The kinetics in mathematical models on segmentation clock genes in zebrafish

Abstract

Somitogenesis is the process for the development of somites in vertebrate embryos. This process is timely regulated by synchronous oscillatory expression of the segmentation clock genes. Mathematical models expressed by delay equations or ODEs have been proposed to depict the kinetics of these genes in interacting cells. Through mathematical analysis, we investigate the parameter regimes for synchronous oscillations and oscillation-arrested in an ODE model and a model with transcriptional and translational delays, both with Michaelis–Menten type degradations. Comparisons between these regimes for the two models are made. The delay model has larger capacity to accommodate synchronous oscillations. Based on the analysis and numerical computations extended from the analysis, we explore how the periods and amplitudes of the oscillations vary with the degradation rates, synthesis rates, and coupling strength. For typical parameter values, the period and amplitude increase as some synthesis rate or the coupling strength increases in the ODE model. Such variational properties of oscillations depend also on the magnitudes of time delays in delay model. We also illustrate the difference between the dynamics in systems modeled with linear degradation and the ones in systems with Michaelis–Menten type reactions for the degradation. The chief concerns are the connections between the dynamics in these models and the mechanism for the segmentation clocks, and the pertinence of mathematical modeling on somitogenesis in zebrafish.

Appendices

Appendix 1: Proof of Theorem 2

Under the conditions of the theorem, the characteristic equation (30) has a pair of complex roots $R(\mu )\pm iI(\mu )$ for $\mu$ near $\mu ^*$. At $\mu = \mu ^*$, this pair becomes purely imaginary, and the other roots have negative real parts, i.e.,

$$\begin{aligned} R(\mu ^{*})=0, ~I(\mu ^{*})>0, \end{aligned}$$

according to Proposition 6. Substituting $\lambda (\mu )=R(\mu )+iI(\mu )$ into $\varDelta _{+}$, and collecting the real and imaginary parts, we have

$$\begin{aligned}&R^4 +\alpha _1R^3 +\alpha _2R^2 +\alpha _3 R +\alpha _4^{+} - (6R^2 +3\alpha _1R +\alpha _2)I^2 +I^4 =0, \end{aligned}$$

(53)

$$\begin{aligned}&I[-(4 R+\alpha _1)I^2 + 4R^3 +3\alpha _1 R^2 +2\alpha _2R +\alpha _3] = 0, \end{aligned}$$

(54)

where $R=R(\mu )$, $I=I(\mu )$. If (54) has a solution with $I(\mu )\ne 0$, then

$$\begin{aligned} -(4R+\alpha _1)I^2 + 4R^3 +3\alpha _1R^2 +2\alpha _2R +\alpha _3=0. \end{aligned}$$

Thus,

$$\begin{aligned} I^2 = \frac{4R^3 +3\alpha _1 R^2 +2\alpha _2 R +\alpha _3}{4R+\alpha _1}. \end{aligned}$$

(55)

Substituting (55) into (53), using $\alpha _1(\mu ^{*})\alpha _2(\mu ^{*})\alpha _3(\mu ^{*})-\alpha _3^2(\mu ^{*})-\alpha _1^2(\mu ^{*})\alpha _4^{+}(\mu ^{*})=0$, differentiating with respect to $\mu$, and utilizing $R(\mu ^{*})=0$ and $I(\mu ^{*})\ne 0$, we obtain

$$\begin{aligned} \frac{d R}{d\mu }(\mu ^{*})=\frac{\frac{d}{d\mu }[\alpha _1(\mu )\alpha _2(\mu )\alpha _3(\mu )-\alpha _3^2(\mu )-\alpha _1^2(\mu )\alpha _4^{+}(\mu )]\vert _{\mu =\mu ^{*}}}{ -2\alpha _1(\mu ^{*})[\alpha _1(\mu ^{*})\alpha _3(\mu ^{*})+(\frac{2\alpha _3(\mu ^{*})}{\alpha _1(\mu ^{*})}-\alpha _2(\mu ^{*}))^2]} \ne 0, \end{aligned}$$

by the assumption of the theorem. Thus the transversality condition is met. We conclude that system (8) undergoes a Hopf bifurcation at $\bar{\mathbb {X}}$ and $\mu =\mu ^{*}$. $\square$

Appendix 2: Stability of bifurcating periodic solution, ODE case

We express system (26) into the form:

$$\begin{aligned} \dot{\mathbb {X}} = A(\mu )\mathbb {X} + \mathbf{f}(\mathbb {X},\mu ), \end{aligned}$$

where $\mathbb {X}=(x_1,\ldots ,x_4,y_1,\ldots ,y_4)$, $A(\mu )$ is the linear part, and $\mathbf{f}=(f_1,\ldots ,f_8)$ is the nonlinear term. At $\mu =\mu ^{*}$, $A(\mu ^*)$ has a pair of purely imaginary eigenvalues $\lambda (\mu ^{*})=\pm i\omega _0$. Let us make a transformation so that the linear part is in normal form. We consider the change of variables $\mathbb {X}=P\mathbb {Z}$, where

$$\begin{aligned} P= \left[ \begin{array}{ccccc} ~\mathrm{Im}(\mathbf{u})&\mathrm{Re}(\mathbf{u})&\mathbf{v}_3&\cdots&\mathbf{v}_8 \end{array} \right] _{8 \times 8}, \end{aligned}$$

and $\mathbf{u}\in \mathbb {R}^8$ is the eigenvector of $A(\mu ^{*})$ corresponding to the eigenvalue $i\omega _0$ and $\mathbf{v}_3,\ldots ,\mathbf{v}_8$ are the generalized eigenvectors for the remaining eigenvalues. Thus

$$\begin{aligned} P^{-1}A(\mu ^{*})P = \left[ \begin{array}{ccc} ~0 &{} -\omega _0 &{} \vdots ~ \\ ~\omega _0 &{} 0 &{} \vdots ~ \\ ~\cdots &{} \cdots &{} D \end{array} \right] , \end{aligned}$$

