E when τ=0.

    Then, according to Theorem 3.2, when α1=0.8, α2=0.95 and α3=0.97, it can be calculated that τ0=3.7286. As a consequence, Hopf bifurcation occurs at τ0=3.7286. As shown in Figure 2, when τ=3<τ0, the positive equilibrium point is locally asymptotically stable.

    Figure 2.  Effects of τ on E when τ=3.

    When τ=3.8>τ0, it becomes unstable, as shown in Figure 3.

    Figure 3.  Effects of τ on E when τ=3.8.

    In order to show the effects of incommensurate orders on Hopf bifurcation, let α1=0.8, α2=0.95 and τ=3.8; when α3=0.92, we obtain τ0=4.236>3.8; when α3=0.97, we obtain τ0=3.6285<3.8. As Figure 4 shows, when α3 changes from 0.92 to 0.97, the equilibrium point becomes stable. It can be seen that the change of fractional-order will also affect the change of system stability.

    Figure 4.  Effects on E when τ=3.8,α3=0.92.

    Next, consider a system that adds an extended feedback controller:

    Example 4.2.

    {Dα1tS(t)=720(32S)15SX7,S(0)=0.7,Dα2tX(t)=32SX720X75XY8+h[X(t)X(tv)],X(0)=0.1,Dα3tY(t)=3X(tτ)Y(tτ)720Y,Y(0)=0.08. (4.2)

    According to Theorem 3.4, when h=0.5, it is calculated that ν0=3.3769. As can be seen from Figures 5 and 6, when ν=3<ν0, the positive equilibrium point of System (4.2) is locally asymptotically stable; when ν=4>ν0, the system is unstable.

    Figure 5.  Effects of υ on E when υ=3.
    Figure 6.  Effects of υ on E when υ=4.

    In this work, a controller was added to a fractional-order chemostat model with incommensurate delay to study its effect on system stability and bifurcation. We first considered some basic results for the positive equilibrium E in the absence of a controller. Then the influence of the controller on the bifurcation and stability of the system was analyzed in detail. Considering the case of incommensurate order, given certain conditions, the corresponding control bifurcation parameters were obtained accurately. Finally, in order to support the theoretical analysis results, numerical simulations were carried out. The results show that the stability and bifurcation of the system were effectively controlled.

    This research was funded by NSFC grant number 11571207, SDNSF grant number ZR2021MA064 and the Taishan Scholar project.

    The authors declare that there is no conflict of interest.



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