1 Introduction

Genetic regulatory networks (GRNs) describe interactions between DNA, RNA, proteins and small molecules in living cells and play fundamental roles in many life processes (Parmar et al. 2015; Ling et al. 2017). Because the processes of gene transcription and messenger RNA translation are not co-located in space and consequently not completed instantaneously, time delays are inevitable in GRNs. The effect of time delay on the dynamic behavior of a GRN model has attracted extensive attention of many scholars, and some research results have been obtained (Parmar et al. 2015; Ling et al. 2017; Wu and Eshete 2011; Bodnar and Bartłomiejczyk 2012; Lewis 2003; Wu 2011; Wang et al. 2010; Wu 2011; Zhang et al. 2017; Lai 2018; Monk 2003; Verdugo and Rand 2008; Sun et al. 2018; Yue et al. 2017; Huang et al. 2016). In Wu and Eshete (2011), the authors considered a model of gene express with two delays and showed effect of time delays on the model. In Bodnar and Bartłomiejczyk (2012), the Hes 1 genetic oscillator system was presented, and the authors showed that the stability of the steady state depends on the sum of time delays. The conditions for the occurrence of Hopf bifurcation and the stability of equilibrium point were proved. Zhang et al. (2017) investigated oscillatory expression in Escherichia coli. The result of their research indicates that the effects of transcriptional and translational delays are two important factors for designing or controlling GRNs and that the effect of diffusion must be taken into account. In Lai (2018), the stability and bifurcation of delayed bidirectional GRNs with negative feedback loops were invested.

In recent years, the cyclic GRNs (CGRNs) have attracted the attention of scholars, and many research results have been achieved (Lewis 2003; Wu 2011; Qiu 2010; Bar-Or et al. 2000; Ling et al. 2015). CGRNs are a kind of GRNs. All the nodes of CGRNs are arranged in a ring structure. In Lewis (2003), Lewis showed, using mathematical simulation, that the direct auto-repression of a gene by its own product could generate oscillations. The period of oscillation was found to be determined by the transcriptional and translational delay. In Wu (2011), Wu first presented conditions for delay-independent local stability of CGRNs and then proceeded to study the bifurcation of ring-structured GRNs with time delays. In Ling et al. (2015), the stability and bifurcation of CGRNs with mixed time delays were studied. Due to the presence of time delays, even if the GRN model is simple, a positive equilibrium point of the system will lose its stability. Sustainable periodic oscillation will occur through a Hopf bifurcation (Parmar et al. 2015; Suzuki et al. 2016). Delay is not considered in the CGRN models in Xi and Turcotte (2015). We consider the effect of time delay on the canonical two-gene binary switch model (see Fig. 1), which is described by the following delay differential equations.

$$\begin{aligned} \left\{ \begin{array}{l} {{M'}_1}\left( t \right) = - {r_1}{M_1}\left( t \right) + a \cdot {a_{1}}\frac{{P_1^n\left( {t - {\tau _2}} \right) }}{{{s^n} + P_1^n\left( {t - {\tau _2}} \right) }} + b \cdot {b_{21}}\frac{{{s^n}}}{{{s^n} + P_2^n\left( {t - {\tau _2}} \right) }}\\ {{M'}_2}\left( t \right) = - {r_2}{M_2}\left( t \right) + a\frac{{P_2^n\left( {t - {\tau _2}} \right) }}{{{s^n} + P_2^n\left( {t - {\tau _2}} \right) }} + b\frac{{{s^n}}}{{{s^n} + P_1^n\left( {t - {\tau _2}} \right) }}\\ {{P'}_1}\left( t \right) = - {c_1}{P_1}\left( t \right) + {d_1}{M_1}\left( {t - {\tau _1}} \right) \\ {{P'}_2}\left( t \right) = - {c_2}{P_2}\left( t \right) + {d_2}{M_2}\left( {t - {\tau _1}} \right) \end{array} \right. \end{aligned}$$
(1)

Let \(\tau = \max \left\{ {{\tau _1},{\tau _2}} \right\} \) and then the initial conditions of the model are given by:

$$\begin{aligned} {M_1}\left( s \right) = {\phi _1}\left( s \right) ,\quad {M_2}\left( s \right) = {\phi _2}\left( s \right) ,\quad {P_1}\left( s \right) = {\phi _3}\left( s \right) ,\quad {P_2}\left( s \right) = {\phi _4}\left( s \right) , \end{aligned}$$

where \(s \in \left[ { - \tau ,0} \right] ,{\phi _i}\left( s \right) \in C\left( {\left[ { - \tau ,0} \right] ,R} \right) \left( {i = 1,2,3,4,{\phi _i}\left( s \right) \ge 0} \right) \). Here  \(C\left( {\left[ { - \tau ,0} \right] ,R} \right) \) is the Banach space. \({M_i}\left( t \right) \left( {i = 1,2} \right) \) and \({P_i}\left( t \right) \left( {i = 1,2} \right) \) describe the concentration of mRNA and protein of genes \(X_{1}\) and \(X_{2}\), respectively. \({r_i}\left( {i = 1,2} \right) \) and \({c_i}\left( {i = 1,2} \right) \) are degradation rates of mRNA and protein, respectively. \({d_i}\left( {i = 1,2} \right) \) are translation rates. Parameters a and b are transcription rates, n is the Hill coefficient. \({a_1}\) and bcommon are parameter asymmetry providers. All the parameters of the model are positive real numbers.

Fig. 1
figure 1

Two-gene mutually repressive, self-promoting model (Xi and Turcotte 2015). The nodes are genes \(X_{1}\) and \(X_{2}\). Parameter asymmetry provider \(\hbox {bcommon}=b_{21}\)

Full size image

The organization of the paper is as follows: In Sect. 2, we study the stability and existence of Hopf bifurcation of system (1) in five cases. The explicit formula is given to determine the direction and stability of bifurcating periodic solutions by using the normal form method and the center manifold theorem in Sect. 3. In Sect. 4, numerical simulations are presented that support the analytical results we obtained. In Sect. 5, using time-delayed stochastic simulations, we study the impact of biological fluctuations (noise) on the dynamics of the system. In Sect. 6, characterization of the orbit is discussed. Finally, in Sect. 7, we discuss the significance of our results and present our conclusions.

2 The Stability and Existence of Hopf Bifurcation

Let \({E_ * } = E(M_1^ * ,M_2^ * ,P_1^ * ,P_2^ * )\) be the positive equilibrium of system (1), then

$$\begin{aligned} \left\{ \begin{array}{l} - {r_1}M_1^ * + a \cdot {a_{1}}\frac{{{{\left( {P_1^ * } \right) }^n}}}{{{s^n} + {{\left( {P_1^ * } \right) }^n}}} + b \cdot {b_{21}}\frac{{{s^n}}}{{{s^n} + {{\left( {P_2^ * } \right) }^n}}} = 0,\\ - {r_2}M_2^ * + a\frac{{{{\left( {P_2^ * } \right) }^n}}}{{{s^n} + {{\left( {P_2^ * } \right) }^n}}} + b\frac{{{s^n}}}{{{s^n} + {{\left( {P_1^ * } \right) }^n}}} = 0,\\ - {c_1}P_1^ * + {d_1}M_1^ * = 0,\\ - {c_2}P_2^ * + {d_2}M_2^ * = 0. \end{array} \right. \end{aligned}$$

Thus, we can get \({E_ * }\), where \(P_1^ * ,M_1^ * ,P_2^ * ,M_2^ *\) are defined in Appendix.

By the linear transform,

$$\begin{aligned} {u_1}\left( t \right)= & {} {M_1}\left( t \right) - M_1^ * ,\quad {u_2}\left( t \right) = {M_2}\left( t \right) - M_2^ * ,\quad {u_3}\left( t \right) = {P_1}\left( t \right) - P_1^ * , \\ {u_4}\left( t \right)= & {} {P_2}\left( t \right) - P_2^ * \end{aligned}$$

We have

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{u}_1}\left( t \right) = {a_{11}}{u_1}\left( t \right) + {a_{13}}{u_3}\left( {t - {\tau _2}} \right) + {a_{14}}{u_4}\left( {t - {\tau _2}} \right) + {F_1},\\ {\dot{u}_2}\left( t \right) = {a_{22}}{u_2}\left( t \right) + {a_{23}}{u_3}\left( {t - {\tau _2}} \right) + {a_{24}}{u_4}\left( {t - {\tau _2}} \right) + {F_2},\\ {\dot{u}_3}\left( t \right) = {a_{31}}{u_1}\left( {t - {\tau _1}} \right) + {a_{33}}{u_3}\left( t \right) + {F_3},\\ {\dot{u}_4}\left( t \right) = {a_{42}}{u_2}\left( {t - {\tau _1}} \right) + {a_{44}}{u_4}\left( t \right) + {F_4}, \end{array} \right. \end{aligned}$$
(2)

where

$$\begin{aligned} {f^{(1)}}&= - {r_1}{M_1} + a \cdot {a_1}\frac{{P_1^n\left( {t - {\tau _2}} \right) }}{{{s^n} + P_1^n\left( {t - {\tau _2}} \right) }} + b \cdot {b_{21}}\frac{{{s^n}}}{{{s^n} + P_2^n\left( {t - {\tau _2}} \right) }},\\ {f^{(2)}}&= - {r_2}{M_2} + a\frac{{P_2^n\left( {t - {\tau _2}} \right) }}{{{s^n} + P_2^n\left( {t - {\tau _2}} \right) }} + b\frac{{{s^n}}}{{{s^n} + P_1^n\left( {t - {\tau _2}} \right) }},\\ {f^{(3)}}&= - {c_1}{P_1}\left( t \right) + {d_1}{M_1}\left( {t - {\tau _1}} \right) ,\\ {f^{(4)}}&= - {c_2}{P_2}\left( t \right) + {d_2}{M_2}\left( {t - {\tau _1}} \right) ,\\ f_{ijk}^{(1)}&= {\left. {\frac{{{\partial ^{i + j + k}}{f^{(1)}}}}{{\partial M_1^i\left( t \right) \partial P_1^j(t - {\tau _2})\partial P_2^k(t - {\tau _2})}}} \right| _{(M_1^ * ,M_2^ * ,P_1^ * ,P_2^ * )}},\quad i,j,k \ge 0,\\ f_{ijk}^{(2)}&= {\left. {\frac{{{\partial ^{i + j + k}}{f^{(2)}}}}{{\partial M_2^i\left( t \right) \partial P_1^j(t - {\tau _2})\partial P_2^k(t - {\tau _2})}}} \right| _{(M_1^ * ,M_2^ * ,P_1^ * ,P_2^ * )}},\quad i,j,k \ge 0,\\ f_{ij}^{(3)}&= {\left. {\frac{{{\partial ^{i + j}}{f^{(3)}}}}{{\partial M_1^i\left( {t - {\tau _1}} \right) \partial P_1^j(t)}}} \right| _{(M_1^ * ,M_2^ * ,P_1^ * ,P_2^ * )}},\quad i,j \ge 0,\\ f_{ij}^{(4)}&= {\left. {\frac{{{\partial ^{i + j}}{f^{(4)}}}}{{\partial M_2^i\left( {t - {\tau _1}} \right) \partial P_2^j(t)}}} \right| _{(M_1^ * ,M_2^ * ,P_1^ * ,P_2^ * )}},\quad i,j \ge 0,\\ {F_1}&= {\sum _{i + j + k \ge 2}}\frac{1}{{i!j!k!}}f_{ijk}^{(1)}u_1^i\left( t \right) u_3^j\left( {t - {\tau _2}} \right) u_4^k\left( {t - {\tau _2}} \right) ,\\ {F_2} =&{\sum _{i + j + k \ge 2}}\frac{1}{{i!j!k!}}f_{ijk}^{(2)}u_2^i\left( t \right) u_3^j\left( {t - {\tau _2}} \right) u_4^k\left( {t - {\tau _2}} \right) ,\\ {F_3}&= {\sum _{i + j \ge 2}}\frac{1}{{i!j!}}f_{ij}^{(3)}u_1^i\left( {t - {\tau _1}} \right) u_3^j\left( t \right) ,\\ {F_4}&= {\sum _{i + j \ge 2}}\frac{1}{{i!j!}}f_{ij}^{(4)}u_2^i\left( {t - {\tau _1}} \right) u_4^j\left( t \right) ,\\ {a_{11}}&= - {r_1},\quad {a_{12}} = 0,\quad {a_{13}} = \frac{{a{a_{11}}n{{\left( {P_1^ * } \right) }^{n - 1}}{s^n}}}{{{{\left( {{s^n} + {{\left( {P_1^ * } \right) }^n}} \right) }^2}}},\quad {a_{14}} = - \frac{{b{b_{21}}n{{\left( {P_2^ * } \right) }^{n - 1}}{s^n}}}{{{{\left( {{s^n} + {{\left( {P_2^ * } \right) }^n}} \right) }^2}}},\\ {a_{21}}&= 0,\quad {a_{22}} = - {r_2},\quad {a_{23}} = - \frac{{bn{{\left( {P_1^ * } \right) }^{n - 1}}{s^n}}}{{{{\left( {{s^n} + {{\left( {P_1^ * } \right) }^n}} \right) }^2}}},\quad {a_{24}} = \frac{{an{{\left( {P_2^ * } \right) }^{n - 1}}{s^n}}}{{{{\left( {{s^n} + {{\left( {P_2^ * } \right) }^n}} \right) }^2}}},\\ {a_{31}}&= {d_1},\quad {a_{32}} = 0,\quad {a_{33}} = - {c_1},\quad {a_{34}} = 0,\\ {a_{41}}&= 0,\quad {a_{42}} = {d_2},\quad {a_{43}} = 0,\quad {a_{44}} = - {c_2}. \end{aligned}$$