(56)

where D is an $6\times 6$ matrix. Then the transformed system becomes

$$\begin{aligned} \dot{\mathbb {Z}}=P^{-1}A(\mu )P\mathbb {Z} + \mathbf{F}(\mathbb {Z},\mu ), \end{aligned}$$

where $\mathbb {Z}=(z_1,\ldots , z_8)$ and $\mathbf{F}(\mathbb {Z},\mu )=P^{-1}{} \mathbf{f}(P\mathbb {Z},\mu )$ with $\mathbf{F}=(F_1,\ldots ,F_8)$. At $\mu =\mu ^{*}$, $\mathbb {Z}=\mathbf{0}$, we define

$$\begin{aligned} g_{11}= & {} \frac{1}{4}\left[ \left( \frac{\partial ^2 F_1}{\partial z_1^2}+\frac{\partial ^2 F_1}{\partial z_2^2}\right) +i\left( \frac{\partial ^2 F_2}{\partial z_1^2}+\frac{\partial ^2 F_2}{\partial z_2^2}\right) \right] , \\ g_{02}= & {} \frac{1}{4}\left[ \left( \frac{\partial ^2 F_1}{\partial z_1^2}-\frac{\partial ^2 F_1}{\partial z_2^2}-2\frac{\partial ^2 F_2}{\partial z_1 \partial z_2}\right) +i\left( \frac{\partial ^2 F_2}{\partial z_1^2}-\frac{\partial ^2 F_2}{\partial z_2^2}+2\frac{\partial ^2 F_1}{\partial z_1 \partial z_2}\right) \right] , \\ g_{20}= & {} \frac{1}{4}\left[ \left( \frac{\partial ^2 F_1}{\partial z_1^2}-\frac{\partial ^2 F_1}{\partial z_2^2}+2\frac{\partial ^2 F_2}{\partial z_1 \partial z_2}\right) +i\left( \frac{\partial ^2 F_2}{\partial z_1^2}-\frac{\partial ^2 F_2}{\partial z_2^2}-2\frac{\partial ^2 F_1}{\partial z_1 \partial z_2}\right) \right] , \\ G_{21}= & {} \frac{1}{8}\left[ \left( \frac{\partial ^3 F_1}{\partial z_1^3}+\frac{\partial ^3 F_1}{\partial z_1 \partial z_2^2}+\frac{\partial ^3 F_2}{\partial z_1^2 \partial z_2}+\frac{\partial ^3 F_2}{\partial z_2^3}\right) \right. \\&\left. +\,i\left( \frac{\partial ^3 F_2}{\partial z_1^3}+\frac{\partial ^3 F_2}{\partial z_1 \partial z_2^2}-\frac{\partial ^3 F_1}{\partial z_1^2 \partial z_2}-\frac{\partial ^3 F_1}{\partial z_2^3}\right) \right] . \end{aligned}$$

Next, for $k=3,\ldots ,8$, we set

$$\begin{aligned} h^{k-2}_{11}= & {} \frac{1}{4}\left( \frac{\partial ^2 F_k}{\partial z_1^2} + \frac{\partial ^2 F_k}{\partial z_2^2}\right) , \\ h^{k-2}_{20}= & {} \frac{1}{4}\left[ \left( \frac{\partial ^2 F_k}{\partial z_1^2} - \frac{\partial ^2 F_k}{\partial z_2^2}\right) -2i\left( \frac{\partial ^2 F_k}{\partial z_1 \partial z_2}\right) \right] , \end{aligned}$$

and let $w^{k-2}_{11}$, $w^{k-2}_{20} \in {\mathbb C}^{6}$ be the solutions of

$$\begin{aligned} Dw^{k-2}_{11}=-h^{k-2}_{11},~~(D-2i\omega _0I)w^{k-2}_{20}=-h^{k-2}_{20}, \end{aligned}$$

where D is defined in (56). For $k=3,\ldots , 8$, let

$$\begin{aligned} G^{k-2}_{110}= & {} \frac{1}{2}\left[ \left( \frac{\partial ^2 F_1}{\partial z_1 \partial z_k} + \frac{\partial ^2 F_2}{\partial z_2 \partial z_k}\right) +i\left( \frac{\partial ^2 F_2}{\partial z_1 \partial z_k} - \frac{\partial ^2 F_1}{\partial z_2 \partial z_k}\right) \right] , \\ G^{k-2}_{101}= & {} \frac{1}{2}\left[ \left( \frac{\partial ^2 F_1}{\partial z_1 \partial z_k} - \frac{\partial ^2 F_2}{\partial z_2 \partial z_k}\right) +i\left( \frac{\partial ^2 F_2}{\partial z_1 \partial z_k} + \frac{\partial ^2 F_1}{\partial z_2 \partial z_k}\right) \right] . \end{aligned}$$

Then we define

$$\begin{aligned} g_{21} = G_{21} + \sum ^{6}_{k=1} (2G^k_{110}w^k_{11} + G^k_{101}w^k_{20}). \end{aligned}$$

Appendix 3: Sketch of steps (I)–(III) in Sect. 4.2

Step (I): For any fixed $\tau _4 \ge 0$, we substitute $\lambda =i\omega$ with $\omega >0$ into the characteristic equation (42), i.e., $\triangle _{\pm }(i\omega ,r,\tau _4)=\tilde{R}_{\pm }(i\omega ,r,\tau _4)+i \tilde{I}_{\pm }(i\omega ,r,\tau _4)=0,$ where the real and imaginary parts are respectively

$$\begin{aligned} \left\{ \begin{array}{lll} \tilde{R}_{\pm }(i\omega ,r,\tau _4)&{}=&{}\omega ^4-\beta _{2}\omega ^{2}+\beta _{4}+ \nu _{3}\nu _{5}\gamma _{1}w \sin {(r \omega )}+d_{4}\nu _{3}\nu _{5}\gamma _{1} \cos {(r \omega )}\\ &{} &{} \pm \nu _{3}\nu _{5}\gamma _{2}\gamma _{3} \cos {((r+\tau _4)\omega )}=0\\ \tilde{I}_{\pm }(i\omega ,r,\tau _4)&{}=&{} -\beta _{1}\omega ^{3}+\beta _{3}\omega +\nu _{3}\nu _{5}\gamma _{1}\omega \cos {(r \omega )}-d_{4}\nu _{3}\nu _{5}\gamma _{1}\sin {(r \omega )}\\ &{} &{} \mp \nu _{3}\nu _{5}\gamma _{2}\gamma _{3}\sin {((r+\tau _4) \omega )}=0. \end{array}\right. \end{aligned}$$

(57)

By the properties of trigonometric functions, (57) can be written as

$$\begin{aligned} \left\{ \begin{array}{l} \sqrt{L_{\pm }(\omega )}\cdot \sin {(\phi _{\pm }+r\omega )}= -\omega ^{4}+\beta _{2}\omega ^{2}-\beta _{4}\\ \sqrt{L_{\pm }(\omega )} \cdot \cos {(\phi _{\pm }+r \omega )} = \beta _{1}\omega ^{3}-\beta _{3}\omega , \end{array}\right. \end{aligned}$$

(58)

where $L_{\pm }(\omega ):=[\nu _{3}\nu _{5}(d_{4}\gamma _{1}\pm \gamma _{2}\gamma _{3} \cos {(\tau _4 \omega )})]^2+[\nu _{3}\nu _{5}(\gamma _{1} \omega \mp \gamma _{2}\gamma _{3} \sin {(\tau _4 \omega )})]^{2} >0$, if $\omega$ is a solution of (58), and $\phi _{\pm }\in [0, 2\pi )$ with

$$\begin{aligned} \begin{array}{l} \sin {(\phi _{\pm })}=\nu _{3}\nu _{5}[d_{4}\gamma _{1}\pm \gamma _{2}\gamma _{3} \cos {(\tau _4 \omega )}]/\sqrt{L_{\pm }(\omega )},\\ \nonumber \cos {(\phi _{\pm })}=\nu _{3}\nu _{5}[\gamma _{1}w \mp \gamma _{2}\gamma _{3} \sin {(\tau _4 \omega )}]/\sqrt{L_{\pm }(\omega )}.\nonumber \end{array} \end{aligned}$$

