# Stability and Hopf Bifurcation Analysis in a Delayed Myc/E2F/miR-17-92 Network Involving Interlinked Positive and Negative Feedback Loops

Authors: Guiyuan Wang; Zhuoqin Yang
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

MiR-17-92 plays an important role in regulating the levels of the Myc/E2F protein. In this paper, we consider a coupling network between Myc/E2F/miR-17-92 delayed negative feedback loop and Myc/E2F positive feedback loop described by a two-dimensional delay differential equation. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing time delay as well as negative feedback strength associated with oscillations as bifurcation parameters, respectively. Furthermore, direction and stability of Hopf bifurcation of time delay are studied by using the normal form method and center manifold theorem. Finally, several numerical simulations are performed to verify the results we obtained.


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## Body

## 1. Introduction

Due to the development of large-scale experimental and computational techniques, a posttranscriptional regulation by small noncoding microRNAs (miRNAs) has been discovered in many cellular processes, including cell growth, development, differentiation, and apoptosis [1–4]. The miR-17-92 cluster as a polycistronic gene located in human chromosome 13 ORF 25 (c13orf25) is composed of 7 mature miRNAs [1]. MiRNAs play critical roles in biological processes, as posttranscriptional regulators of gene expression [2]. MicroRNA-based regulation has been simulated by specially designed mathematical models [5–7]. The transcription factors E2F and Myc act as tumor suppressors or oncogenes and participate in the control of cell proliferation and apoptosis [1]. Aguda et al. proposed a simple model involving miR-17-92, E2F, and Myc, which is composed of a positive feedback (E2F/Myc) and a negative feedback (Myc/E2F/miR-17-92) [1]. It presents a bistable switch behavior and a one-way switch in the network, which corresponds to the bistability and monostability, respectively [1, 2]. Subsequently, Li et al. illustrated an abstract model of the network presented by Aguda et al. and focused on the physiological significance of miRNAs [2]. It was found that the existence of miRNAs improves the ability of the bistable switches in the network [2]. Zhang et al. further analyzed this abstract model and suggested that the interlinked positive and negative feedback loops buffer noise effects rather than only amplifying or suppressing the noise [3].Dynamical analysis of the system with time delay is an essential topic in many fields, especially for the models of gene expression (see [8–11]). Time delay is inevitable in Myc/E2F/miR-17-92 network since feedback loops involve many intermediate processes such as transcription, translation, and posttranslational modifications [12–14]. Time delay influences the time-dependent dynamics even for the simplest circuits with one and two gene elements, which can give rise to rich dynamical behaviors such as periodic and chaotic dynamics [15].In this work, we consider the effect of time delay in the delayed Myc/E2F /miR-17-92 network. As negative feedback is often used in biochemistry to generate oscillations to achieve homeostasis, the steady state may lose stability and be replaced by oscillations [16]. So an inhibition efficiency parameter Γ2 in the network is incorporated into ordinary differential equations to give rise to periodic solutions at certain critical values.The organization of the paper is as follows: we describe the Myc/E2F/miR-17-92 network with time delay in Section2. In Section 3.1, we discuss local stability of the solution and Hopf bifurcation of the network with respect to negative feedback strength Γ2. Under different values of Γ2, local stability and Hopf bifurcation of time delay are studied in Section 3.2. In Section 3.3, a formula is given to determine the direction and stability of the Hopf bifurcation of the time delay. In Section 4, four examples on influence of time delay on different states of negative feedback strength Γ2 are given to support our results. Finally, the conclusions are drawn in Section 5.


## 2. Model Description

An illustration of network involving miR-17-92, E2F, and Myc and abstract structure of this network are depicted in Figure1, where P is the protein module and M denotes the miRNA cluster module. The positive feedback in module P (E2Fs and Myc) as an autocatalytic process can lead to the transcription of M but be inhibited by M.Figure 1
An illustration of the network involving miR-17-92, E2F and Myc (a) and its abstract model (b), where P and M represent the protein module (E2Fs and Myc) and miR-17-92 cluster, respectively.
(a) (b)Aguda B D et al. presented a model of interaction between miR-17-92, E2F and Myc, which is described by the following differential equations [1]:(1)dPdt=αp+kpP2Γ1+P2+Γ2M-βpP,dMdt=αM+kMP-βMM,where [P] and [M] represent the concentrations of P and M, respectively. αM denotes the P-independent constitutive transcription of M, and αp describes the constitutive protein expression. Parameters βp and βM are the degradation rates of P and M, respectively. Γ1 is the coefficient of protein expression and Γ2 is the inhibition efficiency parameter. kM stands for the rate constant, and kP represents the constant of protein expression.For this model, we consider the inhibition efficiency parameterΓ2 to explore stability of equilibria and oscillatory behaviors via Hopf bifurcation through its critical value as negative feedback usually leads to oscillation. Moreover, we investigate the effect of the time delay in the network when the concentrations of P and M are described by the following delayed differential equations (DDEs):(2)dPdt=αp+kpPt-τ2Γ1+Pt-τ2+Γ2Mt-τ-βpP,dMdt=αM+kMP-βMM.For the convenience of analysis, we define(3)P=PandM=M.With these substitutions, system (2) can be rewritten as(4)dPdt=αp+kpP2t-τΓ1+P2t-τ+Γ2Mt-τ-βpP,dMdt=αM+kMP-βMM.The initial values for system (4) take the form of(5)Pθ=φ1θ≥0,Mθ=φ2θ≥0,θ∈-τ,0,φ10=P0>0,φ20=M0>0,where (φ1,φ2)∈C([-τ,0],0,+∞2), and all the parameters are positive.