Then the linear part of system (2) is

$$\begin{aligned} \left\{ \begin{array}{l} {\dot{u}_1}\left( t \right) = {a_{11}}{u_1}\left( t \right) + {a_{13}}{u_3}\left( {t - {\tau _2}} \right) + {a_{14}}{u_4}\left( {t - {\tau _2}} \right) , \\ {\dot{u}_2}\left( t \right) = {a_{22}}{u_2}\left( t \right) + {a_{23}}{u_3}\left( {t - {\tau _2}} \right) + {a_{24}}{u_4}\left( {t - {\tau _2}} \right) , \\ {\dot{u}_3}\left( t \right) = {a_{31}}{u_1}\left( {t - {\tau _1}} \right) + {a_{33}}{u_3}\left( t \right) , \\ {\dot{u}_4}\left( t \right) = {a_{42}}{u_2}\left( {t - {\tau _1}} \right) + {a_{44}}{u_4}\left( t \right) . \\ \end{array} \right. \end{aligned}$$
(3)

Thus, the characteristic equation of system (3) is obtained:

$$\begin{aligned} \det \left[ {\begin{array}{cccc} {\lambda - {a_{11}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - \lambda {\tau _2}}}}&{}\quad { - {a_{14}}{e^{ - \lambda {\tau _2}}}} \\ 0&{}\quad {\lambda - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - \lambda {\tau _2}}}}&{}\quad { - {a_{24}}{e^{ - \lambda {\tau _2}}}} \\ { - {a_{31}}{e^{ - \lambda {\tau _1}}}}&{}\quad 0&{}\quad {\lambda - {a_{33}}}&{}\quad 0 \\ 0&{}\quad { - {a_{42}}{e^{ - \lambda {\tau _1}}}}&{}\quad 0&{}\quad {\lambda - {a_{44}}} \end{array}} \right] = 0 \end{aligned}$$

which is equivalent to

$$\begin{aligned}&{\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0} + ({B_2}{\lambda ^2} + {B_1}\lambda + {B_0}){e^{ - \lambda \left( {{\tau _1} + {\tau _2}} \right) }} \nonumber \\&\quad + {C_0}{e^{ - 2\lambda \left( {{\tau _1} + {\tau _2}} \right) }} = 0 \end{aligned}$$
(4)

where

$$\begin{aligned} {A_0}= & {} {a_{11}}{a_{22}}{a_{33}}{a_{44}}, \\ {A_1}= & {} - \left( {{a_{11}}{a_{33}}{a_{44}} + {a_{22}}{a_{33}}{a_{44}} + {a_{11}}{a_{22}}{a_{44}} + {a_{11}}{a_{22}}{a_{33}}} \right) , \\ {A_2}= & {} {a_{44}}\left( {{a_{11}} + {a_{22}} + {a_{33}}} \right) + {a_{22}}\left( {{a_{11}} + {a_{33}}} \right) + {a_{11}}{a_{33}}, \\ {A_3}= & {} - \left( {{a_{11}} + {a_{22}} + {a_{33}} + {a_{44}}} \right) , \\ {B_0}= & {} - \left( {{a_{31}}{a_{13}}{a_{22}}{a_{44}} + {a_{11}}{a_{42}}{a_{24}}{a_{33}}} \right) , \\ {B_1}= & {} {a_{42}}{a_{24}}{a_{33}} + {a_{31}}{a_{13}}\left( {{a_{44}} + {a_{22}}} \right) + {a_{11}}{a_{42}}{a_{24}}, \\ {B_2}= & {} - \left( {{a_{42}}{a_{24}} + {a_{31}}{a_{13}}} \right) , \\ {C_0}= & {} {a_{31}}{a_{42}}{a_{13}}{a_{24}} - {a_{31}}{a_{42}}{a_{14}}{a_{23}}. \end{aligned}$$

From the form of Eq. (4), it is explicitly clear that the system’s stability depends on the sum of the delays. For Eq. (4), we consider the following five cases.

Case 1 \(\tau = {\tau _1} + {\tau _2} = 0\) (i.e., \({\tau _1} = {\tau _2} = 0\)).

Equation (4) reduces to

$$\begin{aligned} {\lambda ^4} + {A_3}{\lambda ^3} + \left( {{A_2} + {B_2}} \right) {\lambda ^2} + \left( {{A_1} + {B_1}} \right) \lambda + {A_0} + {B_0} + {C_0} = 0 \end{aligned}$$
(5)

By the Routh–Hurwitz criteria, all the roots of Eq. (5) have negative real parts if and only if

$$\begin{aligned} {V_1}= & {} {A_3} > 0, \end{aligned}$$
(6)
$$\begin{aligned} {V_2}= & {} \left| {\begin{array}{cc} {{A_3}}&{}\quad 1 \\ {{A_1} + {B_1}}&{}\quad {{A_2} + {B_2}} \end{array}} \right| = {A_3}\left( {{A_2} + {B_2}} \right) - {A_1} - {B_1} > 0, \end{aligned}$$
(7)
$$\begin{aligned} {V_3}= & {} \left| {\begin{array}{ccc} {{A_3}}&{}\quad 1&{}\quad 0 \\ {{A_1} + {B_1}}&{}\quad {{A_2} + {B_2}}&{}\quad {{A_3}} \\ 0&{}\quad {{A_0} + {B_0} + {C_0}}&{}\quad {{A_1} + {B_1}} \end{array}} \right| > 0, \end{aligned}$$
(8)
$$\begin{aligned} {V_4}= & {} \left| {\begin{array}{cccc} {{A_3}}&{}\quad 1&{}\quad 0&{}\quad 0 \\ {{A_1} + {B_1}}&{}\quad {{A_2} + {B_2}}&{}\quad {{A_3}}&{}\quad 1 \\ 0&{}\quad {{A_0} + {B_0} + {C_0}}&{}\quad {{A_1} + {B_1}}&{}\quad {{A_2} + {B_2}} \\ 0&{}\quad 0&{}\quad 0&{}\quad {{A_0} + {B_0} + {C_0}} \end{array}} \right| > 0. \end{aligned}$$
(9)

Then system (1) is stable at \({E_ * }\). If \({A_0} + {B_0} + {C_0} \ne 0\), Eq. (5) has no zero roots. Hence, the fold bifurcation does not occur.

Case 2 \(\tau = {\tau _1} + {\tau _2} > 0\).

Equation (4) becomes

$$\begin{aligned}&{\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0} \nonumber \\&\quad +({B_2}{\lambda ^2} + {B_1}\lambda + {B_0}){e^{ - \lambda \tau }} + {C_0}{e^{ - 2\lambda \tau }} = 0. \end{aligned}$$
(10)

We take the time delay \(\tau = {\tau _1} + {\tau _2}\) as bifurcation parameters and investigate dynamics of system (1).

\(i\omega \left( {\omega > 0} \right) \) is the root of Eq. (10) if and only if \(i\omega \) satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \left( {{\omega ^4} - {A_2}{\omega ^2} + {A_0} + {C_0}} \right) \cos \left( {\omega \tau } \right) + \left( {{A_3}{\omega ^3} - {A_1}\omega } \right) {\text {sin}}\left( {\omega \tau } \right) = {B_2}{\omega ^2} - {B_0} \\ \left( {{A_1}\omega - {A_3}{\omega ^3}} \right) \cos \left( {\omega \tau } \right) + \left( {{\omega ^4} - {A_2}{\omega ^2} + {A_0} - {C_0}} \right) {\text {sin}}\left( {\omega \tau } \right) = - {B_1}\omega \\ \end{array} \right. \end{aligned}$$
(11)

By Eq. (11), we get

$$\begin{aligned} \cos \left( {\omega \tau } \right)= & {} \frac{{{e_5}{\omega ^6} + {e_6}{\omega ^4} + {e_7}{\omega ^2} + {e_8}}}{{{\omega ^8} + {e_1}{\omega ^6} + {e_2}{\omega ^4} + {e_3}{\omega ^2} + {e_4}}}, \\ sin\left( {\omega \tau } \right)= & {} \frac{{\omega \left( {{e_9}{\omega ^4} + {e_{10}}{\omega ^2} + {e_{11}}} \right) }}{{{\omega ^8} + {e_1}{\omega ^6} + {e_2}{\omega ^4} + {e_3}{\omega ^2} + {e_4}}}, \end{aligned}$$

where

$$\begin{aligned} {e_1}= & {} A_3^2 - 2{A_2}, \\ {e_2}= & {} A_2^2 - 2{A_1}{A_3} + 2{A_0}, \\ {e_3}= & {} A_1^2 - 2{A_0}{A_2}, \\ {e_4}= & {} {A_0}^2 - {C_0}^2, \\ {e_5}= & {} {B_2}, \\ {e_6}= & {} {B_1}{A_3} - {A_2}{B_2} - {B_0}, \\ {e_7}= & {} {A_2}{B_0} + {A_0}{B_2} - {A_1}{B_1} - {C_0}{B_2}, \\ {e_8}= & {} {C_0}{B_0} - {A_0}{B_0}, \\ {e_9}= & {} {A_3}{B_2} - {B_1}, \\ {e_{10}}= & {} {A_2}{B_1} - {A_3}{B_0} - {A_1}{B_2}, \\ {e_{11}}= & {} {A_1}{B_0} - {C_0}{B_1} - {A_0}{B_1}. \end{aligned}$$

Since \({\cos ^2}\left( {\omega \tau } \right) + si{n^2}\left( {\omega \tau } \right) = 1\), we obtain

$$\begin{aligned} {\omega ^{16}} + {k_7}{\omega ^{14}} + {k_6}{\omega ^{12}} + {k_5}{\omega ^{10}} + {k_4}{\omega ^8} + {k_3}{\omega ^6} + {k_2}{\omega ^4} + {k_1}{\omega ^2} + {k_0} = 0 \end{aligned}$$
(12)

where

$$\begin{aligned} {k_0}= & {} {e_4}^2 - {e_8}^2, \\ {k_1}= & {} 2{e_3}{e_4} - 2{e_7}{e_8} - {e_{11}}^2, \\ {k_2}= & {} {e_3}^2 - 2{e_{10}}{e_{11}} + 2{e_2}{e_4} - 2{e_6}{e_8} - {e_7}^2, \\ {k_3}= & {} 2{e_2}{e_3} - 2{e_9}{e_{11}} - 2{e_6}{e_7} - {e_{10}}^2 - 2{e_5}{e_8} + 2{e_1}{e_4}, \\ {k_4}= & {} {e_2}^2 + 2{e_1}{e_3} - 2{e_5}{e_7} - 2{e_9}{e_{10}} + 2{e_4} - {e_6}^2, \\ {k_5}= & {} 2 {e_3} + 2{e_1}{e_2} - 2{e_5}{e_6} - {e_9}^2, \\ {k_6}= & {} {e_1}^2 + 2{e_2} - {e_5}^2, \\ {k_7}= & {} 2{e_1}. \end{aligned}$$