In order to find solution $\omega$ to (58), we sum up the square of equations (58) and obtain

$$\begin{aligned} Q_{\pm }(\omega )=\nu _{3}^{2}\nu _{5}^{2}(d_{4}^{2}\gamma _{1}^{2}+\gamma _{2}^{2}\gamma _{3}^{2})=:\varGamma , \end{aligned}$$

where

$$\begin{aligned} Q_{\pm }(\omega )= & {} \omega ^{8}+(\beta _{1}^{2}-2\beta _{2})\omega ^{6}+ (\beta _{2}^{2}-2 \beta _{1}\beta _{3}+2\beta _{4})\omega ^{4}\\&\quad + (\beta _{3}^{2}-2\beta _{2}\beta _{4}-\nu _{3}^{2} \nu _{5}^{2}\gamma _{1}^{2})\omega ^{2}\\&\quad \pm 2 \nu _{3}^{2}\nu _{5}^{2} \gamma _{1}\gamma _{2}\gamma _{3} \sin {(\tau _4 \omega )}\omega + \beta _{4}^{2}\mp 2 d_{4}\nu _{3}^{2}\nu _{5}^{2}\gamma _{1}\gamma _{2}\gamma _{3} \cos {(\tau _4 \omega )} . \end{aligned}$$

Since $Q_{\pm }(0)=\beta _{4}^{2}\mp 2 d_{4}\nu _{3}^{2}\nu _{5}^{2}\gamma _{1}\gamma _{2}\gamma _{3}$ and both $Q_{+}(\omega )$ and $Q_{-}(\omega )$ are strictly increasing eventually and blow up as $\omega \rightarrow \infty$, we thus establish Proposition 7.

Proposition 7 guarantees the existence of the minimal bifurcation value $r_c$ and its corresponding purely imaginary eigenvalue $i \omega _c$. In the sequel, we denote by $\omega _+$ (resp., $\omega _-$) a positive solution to $Q_{+}(\cdot )=\varGamma$ (resp., $Q_{-}(\cdot )=\varGamma$). By using (58), we have

$$\begin{aligned}&\tan {(\phi _{\pm }+r \omega )}=S(\omega )/C(\omega ), \nonumber \\&S(\omega ):=-\omega ^{4}+\beta _{2}\omega ^{2}-\beta _{4}, \nonumber \\&C(\omega ):=\beta _{1}\omega ^{3}-\beta _{3}\omega . \end{aligned}$$

(59)

Let $\sigma =+$ or −. For each solution $\omega _{\sigma }$ to $Q_{\sigma }$, there exists a sequence $\{r_{\sigma }^{(k)}(\omega _{\sigma })\}_{k \in \mathbb {Z}}$:

$$\begin{aligned} r_{\sigma }^{(k)}(\omega _{\sigma }):=\left\{ \begin{array}{ll} \frac{1}{\omega _{\sigma }}\left[ \tan ^{-1}\left( {\frac{S(\omega _{\sigma })}{C(\omega _{\sigma })}}\right) - \phi _{\sigma }+2k\pi \right] , &{} \text{ if } C(\omega _{\sigma })>0~,\\ \frac{1}{\omega _{\sigma }}\left[ \tan ^{-1}\left( {\frac{S(\omega _{\sigma })}{C(\omega _{\sigma })}}\right) - \phi _{\sigma }+(2k-1)\pi \right] , &{} \text{ if } C(\omega _{\sigma })<0~,\\ \frac{1}{\omega _{\sigma }}\left[ \frac{3 \pi }{2}-\phi _{\sigma }+2k\pi \right] , &{} \text{ if } C(\omega _{\sigma })=0,~S(w_{\sigma })<0,\\ \frac{1}{\omega _{\sigma }}\left[ \frac{\pi }{2}-\phi _{\sigma }+2k\pi \right] , &{} \text{ if } C(\omega _{\sigma })=0,~S(\omega _{\sigma })>0, \end{array}\right. \end{aligned}$$

such that $\triangle _{\sigma }( i \omega _{\sigma },r_{\sigma }^{(k)}(\omega _{\sigma }),\tau _4)=0$. To simplify the notation, we denote $r_{\sigma }^{(k)}:= r_{\sigma }^{(k)}(\omega _{\sigma })$ as the bifurcation value. In particular, we shall take into account the case that $r_{\sigma }^{(k)}>0$ in the following discussions. Accordingly, a positive solution $\omega _{+}$ (resp., $\omega _{-}$) of $Q_{+}(\cdot )=\varGamma$ (resp., $Q_{-}(\cdot )=\varGamma$) corresponds to a pair of purely imaginary roots $\pm i \omega _{+}$ (resp., $\pm i \omega _{-}$) of $\triangle _{+}(\cdot ,r_{+}^{(k)},\tau _4)=0$ (resp., $\triangle _{-}(\cdot ,r_-^{(k)},\tau _4)=0$). This completes the first step.

Step (II): Since every solution of (58) is also a solution to $Q_+$ or $Q_-$, if $Q^{'}_{\sigma }(\omega _{\sigma })\ne 0$, and all other positive solutions to $Q_{+}(\cdot )=\varGamma$ and $Q_{-}(\cdot )=\varGamma$ are not integer multiples of $\omega _{\sigma }$, then $i \omega _{\sigma }$ is a simple purely imaginary eigenvalue, with $\sigma =+$ or −.

Step (III): The following conditions guarantee the transversality property:

$$\begin{aligned}&[R_{\sigma }(\omega _{\sigma },r_{\sigma }^{(k)})]^{2}+[I_{\sigma }(\omega _{\sigma },r_{\sigma }^{(k)})]^{2}\ne 0,~ W_{1,\sigma }^{2}(\omega _{\sigma },r_{\sigma }^{(k)})+W_{2,\sigma }^{2}(\omega _{\sigma },r_{\sigma }^{(k)})\ne 0,\\&Q_{1,\sigma }(\omega _{\sigma },r_{\sigma }^{(k)})W_{1,\sigma }(\omega _{\sigma },r_{\sigma }^{(k)}) +Q_{2,\sigma }(\omega _{\sigma },r_{\sigma }^{(k)})W_{2,\sigma }(\omega _{\sigma },r_{\sigma }^{(k)})\ne 0, \end{aligned}$$