## 3. Main Results

System (4) exhibits dynamics behaviors of the steady state and periodic phenomenon in the different parameter regimes. Here, we focus especially on theoretical analysis on the stability and existence of Hopf bifurcation of system (4).


### 3.1. Local Stability and Hopf Bifurcation of the Negative Feedback StrengthΓ2

Firstly, at the time delayτ=0, we consider Hopf bifurcation of the negative feedback strength Γ2 as the measure of the miRNA inhibition through the critical values and local stability of equilibrium before the bifurcation in system (4).LetE∗(P∗,M∗) be the equilibrium of system (4), and then(6)αp+kpP∗2Γ1+P∗2+Γ2M∗-βpP∗=0,αM+kMP∗-βMM∗=0.Eliminating M∗ from above equations, we get the following equation on P∗:(7)βMβpP∗3+kMΓ2βp-βMαp+kpP∗2+βMΓ1βp+Γ2αMβp-kMαpP∗-αpβMΓ1+Γ2αM=0.Obviously, the equation has at most three positive real roots. Setting E∗(P∗,M∗) is one of the three roots.Secondly, we translate the equilibriumE∗ to the origin. By the linear transform(8)xt=Pt-P∗yt=Mt-M∗system (4) becomes(9)x˙t=mxt-τ-nyt-τ-βpx+∑i+j≥21i!j!fij1xit-τyjt-τ,y˙t=kMxt-βMyt,where(10)m=2kpP∗Γ1+Γ2M∗Γ1+P∗2+Γ2M∗2,n=kpP∗2Γ2Γ1+P∗2+Γ2M∗2,f1=αp+kpP2t-τΓ1+P2t-τ+Γ2Mt-τ-βpP,f2=αM+kMP-βMM,fij1=∂i+jf1∂xi∂yjP∗,M∗,i,j≥0.From (9), it is easy to get the following characteristic equation. That is,(11)λ-m+βpn-kMλ+βM=0;thus, the two degree polynomial equation is obtained(12)λ2+βM+βp-mλ+βMβp-m+nkM=0.IfH1βM+βp-m>0,H2βMβp-m+nkM>0,then all the roots of (12) have negative real parts. So the equilibrium E∗ of system (4) is locally stable.On the other hand, we assumeH3βM+βp-m=0and λ1,2(μ)=α(μ)±iω(μ) are the roots of (12).Viewing the negative feedback strengthΓ2 as a bifurcation parameter, the sufficient conditions for the occurrence of the Hopf bifurcation of system (4) are obtained in the following results.Theorem 1.
IfH2 and H3 hold, there exist Γ2=Γ20, such that(13)αΓ20=0,ωΓ20>0,dαΓ20dΓ2≠0,then system (4) can exhibit Hopf bifurcation.Time delay is inevitable and plays an important role in negative feedback loop of the Myc/E2F/miR-17-92 network due to the transcription and translation. So time delay on this network is considered in the next sections.


### 3.2. Local Stability and Hopf Bifurcation of Time Delay

In this section, we take the time delayτ as a bifurcating parameter to investigate the stability and existence of Hopf bifurcation of time delay in system (4). For convenience, let k1=kp,k2=βp, k3=kM,k4=βM, then system (9) becomes(14)x˙t=mxt-τ-nyt-τ-k2x+∑i+j≥21i!j!fij1xit-τyjt-τ,y˙t=k3x-k4y,where(15)m=k1n1,n=k1n2,n1=2x∗Γ1+Γ2y∗Γ1+x∗2+Γ2y∗2,n2=Γ2x∗2Γ1+x∗2+Γ2y∗2.Then we obtain the characteristic equation of system (14).That is,(16)λ2+k2+k4λ-k1n1e-λτλ+k2k4+k1k3n2-k1k4n1e-λτ=0.When τ=0, it becomes(17)λ2+k2+k4-k1n1λ+k2k4+k1k3n2-k1k4n1=0.i ω   ω > 0 is the root of (16) if and only if iω satisfies(18)-ω2+k2+k4iω-k1n1cos⁡ωτ-isin⁡ωτiω+k2k4+k1k3n2-k1k4n1cos⁡ωτ-isin⁡ωτ=0.Separating the real and imaginary parts, we get(19)k2+k4ω=k1n1ωcos⁡ωτ+k1k3n2-k1k4n1sin⁡ωτ,k2k4-ω2=k1n1ωsin⁡ωτ-k1k3n2-k1k4n1cos⁡ωτ.which is equivalent to(20)ω4-k12n12-k22-k42ω2+k22k42-k12k3n2-k4n12=0.Let z=ω2, then it is transformed to be(21)z2-pz+q=0where(22)p=k12n12-k22-k42,q=k22k42-k12k3n2-k4n12.Assume (H4) equation (21) has at least one positive real root.From (21), we can obtain(23)z1,2=12p±12p2-4q.Thus, ifH5p<0,q>0orp2-4q<0then none of z1,z2 is positive. So (21) has no positive roots. It means that there are no imaginary roots in characteristic equation (16).From those discussed above, we obtain(24)cos⁡ωτ=k2k42n1+k3n2+k2n1ω2-k2k3k4n2k1k3n2-k4n12+n12ω2,sin⁡ωτ=ωk2k3n2+k3n2k4-n1ω2-n1k42k1k3n2-k4n12+n12ω2.Without loss of generality, (21) is assumed to have two positive roots, defined by z1,z2, respectively. Thus, (20) has two positive roots ω1=z1,ω2=z2.So we have(25)cos⁡ωkτk=k2k42n1+k3n2+k2n1ωk2-k2k3k4n2k1k3n2-k4n12+n12ωk2.It follows that(26)τkj=1ωkarccosk2k42n1+k3n2+k2n1ωk2-k2k3k4n2k1k3n2-k4n12+n12ωk2+2jπωk,where k=1,2,j=0,1,2,….Define(27)τ0=minτ10,τ20.Taking the derivative of λ with respect to τ, we can get(28)dλdτ-1=2λ+k2+k4-k1n1e-λτk1λe-λτk3n2-k4n1-n1λ-τλ.Then,(29)Redλdττ=τ0-1=c+da2+b2,where(30)a=k1k3n2-k1k4n1,b=k1n1ω0,c=2cosω0τ0a-sinω0τ0b-k12n12,d=k2+k4k1n1cos⁡ω0τ0+1ω0asin⁡ω0τ0.If the conditionH6c+d≠0holds, then(31)Redλdττ=τ0-1≠0.That is, the transversality condition is satisfied.Based on the analysis above, the following results are obtained.Theorem 2.
Letτ=τkj be defined by (26); one has
(i) ifH1, H2, and H5 hold, then the equilibrium E∗ of system (4) is asymptotically stable for all τ≥0;
(ii) ifH1, H2, and H4 hold, then the equilibrium E∗ is locally asymptotically stable for 0≤τ<τ0 and unstable for τ>τ0; moreover, if c+d≠0, then system (4) undergoes a Hopf bifurcation at the equilibrium E∗ when τ=τkj(k=1,2,j=0,1,2,…).