Denote \(z = {\omega ^2}\), then Eq. (12) becomes

$$\begin{aligned} {z^8} + {k_7}{z^7} + {k_6}{z^6} + {k_5}{z^5} + {k_4}{z^4} + {k_3}{z^3} + {k_2}{z^2} + {k_1}z + {k_0} = 0 \end{aligned}$$
(13)

Let \(h(z) = {z^8} + {k_7}{z^7} + {k_6}{z^6} + {k_5}{z^5} + {k_4}{z^4} + {k_3}{z^3} + {k_2}{z^2} + {k_1}z + {k_0}\), it is obvious Eq. (13) has no positive root if \({k_i} > 0,i = 0 \ldots 7\). It means that system (1) is stable for all \(\tau \ge 0\). On the other hand, since  \({\mathop {\lim }\limits _{z \rightarrow \infty }} {h}\left( z \right) = + \infty \), we know that Eq. (13) has at least one positive root if \({k_0} < 0\). Without loss of generality, we assume that it has eight positive real roots, defined by \({z_k},k = 1, \ldots 8\), respectively. From Eq. (11), we have

$$\begin{aligned} \cos \left( {{\omega _k}{\tau _k}} \right) = \frac{{{e_5}\omega _k^6 + {e_6}\omega _k^4 + {e_7}\omega _k^2 + {e_8}}}{{\omega _k^8 + {e_1}\omega _k^6 + {e_2}\omega _k^4 + {e_3}\omega _k^2 + {e_4}}}. \end{aligned}$$

It follows that

$$\begin{aligned} \tau _k^j = \frac{1}{{{\omega _k}}}\arccos \left( {\frac{{{e_5}\omega _k^6 + {e_6}\omega _k^4 + {e_7}\omega _k^2 + {e_8}}}{{\omega _k^8 + {e_1}\omega _k^6 + {e_2}\omega _k^4 + {e_3}\omega _k^2 + {e_4}}}} \right) + \frac{{2j\pi }}{{{\omega _k}}}, \end{aligned}$$

where \(k = 1, \ldots 8;j = 0,1 \ldots \).

Define

$$\begin{aligned} {\tau _0} = {\tau _{{k_0}}} = \mathop {\min }\limits _{k \in \left\{ {1, \ldots ,8} \right\} } \left\{ {\tau _k^0} \right\} ,\quad {\omega _0} = {\omega _{{k_0}}}. \end{aligned}$$

For convenience, let

$$\begin{aligned} \bigcup \limits _{k = 1}^8 {\left\{ {\tau _k^j} \right\} _{j = 0}^{ + \infty }} = \left\{ {{\tau _i}} \right\} _{i = 0}^{ + \infty } \end{aligned}$$
(14)

such that

$$\begin{aligned} {\tau _0}< {\tau _1}< {\tau _2}< \cdots< {\tau _j} < \cdots \end{aligned}$$
(15)

Multiplying \({e^{\lambda \tau }}\) on both sides of Eq. (10) and taking the derivative of \(\lambda \left( \tau \right) \) with respect to \(\tau \), we get

$$\begin{aligned}&{\text {sgn}} \left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) }_{\tau = {\tau _0}}}} \right] \\&\quad = {\text {sgn}} \left\{ {{{\left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) }^{ - 1}}} \right] }_{\tau = {\tau _0}}}} \right\} \\&\quad = {\text {sgn}} \left( {{\text {Re}}\, {{\left( {\frac{{\left( {4{\lambda ^3} + 3{A_3}{\lambda ^2} + 2{A_2}\lambda + {A_1}} \right) {e^{\lambda \tau }} + 2{B_2}\lambda + {B_1}}}{{{C_0}\lambda {e^{ - \lambda \tau }} - \left( {{\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0}} \right) \lambda {e^{\lambda \tau }}}} - \frac{\tau }{\lambda }} \right) }_{\tau = {\tau _0}}}} \right) \\&\quad = {\text {sgn}} \left( {\frac{{{P_1}{P_3} + {P_2}{P_4}}}{{{P_1}^2 + {P_2}^2}}} \right) , \end{aligned}$$

where

$$\begin{aligned} {P_1}= & {} 2 {C_0}\sin \left( {{\omega _0}{\tau _0}} \right) - {B_1}\omega _0^2 ,\\ {P_2}= & {} {B_0}{\omega _0} - {B_2}\omega _0^3 + 2{C_0}{\omega _0}\cos \left( {{\omega _0}{\tau _0}} \right) ,\\ {P_3}= & {} \left( {{A_1} - 3{A_3}\omega _0^2} \right) \cos \left( {{\omega _0}{\tau _0}} \right) + \left( {4\omega _0^3 - 2{A_2}{\omega _0}} \right) \sin \left( {{\omega _0}{\tau _0}} \right) + {B_1} ,\\ {P_4}= & {} \left( {2{A_2}{\omega _0} - 4\omega _0^3} \right) \cos \left( {{\omega _0}{\tau _0}} \right) + \left( {{A_1} - 3{A_3}\omega _0^2} \right) \sin \left( {{\omega _0}{\tau _0}} \right) {\text { + 2}}{B_2}{\omega _0}. \end{aligned}$$

If condition (H3) \({P_1}{P_3} + {P_2}{P_4} \ne 0\) holds, then \({\text {Re}}\, \left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) _{\tau = {\tau _0}}^{ - 1} \ne 0\).

In order to obtain our main results, it is necessary to give the following conditions.

  1. (H1)

    Formulas (6)–(9) hold;

  2. (H2)

    Equation (13) has at least one positive real root;

  3. (H3)

    \({P_1}{P_3} + {P_2}{P_4} \ne 0\).

In summary, we have the following results.

Theorem 1

For system (1), the following results hold.

  1. (i)

     If \({k_i} > 0,i = 0 \ldots 7,{A_0} + {B_0} + {C_0} \ne 0\) and (H1) hold, then the positive equilibrium  \({E_ * }\) is asymptotically stable for all  \(\tau \ge 0\). In other words, Hopf bifurcation does not occur for all \(\tau \ge 0\).

  2. (ii)

     If conditions (H1), (H2) and (H3) hold, then the positive equilibrium \({E_ * }\) of System (1) is asymptotically stable when \(0 \le \tau < {\tau _0}\) and unstable when \(\tau > {\tau _0}\). Furthermore, System (1) undergoes a Hopf bifurcation at  \({E_ * }\) when \(\tau = {\tau _j},j = 0,1,2, \ldots \).

Proof

  1. (i)

    When \(\tau = 0\), characteristic Eq. (10) becomes

    $$\begin{aligned} {\lambda ^4} + {A_3}{\lambda ^3} + \left( {{A_2} + {B_2}} \right) {\lambda ^2} + \left( {{A_1} + {B_1}} \right) \lambda + {A_0} + {B_0} + {C_0} = 0. \end{aligned}$$

    According to Routh–Hurwitz criterion, all the roots of above equation have negative real parts if and only if (H1) holds. Hence, all the roots of Eq. (10) have negative real parts when \(\tau = 0\).

    On the other hand, from \({k_i} > 0,i = 0 \ldots 7\), we obtain \(h\left( 0 \right) = {k_0} > 0\) and \(h'\left( z \right) = 8{z^7} + 7{k_7}{z^6} + 6{k_6}{z^5} + 5{k_5}{z^4} + 4{k_4}{z^3}+ 3{k_3}{z^2} + 2{k_2}z + {k_1}\). Thus, we have Eq. (13) has no real root, and hence Eq. (10) has no purely imaginary root. Since \({A_0} + {B_0} + {C_0} \ne 0\), \(\lambda = 0\) is not a root of Eq. (10). To sum up, we can obtain that Eq. (10) has no root with zero real part for all \(\tau > 0\). Therefore, all the roots of Eq. (10) have negative real parts for all \(\tau \ge 0\). This completes the proof of (i).

  2. (ii)

    The definition of \({\tau _0}\) implies that all the roots of Eq. (10) have negative real parts when \(0 \le \tau < {\tau _0}\). Hence, the positive equilibrium \({E_ * }\) is asymptotically stable for \(0 \le \tau < {\tau _0}\). If the condition (H2) holds, Eq. (13) has at least one real root, which indicates that the positive equilibrium \({E_ * }\) is unstable for \(\tau > {\tau _0}\). Moreover, \({P_1}{P_3} + {P_2}{P_4} \ne 0\) implies that the transversality condition for Hopf bifurcations is satisfied under the given assumption (H3). Therefore, System (1) undergoes a Hopf bifurcation at  \({E_ * }\) when \(\tau = {\tau _j},j = 0,1,2, \ldots \).

Case 3 \({\tau _2}{{ > 0,}}{\tau _1} = 0\). Equation (4) becomes

$$\begin{aligned} {\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0} + ({B_2}{\lambda ^2} + {B_1}\lambda + {B_0}){e^{ - \lambda {\tau _2}}} + {C_0}{e^{ - 2\lambda {\tau _2}}} = 0 \end{aligned}$$

It is same as the characteristic equation in case 2, and similar results can be obtained. The difference is that case 2 takes the sum of time delays as the bifurcation parameter. Here, only the time delay \({\tau _2}\) is used as the bifurcation parameter. For the case \({\tau _1} > 0,{\tau _2} = 0\), with \({\tau _1}\) as the bifurcation parameter, the characteristic equation remains unchanged, and similar results will be obtained.

Case 4 \({\tau _1} = {\tau _2} = \tau \ne 0\). Equation (4) becomes

$$\begin{aligned} {\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0}+ ({B_2}{\lambda ^2} + {B_1}\lambda + {B_0}){e^{ - 2\lambda \tau }} + {C_0}{e^{ - 4\lambda \tau }} = 0 \end{aligned}$$
(16)

\(i\omega \left( {\omega > 0} \right) \) is the root of Eq. (16) if and only if \(i\omega \) satisfies

$$\begin{aligned} \left\{ \begin{array}{l} \left( {{\omega ^4} - {A_2}{\omega ^2} + {A_0} + {C_0}} \right) \cos \left( {2\omega \tau } \right) + \left( {{A_3}{\omega ^3} - {A_1}\omega } \right) {\text {sin}}\left( {2\omega \tau } \right) = {B_2}{\omega ^2} - {B_0} \\ \left( {{A_1}\omega - {A_3}{\omega ^3}} \right) \cos \left( {2\omega \tau } \right) + \left( {{\omega ^4} - {A_2}{\omega ^2} + {A_0} - {C_0}} \right) {\text {sin}}\left( {2\omega \tau } \right) = - {B_1}\omega \\ \end{array} \right. \nonumber \\ \end{aligned}$$
(17)

By Eq. (17), we get

$$\begin{aligned} \cos \left( {2\omega \tau } \right)= & {} \frac{{{e_5}{\omega ^6} + {e_6}{\omega ^4} + {e_7}{\omega ^2} + {e_8}}}{{{\omega ^8} + {e_1}{\omega ^6} + {e_2}{\omega ^4} + {e_3}{\omega ^2} + {e_4}}}, \nonumber \\ sin\left( {2\omega \tau } \right)= & {} \frac{{\omega \left( {{e_9}{\omega ^4} + {e_{10}}{\omega ^2} + {e_{11}}} \right) }}{{{\omega ^8} + {e_1}{\omega ^6} + {e_2}{\omega ^4} + {e_3}{\omega ^2} + {e_4}}}. \end{aligned}$$
(18)

Since \({\cos ^2}\left( {2\omega \tau } \right) + si{n^2}\left( {2\omega \tau } \right) = 1\), we get

$$\begin{aligned} {\omega ^{16}} + {k_7}{\omega ^{14}} + {k_6}{\omega ^{12}} + {k_5}{\omega ^{10}} + {k_4}{\omega ^8} + {k_3}{\omega ^6} + {k_2}{\omega ^4} + {k_1}{\omega ^2} + {k_0} = 0 \end{aligned}$$
(19)

Denote \(z = {\omega ^2}\), then Eq. (19) becomes

$$\begin{aligned} {z^8} + {k_7}{z^7} + {k_6}{z^6} + {k_5}{z^5} + {k_4}{z^4} + {k_3}{z^3} + {k_2}{z^2} + {k_1}z + {k_0} = 0 \end{aligned}$$
(20)

This equation is the same as the case two.