where

$$\begin{aligned} R_{\pm }(\omega ,r):= & {} -3\beta _{1}\omega ^{2}+\beta _{3}+\nu _{3}\nu _{5}[(1-d_{4}r)\gamma _{1}\cos {(r\omega )} \\&\mp \gamma _{2}\gamma _{3}(r+\tau _4) \cos {((r+\tau _4)\omega )}-\gamma _{1}r \omega \sin {(r\omega )}],\\ I_{\pm }(\omega ,r):= & {} -4 \omega ^{3}+2\beta _{2}\omega -\nu _{3}\nu _{5}[\gamma _{1}r \omega \cos {(r\omega )}+(1-d_{4}r)\gamma _{1}\sin {(r\omega )}\\&\mp \gamma _{2}\gamma _{3}(r+\tau _4)\sin {((r+\tau _4)\omega )}],\\ Q_{1,\pm }(\omega ,r):= & {} \nu _{3}\nu _{5}\omega [-\gamma _{1}\omega \cos {(r\omega )}+d_{4}\gamma _{1}\sin {(r\omega )}\pm \gamma _{2}\gamma _{3} \sin {((r+\tau _4)\omega )}], \\ Q_{2,\pm }(\omega ,r):= & {} \nu _{3}\nu _{5}\omega [d_{4} \gamma _{1} \cos {(r\omega )}+\gamma _{1}\omega \sin {(r\omega )}\pm \gamma _{2}\gamma _{3} \cos {((r+\tau _4)\omega )}], \\ W_{1,\pm }(\omega ,r):= & {} -3\beta _{1}\omega ^{2}+\beta _{3}+\nu _{3}\nu _{5}[(1-d_{4}r)\gamma _{1}\cos {(r\omega )}\\&\mp \gamma _{2}\gamma _{3}(r+\tau _4)\cos {((r+\tau _4)\omega )} -\gamma _{1}r \omega \sin {(r\omega )}],\\ W_{2,\pm }(\omega ,r):= & {} -4\omega ^{3}+2\beta _{2}\omega -\nu _{3}\nu _{5}[\gamma _{1}r\omega \cos {(r\omega )}+(1-d_{4}r)\gamma _{1}\sin {(r\omega )}\\&\mp \gamma _{2}\gamma _{3}(r+\tau _4)\sin {((r+\tau _4)\omega )}]. \end{aligned}$$

Appendix 4: Sketch of steps (IV) and (V) in Sect. 4.2

Step (IV): Let us consider the characteristic equation (41) with $r=0$:

$$\begin{aligned} \triangle _{+}(\lambda ,0,\tau _4) \cdot \triangle _{-}(\lambda ,0,\tau _4)=0. \end{aligned}$$

(60)

Our goal is to derive the condition for parameters under which all roots of (60) have negative real parts for all $\tau _4\ge 0$. If so, then the stability switch will not occur for any $\tau _4 \ge 0$ and $r=0$. Moreover, for any fixed $\tau _4 \ge 0$, there is also no stability switch when $r<r_{c}$. Combining these properties, we can then confirm the asymptotical stability of the origin for system (17) for any $\tau _4 \ge 0$ and $r <r_{c}$.

We have seen that all characteristic values of (60) have negative real parts when $\tau _4=0$, under condition (33) or (34). Therefore, it suffices to show that all characteristic values of (60) remain to have negative real parts for any $\tau _4 \ge 0$ and $r=0$.

Proposition 8

For any $\tau _4 \ge 0$, all characteristic values of (60) have negative real parts if

$$\begin{aligned}&E_{1}>0, E_{3}>0, E_{5}>0, \end{aligned}$$

(61)

$$\begin{aligned}&\beta _{4}> \nu _{3}\nu _{5}(\gamma _{2}\gamma _{3}-d_{4}\gamma _{1}), \end{aligned}$$

(62)

where

$$\begin{aligned} E_{1}= & {} 2[d_{1}^{2}d_{2}^{2}(d_{3}^{2}+d_{4}^{2})+(d_{1}^{2}+d_{2}^{2})d_{3}^{2}d_{4}^{2}+2(d_{1}d_{2}d_{3} -(d_{1}\nonumber \\&+d_{2}+d_{3})d_{4}^{2})\nu _{3}\nu _{5}\gamma _{1}+\nu _{3}^{2}\nu _{5}^{2}\gamma _{1}^{2}], \nonumber \\ E_{3}= & {} 4[d_{1}^{2}(d_{2}^{2}+d_{3}^{2}+d_{4}^{2})+d_{2}^{2}(d_{3}^{2}+d_{4}^{2})+d_{3}^{2}d_{4}^{2}\nonumber \\&-\,2(d_{1} +d_{2}+d_{3})\nu _{3}\nu _{5}\gamma _{1}], \nonumber \\ E_{5}= & {} 6(d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}). \end{aligned}$$

(63)

Proof

We substitute $\lambda =i\nu$ with $\nu \ge 0$ into $\triangle _{\pm }(\lambda ,0,\tau _4)=0$ and collect the real and imaginary parts to obtain

$$\begin{aligned} \left\{ \begin{array}{lll} \mp \nu _{3}\nu _{5}\gamma _{2}\gamma _{3} \cos {(r\nu )}&{}=&{} \nu ^{4} -\beta _{2}\nu ^{2}+\beta _{4}+d_{4}\nu _{3}\nu _{5}\gamma _{1}\\ \pm \nu _{3}\nu _{5}\gamma _{2}\gamma _{3} \sin {(r\nu )}&{}=&{} -\beta _{1}\nu ^{3}+\nu (\nu _{3}\nu _{5}\gamma _{1}+\beta _{3}). \end{array}\right. \end{aligned}$$

(64)

Next, we sum up the square of equations (64) to get

$$\begin{aligned} E(\nu )=(\nu _{3}\nu _{5}\gamma _{2}\gamma _{3})^{2}, \end{aligned}$$

(65)

where

$$\begin{aligned} E(\nu ):= & {} \nu ^{8}+(\beta _{1}^{2}-2\beta _{2})\nu ^{6}+[2d_{4}\nu _{3}\nu _{5}\gamma _{1}+\beta _{2}^{2}-2\beta _{1}(\nu _{3}\nu _{5}\gamma _{1}+\beta _{3})+2\beta _{4}]\nu ^{4}\\&+\,[\nu _{3}^{2}\nu _{5}^{2}\gamma _{1}^{2}+\beta _{3}^{2}+2\nu _{3}\nu _{5}\gamma _{1}(-d_{4}\beta _{2} +\beta _{3})-2\beta _{2}\beta _{4}]\nu ^{2}+(d_{4}\nu _{3}\nu _{5}\gamma _{1}+\beta _{4})^{2}. \end{aligned}$$

Then $E'(\nu )=8\nu ^7+E_5 \nu ^5+E_3 \nu ^3+E_1 \nu$, where $E_1,~E_3,~E_5$ are defined in (63). Thus $E(\nu )$ is strictly increasing on $[0, \infty )$, under condition (61). Moreover, it is straightforward to verify that $E(0)=(d_{4}\nu _{3}\nu _{5}\gamma _{1}+\beta _{4})^{2}> (\nu _{3}\nu _{5}\gamma _{2}\gamma _{3})^{2}$ if (62) holds. Consequently, $E(\nu )>(\nu _{3}\nu _{5}\gamma _{2}\gamma _{3})^{2}$, for all $\nu \ge 0$, and hence for all $\nu \in {\mathbb R}$, as E is an even function. Therefore, there does not exist any real solution to (65) under conditions (61) and (62). The assertion thus follows from continuity of characteristic values. This completes the proof.