### 3.3. Direction and Stability of the Hopf Bifurcation at Time Delayτ=τ0

In this section, we further consider direction and stability of Hopf bifurcation assuming atτ=τ0 by using the methods in Hassard et al. [17]. For convenience, let τ=τ0+μ,μ∈R,s=t/τ and denote s=t, then system (14) becomes(32)x˙t=τ0+μmxt-1-nyt-1-k2x+∑i+j≥21i!j!fij1xit-1yjt-1,y˙t=τ0+μk3xt-k4yt.Clearly, μ=0 is a Hopf bifurcation value of system (32).Its linear part is given by(33)x˙t=τ0+μmxt-1-nyt-1-k2xt,y˙t=τ0+μk3xt-k4yt.and nonlinear part is(34)fμ,ut=τ0+μ∑i+j≥21i!j!fij1xit-1yjt-1]0.For convenience of studying the problems of Hopf bifurcation, we rewrite system (32) as(35)u˙t=Lμut+fμ,ut,where u(t)=(x(t),y(t))T and ut(θ)=u(t+θ) for θ∈[-τ,0]; Lμ:C→R,f:R×C→R are given, respectively, by(36)Lμφ=τ0+μ-k20k3-k4φ10φ20+τ0+μm-n00φ1-1φ2-1,fμ,ut=τ0+μ∑i+j≥21i!j!fij1xit-1yjt-1]0=τ0+μ12f201φ12-1+f111φ1-1φ2-1+12f021φ22-1+h.o.t.0.Here,h.o.t. denotes higher order terms.For convenience analysis, let(37)L20=12f201,L11=f111,L02=12f021,where f20(1)=2k1Γ1+Γ2y∗Γ1-3x∗2+Γ2y∗/Γ1+x∗2+Γ2y∗3; f11(1)=-2k1x∗Γ2Γ1-x∗2+Γ2y∗/Γ1+x∗2+Γ2y∗3, f02(1)=2k1x∗2Γ22/Γ1+x∗2+Γ2y∗3.Then,(38)fμ,ut=τ0+μL20φ12-1+L11φ1-1φ2-1+L02φ22-1+h.o.t.0.According to the Riesz representation theorem, there exists a 2 × 2 matrix functionη(θ,μ),θ∈-1,0, such that(39)Lμφ=∫-10dηθ,μφθ,φ∈C1-1,0,R2.Here, we choose(40)ηθ,μ=τ0+μ-k20k3-k4δθ-τ0+μm-n00δθ+1,where δθ is defined by (41)δθ=0,θ≠01,θ=0.Forφ∈C1[-1,0],R2, define(42)Aμφ=dφθdθ,-1≤θ<0,∫-10dηθ,μφθ,θ=0,and(43)Rμφ=00,-1≤θ<0,τ0+μ∑i+j≥21i!j!fij1xit-1yjt-10,θ=0.For ψ∈C1[0,1],(R2)∗, the adjoint operator A∗ of A is defined by(44)A∗ψs=-dψsds,0<s≤1,∫-10dηTs,μψ-s,s=0.And define a bilinear inner product as follows:(45)ψs,φθ=ψ-0φ0-∫-10∫ξ=0θψ-ξ-θdηθφξdξ,where η(θ)=η(θ,0).From the discussion in Section3.2, we obtain that ±iτ0ω0 are eigenvalues of A(0). At the same time, they are also eigenvalues of A∗ since A and A∗ are adjoint operator. In the following, we calculate the eigenvectors of A(0) and A∗ corresponding to iτ0ω0 and -iτ0ω0, respectively. Suppose q(θ)=(1,β)Τeiτ0ω0θ,θ∈-1,0 is eigenvector of A(0). Through the discussion above, we have(46)A0q0=iτ0ω0q0.That is,(47)τ0iω0+k2-me-iτ0ω0ne-iτ0ω0-k3iω0+k4q0=00.We can obtain(48)β=k3iω0+k4orβ=me-iτ0ω0-iω0-k2ne-iτ0ω0.Thus, q(0)=(1,β)Τ=(1,k3/iω0+k4)Τ or q(0)=(1,me-iτ0ω0-iω0-k2/ne-iτ0ω0)Τ.On the other hand, we assumeq∗(s)=D(1,β∗)Τeiτ0ω0s,s∈0,1 is the eigenvector of A∗ and then obtain(49)τ0-iω0+k2-me-iτ0ω0-k3ne-iτ0ω0-iω0+k4q∗0=00.and(50)q∗s=D1,β∗Τ=D1,ne-iτ0ω0iω0-k4orq∗s=D1,k2-iω0-me-iτ0ω0k3.Then, normalizeq and q∗ by the following conditions:(51)q∗,q=1,q∗,q-=0.In order to get 〈q∗(s),q(θ)〉=1, we need to determine the value of D.(52)q∗s,qθ=q-∗0·q0-∫θ=-10∫ξ=0θq-∗Τξ-θdηθqξdξ=D-1,β-∗1,βΤ-∫-10∫ξ=0θD-1,β-∗e-iξ-θτ0ω0dηθ1,βΤe-iξτ0ω0dξ=D-1+ββ-∗-∫-101,β-∗θeiθτ0ω0dηθ1,βΤ=D-1+ββ-∗-τ0nβ-me-iτ0ω0=1.Thus, we choose D as(53)D=11+β-β∗-τ0nβ-meiτ0ω0such that 〈q∗,q〉=1,〈q∗,q-〉=0.Next, we employ the algorithms in [17] and a computation process similar to [18–20] to calculate some important coefficients as follows:(54)g20=2τ0D-L20q1-12+L02q2-12+L11q1-1q2-1,g11=τ0D-2L20q1-1q-1-1+2L02q2-1q-2-1+L11A11,g02=2τ0D-L20q-1-12+L02q-2-12+L11q-1-1q-2-1,g21=τ0D-2L20W201-1q-1-1+2q1-1W111-1+L02A02+L11a11,where(55)A11=q1-1q-2-1+q-1-1q2-1,A02=W202-1q-2-1+2q2-1W112-1,a11=W201-1q-2-1+2W111-1q2-1+2q1-1W112-1+W202-1q-1-1,qθ=q1θ,q2θT,q1-1=e-iτ0ω0,q2-1=βe-iτ0ω0.The