Similar to case 2, we know that Eq. (20) has at least one positive root if \({k_0} < 0\). Without loss of generality, we assume that it has eight positive real roots, defined by \({z_k},k=1,2,\ldots ,8\), respectively. From (18), we have

$$\begin{aligned} \tau _{k}^j = \frac{1}{{2{\omega _k}}}\arccos \left( {\frac{{{e_5}\omega _k^6 + {e_6}\omega _k^4 + {e_7}\omega _k^2 + {e_8}}}{{\omega _k^8 + {e_1}\omega _k^6 + {e_2}\omega _k^4 + {e_3}\omega _k^2 + {e_4}}}} \right) + \frac{{j\pi }}{{{\omega _k}}} \end{aligned}$$
(21)

where \(k = 1, \ldots 8;j = 0,1 \ldots \).

Define

$$\begin{aligned} {\tau _{0}} = {\tau _{{k_0}}} = \mathop {\min }\limits _{k \in \left\{ {1, \ldots ,8} \right\} } \left\{ {\tau _k^0} \right\} ,{\omega _{0}} = {\omega _{{k_0}}}. \end{aligned}$$

For convenience, let

$$\begin{aligned} \bigcup \limits _{k = 1}^8 {\left\{ {\tau _k^j} \right\} _{j = 0}^{ + \infty }} = \left\{ {{\tau _i}} \right\} _{i = 0}^{ + \infty } \end{aligned}$$
(22)

such that

$$\begin{aligned} {\tau _0}< {\tau _1}< {\tau _2}< \cdots< {\tau _j} < \cdots \end{aligned}$$
(23)

Multiplying \({e^{2\lambda \tau }}\) on both sides of Eq. (16) and taking the derivative of \(\lambda \left( \tau \right) \) with respect to \(\tau \), we have

$$\begin{aligned}&{\text {sgn}} \left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) }_{\tau = {\tau _0}}}} \right] \\&\quad = {\text {sgn}} \left\{ {{{\left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) }^{ - 1}}} \right] }_{\tau = {\tau _0}}}} \right\} \\&\quad = {\text {sgn}} \left( {{\text {Re}}\, {{\left( {\frac{{\left( {4{\lambda ^3} + 3{A_3}{\lambda ^2} + 2{A_2}\lambda + {A_1}} \right) {e^{2\lambda \tau }} + 2{B_2}\lambda + {B_1}}}{{2{C_0}\lambda {e^{ - 2\lambda \tau }} - 2\left( {{\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0}} \right) \lambda {e^{2\lambda \tau }}}} - \frac{\tau }{\lambda }} \right) }_{\tau = {\tau _0}}}} \right) \\&\quad = {\text {sgn}} \left( {\frac{{{Q_1}{Q_3} + {Q_2}{Q_4}}}{{{Q_1}^2 + {Q_2}^2}}} \right) . \end{aligned}$$

where

$$\begin{aligned} x= & {} \cos (2{\omega _0}{\tau _0}), \\ y= & {} \sin \left( {2{\omega _0}{\tau _0}} \right) , \\ {Q_1}= & {} 2{\omega _0}\left( {{C_0}y - \omega _0^4y - \omega _0^3{A_3}x - \omega _0^2{A_2}y + {\omega _0}{A_1}x + {A_0}y} \right) , \\ {Q_2}= & {} 2{\omega _0}\left( {{C_0}x - \omega _0^4x - \omega _0^3{A_3}y + \omega _0^2{A_2}x + {\omega _0}{A_1}y - {A_0}x} \right) , \\ {Q_3}= & {} 4\omega _0^3y - 3{A_3}\omega _0^2x - 2{A_2}{\omega _0}y + {A_1}x + {B_1}, \\ {Q_4}= & {} - 4\omega _0^3x - 3{A_3}\omega _0^2y + 2{A_2}{\omega _0}x + {A_1}y + 2{B_2}{\omega _0}. \end{aligned}$$

If condition (H4) \({Q_1}{Q_3} + {Q_2}{Q_4} \ne 0\) holds, then \({\mathrm{Re}} \left( {\frac{{d\lambda \left( \tau \right) }}{{d\tau }}} \right) _{\tau = {\tau _0}}^{ - 1} \ne 0\). In order to obtain the main results, we give the following assumption.

  1. (H4)

    \({Q_1}{Q_3} + {Q_2}{Q_4} \ne 0\);

  2. (H5)

    Equation (20) has at least one positive real root.

Applying Hopf bifurcation theorem in Hassard et al. (1981), we have the following result. \(\square \)

Theorem 2

For System (1), suppose that (H1), (H4) and (H5) hold, then the positive equilibrium \({E_*}\) of System (1) is asymptotically stable for \(0 \le \tau < {\tau _0}\) and unstable for \(\tau > {\tau _0}\). Furthermore, System(1) undergoes a Hopf bifurcation at \({E_ *}\) when \(\tau = {\tau _j},j = 0,1,2, \ldots \), where \(\tau ={\tau _j}\) as defined in (23).

Case 5 \({\tau _2} > 0,{\tau _1} \in \left[ {0,{\tau _{10}}} \right) \),and \({\tau _1} \ne {\tau _2}\).

We consider Eq. (4) with \({\tau _1}\) in its stable interval; \({\tau _2}\) is regarded as a bifurcation parameter, where \({\tau _{10}}\) is the first bifurcation point when \({\tau _2} = 0\). \(i{\omega _2}\left( {{\omega _2} > 0} \right) \) is the root of Eq. (4) if and only if \(i{\omega _2}\) satisfies

$$\begin{aligned} \left\{ \begin{array}{l} {E_{11}}\cos \left( {{\omega _2}{\tau _2}} \right) + {E_{12}}{\text {sin}}\left( {{\omega _2}{\tau _2}} \right) = {E_{13}}, \\ {E_{14}}\cos \left( {{\omega _2}{\tau _2}} \right) + {E_{15}}{\text {sin}}\left( {{\omega _2}{\tau _2}} \right) = {E_{16}}, \\ \end{array} \right. \end{aligned}$$
(24)

where

$$\begin{aligned} {E_{11}}= & {} {\omega _2}^4 - {A_2}{\omega _2}^2 + {A_0} + {C_0}, \\ {E_{12}}= & {} {A_3}{\omega _2}^3 - {A_1}{\omega _2}, \\ {E_{13}}= & {} \left( {{B_2}{\omega _2}^2 - {B_0}} \right) \cos \left( {{\omega _2}{\tau _1}} \right) - {B_1}{\omega _2}sin\left( {{\omega _2}{\tau _1}} \right) , \\ {E_{14}}= & {} {A_1}{\omega _2} - {A_3}{\omega _2}^3, \\ {E_{15}}= & {} {\omega _2}^4 - {A_2}{\omega _2}^2 + {A_0} - {C_0}, \\ {E_{16}}= & {} - {B_1}{\omega _2}\cos \left( {{\omega _2}{\tau _1}} \right) + \left( {{B_0} - {B_2}{\omega _2}^2} \right) sin\left( {{\omega _2}{\tau _1}} \right) . \end{aligned}$$

From Eq. (24), we get

$$\begin{aligned} \cos \left( {{\omega _2}{\tau _2}} \right)= & {} - \frac{{{E_{12}}{E_{16}} - {E_{13}}{E_{15}}}}{{{E_{11}}{E_{15}} - {E_{14}}{E_{12}}}}, \nonumber \\ sin\left( {{\omega _2}{\tau _2}} \right)= & {} \frac{{{E_{11}}{E_{16}} - {E_{14}}{E_{13}}}}{{{E_{11}}{E_{15}} - {E_{14}}{E_{12}}}}. \end{aligned}$$
(25)

From (25), we obtain

$$\begin{aligned}&{\omega _2}^{16} + {n_1}{\omega _2}^{14} + {n_2}{\omega _2}^{12} + {n_3}{\omega _2}^{10} + {n_4}{\omega _2}^8 + {n_5}{\omega _2}^6 + {n_6}{\omega _2}^4 + {n_7}{\omega _2}^2 + {n_8} \nonumber \\&\quad +\left( {{m_0} + {m_2}{\omega _2}^2 + {m_4}{\omega _2}^4 + {m_6}{\omega _2}^6 + {m_8}{\omega _2}^8} \right) {\cos ^2}\left( {{\omega _2}{\tau _1}} \right) \nonumber \\&\quad +\left( {{m_1}{\omega _2} + {m_3}{\omega _2}^3 + {m_5}{\omega _2}^5 + {m_7}{\omega _2}^7} \right) sin\left( {2{\omega _2}{\tau _1}} \right) = 0, \end{aligned}$$
(26)

where \({n_i}\left( {i = 1 \ldots 8} \right) ,{m_j}\left( {j = 0 \ldots 8} \right) \) are defined in Appendix.

In order to get the main result, we give the following assumption.

  1. (H6)

    Equation (26) has at least finite positive roots, defined by \(\omega _2^{\left( 1 \right) },\omega _2^{\left( 2 \right) }, \ldots \omega _2^{\left( k \right) }\). For every fixed \(\omega _2^{\left( i \right) },i = 1,2 \ldots k\), there exists a sequence \(\tau _{2i}^j\) such that Eq. (26) holds.

From (25), we have

$$\begin{aligned} \tau _{2i}^j = \frac{1}{{\omega _2^{\left( i \right) }}}\arccos \left( {\frac{{{E_{13}}{E_{15}} - {E_{12}}{E_{16}}}}{{{E_{11}}{E_{15}} - {E_{14}}{E_{12}}}}} \right) + \frac{{2\pi j}}{{\omega _2^{\left( i \right) }}} \end{aligned}$$

where \(i = 1,2 \ldots k;j = 0,1,2, \ldots \).

Define

$$\begin{aligned} {\tau _{20}} = \min \left\{ {\tau _{2i}^0\left| {i = 1,2,...,k;j = 0,1,2,....} \right. } \right\} \end{aligned}$$

Therefore, Eq. (26) has a pair of purely imaginary roots \(\pm i{\omega _{20}}\) when \({\tau _2} = {\tau _{20}}\). Multiplying \({e^{\lambda {\tau _2}}}\) on both sides of Eq. (4) and taking the derivative of \(\lambda \) with respect to \({\tau _2}\), we get

$$\begin{aligned}&{\text {sgn}} \left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda }}{{d{\tau _2}}}} \right) }_{{\tau _2} = {\tau _{20}}}}} \right] \\&\quad = {\text {sgn}} \left\{ {{{\left[ {{\text {Re}}\, {{\left( {\frac{{d\lambda }}{{d{\tau _2}}}} \right) }^{ - 1}}} \right] }_{{\tau _2} = {\tau _{20}}}}} \right\} \\&\quad = {\text {sgn}} \left( {{\text {Re}}\, {{\left( {\frac{{{E_{21}}{e^{\lambda {\tau _2}}} + {E_{24}}{e^{ - \lambda {\tau _1}}} - {E_{22}}{\tau _1}{e^{ - \lambda {\tau _1}}} - 2{C_0}{\tau _1}{e^{ - \lambda \left( {2{\tau _1} + {\tau _2}} \right) }}}}{{{C_0}\lambda {e^{ - \lambda \left( {2{\tau _1} + {\tau _2}} \right) }} - {E_{23}}\lambda {e^{\lambda {\tau _2}}}}} - \frac{{{\tau _2}}}{\lambda }} \right) }_{{\tau _2} = {\tau _{20}}}}} \right) \\&\quad = {\text {sgn}} \left( {\frac{{{T_1}{T_3} + {T_2}{T_4}}}{{{T_1}^2 + {T_2}^2}}} \right) , \end{aligned}$$