Step (V): Setting $\sigma =+$ (resp., $\sigma =-$) if $Q_+(\omega _c)=\varGamma$ (resp., $Q_-(\omega _c)=\varGamma )$, the following conditions ensure that the Hopf bifurcation occurs at $r=r_c$ and $\lambda =i \omega _c$:

• Condition (B1): $Q^{'}_\sigma (\omega _c)\ne 0$, and all other positive solutions to $Q_{+}(\cdot )=\varGamma$ and $Q_{-}(\cdot )=\varGamma$ satisfy $\omega \ne m \omega _c$ for any integer m;

• Condition (B2): $[R_\sigma (\omega _c,r_c)]^{2}+[I_\sigma (\omega _c,r_c)]^{2}\ne 0$, $W^{2}_{1,\sigma }(\omega _c,r_c)+W^{2}_{2,\sigma }(\omega _c,r_c)\ne 0$, and $Q_{1,\sigma }(\omega _c,r_c)W_{1,\sigma }(\omega _c,r_c)+Q_{2,\sigma }(\omega _c,r_c)W_{2,\sigma }(\omega _c,r_c) \ne 0$.

Appendix 5: Stability of bifurcating periodic solution, delay case

Step (VI): The stability of the bifurcating periodic solution is determined by the nonlinear terms of the equations. The condition for the stability can be expressed after the equations are put in a suitable form, under the theory of center manifold and normal form. Let us outline the main process for this step, following Hassard et al. (1981), see also Liao et al. (2012) and Yu and Cao (2006).

We write system (17) into the form:

$$\begin{aligned} \dot{{\mathbb X}}(t)=L_{\mu }{\mathbb X}_{t}+G_{\mu }({\mathbb X}_{t}), \end{aligned}$$

(66)

where ${\mathbb X}(t)= (x_{1}(t), \ldots , x_{4}(t),y_{1}(t), \ldots , y_{4}(t))^T$, ${\mathbb X}_{t}(\theta )={\mathbb X}(t+\theta ), \theta \in [-\tau _M, 0]$, $L_{\mu }:\mathscr {C}\rightarrow \mathbb {R}^{8}$ is a linear operator, and $G_{\mu }:\mathscr {C}\rightarrow \mathbb {R}^{8}$ is a nonlinear operator, and the phase space $\mathscr {C}:=\mathscr {C}([-\tau _{M},0],\mathbb {R}_+^{8})$ is a Banach space under the supremum norm $\Vert \phi \Vert =\sup \{|\phi (\theta )|: {-\tau _{M}\le \theta \le 0}\}$. We label these operators by $\mu :=r-r_{c}=\tau _{1}+\tau _{2}-r_{c}$, for a fixed minimal bifurcation value $r_c$. Herein, $L_{\mu }$ is defined by

$$\begin{aligned} L_{\mu }\phi= & {} M_{0}\phi (0)+M_{1}\phi (-\tau _{1})+M_{2}\phi (-\tau _{2})+M_{4}\phi (-\tau _{4}), \end{aligned}$$

where

$$\begin{aligned} (M_{0})_{ij}= & {} \left\{ \begin{array}{ll} -d_{1}, &{}\quad \text{ if } ij=11,55\\ -d_{2}, &{}\quad \text{ if } ij=22,66\\ -d_{3}, &{}\quad \text{ if } ij=33,77\\ -d_{4}, &{}\quad \text{ if } ij=44,88\\ \nu _{5}, &{}\quad \text{ if } ij=32, \text{ and } ij=76\\ 0, &{}\quad \text{ otherwise }, \end{array}\right. \\ (M_{1})_{ij}= & {} \left\{ \begin{array}{ll} -\gamma _1, &{}\quad \text{ if } ij=13, \text{ and } ij=57\\ \gamma _2, &{} \quad \text{ if } ij=18, \text{ and } ij=54\\ 0, &{}\quad \text{ otherwise }, \end{array}\right. \\ (M_{2})_{ij}= & {} \left\{ \begin{array}{ll} \nu _{3}, &{}\quad \text{ if } ij=21, \text{ and } ij=65\\ 0, &{}\quad \text{ otherwise }, \end{array}\right. \\ (M_{4})_{ij}= & {} \left\{ \begin{array}{ll} -\gamma _3, &{}\quad \text{ if } ij=43 \text{ and } ij=87\\ 0, &{}\quad \text{ otherwise }, \end{array}\right. \end{aligned}$$

as expressed in (40), with $d_{1},d_{2},d_{3},d_{4}$ and $\gamma _1, \gamma _2, \gamma _3$ defined in (28) and (29), respectively. In fact, according to the Riesz representation theorem, by choosing

$$\begin{aligned} \eta (\theta ,\mu )= M_{0}\delta (\theta )+M_{1}\delta (\theta +\tau _{1})+ M_{2}\delta (\theta +\tau _{2})+M_{4}\delta (\theta +\tau _{4}), \end{aligned}$$

we see that

$$\begin{aligned} L_{\mu }\phi =\int _{-\tau _{M}}^{0}d\eta (\theta ,\mu )\phi (\theta ), \end{aligned}$$

where $\delta (\cdot )$ is the Dirac delta function. For operator $G_{\mu }:\mathscr {C}\rightarrow \mathbb {R}^{8}$, its eight components are