coefficientsg20,g11, and g02 can be obtained if the parameters in system (4) are given, while the coefficient g21 needs to compute W11(θ) and W20(θ) by a calculation process similar to [18–20] as follows:(56)W20θ=ig20τ0ω0q0eiτ0ω0θ+ig-023τ0ω0q-0e-iτ0ω0θ+E1e2iτ0ω0θ,and(57)W11θ=-ig11τ0ω0q0eiτ0ω0θ+ig-11τ0ω0q-0e-iτ0ω0θ+E2,where E1=E1(1),E1(2)Τ,E2=E2(1),E2(2)Τ are determined by the following equations, respectively:(58)2iω0+k2-me-2iτ0ω0ne-2iτ0ω0-k32iω0+k4E1=h10,k2-mn-k3k4E2=h20,where(59)h1=2L20q1-12+L02q2-12+L11q1-1q2-1,h2=2L20q1-1q-1-1+2L02q2-1q-2-1+L11q1-1q-2-1+q-1-1q2-1.It follows that(60)E11=1Eh1ne-2iτ0ω002iω0+k4,E12=1E2iω0+k2-me-2iτ0ω0h1-k30,where(61)E=2iω0+k2-me-2iτ0ω0ne-2iτ0ω0-k32iω0+k4.Similarly, we can get(62)E21=1Fh2n0k4,E22=1Fk2-mh2-k30,where(63)F=k2-mn-k3k4.From (56), (57), and (58), we can get g21. Furthermore, we can obtain the following values:(64)c10=i2τ0ω0g11g20-2g112-g0223+g212,μ2=-Rec10Reλ′τ0,β2=2Re⁡c10,T2=-Imc10+μ2Imλ′τ0τ0ω0.Through the discussion above, we obtain the following result.Theorem 3.
For system (4), the coefficient μ2 determines the direction of Hopf bifurcation; it is supercritical (subcritical) if μ2>0(μ2<0); the periodic solutions are stable (unstable) on the manifold if β2<0(β2>0); and the period increases (decreases) if T2>0(T2<0).


## 3.1. Local Stability and Hopf Bifurcation of the Negative Feedback StrengthΓ2

Firstly, at the time delayτ=0, we consider Hopf bifurcation of the negative feedback strength Γ2 as the measure of the miRNA inhibition through the critical values and local stability of equilibrium before the bifurcation in system (4).LetE∗(P∗,M∗) be the equilibrium of system (4), and then(6)αp+kpP∗2Γ1+P∗2+Γ2M∗-βpP∗=0,αM+kMP∗-βMM∗=0.Eliminating M∗ from above equations, we get the following equation on P∗:(7)βMβpP∗3+kMΓ2βp-βMαp+kpP∗2+βMΓ1βp+Γ2αMβp-kMαpP∗-αpβMΓ1+Γ2αM=0.Obviously, the equation has at most three positive real roots. Setting E∗(P∗,M∗) is one of the three roots.Secondly, we translate the equilibriumE∗ to the origin. By the linear transform(8)xt=Pt-P∗yt=Mt-M∗system (4) becomes(9)x˙t=mxt-τ-nyt-τ-βpx+∑i+j≥21i!j!fij1xit-τyjt-τ,y˙t=kMxt-βMyt,where(10)m=2kpP∗Γ1+Γ2M∗Γ1+P∗2+Γ2M∗2,n=kpP∗2Γ2Γ1+P∗2+Γ2M∗2,f1=αp+kpP2t-τΓ1+P2t-τ+Γ2Mt-τ-βpP,f2=αM+kMP-βMM,fij1=∂i+jf1∂xi∂yjP∗,M∗,i,j≥0.From (9), it is easy to get the following characteristic equation. That is,(11)λ-m+βpn-kMλ+βM=0;thus, the two degree polynomial equation is obtained(12)λ2+βM+βp-mλ+βMβp-m+nkM=0.IfH1βM+βp-m>0,H2βMβp-m+nkM>0,then all the roots of (12) have negative real parts. So the equilibrium E∗ of system (4) is locally stable.On the other hand, we assumeH3βM+βp-m=0and λ1,2(μ)=α(μ)±iω(μ) are the roots of (12).Viewing the negative feedback strengthΓ2 as a bifurcation parameter, the sufficient conditions for the occurrence of the Hopf bifurcation of system (4) are obtained in the following results.Theorem 1.
IfH2 and H3 hold, there exist Γ2=Γ20, such that(13)αΓ20=0,ωΓ20>0,dαΓ20dΓ2≠0,then system (4) can exhibit Hopf bifurcation.Time delay is inevitable and plays an important role in negative feedback loop of the Myc/E2F/miR-17-92 network due to the transcription and translation. So time delay on this network is considered in the next sections.