where

$$\begin{aligned} {E_{21}} =&4{\lambda ^3} + 3{A_3}{\lambda ^2} + 2{A_2}\lambda + {A_1},\\ {E_{22}}=&{B_2}{\lambda ^2} + {B_1}\lambda + {B_0},\\ {E_{23}}=&{\lambda ^4} + {A_3}{\lambda ^3} + {A_2}{\lambda ^2} + {A_1}\lambda + {A_0},\\ {E_{24}}=&2{B_2}\lambda + {B_1},\\ {T_1} =&\cos \left( {{\tau _1}{\omega _{20}}} \right) \left( {{B_1} - {B_0}{\tau _1} + {B_2}{\tau _1}\omega _{20}^2} \right) + \cos \left( {{\tau _1}{\omega _{20}}} \right) \left( {{A_1} - 3{A_3}\omega _{20}^2} \right) \\&-2{\tau _1}{C_0}\cos \left( {\left( {2{\tau _1} + {\tau _2}} \right) {\omega _{20}}} \right) + {\omega _{20}}sin\left( {{\tau _1}{\omega _{20}}} \right) \left( {2{B_2} - {\tau _1}{B_1}} \right) \\&+sin\left( {{\tau _{20}}{\omega _{20}}} \right) \left( {4\omega _{20}^3 - 2{A_2}{\omega _{20}}} \right) ,\\ {T_2} =&\left( {2{B_2}{\omega _{20}} - {\tau _1}{B_1}} \right) \cos \left( {{\tau _1}{\omega _{20}}} \right) + \left( {2{A_2}{\omega _{20}} - 4\omega _{20}^3} \right) \cos \left( {{\tau _{20}}{\omega _{20}}} \right) \\&+\left( {{B_0}{\tau _1} - {B_1} - {\tau _1}{B_2}\omega _{20}^2} \right) sin\left( {{\tau _1}{\omega _{20}}} \right) + \left( {{A_1} - 3{A_3}\omega _{20}^2} \right) sin\left( {{\tau _{20}}{\omega _{20}}} \right) \\&+ 2{\tau _1}{C_0}sin\left( {\left( {2{\tau _1} + {\tau _2}} \right) {\omega _{20}}} \right) ,\\ {T_3} =&\left( {{A_1}\omega _{20}^2 - {A_3}\omega _{20}^4} \right) \cos \left( {{\tau _{20}}{\omega _{20}}} \right) + \left( {{A_0}{\omega _{20}} - {A_2}\omega _{20}^3 + \omega _{20}^5} \right) sin\left( {{\tau _{20}}{\omega _{20}}} \right) \\&+ {C_0}{\omega _{20}}sin\left( {\left( {2{\tau _1} + {\tau _2}} \right) {\omega _{20}}} \right) ,\\ {T_4} =&\left( {{A_2}\omega _{20}^3 - {A_0}{\omega _{20}} - \omega _{20}^5} \right) \cos \left( {{\tau _{20}}{\omega _{20}}} \right) + {C_0}{\omega _{20}}\cos \left( {\left( {2{\tau _1} + {\tau _2}} \right) {\omega _{20}}} \right) \\&+\left( {{A_1}\omega _{20}^2 - {A_3}\omega _{20}^4} \right) sin\left( {{\tau _{20}}{\omega _{20}}} \right) . \end{aligned}$$

If condition (H7) \({T_1}{T_3} + {T_2}{T_4} \ne 0\)  holds, then \({\text {Re}}\, \left( {\frac{{d\lambda \left( \tau \right) }}{{d{\tau _2}}}} \right) _{{\tau _2} = {\tau _{20}}}^{ - 1} \ne 0\).

By the discussion above, we get the following theorem.

Theorem 3

For system (1), \({\tau _2} > 0,{\tau _1} \in \left[ {0,{\tau _{10}}} \right) \), and \({\tau _1} \ne {\tau _2}\). If conditions (H1),(H6) and (H7) hold, then the positive equilibrium \({E_ * }\) of system (1) is locally asymptotically stable when \(0 \le {\tau _2} < {\tau _{20}}\) and unstable when \({\tau _2} > {\tau _{20}}\). Furthermore, system (1) undergoes a Hopf bifurcation at \({E_ * }\) when  \({\tau _2} = {\tau _{20}}\).

For the case \({\tau _1} > 0,{\tau _2} \in \left[ {0,{\tau _{20}}} \right) \), and \({\tau _1} \ne {\tau _2}\), we can obtain similar results as those in Theorem 3.

3 Direction and Stability of Hopf Bifurcation

In this section, we consider the direction and stability of Hopf bifurcation with respect to \({\tau _2}\) and \({\tau _1} \in \left[ {0,{\tau _{10}}} \right) \) by using the methods in Hassard et al. (1981). Without loss of generality, we assume system (2) undergoes a Hopf bifurcation at \({\tau _2} = {\tau _{20}},{\tau _1} \in \left[ {0,{\tau _{10}}} \right) \). For convenience, letting \({\bar{u}_i}(t) = {u_i}(\tau t),i = 1,2,3,{\tau _2} = {\tau _{20}} + \mu ,\mu \in R\) and dropping the bars, then system (2) becomes a functional differential equation in \(C = {C^1}\left( {[ - 1,0],{R^3}} \right) \):

$$\begin{aligned} \dot{u}(t) = {L_\mu }{u_t}\left( \theta \right) + f(\mu ,{u_t}\left( \theta \right) ) \end{aligned}$$

where \(u(t) = {({u_1}(t),{u_2}(t),{u_3}(t),{u_4}(t))^T}\), \({u_t}(\theta ) = u(t + \theta )(\theta \in [ - \tau ,0] \) for \(\theta \in [ - \tau ,0]\), and \({L_\mu }:C \rightarrow R,\,f:R \times C \rightarrow R\) are given, respectively, by

$$\begin{aligned} {L_\mu }(\varphi ) = ({\tau _{20}} + \mu )\hat{A}\varphi \left( 0 \right) + ({\tau _{20}} + \mu )\hat{B}\varphi \left( { - \frac{{{\tau _1}}}{{{\tau _{20}}}}} \right) + ({\tau _{20}} + \mu )\hat{C}\varphi \left( { - 1} \right) \end{aligned}$$

and

$$\begin{aligned} f\left( {\mu ,{u_t}\left( \theta \right) } \right) = ({\tau _{20}} + \mu ){\left( {{f_1},{f_2},{f_3}} \right) ^T}, \end{aligned}$$

here

$$\begin{aligned} \hat{A}&= \left[ {\begin{array}{cccc} {{a_{11}}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad {{a_{22}}}&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right] ,\quad \hat{B} = \left[ {\begin{array}{cccc} 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ {{a_{31}}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad 0 \end{array}} \right] ,\quad \\ \hat{C}&= \left[ {\begin{array}{cccc} 0&{}\quad 0&{}\quad {{a_{13}}}&{}\quad {{a_{14}}}\\ 0&{}\quad 0&{}\quad {{a_{23}}}&{}\quad {{a_{24}}}\\ 0&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad 0 \end{array}} \right] ,\\ \varphi \left( \theta \right)&= {\left( {{\varphi _1}\left( \theta \right) ,{\varphi _2}\left( \theta \right) ,{\varphi _3}\left( \theta \right) ,{\varphi _4}\left( \theta \right) } \right) ^T} \in C,\\ {f_1}&= {k_{11}}\varphi _3^2\left( { - 1} \right) + {k_{12}}\varphi _4^2\left( { - 1} \right) + {k_{13}}\varphi _3^3\left( { - 1} \right) + {k_{14}}\varphi _4^3\left( { - 1} \right) ,\\ {f_2}&= {k_{21}}\varphi _3^2\left( { - 1} \right) + {k_{22}}\varphi _4^2\left( { - 1} \right) + {k_{23}}\varphi _3^3\left( { - 1} \right) + {k_{24}}\varphi _4^3\left( { - 1} \right) ,\\ {f_3}&= 0,\\ {f_4}&= 0,\\ {k_{11}}&= - \frac{{a{a_{11}}n\left( {{{\left( {P_1^ * } \right) }^{2n - 2}}{s^n}\left( {n + 1} \right) + {{\left( {P_1^ * } \right) }^{n - 2}}{s^{2n}}\left( {1 - n} \right) } \right) }}{{{{\left( {{s^n} + {{\left( {P_1^ * } \right) }^n}} \right) }^3}}},\\ {k_{12}}&= - \frac{{b{b_{21}}n\left( {{{\left( {P_2^ * } \right) }^{2n - 2}}{s^n}\left( {n + 1} \right) + {{\left( {P_2^ * } \right) }^{n - 2}}{s^{2n}}\left( {1 - n} \right) } \right) }}{{{{\left( {{s^n} + {{\left( {P_2^ * } \right) }^n}} \right) }^3}}},\\ {k_{21}}&= - \frac{{an\left( {{{\left( {P_2^ * } \right) }^{2n - 2}}{s^n}\left( {n + 1} \right) + {{\left( {P_2^ * } \right) }^{n - 2}}{s^{2n}}\left( {1 - n} \right) } \right) }}{{{{\left( {{s^n} + {{\left( {P_2^ * } \right) }^n}} \right) }^3}}},\\ {k_{22}}&= - \frac{{bn\left( {{{\left( {P_1^ * } \right) }^{2n - 2}}{s^n}\left( {n + 1} \right) + {{\left( {P_1^ * } \right) }^{n - 2}}{s^{2n}}\left( {1 - n} \right) } \right) }}{{{{\left( {{s^n} + {{\left( {P_1^ * } \right) }^n}} \right) }^3}}}. \end{aligned}$$

By the Riesz representation theorem, there exists a matrix function \(\eta (\theta ,\mu ),\theta \in \left[ { - 1,0} \right] \), such that

$$\begin{aligned} {L_\mu }(\varphi ) = \int _{ - 1}^0 {d\eta (\theta ,\mu )} \varphi (\theta ),{\text { }}\varphi \in {C^1}\left( {[ - 1,0],{R^3}} \right) . \end{aligned}$$

Here, we choose

$$\begin{aligned} \eta (\theta ,\mu ) = \left\{ \begin{array}{l} ({\tau _{20}} + \mu )\left( {\hat{A} + \hat{B} + \hat{C}} \right) ,\theta \in 0, \\ ({\tau _{20}} + \mu )\left( {\hat{B} + \hat{C}} \right) ,\theta \in \left[ { - \frac{{{\tau _1}}}{{{\tau _{20}}}},0} \right) , \\ ({\tau _{20}} + \mu )\hat{C},{\text { }}\theta \in \left( { - 1, - \frac{{{\tau _1}}}{{{\tau _{20}}}}} \right) , \\ 0,\,\,\theta = - 1. \\ \end{array} \right. \end{aligned}$$

For \(\varphi \in {C^1}\left( {[ - 1,0],{R^3}} \right) \), define

$$\begin{aligned} {A_\mu }(\varphi ) = \left\{ \begin{array}{ll} \frac{{d\varphi (\theta )}}{{d\theta }},\, - 1 \le \theta < 0,\\ \int _{ - 1}^0 {d\eta (\theta ,\mu )} \varphi (\theta ),\, \theta = 0, \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} {R_\mu }(\varphi ) = \left\{ \begin{array}{l} {\left( {\begin{array}{ccc} 0&{}0&{}0 \end{array}} \right) ^T},\,- 1 \le \theta < 0,\\ f\left( {\mu ,\varphi } \right) ,\,\theta = 0. \end{array} \right. \end{aligned}$$

For \(\psi \in {C^1}\left( {[0,1],{{({R^3})}^ * }} \right) \), the adjoint operator \({A^ * }\) of A is defined by

$$\begin{aligned} {A^ * }(\mu )\psi = \left\{ \begin{array}{ll} - \frac{{d\psi (s)}}{{ds}},\, 0 < s \le 1,\\ \int _{ - 1}^0 {d{\eta ^T}(s,\mu )} \psi ( - s),\, s = 0. \end{array} \right. \end{aligned}$$

Next, we define a bilinear form:

$$\begin{aligned} \langle \psi (s),\varphi (\theta )\rangle = \bar{\psi } (0)\varphi (0) - \int _{ - 1}^0 {\int _{\xi = 0}^\theta {\bar{\psi } (\xi - \theta )} } d\eta (\theta )\varphi (\xi )d\xi , \end{aligned}$$

where \(\eta (\theta ) = \eta (\theta ,0)\).