$$\begin{aligned}&m_{11}\phi _{3}^{2}(-\tau _{1})+m_{12}\phi _{3}(-\tau _{1}) \phi _{8}(-\tau _{1})+m_{13}\phi _{8}^{2}(-\tau _{1})+m_{14}\phi _{3}^{3}(-\tau _{1})\\&\quad +\,m_{15}\phi _{3}^{2}(-\tau _{1})\phi _{8}(-\tau _{1}) +m_{16}\phi _{3}(-\tau _{1})\phi _{8}^{2}(-\tau _{1})\\&\quad +\,m_{17}\phi _{8}^{3}(-\tau _{1})+m_{18}\phi _{1}^{2}(0)+m_{19}\phi _{1}^{3}(0)+\mathrm{h.o.t.}, \\&m_{21}\phi _{2}^{2}(0)+m_{22}\phi _{2}^{3}(0)+\mathrm{h.o.t.}, \\&m_{31}\phi _{3}^{2}(0)+m_{32}\phi _{3}^{3}(0)+\mathrm{h.o.t.}, \\&m_{41}\phi _{3}^{2}(-\tau _{4})+m_{42}\phi _{3}^{3}(-\tau _{4})+m_{43}\phi _{4}^{2}(0)+m_{44}\phi _{4}^{3}(0)+ \mathrm{h.o.t.},\\&m_{11}\phi _{7}^{2}(-\tau _{1})+m_{12}\phi _{7}(-\tau _{1})\phi _{4}(-\tau _{1})+m_{13}\phi _{4}^{2}(-\tau _{1})\\&\quad +\,m_{14}\phi _{7}^{3}(-\tau _{1})+m_{15}\phi _{7}^{2}(-\tau _{1})\phi _{4}(-\tau _{1})\\&\quad +\,m_{16}\phi _{7}(-\tau _{1})\phi _{4}^{2}(-\tau _{1}) +m_{17}\phi _{4}^{3}(-\tau _{1})+m_{18}\phi _{5}^{2}(0)+m_{19}\phi _{5}^{3}(0)+\mathrm{h.o.t.},\\&m_{21}\phi _{6}^{2}(0)+m_{22}\phi _{6}^{3}(0)+\mathrm{h.o.t.},\\&m_{31}\phi _{7}^{2}(0)+m_{32}\phi _{7}^{3}(0)+\mathrm{h.o.t.},\\&m_{41}\phi _{7}^{2}(-\tau _{4})+m_{42}\phi _{7}^{3}(-\tau _{4})+m_{43}\phi _{8}^{2}(0)+m_{44}\phi _{8}^{3}(0)+ \mathrm{h.o.t.}, \end{aligned}$$

successively, where $\phi =(\phi _1, \ldots , \phi _8)$, and

$$\begin{aligned} m_{11}:= & {} \frac{1}{2}\frac{\partial ^{2}g_{H}}{\partial u^{2}}(\bar{x}_{3},\bar{x}_{4}), m_{12}:=\frac{\partial ^{2}g_{H}}{\partial u \partial v}(\bar{x}_{3},\bar{x}_{4}),\\ m_{13}:= & {} \frac{1}{2}\frac{\partial ^{2}g_{H}}{\partial v^{2}}(\bar{x}_{3},\bar{x}_{4}), m_{14}:=\frac{1}{6}\frac{\partial ^{3}g_{H}}{\partial u^{3}}(\bar{x}_{3},\bar{x}_{4}),\\ m_{15}:= & {} \frac{1}{2}\frac{\partial ^{3}g_{H}}{\partial u^{2}\partial v}(\bar{x}_{3},\bar{x}_{4}), m_{16}:=\frac{1}{2}\frac{\partial ^{3}g_{H}}{\partial u \partial v^{2}}(\bar{x}_{3},\bar{x}_{4}),\\ m_{17}:= & {} \frac{1}{6}\frac{\partial ^{3}g_{H}}{\partial v^{3}}(\bar{x}_{3},\bar{x}_{4}), m_{18}:=-\frac{1}{2}f_{1}''(\bar{x}_{1}),\\ m_{19}:= & {} -\frac{1}{6}f_{1}'''(\bar{x}_{1}), m_{21}:=-\frac{1}{2}f_{2}''(\bar{x}_{2}), \\ m_{22}:= & {} -\frac{1}{6}f_{2}'''(\bar{x}_{2}), m_{31}:=-\frac{1}{2}f_{3}''(\bar{x}_{3}),\\ m_{32}:= & {} -\frac{1}{6}f_{3}'''(\bar{x}_{3}), m_{41}:=\frac{1}{2}g_{D}''(\bar{x}_{3}), \\ m_{42}:= & {} \frac{1}{6}g_{D}'''(\bar{x}_{3}), m_{43}:=-\frac{1}{2}f_{4}''(\bar{x}_{4}), \\ m_{44}:= & {} -\frac{1}{6}f_{4}'''(\bar{x}_{4}). \end{aligned}$$

To put (66) in a suitable form, we define two operators on $\mathscr {C}^1 := \mathscr {C}^1([-\tau _{M},0],\mathbb {R}^{8})$:

$$\begin{aligned} (A_{\mu }\phi )(\theta )= & {} \left\{ \begin{array}{ll} d\phi (\theta )/d\theta , &{}\quad \theta \in [-\tau _{M},~0), \\ \int _{-\tau _{M}}^{0}d\eta (\zeta ,\mu ) \phi (\zeta ), &{}\quad \theta =0, \end{array}\right. \\ (R_{\mu }\phi )(\theta )= & {} \left\{ \begin{array}{ll} 0, &{}\quad \theta \in [-\tau _{M},~0), \\ G_{\mu }(\phi ), &{}\quad \theta =0. \end{array}\right. \end{aligned}$$

Then (66) can be recast into

$$\begin{aligned} \dot{{\mathbb X}}_{t}=A_{\mu }{\mathbb X}_{t}+R_{\mu }{\mathbb X}_{t}. \end{aligned}$$

(67)

The adjoint operator $A_\mu ^{*}$ of $A_\mu$ can be computed as

$$\begin{aligned} (A^{*}_{\mu }\psi )(\theta ^{*})= \left\{ \begin{array}{ll} -d\psi (\theta ^{*})/d\theta ^{*}, &{} \theta ^{*}\in (0,~\tau _{M}], \\ \int _{-\tau _{M}}^{0}d\eta ^{T}(\zeta ,\mu )\psi (-\zeta ), &{} \theta ^{*}=0, \end{array}\right. \end{aligned}$$

where $\psi \in \mathscr {C}^1([0,\tau _{M}],\mathbb {R}^{8})$. In the following computation, for convenience, we allow functions to take values in $\mathbb {C}^{8}$. We use the bilinear form

$$\begin{aligned} \langle \psi ,\phi \rangle =\overline{\psi }^T(0) \phi (0)-\int _{\theta =-\tau _{M}}^{0}\int _{\xi =0}^{\theta }\overline{\psi }^{T}(\xi -\theta )d\eta (\theta ) \phi (\xi )d\xi , \end{aligned}$$

for $\phi \in \mathscr {C}([-\tau _{M},0],\mathbb {C}^{8})$, $\psi \in \mathscr {C}([0,\tau _{M}],\mathbb {C}^{8})$, to determine the coordinates of the center manifold near the origin of (66), where $\eta (\theta ):=\eta (\theta ,0)$.