## 3.2. Local Stability and Hopf Bifurcation of Time Delay

In this section, we take the time delayτ as a bifurcating parameter to investigate the stability and existence of Hopf bifurcation of time delay in system (4). For convenience, let k1=kp,k2=βp, k3=kM,k4=βM, then system (9) becomes(14)x˙t=mxt-τ-nyt-τ-k2x+∑i+j≥21i!j!fij1xit-τyjt-τ,y˙t=k3x-k4y,where(15)m=k1n1,n=k1n2,n1=2x∗Γ1+Γ2y∗Γ1+x∗2+Γ2y∗2,n2=Γ2x∗2Γ1+x∗2+Γ2y∗2.Then we obtain the characteristic equation of system (14).That is,(16)λ2+k2+k4λ-k1n1e-λτλ+k2k4+k1k3n2-k1k4n1e-λτ=0.When τ=0, it becomes(17)λ2+k2+k4-k1n1λ+k2k4+k1k3n2-k1k4n1=0.i ω   ω > 0 is the root of (16) if and only if iω satisfies(18)-ω2+k2+k4iω-k1n1cos⁡ωτ-isin⁡ωτiω+k2k4+k1k3n2-k1k4n1cos⁡ωτ-isin⁡ωτ=0.Separating the real and imaginary parts, we get(19)k2+k4ω=k1n1ωcos⁡ωτ+k1k3n2-k1k4n1sin⁡ωτ,k2k4-ω2=k1n1ωsin⁡ωτ-k1k3n2-k1k4n1cos⁡ωτ.which is equivalent to(20)ω4-k12n12-k22-k42ω2+k22k42-k12k3n2-k4n12=0.Let z=ω2, then it is transformed to be(21)z2-pz+q=0where(22)p=k12n12-k22-k42,q=k22k42-k12k3n2-k4n12.Assume (H4) equation (21) has at least one positive real root.From (21), we can obtain(23)z1,2=12p±12p2-4q.Thus, ifH5p<0,q>0orp2-4q<0then none of z1,z2 is positive. So (21) has no positive roots. It means that there are no imaginary roots in characteristic equation (16).From those discussed above, we obtain(24)cos⁡ωτ=k2k42n1+k3n2+k2n1ω2-k2k3k4n2k1k3n2-k4n12+n12ω2,sin⁡ωτ=ωk2k3n2+k3n2k4-n1ω2-n1k42k1k3n2-k4n12+n12ω2.Without loss of generality, (21) is assumed to have two positive roots, defined by z1,z2, respectively. Thus, (20) has two positive roots ω1=z1,ω2=z2.So we have(25)cos⁡ωkτk=k2k42n1+k3n2+k2n1ωk2-k2k3k4n2k1k3n2-k4n12+n12ωk2.It follows that(26)τkj=1ωkarccosk2k42n1+k3n2+k2n1ωk2-k2k3k4n2k1k3n2-k4n12+n12ωk2+2jπωk,where k=1,2,j=0,1,2,….Define(27)τ0=minτ10,τ20.Taking the derivative of λ with respect to τ, we can get(28)dλdτ-1=2λ+k2+k4-k1n1e-λτk1λe-λτk3n2-k4n1-n1λ-τλ.Then,(29)Redλdττ=τ0-1=c+da2+b2,where(30)a=k1k3n2-k1k4n1,b=k1n1ω0,c=2cosω0τ0a-sinω0τ0b-k12n12,d=k2+k4k1n1cos⁡ω0τ0+1ω0asin⁡ω0τ0.If the conditionH6c+d≠0holds, then(31)Redλdττ=τ0-1≠0.That is, the transversality condition is satisfied.Based on the analysis above, the following results are obtained.Theorem 2.
Letτ=τkj be defined by (26); one has
(i) ifH1, H2, and H5 hold, then the equilibrium E∗ of system (4) is asymptotically stable for all τ≥0;
(ii) ifH1, H2, and H4 hold, then the equilibrium E∗ is locally asymptotically stable for 0≤τ<τ0 and unstable for τ>τ0; moreover, if c+d≠0, then system (4) undergoes a Hopf bifurcation at the equilibrium E∗ when τ=τkj(k=1,2,j=0,1,2,…).