Through the discussion in Sect. 2, we know \(\pm i{\tau _{20}}{\omega _{20}}\) are eigenvalues of A(0). They are also eigenvalues of \({A^ * }\). In the following, we compute the eigenvectors of A(0) and \({A^ * }\) corresponding to \(i{\tau _{20}}{\omega _{20}}\) and \(- i{\tau _{20}}{\omega _{20}}\), respectively. Assume \(q(\theta ) = {(1,\alpha ,\beta ,\gamma )^{\mathrm{T}}}{e^{i{\tau _{20}}{\omega _{20}}\theta }},\theta \in \left[ { - 1,0} \right] \) is eigenvector of A(0). Then, we know

$$\begin{aligned} A(0)q(0) = i{\tau _{20}}{\omega _{20}}q(0). \end{aligned}$$

That is,

$$\begin{aligned} {\tau _{20}}\left[ {\begin{array}{cccc} {i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - i{\tau _{20}}{\omega _{20}}}}}&{}\quad { - {a_{14}}{e^{ - i{\tau _{20}}{\omega _{20}}}}}\\ 0&{}\quad {i{\omega _{20}} - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - i{\tau _{20}}{\omega _{20}}}}}&{}\quad { - {a_{24}}{e^{ - i{\tau _{20}}{\omega _{20}}}}}\\ { - {a_{31}}{e^{ - i{\tau _1}{\omega _{20}}}}}&{}\quad 0&{}\quad {i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - i{\tau _1}{\omega _{20}}}}}&{}\quad 0&{}\quad {i{\omega _{20}} - {a_{44}}} \end{array}} \right] q(0) = \left[ {\begin{array}{c} 0\\ 0\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$

By calculation, we get

$$\begin{aligned} q(0) = {\left( 1,\frac{{\left( {i{\omega _{20}} - {a_{44}}} \right) {\Delta _1}}}{{{\Delta _2}}},\frac{{{a_{31}}{e^{ - i{\tau _1}{\omega _{20}}}}}}{{i{\omega _{20}} - {a_{33}}}},\frac{{{a_{42}}{e^{ - i{\tau _1}{\omega _{20}}}}{\Delta _1}}}{{{\Delta _2}}}\right) ^{\mathrm{T}}}, \end{aligned}$$

where

$$\begin{aligned} {\Delta _1} = \frac{{{a_{23}}{a_{31}}{e^{ - i{\omega _{20}}\left( {{\tau _{20}} + {\tau _1}} \right) }}}}{{i{\omega _{20}} - {a_{33}}}},{\Delta _2} = \left( {i{\omega _{20}} - {a_{44}}} \right) \left( {i{\omega _{20}} - {a_{22}}} \right) - {a_{24}}{a_{42}}{e^{ - i{\omega _{20}}\left( {{\tau _{20}} + {\tau _1}} \right) }}. \end{aligned}$$

On the other hand, we assume \({q^ * }(s) = D(1,{\alpha ^ * },{\beta ^ * },{\gamma ^ * }){e^{i{\tau _{20}}{\omega _{20}}s}},s \in \left[ {0,1} \right] \) is eigenvector of \({A^ * }\). Thus, we have:

$$\begin{aligned} {\tau _{20}}\left[ {\begin{array}{cccc} { - i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad { - {a_{31}}{e^{i{\tau _1}{\omega _{20}}}}}&{}\quad 0\\ 0&{}\quad { - i{\omega _{20}} - {a_{22}}}&{}\quad 0&{}\quad { - {a_{42}}{e^{i{\tau _1}{\omega _{20}}}}}\\ { - {a_{13}}{e^{i{\tau _{20}}{\omega _{20}}}}}&{}\quad { - {a_{23}}{e^{i{\tau _{20}}{\omega _{20}}}}}&{}\quad { - i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ { - {a_{14}}{e^{i{\tau _{20}}{\omega _{20}}}}}&{}\quad { - {a_{24}}{e^{i{\tau _{20}}{\omega _{20}}}}}&{}\quad 0&{}\quad { - i{\omega _{20}} - {a_{44}}} \end{array}} \right] {\left( {{q^ * }(0)} \right) ^{\mathrm{T}}} = \left[ {\begin{array}{c} 0\\ 0\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$

It is not hard to get

$$\begin{aligned} {q^ * }(0) = D(1,{\alpha ^ * },{\beta ^ * },{\gamma ^ * }) = D(1,{\Delta _3}, - \frac{{i{\omega _{20}} + {a_{11}}}}{{{a_{31}}{e^{i{\tau _1}{\omega _{20}}}}}}, - \frac{{\left( {i{\omega _{20}} + {a_{22}}} \right) {\Delta _3}}}{{{a_{42}}{e^{i{\tau _1}{\omega _{20}}}}}}) \end{aligned}$$

where

$$\begin{aligned} {\Delta _3} = \frac{{{a_{13}}}}{{{a_{23}}}} - \frac{{\left( {i{\omega _{20}} + {a_{11}}} \right) \left( {i{\omega _{20}} + {a_{33}}} \right) }}{{{a_{23}}{a_{31}}{e^{i{\omega _{20}}\left( {{\tau _1} + {\tau _{20}}} \right) }}}}. \end{aligned}$$

Next, normalizing q and \({q^ * }\) by the condition \(\langle {q^ * },q\rangle = 1, \langle {q^ * },\bar{q}\rangle = 0\). In order to obtain \(\langle {q^ * }(s),q(\theta )\rangle = 1\), it is necessary to compute the value of D.

$$\begin{aligned}&\left\langle {{q^ * }\left( s \right) ,q\left( \theta \right) } \right\rangle \\&\quad = {{\bar{q}}^ * }\left( 0 \right) \cdot q\left( 0 \right) - \int _{\theta = - 1}^0 {\int _{\xi = 0}^\theta {{{\bar{q}}^ * }\left( {\xi - \theta } \right) } } d\eta \left( \theta \right) q\left( \xi \right) d\xi \\&\quad = \bar{D}\left( {1,{{\bar{\alpha } }^ * },{{\bar{\beta } }^ * },{{\bar{\gamma } }^ * }} \right) {\left( {1,\alpha ,\beta ,\gamma } \right) ^{\mathrm{T}}} \\&\quad \quad -\int _{ - 1}^0 {\int _{\xi = 0}^\theta {\bar{D}\left( {1,{{\bar{\alpha } }^ * },{{\bar{\beta } }^ * },{{\bar{\gamma } }^ * }} \right) } } {e^{ - i\left( {\xi - \theta } \right) {\tau _{20}}{\omega _{20}}}}d\eta \left( \theta \right) {\left( {1,\alpha ,\beta ,\gamma } \right) ^{\mathrm{T}}}{e^{i\xi {\tau _{20}}{\omega _{20}}}}d\xi \\&\quad = \bar{D}\{ 1 + \alpha {{\bar{\alpha } }^ * } + \beta {{\bar{\beta } }^ * } + \gamma {{\bar{\gamma } }^ * } \\&\quad \quad - \int _{ - 1}^0 {\left( {1,{{\bar{\alpha } }^ * },{{\bar{\beta } }^ * },{{\bar{\gamma } }^ * }} \right) } \theta {e^{i\theta {\tau _{20}}{\omega _{20}}}}d\eta \left( \theta \right) {\left( {1,\alpha ,\beta ,\gamma } \right) ^{\mathrm{T}}}\} \\&\quad = \bar{D}\{ 1 + \alpha {{\bar{\alpha } }^ * } + \beta {{\bar{\beta } }^ * } + \gamma {{\bar{\gamma } }^ * } + {\tau _1}\left( {{a_{31}}{{\bar{\beta } }^ * } + \alpha {a_{42}}{{\bar{\gamma } }^ * }} \right) {e^{ - i{\tau _1}{\omega _{20}}}} \\&\qquad + {\tau _{20}}\left( {\beta \left( {{a_{13}} + {a_{23}}{{\bar{\alpha } }^ * }} \right) + \gamma \left( {{a_{14}} + {a_{24}}{{\bar{\alpha } }^ * }} \right) } \right) {e^{ - i{\tau _{20}}{\omega _{20}}}}\}, \end{aligned}$$

here, we choose

$$\begin{aligned} D = \frac{1}{{m + n}} \end{aligned}$$

such that \(\langle {q^ * }\left( s \right) ,q\left( \theta \right) \rangle = 1,\langle {q^ * }\left( s \right) ,\bar{q}\left( \theta \right) \rangle = 0\), where \(m = 1 + \bar{\alpha } {\alpha ^ * } + \bar{\beta } {\beta ^ * } + \bar{\gamma } {\gamma ^ * } + {\tau _1}({a_{31}}{\beta ^ * } + \alpha {a_{42}}{\gamma ^ * }){e^{i{\tau _1}{\omega _{20}}}}\), \(n = {\tau _{20}}\left( {\beta \left( {{a_{13}} + {a_{23}}{\alpha ^ * }} \right) + \gamma \left( {{a_{14}} + {a_{24}}{\alpha ^ * }} \right) } \right) {e^{i{\tau _{20}}{\omega _{20}}}}\).

Next, we employ a computation process similar to Wang et al. (2019), Deng et al. (2014), Wang and Yang (2018) to compute the coefficients as follows,

$$\begin{aligned} {g_{20}} =&2{\tau _{20}}\bar{D}\left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2} + {\alpha ^ * }\left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) } \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}},\\ {g_{11}} =&2{\tau _{20}}\bar{D}\left( {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2} + {\alpha ^ * }\left( {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}} \right) } \right) ,\\ {g_{02}} =&2{\tau _{20}}\bar{D}\left( {{k_{11}}{{\bar{\beta } }^2} + {k_{12}}{{\bar{\gamma } }^2} + {\alpha ^ * }\left( {{k_{21}}{{\bar{\beta } }^2} + {k_{22}}{{\bar{\gamma } }^2}} \right) } \right) {e^{2{\tau _{20}}{\omega _{20}}i}},\\ {g_{21}} =&{\tau _{20}}\bar{D}[{k_{11}}{M_1} + {k_{12}}{M_2} + {k_{13}}{M_3} + {k_{14}}{M_4} \\&+{\alpha ^ * }\left( {{k_{21}}{M_1} + {k_{22}}{M_2} + {k_{23}}{M_3} + {k_{24}}{M_4}} \right) . \end{aligned}$$

where

$$\begin{aligned} {M_1}= & {} 2\beta W_{11}^{(3)}( - 1){e^{ - {\tau _{20}}{\omega _{20}}i}} + \bar{\beta } W_{20}^{(3)}( - 1){e^{{\tau _{20}}{\omega _{20}}i}},\\ {M_2}= & {} 2\gamma W_{11}^{(4)}( - 1){e^{ - {\tau _{20}}{\omega _{20}}i}} + \bar{\gamma } W_{20}^{(4)}( - 1){e^{{\tau _{20}}{\omega _{20}}i}},\\ {M_3}= & {} 3{\beta ^2}\bar{\beta } {e^{ - {\tau _{20}}{\omega _{20}}i}},\\ {M_4}= & {} 3{\gamma ^2}\bar{\gamma } {e^{ - {\tau _{20}}{\omega _{20}}i}}. \end{aligned}$$

The coefficients \({g_{20}},{g_{11}}\) and \({g_{02}}\) can be computed when the parameters and delay are determined in system (1), while the coefficient \({g_{21}}\) cannot be obtained since there are \({W_{11}}(\theta )\) and \({W_{20}}(\theta )\) in \({g_{21}}\). Next, we employ a calculation process similar to Wang et al. (2019), Deng et al. (2014), Wang and Yang (2018) to compute \({W_{11}}(\theta )\) and \({W_{20}}(\theta )\) as follows

$$\begin{aligned} {W_{20}}(\theta )= & {} \frac{{i{g_{20}}}}{{{\tau _{20}}{\omega _{20}}}}q(0){e^{i{\tau _{20}}{\omega _{20}}\theta }} + \frac{{i{{\bar{g}}_{02}}}}{{3{\tau _{20}}{\omega _{20}}}}\bar{q}(0){e^{ - i{\tau _{20}}{\omega _{20}}\theta }} + {E_1}{e^{2i{\tau _{20}}{\omega _{20}}\theta }},\\ {W_{11}}(\theta )= & {} \frac{{ - i{g_{11}}}}{{{\tau _{20}}{\omega _{20}}}}q(0){e^{i{\tau _{20}}{\omega _{20}}\theta }} + \frac{{i{{\bar{g}}_{11}}}}{{{\tau _{20}}{\omega _{20}}}}\bar{q}(0){e^{ - i{\tau _{20}}{\omega _{20}}\theta }} + {E_2}, \end{aligned}$$

where \({E_1} = {\left( {E_1^{(1)},E_1^{(2)},E_1^{(3)},E_1^{\left( 4 \right) }} \right) ^{\mathrm{T}}},{E_2} = {\left( {E_2^{(1)},E_2^{(2)},E_2^{(3)},E_2^{(4)}} \right) ^{\mathrm{T}}}\) can be obtained by the following equations, respectively.

$$\begin{aligned}&\left[ {\begin{array}{cccc} {2i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{14}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad {2i{\omega _{20}} - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{24}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ { - {a_{31}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{44}}} \end{array}} \right] {E_1} = 2\left[ \begin{array}{l} \begin{array}{*{20}{c}} {{R_1}}\\ \begin{array}{l} {R_2}\\ 0 \end{array} \end{array}\\ 0 \end{array} \right] ,\\&\left[ {\begin{array}{cccc} {{a_{11}}}&{}\quad 0&{}\quad {{a_{13}}}&{}\quad {{a_{14}}}\\ 0&{}\quad {{a_{22}}}&{}\quad {{a_{23}}}&{}\quad {{a_{24}}}\\ {{a_{31}}}&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right] {E_2} = - 2\left[ {\begin{array}{c} {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2}}\\ {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}}\\ 0\\ 0 \end{array}} \right] . \end{aligned}$$

where \({R_1} = \left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}\), \({R_2} = \left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}\).