Next, denote by $q(\theta )$ the eigenvector of $A:=A_{0}$, and $q^{*}(\theta ^{*})$ of $A^{*}:=A^{*}_{0}$ corresponding to purely imaginary eigenvalues $i \omega _{c}$ and $-i\omega _{c}$, respectively, namely,

$$\begin{aligned} Aq(\theta )=i\omega _{c}q(\theta ), \text{ and } A^{*}q^{*}(\theta ^{*})=-i\omega _{c}q^{*}(\theta ^{*}). \end{aligned}$$

(68)

We also impose the normalized condition $\langle q^{*},q \rangle =1$ and $\langle q^{*},\bar{q} \rangle =0$. To this end, we assume that

$$\begin{aligned} q(\theta )=q(0)e^{i\omega _{c}\theta },~q^{*}(\theta ^{*})=q^{*}(0)e^{i\omega _{c}\theta ^{*}}, \end{aligned}$$

(69)

for $\theta \in [- \tau _{M},0)$, $\theta ^{*}\in (0, \tau _{M}]$, and $q(0)=(q_{1}, \ldots , q_{8})^{T},~ q^{*}(0)=\frac{1}{\rho }(q_{1}^{*}, \ldots , q^{*}_{8})^{T}$, where $\rho$, $q_{i}$ and $q^{*}_{i}$, $i=1,\ldots ,8$, are to be determined. Substituting (69) into (68) and evaluating at $\theta =0$, we obtain

$$\begin{aligned}&q_{1}=1,~q_{2}=\frac{\nu _{3}U_{2}}{V_{2}},~ q_{3}=\frac{\nu _{3}\nu _{5}U_{2}}{V_{2}V_{3}},~ q_{4}=\frac{\nu _{3}\nu _{5}\gamma _3 U_{2}U_{4}}{V_{2}V_{3}V_{4}}, \nonumber \\&q_{5}=\frac{\overline{U}_{4}V_{4}(\nu _{3}\nu _{5}\gamma _1+\overline{U}_{1}\overline{U}_{2}V_{1}V_{2}V_{3})}{-\nu _{3}\nu _{5}\gamma _2\gamma _3},~ q_{6}=\frac{\overline{U}_{4}V_{3}V_{4}(\overline{U}_{1}V_{1}V_{2}V_{3}+\nu _{3}\nu _{5}\gamma _1U_{2})}{-\nu _{5}\gamma _2\gamma _3V_{2}V_{3}}, \nonumber \\&q_{7}=\frac{\overline{U}_{4}V_{4}(\overline{U}_{1}V_{1}V_{2}V_{3}+\nu _{3}\nu _{5}\gamma _1U_{2})}{-\gamma _2\gamma _3V_{2}V_{3}},~ q_{8}=\frac{\overline{U}_{1}V_{1}V_{2}V_{3}+\nu _{3}\nu _{5}\gamma _1U_{2}}{\gamma _2V_{2}V_{3}},\nonumber \\&q_{1}^{*}=1,~q_{2}^{*}=\frac{U_{2}\overline{V}_{1}}{\nu _{3}},~ q_{3}^{*}=\frac{U_{2}\overline{V}_{1}\overline{V}_{2}}{\nu _{3}\nu _{5}},\nonumber \\&q_{4}^{*}=\frac{\nu _{3}\nu _{5}\gamma _1\overline{U}_{1}U_{4}+U_{2}U_{4}\overline{V}_{1}\overline{V}_{2}\overline{V}_{3}}{-\nu _{3}\nu _{5}\gamma _3},~ q_{5}^{*}=\frac{\nu _{3}\nu _{5}\gamma _1U_{4}\overline{V}_{4}+U_{1}U_{2}U_{4}\overline{V}_{1}\overline{V}_{2}\overline{V}_{3}\overline{V}_{4}}{-\nu _{3}\nu _{5}\gamma _2\gamma _3},\nonumber \\&q_{6}^{*}=-\frac{U_{1}U_{2}^{2}U_{4}\overline{V}_{4}(-\nu _{3}\nu _{5}\gamma _1\overline{U}_{1}\overline{U}_{2}\overline{V}_{1}- \overline{V}_{1}^{2}\overline{V}_{2}\overline{V}_{3})}{-\nu _{3}^{2}\nu _{5}\gamma _2\gamma _3},\nonumber \\&q_{7}^{*}=\frac{U_{1}U_{2}^{2}U_{4}\overline{V}_{1}\overline{V}_{2}\overline{V}_{4}(\nu _{3}\nu _{5}\gamma _1\overline{U}_{1}\overline{U}_{2} +\overline{V}_{1}\overline{V}_{2}\overline{V}_{3})}{-\nu _{3}^{2}\nu _{5}^{2}\gamma _2\gamma _3},~ q_{8}^{*}=\frac{\gamma _2\overline{U}_{1}}{\overline{V}_{4}}, \end{aligned}$$

where $U_{j}=e^{-i\omega _{c}\tau _{j}}$, $j=1, 2, 4$, and $V_{j}=i\omega _{c}+d_{j}$, for $j=1,\ldots , 4$. Notice that $\langle q^{*},q\rangle =1$ and $\langle q^{*},\bar{q}\rangle =0$, if we set

$$\begin{aligned} \bar{\rho }= & {} (q_{1}\overline{q_{1}^{*}}+q_{2}\overline{q_{2}^{*}}+\cdots +q_{8}\overline{q_{8}^{*}})\\&+J_{1}\tau _{1}e^{-i\omega _{c}\tau _{1}}+J_{2}\tau _{2}e^{-i\omega _{c}\tau _{2}}+J_{4}\tau _{4}e^{-i\omega _{c}\tau _{4}}, \end{aligned}$$

where $J_{1}:=-\gamma _1(q_{3}\overline{q}_{1}^{*}+q_{7}\overline{q}_{5}^{*}) +\gamma _2(q_{8}\overline{q}_{1}^{*}+q_{4}\overline{q}_{5}^{*})$, $J_{2}:= \nu _{3}(q_{1}\overline{q}_{2}^{*}+q_{5}\overline{q}_{6}^{*})$, and $J_{4}:=-\gamma _3(q_{3}\overline{q}_{4}^{*}+q_{7}\overline{q}_{8}^{*})$.

Now, we use q and $q^{*}$ to construct a coordinate on the center manifold $\mathscr {M}_{0}$ at $\mu =0$. For each $\phi \in \mathscr {C}([-\tau _{M},0],\mathbb {C}^{8})$, we associate a pair (z, w), with $z=\langle q^{*}, \phi \rangle$, $w=\phi -z q -\overline{z} \overline{q}= \phi -2 \mathrm{Re}(zq)$. Let ${\mathbb X}_{t}={\mathbb X}_{t}(\theta )=(x_{1,t}(\theta ), \ldots ,x_{4,t}(\theta ),y_{1,t}(\theta ), \ldots , y_{4,t}(\theta ))^T$ be a solution of (67), and let

$$\begin{aligned} z(t):= & {} \langle q^{*},~{\mathbb X}_{t}\rangle ,\\ W(t,\theta ):= & {} {\mathbb X}_{t}(\theta )-2 \mathrm{Re}(z(t)q(\theta )). \end{aligned}$$

On the center manifold $\mathscr {M}_{0}$, $W(t,\theta )= w(z(t),\bar{z}(t),\theta )$, where

$$\begin{aligned} w(z(t),\bar{z}(t),\theta )=w_{20}(\theta )\frac{z^{2}(t)}{2}+w_{11}(\theta )z(t)\bar{z}(t)+w_{02}(\theta )\frac{\bar{z}^{2}(t)}{2}+\cdots . \end{aligned}$$

Herein, z and $\bar{z}$ are the local coordinates of the center manifold $\mathscr {M}_{0}$ in directions $q^{*}$ and $\overline{{q}^{*}}$, respectively.