## 3.3. Direction and Stability of the Hopf Bifurcation at Time Delayτ=τ0

In this section, we further consider direction and stability of Hopf bifurcation assuming atτ=τ0 by using the methods in Hassard et al. [17]. For convenience, let τ=τ0+μ,μ∈R,s=t/τ and denote s=t, then system (14) becomes(32)x˙t=τ0+μmxt-1-nyt-1-k2x+∑i+j≥21i!j!fij1xit-1yjt-1,y˙t=τ0+μk3xt-k4yt.Clearly, μ=0 is a Hopf bifurcation value of system (32).Its linear part is given by(33)x˙t=τ0+μmxt-1-nyt-1-k2xt,y˙t=τ0+μk3xt-k4yt.and nonlinear part is(34)fμ,ut=τ0+μ∑i+j≥21i!j!fij1xit-1yjt-1]0.For convenience of studying the problems of Hopf bifurcation, we rewrite system (32) as(35)u˙t=Lμut+fμ,ut,where u(t)=(x(t),y(t))T and ut(θ)=u(t+θ) for θ∈[-τ,0]; Lμ:C→R,f:R×C→R are given, respectively, by(36)Lμφ=τ0+μ-k20k3-k4φ10φ20+τ0+μm-n00φ1-1φ2-1,fμ,ut=τ0+μ∑i+j≥21i!j!fij1xit-1yjt-1]0=τ0+μ12f201φ12-1+f111φ1-1φ2-1+12f021φ22-1+h.o.t.0.Here,h.o.t. denotes higher order terms.For convenience analysis, let(37)L20=12f201,L11=f111,L02=12f021,where f20(1)=2k1Γ1+Γ2y∗Γ1-3x∗2+Γ2y∗/Γ1+x∗2+Γ2y∗3; f11(1)=-2k1x∗Γ2Γ1-x∗2+Γ2y∗/Γ1+x∗2+Γ2y∗3, f02(1)=2k1x∗2Γ22/Γ1+x∗2+Γ2y∗3.Then,(38)fμ,ut=τ0+μL20φ12-1+L11φ1-1φ2-1+L02φ22-1+h.o.t.0.According to the Riesz representation theorem, there exists a 2 × 2 matrix functionη(θ,μ),θ∈-1,0, such that(39)Lμφ=∫-10dηθ,μφθ,φ∈C1-1,0,R2.Here, we choose(40)ηθ,μ=τ0+μ-k20k3-k4δθ-τ0+μm-n00δθ+1,where δθ is defined by (41)δθ=0,θ≠01,θ=0.Forφ∈C1[-1,0],R2, define(42)Aμφ=dφθdθ,-1≤θ<0,∫-10dηθ,μφθ,θ=0,and(43)Rμφ=00,-1≤θ<0,τ0+μ∑i+j≥21i!j!fij1xit-1yjt-10,θ=0.For ψ∈C1[0,1],(R2)∗, the adjoint operator A∗ of A is defined by(44)A∗ψs=-dψsds,0<s≤1,∫-10dηTs,μψ-s,s=0.And define a bilinear inner product as follows:(45)ψs,φθ=ψ-0φ0-∫-10∫ξ=0θψ-ξ-θdηθφξdξ,where η(θ)=η(θ,0).From the discussion in Section3.2, we obtain that ±iτ0ω0 are eigenvalues of A(0). At the same time, they are also eigenvalues of A∗ since A and A∗ are adjoint operator. In the following, we calculate the eigenvectors of A(0) and A∗ corresponding to iτ0ω0 and -iτ0ω0, respectively. Suppose q(θ)=(1,β)Τeiτ0ω0θ,θ∈-1,0 is eigenvector of A(0). Through the discussion above, we have(46)A0q0=iτ0ω0q0.That is,(47)τ0iω0+k2-me-iτ0ω0ne-iτ0ω0-k3iω0+k4q0=00.We can obtain(48)β=k3iω0+k4orβ=me-iτ0ω0-iω0-k2ne-iτ0ω0.Thus, q(0)=(1,β)Τ=(1,k3/iω0+k4)Τ or q(0)=(1,me-iτ0ω0-iω0-k2/ne-iτ0ω0)Τ.On the other hand, we assumeq∗(s)=D(1,β∗)Τeiτ0ω0s,s∈0,1 is the eigenvector of A∗ and then obtain(49)τ0-iω0+k2-me-iτ0ω0-k3ne-iτ0ω0-iω0+k4q∗0=00.and(50)q∗s=D1,β∗Τ=D1,ne-iτ0ω0iω0-k4orq∗s=D1,k2-iω0-me-iτ0ω0k3.Then, normalizeq and q∗ by the following conditions:(51)q∗,q=1,q∗,q-=0.In order to get 〈q∗(s),q(θ)〉=1, we need to determine the value of D.(52)q∗s,qθ=q-∗0·q0-∫θ=-10∫ξ=0θq-∗Τξ-θdηθqξdξ=D-1,β-∗1,βΤ-∫-10∫ξ=0θD-1,β-∗e-iξ-θτ0ω0dηθ1,βΤe-iξτ0ω0dξ=D-1+ββ-∗-∫-101,β-∗θeiθτ0ω0dηθ1,βΤ=D-1+ββ-∗-τ0nβ-me-iτ0ω0=1.Thus, we choose D as(53)D=11+β-β∗-τ0nβ-meiτ0ω0such that 〈q∗,q〉=1,〈q∗,q-〉=0.Next, we employ the algorithms in [17] and a computation process similar to [18–20] to calculate some important coefficients as follows:(54)g20=2τ0D-L20q1-12+L02q2-12+L11q1-1q2-1,g11=τ0D-2L20q1-1q-1-1+2L02q2-1q-2-1+L11A11,g02=2τ0D-L20q-1-12+L02q-2-12+L11q-1-1q-2-1,g21=τ0D-2L20W201-1q-1-1+2q1-1W111-1+L02A02+L11a11,where(55)A11=q1-1q-2-1+q-1-1q2-1,A02=W202-1q-2-1+2q2-1W112-1,a11=W201-1q-2-1+2W111-1q2-1+2q1-1W112-1+W202-1q-1-1,qθ=q1θ,q2θT,q1-1=e-iτ0ω0,q2-1=βe-iτ0ω0.The coefficientsg20,g11, and g02 can be obtained if the parameters in system (4) are given, while the coefficient g21 needs to compute W11(θ) and W20(θ) by a calculation process similar to [18–20] as follows:(56)W20θ=ig20τ0ω0q0eiτ0ω0θ+ig-023τ0ω0q-0e-iτ0ω0θ+E1e2iτ0ω0θ,and(57)W11θ=-ig11τ0ω0q0eiτ0ω0θ+ig-11τ0ω0q-0e-iτ0ω0θ+E2,where E1=E1(1),E1(2)Τ,E2=E2(1),E2(2)Τ are determined by the following equations, respectively:(58)2iω0+k2-me-2iτ0ω0ne-2iτ0ω0-k32iω0+k4E1=h10,k2-mn-k3k4E2=h20,where(59)h1=2L20q1-12+L02q2-12+L11q1-1q2-1,h2=2L20q1-1q-1-1+2L02q2-1q-2-1+L11q1-1q-2-1+q-1-1q2-1.It follows that(60)E11=1Eh1ne-2iτ0ω002iω0+k4,E12=1E2iω0+k2-me-2iτ0ω0h1-k30,where(61)E=2iω0+k2-me-2iτ0ω0ne-2iτ0ω0-k32iω0+k4.Similarly, we can get(62)E21=1Fh2n0k4,E22=1Fk2-mh2-k30,where(63)F=k2-mn-k3k4.From (56), (57), and (58), we can get g21. Furthermore, we can obtain the following values:(64)c10=i2τ0ω0g11g20-2g112-g0223+g212,μ2=-Rec10Reλ′τ0,β2=2Re⁡c10,T2=-Imc10+μ2Imλ′τ0τ0ω0.Through the discussion above, we obtain the following result.Theorem 3.
For system (4), the coefficient μ2 determines the direction of Hopf bifurcation; it is supercritical (subcritical) if μ2>0(μ2<0); the periodic solutions are stable (unstable) on the manifold if β2<0(β2>0); and the period increases (decreases) if T2>0(T2<0).