Thus, we obtain

$$\begin{aligned} E_1^{(1)} =&\frac{2}{E}\left| {\begin{array}{cccc} {\left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{14}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ {\left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad {2i{\omega _{20}} - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{24}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{44}}} \end{array}} \right| ,\\ E_1^{(2)} =&\frac{2}{E}\left| {\begin{array}{cccc} {2i{\omega _{20}} - {a_{11}}}&{}\quad {\left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{13}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{14}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad {\left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{23}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{24}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ { - {a_{31}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{44}}} \end{array}} \right| ,\\ E_1^{(3)} =&\frac{2}{E}\left| {\begin{array}{cccc} {2i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad {\left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{14}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad {2i{\omega _{20}} - {a_{22}}}&{}\quad {\left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{24}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ { - {a_{31}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{44}}} \end{array}} \right| ,\\ E_1^{(4)} =&\frac{2}{E}\left| {\begin{array}{cccc} {2i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad {\left( {{k_{11}}{\beta ^2} + {k_{12}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad {2i{\omega _{20}} - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad {\left( {{k_{21}}{\beta ^2} + {k_{22}}{\gamma ^2}} \right) {e^{ - 2{\tau _{20}}{\omega _{20}}i}}}\\ { - {a_{31}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad 0 \end{array}} \right| . \end{aligned}$$

where

$$\begin{aligned} E = \left| {\begin{array}{cccc} {2i{\omega _{20}} - {a_{11}}}&{}\quad 0&{}\quad { - {a_{13}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{14}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ 0&{}\quad {2i{\omega _{20}} - {a_{22}}}&{}\quad { - {a_{23}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}&{}\quad { - {a_{24}}{e^{ - {\tau _{20}}{\omega _{20}}i}}}\\ { - {a_{31}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{33}}}&{}\quad 0\\ 0&{}\quad { - {a_{42}}{e^{ - {\tau _1}{\omega _{20}}i}}}&{}\quad 0&{}\quad {2i{\omega _{20}} - {a_{44}}} \end{array}} \right| . \end{aligned}$$

Similarly, we get

$$\begin{aligned} E_2^{(1)} =&\frac{{ - 2}}{F}\left| {\begin{array}{cccc} {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2}}&{}\quad 0&{}\quad {{a_{13}}}&{}\quad {{a_{14}}}\\ {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}}&{}\quad {{a_{22}}}&{}\quad {{a_{23}}}&{}\quad {{a_{24}}}\\ 0&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right| ,\\ E_2^{(2)} =&\frac{{ - 2}}{F}\left| {\begin{array}{cccc} {{a_{11}}}&{}\quad {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2}}&{}\quad {{a_{13}}}&{}\quad {{a_{14}}}\\ 0&{}\quad {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}}&{}\quad {{a_{23}}}&{}\quad {{a_{24}}}\\ {{a_{31}}}&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad 0&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right| ,\\ E_2^{(3)} =&\frac{{ - 2}}{F}\left| {\begin{array}{cccc} {{a_{11}}}&{}\quad 0&{}\quad {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2}}&{}\quad {{a_{14}}}\\ 0&{}\quad {{a_{22}}}&{}\quad {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}}&{}\quad {{a_{24}}}\\ {{a_{31}}}&{}\quad 0&{}\quad 0&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right| ,\\ E_2^{(4)} =&\frac{{ - 2}}{F}\left| {\begin{array}{cccc} {{a_{11}}}&{}\quad 0&{}\quad {{a_{13}}}&{}\quad {{k_{11}}{{\left| \beta \right| }^2} + {k_{12}}{{\left| \gamma \right| }^2}}\\ 0&{}\quad {{a_{22}}}&{}\quad {{a_{23}}}&{}\quad {{k_{21}}{{\left| \beta \right| }^2} + {k_{22}}{{\left| \gamma \right| }^2}}\\ {{a_{31}}}&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad 0 \end{array}} \right| , \end{aligned}$$

where

$$\begin{aligned} F = \left| {\begin{array}{cccc} {{a_{11}}}&{}\quad 0&{}\quad {{a_{13}}}&{}\quad {{a_{14}}}\\ 0&{}\quad {{a_{22}}}&{}\quad {{a_{23}}}&{}\quad {{a_{24}}}\\ {{a_{31}}}&{}\quad 0&{}\quad {{a_{33}}}&{}\quad 0\\ 0&{}\quad {{a_{42}}}&{}\quad 0&{}\quad {{a_{44}}} \end{array}} \right| . \end{aligned}$$

Hence, we can compute \({g_{21}}\). Further, we have the following quantities:

$$\begin{aligned} \left\{ \begin{array}{l} {c_1}(0) = \frac{i}{{2{\tau _{20}}{\omega _{20}}}}\left( {g_{11}}{g_{20}} - 2{\left| {{g_{11}}} \right| ^2} - \frac{{{{\left| {{g_{02}}} \right| }^2}}}{3}\right) + \frac{{{g_{21}}}}{2}, \\ {\mu _2} = - {\frac{{\text {Re}}\, \{{c_1}(0)\}}{{{\text {Re}}\, \{ \lambda '({\tau _{20}})\}}}}, \\ {\beta _2} = 2{\text {Re}}\, \{{c_1}(0)\}, \\ {T_2} = - \frac{{{\text {Im}}\, \{ {c_1}(0)\} + {\mu _2}{\text {Im}}\, \{ \lambda '({\tau _{20}})\} }}{{{\tau _{20}}{\omega _{20}}}}. \\ \end{array} \right. \end{aligned}$$

In summary, we have the following results.

Theorem 4

For system (1), the following results hold.

  1. (i)

    The sign of \({\mu _2}\) determines the direction of Hopf bifurcation: Hopf bifurcation is supercritical (subcritical)if \({\mu _2} > 0({\mu _2} < 0)\);

  2. (ii)

    The sign of \({\beta _2}\) determines the stability of the bifurcating periodic solutions: The periodic solutions are orbitally stable (unstable) if \({\beta _2} < 0~({\beta _2} > 0)\);

  3. (iii)

    The sign of \({T_2}\) determines the period of bifurcating periodic solutions: The period increases (decreases) if \({T_2} > 0~({T_2} < 0)\).

4 Numerical Simulations

In this section, numerical simulations are given to illustrate the analytical results we obtained. We chose parameter \({r_1} = 1,{r_2} = 1,{c_1} = 2,{c_2} = 2,s = 0.5,a = 0.5,b = 0.5,{d_1} = 2,{d_2} = 2,n = 3,{a_1} = 1,{b_{21}} = 1\) Xi and Turcotte (2015); then, system (1) becomes:

$$\begin{aligned} \left\{ \begin{array}{l} {{ M'}_1}\left( t \right) = - {M_1}\left( t \right) + 0.5\frac{{P_1^3\left( {t - {\tau _2}} \right) }}{{{{0.5}^3} + P_1^3\left( {t - {\tau _2}} \right) }} + 0.5\frac{{{{0.5}^3}}}{{{{0.5}^3} + P_2^3\left( {t - {\tau _2}} \right) }}, \\ {{ M'}_2}\left( t \right) = - {M_2}\left( t \right) + 0.5\frac{{P_2^3\left( {t - {\tau _2}} \right) }}{{{{0.5}^3} + P_2^3\left( {t - {\tau _2}} \right) }} + 0.5\frac{{{{0.5}^3}}}{{{{0.5}^3} + P_1^3\left( {t - {\tau _2}} \right) }}, \\ {{P'}_1}\left( t \right) = - 2{P_1}\left( t \right) + 2{M_1}\left( {t - {\tau _1}} \right) , \\ {{P'}_2}\left( t \right) = - 2{P_2}\left( t \right) + 2{M_2}\left( {t - {\tau _1}} \right) . \\ \end{array} \right. \end{aligned}$$
(27)

By computation, it can be concluded that system (27) has three positive equilibrium points \(E_{1}\), \(E_{2}\) and \(E_{3}\), where \({E_1}\left( {0.931827,0.068173,0.931827,0.068173} \right) \) and \({E_3}\left( {0.068173,0.931827,0.068173,0.931827} \right) \) are stable equilibrium points, and \({E_2}\left( {0.5,0.5,0.5,0.5} \right) \) is a saddle point. System (27) will have bistable behavior. Which stable point of system (27) tends to depend on the initial values. For \({\tau _1} = {\tau _2} = \tau \ne 0\), we obtain \({\tau _0} \approx 9.5\), the positive equilibrium points \({E_3}\) lose stability and a Hopf bifurcation occurs when the time delay \(\tau \) passes through the critical value \({\tau _0}\). By Theorem 2, we know system (27) is stable when \(0 \le \tau < {\tau _0}\), and unstable for \(\tau > {\tau _0}\). The corresponding time cources diagram and phase portrait are depicted in Fig. 2.

Fig. 2
figure 2

Time cources diagram of \({P_1}\) and 3D phase portrait of system (27) with \(\tau {{ = 1}}0\). a \(\tau = 0\); b \(\tau {{ = 3}}\); c \(\tau {{ = 10}}\); d the delayed track oscillates between fixed point 1 and 2 that, under no-delay conditions, would be stable. Initial conditions: \({M_1}\left( 0 \right) {{ = 0}}.1,{M_2}\left( 0 \right) {{ = 0}}.35,{P_1}\left( 0 \right) = 0.4,{P_2}\left( 0 \right) = 0.3\)

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When the initial value is selected near the saddle point \({E_2}\),there is no periodic solution. The stability of system (27) is maintained with time delay \(\tau \) increasing (see Fig. 3).

Fig. 3
figure 3

Time cources diagram of \({P_1}\) with different time delay \(\tau \). a \(\tau = 0\); b \(\tau = 10\); c \(\tau = 100\). Initial conditions: \({M_1}(0) = 0.4,{M_2}(0) = 0.4,{P_1}(0) =0.4,{P_2}(0) = 0.4\)

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By computation, we can also obtain \({\tau _1} = {\tau _2} = {\tau _0} \approx 11.5\), and a Hopf bifurcation occurs at the positive equilibrium point \({E_1}\) for \(\tau = {\tau _0}\). By Theorem 2, we know that the positive equilibrium point \({E_1}\) of system (27) is stable when \(0 \le \tau < {\tau _0}\) and undergoes a Hopf bifurcation at \(\tau = {\tau _0} \approx 11.5\) (see Fig. 4).

Fig. 4
figure 4

Time cources diagram of \({P_1}\) with different time delay \(\tau \). a \(\tau =0\); b \(\tau = 5\); c \(\tau = 12\). Initial conditions: \({M_1}(0) = 0.3,{M_2}(0)= 0.35,{P_1}(0) = 0.4,{P_2}(0) = 0.3\)

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Time delay can transform bistable behavior into oscillatory behavior and plays important roles in this model. Through numerical simulation, we can also draw the conclusion that the initial value and time delay both affect the dynamic behavior of system (27). With different initial values, even if the time delay is the same (\({\tau _1} = {\tau _2} = 67\)), the trajectory of the solution of system (27) is different (see Fig. 5).