## 4. Numerical Simulations

In this section, we present four examples to verify our theoretical results. At same time, the effect of time delay at different situations of negative feedback strengthΓ2 is also investigated.Example 1.
By increasing the negative feedback strengthΓ2, system (4) undergoes a transition from monostability to oscillation and then back to monostability. That is, Hopf bifurcation occurs when the parameter Γ2 increases to the critical values. Here, we give an example when some important parameters are taken as follows [1, 2]:(65)αp=0.01,αM=0.01,kp=0.4,Γ1=0.1,βp=0.25,kM=0.4,βM=0.02.Then, system (4) becomes the following ordinary differential equations:(66)x˙=0.01+0.4x20.1+x2+Γ2y-0.25x,y˙=0.01+0.4x-0.02y.Through bifurcation analysis of system (66) in Figure 2(a), we obtain two Hopf bifurcations at Γ2=0.04011 and 0.06554 when H2 and H3 hold, respectively, as well as limit cycles for 0.04011<Γ2<0.06554 between the two Hopf bifurcations when βM+βp-m<0. Also, the equilibrium E∗ is asymptotically stable for Γ2<0.04011 or Γ2>0.06554. The equilibrium of system (66) at Γ2=0.01 is stable (see Figure 2(b)) and oscillations of limit cycles at Γ2=0.05 are presented in Figure 2(c), and then the oscillation behavior vanishes and the equilibrium tends to be stable again at Γ2=0.09 (see Figure 2(d)).Figure 2
(a) Bifurcation diagram of [P] withΓ2 as a bifurcation parameter. The red solid lines depict stable steady states and the black dashed line depicts unstable steady states. The maxima and minima are depicted by the blue dots and green dots for the stable and unstable limit cycle, respectively. The bifurcation points of the subcritical Hopf bifurcation are marked as sub and LPC as fold limit cycle bifurcation points. (b), (c), and (d) are time courses diagram of P and M for the parameter Γ2=0.01(b), 0.05(c), and 0.09 (d), respectively. The initial conditions: P(0)=0.1,M(0)=0.1.
(a) (b) (c) (d)Example 2.
We consider dynamic changes induced by time delayτ≥0 for the negative feedback strength Γ2=0.01 when system (4) has a positive equilibrium E1(1.37151,27.93015) at a high steady state in Example 1 in Figure 2. Other parameters are same as Example 1.
In Theorem2, we get n1=0.20364,n2=0.00368, βM+βp-m=k2+k4-k1n1=0.18854>0, βM(βp-m)+nkM=k2k4+k1k3n2-k1k4n1=0.00396>0, p=k12n12-k22-k42=-0.05626, and q=k22k42-k12(k3n2-k4n1)2=0.00002. Then conditions H1, H2, and H5 hold. Therefore, we obtain that the steady state of system (4) is stable for all τ≥0 (see Figures 3(a), 3(b), 3(c), and 3(d)).Figure 3
(a) Bifurcation diagram of [P] with time delayτ as the bifurcating parameter when fixing the negative feedback strength Γ2=0.01. (b), (c), and (d) are time courses diagram of [P] for the parameter τ=1 (b), 40 (c), and 60 (d), respectively. Initial conditions: P(0)=0.1,M(0)=0.1.
(a) (b) (c) (d)Example 3.
A limit cycle and an equilibrium can coexist in system (4) for τ=0 at the negative feedback strength Γ2=0.04 in Figure 2. So system (4) appears to be periodic solution or steady state depending on different initial conditions.
As time delayτ passes through the critical value τ0≈2.23, the periodic solution transits to steady-state solution via Hopf bifurcation when the initial point is chosen near the limit cycle (see Figure 4). Next, we calculate the values of β2 and μ2 to determine the stability of periodic solutions bifurcating from equilibrium E2(0.72821,15.06414) and direction of the Hopf bifurcation at the critical point τ0. When τ=τ0≈2.23, we can compute Re(c1(0))≈-0.20118,β2≈-0.40236 and c+d≈0.01847>0 by means of software Maple. Further, we can get Re(λ′(τ0))≈17.96347>0,μ2≈0.0112>0. By Theorem 3, we know that the direction of Hopf bifurcation is supercritical, and the periodic solutions are stable on the manifold (see Figures 4(a), 4(b), and 4(c)).Figure 4
(a) Bifurcation diagram of [P] with time delayτ as the bifurcating parameter when fixing the negative feedback strength Γ2=0.04. The stable steady state is depicted by the red solid line, and the maxima and minima are depicted by the blue dots for the stable limit cycle. (b) and (c) are time courses diagram of [P] for the parameter τ=1 (b) and 4.5 (c), respectively. Initial conditions: P(0)=0.1,M(0)=0.1.
(a) (b) (c)Example 4.