Fig. 5
figure 5

Time cources diagram of \({P_1}\) with different initial conditions. a \({M_1}(0) = 1,{M_2}(0) = 1,{P_1}(0) =1,{P_1}(0) =1\), b \({M_1}(0)= 0.1,{M_2}(0) = 1,{P_1}(0) = 1,{P_2}(0) = 1\)

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5 The Effect of Biological Noise

Biological systems are not purely deterministic as they comprise significant molecular noise as an often not just present but, in fact, essential component to establishing phenotype; see the general review Tsimring and Lev (2014) and Çağatay et al. (2009), Xi et al. (2013), Xi et al. (2013) for specific examples in bacteria. Here, by molecular noise, we understand biological random fluctuations of molecular components entirely due to paucity in numbers of some (or all) key species that affect the overall behavior of the system. Hence we were curious to see how our theoretical finding of the time delay-induced bifurcation to oscillations could be affected by the presence of molecular noise.

So, to this end, we used the well-traveled route provided by the Gillespie algorithm (Gillespie 1976) to translate the delayed four-dimensional deterministic model (equation #1) into a set of discrete event reactions. The detailed implementation of the stochastic algorithm for the instantaneous (no-delay) version of this system is given in Xi and Turcotte (2015). Briefly, the system was separated in a set of production and degradation reactions; each reaction corresponds to the individual production terms (positive terms) and degradation terms (negative terms) in the four differential equation system. Thus, in all, the stochastic algorithm comprises four production reactions and four degradation reactions. Throughout the course of a stochastic simulation, reactions are individually fired according to their time-evolving propensities using the Gillespie algorithm. Stochastic simulations thus track the varying numbers of all species in the system as a function of time as discrete event reactions individually fire. Finally, the time delay is implemented by straightforward lookup into the running history of the simulation.

Figure 6 shows the result of two extended time-delayed stochastic simulations (purple and red trajectories) overlapped onto the time-delayed integration of the 4 ODE, or 4D deterministic system (cyan line). For guidance, we also show the instantaneous 2 D nullclines obtained from setting \(\hbox {dmRNA1}/\mathrm{d}t = \hbox {dmRNA2}/\mathrm{d}t = 0\) (solid red and green lines), thus collapsing the four-dimensional system into a two-dimensional system by setting the messenger RNA sub-manifold at rest. We also show the 4D nullcline constructions (red and green circles overlapping the 2D nullclines). Actual methods on how these are computed can be found in Xi and Turcotte (2015) and would not be repeated here for brevity. The intersections of the two 2D nullclines define the locus of the 2-dimensional instantaneous system being a rest on the phase plane (\(\mathrm{d}X_{1}/\mathrm{d}t = \mathrm{d}X_{2}/\mathrm{d}t = 0\)). These three fixed points are labeled #1, #2 and #3. From Xi and Turcotte (2015), points #1 and #3 are stable nodes, and point #2, the middle point, is a saddle node. These fixed points are deterministic delay-free fixed points of the dynamics that we use to pin and contrast the discussion of the delayed stochastic system behavior.

First we notice that the deterministic time-delayed four-dimensional system track projected on the \(X_{1}\) and \(X_{2}\) plane (the cyan line) roughly oscillates between points #1 and #3. This is consistent with our earlier findings that time delay bifurcates an otherwise bistable system into an oscillatory one. The direction of rotation is counterclockwise.

Turning our attention to stochastic effects, the purple stochastic track was started at the same location on the phase plane where the deterministic track (cyan trajectory) originates. We adjusted the level of noise in the system by setting the total number of molecules per unit of concentration in the system to \(\Omega =10{,}000\). \(\Omega \) actually has units of volume and is thus a convenient way to dial-in a desired noise level based on how big the sample is, while maintaining the concentrations the same. An invariant concentration is obtained by scaling both the number of molecules \(N_{\mathrm{molec}}\) and the volume \(V_{\mathrm{system}}\) of the system by the same factor, so the ratio \(N_{\mathrm{molec}}/V_{\mathrm{system}}\) does not change. While concentrations are invariant under any \(\Omega \), the random fluctuations on these concentrations are expected to vary approximately as 1/sqrt(\(N_{\mathrm{molec}}\)). For more details the reader is referred to Tsimring and Lev (2014) and to the supplement of Çağatay et al. (2009). So, for \(\Omega \) increasing, the size of the fluctuations will decrease. Because the number of molecules per unit concentration in the system is high, but not overly high, the observed fluctuations are significant, but they not overwhelming the dynamics; see Fig. 6.

Fig. 6
figure 6

Stochastic behavior. Green and red solid lines are nullclines of the instantaneous (no-delay) deterministic system, as explained in the text. Points #1, #2 and #3 are fixed points of the instantaneous system: stable, saddle and stable, respectively. The cyan line depicts the time-delayed four-dimensional system integration. The direction of rotation is counterclockwise. The purple trajectory is a delayed stochastic simulation initiated at the same starting location as the deterministic simulation (cyan line). The red trajectory is another delayed stochastic simulation, and this one initiated within the basin of attraction of point #3. Initially the red trajectory moves in the direction from 3 to 1 (as seen on the upper branch of the counterclockwise loop) but it soon hops to the lower branch, and progresses in the direction from 1 to 3. Both stochastic trajectories become trapped in the vicinity of their respective destinations (point #1 and point #3) despite the deterministic delayed systems being oscillatory (cyan line). Stochastic simulations are run with \(\Omega =10{,}000\) imparting modest noise to the system (Color figure online)

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The main points conveyed in Fig. 6 are twofold. Firstly, the stochastic simulation, particularly in the early phase of the purple track where the number of molecules is high, closely follows the deterministic track, even in regions of the phase space where the shape of the deterministic parent trajectory is complex. But much more importantly, secondly, in the neighborhood of point #1, we observe the purple track dwelling indefinitely by essentially hopping back and forth from an inward going direction to an outward going direction to point #1. We can understand how this behavior is happening because, in this region of phase space near point #1, the inward and outward flow directions are very close to each other (Fig. 6); in the pinching part of the looping track. Effectively, in vicinity of point 1, the stochastic system hops back and forth in direction inward and outward to point #1. In short, the noisy system is dynamically and stochastically trapped to the neighborhood of point 1.

The second stochastic simulation, the red track, was started closer to and in the basin of attraction of and above point #3. We see that initially this trajectory actually moves away from point #3, settling onto the outbound flow of the oscillating trajectory (upper branch of the loop). But, eventually and randomly, the stochastic system jumps back over onto the deterministic flow (lower branch of the loop) that propel deterministic tracks to move the system toward point #3. As the stochastic simulation progresses further, the inward and outward flows become closer and closer to each other on the phase plane, similar to the case of the neighborhood of point #1, described above. Hence, again, the system essentially dwells in the neighborhood of the fixed point, as it hops back and forth in direction inward and outward of point #3. Hence, the noisy system is again dynamically and stochastically trapped, but this time, to the neighborhood of point 3.

Figure 6 thus demonstrates a key and unexpected behavior imparted by molecular noise: Whereas the delayed deterministic system oscillates beyond the Hopf as shown described earlier, the equivalent stochastic delayed system will nevertheless remain bistable because noise effectively imparts a stabilizing effect through the mechanism explained above. In short, interestingly, molecular noise actually stabilizes an otherwise oscillating system. To our knowledge, the description of this behavior is original.

We have also numerically studied two limits: \(\Omega = \hbox {large}\)(100,000) and \(\Omega =\hbox {small}\)(100). In the small \(\Omega \) limit, the system is quickly overwhelmed by noise (the number of particles in the system is small) and much of the interesting dynamics is blurred away. In the large \(\Omega \) system, we have verified that the stochastic system’s behavior, as it approaches the so-called thermodynamic limit (Gillespie 2007), as expected more and more approximates that of the deterministic system. Large \(\Omega \) simulations require many thousands of hours to perform. Data are omitted for brevity.

6 Characterization of the Orbit

In this section, we briefly consider the effect of nonlinear terms in the normal form of the orbit equation. Following the lead of Molnár et al. (2016), we see that the stability and amplitude of a limit cycle born at the Hopf bifurcation depend on the cubic term in the expansion, but further away from the bifurcation, the possibility that higher-order terms (quintic,\(\ldots \)) would modify the amplitude and/or would even contribute new dynamical behavior such as a fold cannot be excluded; it depends on the details of the problem (Molnár et al. 2016, Molnár 2020). Casting the problem into normal form is beyond the scope of this paper. So in order to study this nonlinear dynamical aspect, and in order to determine its potential impact on our analysis, we generated multiple time curves of the oscillations for \(\tau \) just past the Hopf bifurcation and further away from it, both for short and essentially extremely large time intervals. In all cases, the attained amplitude of oscillations and frequency is the same. The data are not included here, for the sake of saving space, but it is available for perusal if needed. We therefore conclude that, for the problem we studied, such possible buildup of nonlinear effects has no impact.

7 Discussion and Conclusions

In the first part of this paper, we study the effect of two time delays on a mutually inhibitory core gene regulation circuit. In the case of instantaneous regulation (no-delay), we expect the dynamics to be bistable. Since delays are inherent to the spatial and mechanistic regulation of cells, we focus on the influence of time delays in five special cases and thus obtain the conditions for the stability and the bifurcation of system (1) to oscillations. We derive a formula to determine the direction of the Hopf bifurcation and the stability of the periodic solutions by applying the normal form method and the center manifold theorem. In the absence of time delays, bistable behavior occurs in system (1), so the selection of initial values is very important. A separatrix passing through the middle saddle point divides the respective basins of attraction of the two stable fixed points on either side. By adding time delay to the dynamics of the regulation, the bistable behavior can be transformed into oscillatory behavior. Delay and initial values combine to affect the dynamic behavior of the system. Thus, complex dynamic behavior appears. Numerical simulations are performed to illustrate and confirm the analytical results.

In the second part of the paper, we turned out attention to the effect of biological noise on the system. In many cases, noise could be naively expected to destabilize an otherwise stable system. But our main point in the deterministic analysis was that delays induce oscillations via Hopf bifurcation in an otherwise bistable system. The delayed system is already “unstable” through oscillations. So, what then could the effect of unavoidable biological noise on the now oscillatory system be? Somewhat un-intuitively, our numerical simulations, based on the robust Gillespie algorithm augmented to account for delays, showed that the fluctuations actually stabilize the otherwise oscillatory system. By carefully studying the developing stochastic histories, we established the mechanism for this. In fact, near destabilized fixed points of the dynamics, noise allows the delayed stochastic system to essentially hop from inward going flow to outward flow and back. Thus, noise actually re-stabilizes a de-stabilized point. Hence, while the delayed deterministic dynamics bifurcates to the present oscillatory behavior, the stochastic dynamics acts to recover bistability, in effect, negating the effects of delays.

Delays are inherent to transport and other mechanisms in the cell and are thus not only un-avoidable, but because they are part of how biological systems work, they cannot reasonably be ignored. The same is also true about fluctuations in the concentrations of molecular regulators of living systems. In this work we chose perhaps the simplest and best example there can be of a meaningful core biological control circuit—the “bistable switch”—to first demonstrate its inherent fragility to time delays under a deterministic analysis. Bistability is a dynamic feature that enables commitment to one of two possible states. But here, we explicitly show that time delays impact the behavior of the system by breaking this commitment and permitting switching back and forth between otherwise stable states. Further, and perhaps even more importantly, we show that unavoidable biological noise somewhat unintuitively restores the bistable behavior of the system.

In further work, we plan to study in more details the efficiency of the noise stabilization process for it cannot be excluded that noise may allow, with some probability, occasional exit from the re-stabilized point. And clearly the size of the noise will have an impact. We also plan to explore the impact of the number of bifurcations and of another unavoidable feature of realistic biological circuits, timescale separation, in this and particularly in other progressively more complex systems. And of course, propelled by the results we obtained in this analysis, we plan to continue the study of how biological noise impacts dynamical behavior. We expect this work to continue helping shedding light on how and particularly why Evolution selects certain gene regulation circuit topologies over others.