Fixing the negative feedback strengthΓ2=0.07, system (4) has a positive equilibrium E3(0.08174,2.13478) at a low steady state in Figure 2. The periodic solutions bifurcate from the equilibrium via Hopf bifurcation when time delay τ increases to the critical value τ0≈35(see Figure 5).
For the arbitrary negative feedback strengthΓ2∈0,0.04 and the parameters given in Example 1, system (4) is always stable. For example, at Γ2=0.02, we have n1=0.36029, n2=0.00728, βM+βp-m=k2+k4-k1n1=0.12588>0, βM(βp-m)+nkM=0.00328>0, p=k12n12-k22-k42=-0.04213, and q=k22k42-k12(k3n2-k4n1)2=0.000022. It means that conditions H1, H2, and H5 hold. By the Theorem 2, we know that system (4) is stable for all τ≥0. It can explain that delayed negative feedback could not lead to oscillations in the high state in Figure 2. In a word, the time delay leads to oscillation behaviors when system (4) is at the low steady state but not at the high one in Figure 2.Figure 5
(a) Bifurcation diagram of [P] with time delayτ as the bifurcating parameter when fixing the negative feedback strength Γ2=0.07. The red solid line depicts stable steady states and the blue dots depict the maxima and minima for the stable limit cycle. (b) and (c) are time courses diagram of [P] for the parameter τ=1 (b) and 40 (c), respectively. Initial conditions: P(0)=0.1,M(0)=0.1.
(a) (b) (c)


## 5. Conclusions and Discussions

In this paper, the Myc/E2F/miR-17-92 network with time delay is considered. Occurrence of Hopf bifurcations associated with oscillation of the inhibition efficiency parameterΓ2 and further the time delay τ, respectively, are investigated by combined local stability and bifurcation theory with numerical simulations. Furthermore, the direction and stability of Hopf bifurcation are also studied by the center manifold theorem and normal form method as well as a numerical example supporting the results.We find that the time delay has a destabilizing role by choosing appropriate parameters in this network. Besides that, initial condition is important as periodic solution and steady state coexist under some circumstances (see Figure2). If the initial values are chosen near the positive equilibrium, the steady state of system (4) remains unchanged when the parameter τ less than the critical value τ0. However, if the initial values are chosen near the limit cycle, system (4) appears to be periodic solution and then transits to steady-state solution via Hopf bifurcation of time delay τ (see Figure 4). With the negative feedback strength Γ2 increasing, system (4) undergoes a transition from the high steady state to oscillations and then to low steady state (see Figure 2(a)). We consider the effect of time delay on the dynamics of the network at every different state. When the negative feedback strength Γ2 is chosen at the high steady states, there will be no periodic oscillations with increasing time delay τ (see Figure 3). However, when the negative feedback strength Γ2 is chosen between the two Hopf bifurcation points in Figure 2(a), the periodic solutions transit to steady-state solutions via Hopf bifurcation with increasing time delay (see Figure 4). The periodic solutions bifurcate from the equilibrium of system (4) with increasing time delay τ when the negative feedback strength Γ2 is chosen at the low steady states (see Figure 5). The values of the parameters are chosen in the special way above or they need to satisfy the conditions in Theorem 2. If H1, H2, and H5 hold, then the equilibrium of system (4) is asymptotically stable for all τ≥0; if H1, H2, and H4 hold, the equilibrium is locally asymptotically stable for 0≤τ<τ0; moreover, if c+d≠0, system (4) undergoes a Hopf bifurcation at the equilibrium when τ=τ0. At the same time, the direction and stability of Hopf bifurcation can be determined by Theorem 3. This work can further provide a theoretic instruction for exploring the dynamics of the network.Noise is inevitable in gene regulatory networks and plays important roles in circuits’ dynamics. It may induce bistability, oscillations, and bifurcations which are not present in the deterministic model [7, 21, 22]. How does stochastic noise affect dynamics of the network with time delay? This is a very valuable and worth exploring problem. We leave it as the future work.



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