Hence, the solution ${\mathbb X}_{t}\in \mathscr {M}_{0}$ of (67), at $\mu =0$, satisfies

$$\begin{aligned} \dot{z}(t)=i\omega _{c}z(t)+g(z(t),\bar{z}(t)), \end{aligned}$$

where

$$\begin{aligned} g(z,\bar{z})= & {} \overline{q^{*}}^{T}(0) G_{0}(z, \overline{z})=g_{20}\frac{z^{2}}{2}+g_{11}z\bar{z}\\&+ g_{02}\frac{\bar{z}^{2}}{2}+g_{21}\frac{z^{2}\bar{z}}{2}+\cdots , \end{aligned}$$

and

$$\begin{aligned} g_{20}= & {} \frac{2}{\bar{\rho }}\{m_{18}(q_{1}^{2}\overline{q}_{1}^{*}+q_{5}^{2}\overline{q}_{5}^{*}) +\,m_{12}(q_{2}^{2}\overline{q}_{2}^{*}+q_{6}^{2}\overline{q}_{6}^{*})\\&+\,m_{31}(q_{3}^{2}\overline{q}_{3}^{*}+q_{7}^{2}\overline{q}_{7}^{*})+m_{43}(q_{4}^{2}\overline{q}_{4}^{*}+q_{8}^{2}\overline{q}_{8}^{*})\\&+\, e^{-2i \omega _{c}\tau _{1}}[(m_{11}q_{3}^{2}+q_{8}(m_{12}q_{3}+m_{13}q_{8}))\overline{q}_{1}^{*}+(m_{13}q_{4}^{2}\\&+\,q_{7}(m_{12}q_{4}+m_{11}q_{7}))\overline{q}_{5}^{*}] +\, e^{-2i \omega _{c}\tau _{4}}m_{41}(q_{3}^{2}\overline{q}_{4}^{*}+q_{7}^{2}\overline{q}_{8}^{*})\},\\ g_{11}= & {} \frac{1}{\bar{\rho }}\{ [2m_{18}q_{1}\overline{q}_{1}+2m_{11}q_{3}\overline{q}_{3}+m_{12}(q_{8}\overline{q}_{3}+q_{3}\overline{q}_{8})\\&+\,2m_{13}q_{8}\overline{q}_{8} ]\overline{q}_{1}^{*} + (2m_{14}q_{3}\overline{q}_{3}+2m_{43}q_{4}\overline{q}_{4})\overline{q}_{4}^{*}\\&+\,[2m_{13}q_{4}\overline{q}_{4}+2m_{18}q_{5}\overline{q}_{5}+m_{12}(q_{7}\overline{q}_{4}+q_{4}\overline{q}_{7})+2m_{11}q_{7}\overline{q}_{7}]\overline{q}_{5}^{*}\\&+\,(2m_{14}q_{7}\overline{q}_{7}+2m_{43}q_{8}\overline{q}_{8})\overline{q}_{8}^{*} +\,2m_{21}(q_{2}\overline{q}_{2}\overline{q}_{2}^{*}+q_{6}\overline{q}_{6}\overline{q}_{6}^{*})\\&+\,2m_{31}(q_{3}\overline{q}_{3}\overline{q}_{3}^{*}+q_{7}\overline{q}_{7}\overline{q}_{7}^{*})\},\\ g_{02}= & {} \frac{2}{\bar{\rho }}\{ m_{18}(\overline{q}_{1}^{2}\overline{q}_{1}^{*}+\overline{q}_{5}^{2}\overline{q}_{5}^{*}) +\,m_{21}(\overline{q}_{2}^{2}\overline{q}_{2}^{*}+\overline{q}_{6}^{2}\overline{q}_{6}^{*}) +m_{31}(\overline{q}_{3}^{2}\overline{q}_{3}^{*}+\overline{q}_{7}^{2}\overline{q}_{7}^{*}) \\&+\,m_{43}(\overline{q}_{4}^{2}\overline{q}_{4}^{*}+\overline{q}_{8}^{2}\overline{q}_{8}^{*}) +e^{2i \omega _{c}\tau _{1}}[(m_{11}\overline{q}_{3}^{2}+m_{12}\overline{q}_{3}\overline{q}_{8}+m_{13}\overline{q}_{8}^{2})\overline{q}_{1}^{*}\\&+\,(m_{13}\overline{q}_{4}^{2}+m_{12}\overline{q}_{4}\overline{q}_{7}+m_{11}\overline{q}_{7}^{2})\overline{q}_{5}^{*}] +\,e^{2i\omega _{c}\tau _{4}}m_{41}(\overline{q}_{3}^{2}\overline{q}_{4}^{*}+\overline{q}_{7}^{2}\overline{q}_{8}^{*})\}, \end{aligned}$$

$$\begin{aligned} g_{21}= & {} \frac{1}{\bar{\rho }} \{ 6m_{19}(q_{1}^{2}\overline{q}_{1}\overline{q}_{1}^{*}+q_{5}^{2}\overline{q}_{5}\overline{q}_{5}^{*}) +\,6m_{22}(q_{2}^{2}\overline{q}_{2}\overline{q}_{2}^{*}+q_{6}^{2}\overline{q}_{6}\overline{q}_{6}^{*})\\&+\,6m_{32}(q_{3}^{2}\overline{q}_{3}\overline{q}_{3}^{*}+q_{7}^{2}\overline{q}_{7}\overline{q}_{7}^{*}) +\,6m_{44}(q_{4}^{2}\overline{q}_{4}\overline{q}_{4}^{*}+q_{8}^{2}\overline{q}_{8}\overline{q}_{8}^{*})\\&+\,4m_{18}(q_{1}\overline{q}_{1}^{*}w_{11}^{(1)}(0)+q_{5}\overline{q}_{5}^{*}w_{11}^{(5)}(0)) \nonumber \\&+\,4m_{21}(q_{2}\overline{q}_{2}^{*}w_{11}^{(2)}(0)+q_{6}\overline{q}_{6}^{*}w_{11}^{(6)}(0))\\&+\,4m_{31}(q_{3}\overline{q}_{3}^{*}w_{11}^{(3)}(0)+q_{7}\overline{q}_{7}^{*}w_{11}^{(7)}(0)) \nonumber \\&+\,4m_{43}(q_{4}\overline{q}_{4}^{*}w_{11}^{(4)}(0)+q_{8}\overline{q}_{8}^{*}w_{11}^{(8)}(0)) \nonumber \\&+\,e^{-i\omega _{c}\tau _{4}}[6m_{42}q_{3}^{2}\overline{q}_{3}\overline{q}_{4}^{*}+4m_{41}q_{3}\overline{q}_{4}^{*}w_{11}^{(3)}(-\tau _{4})\\&+\,2q_{7}\overline{q}_{8}^{*}(3m_{42}q_{7}\overline{q}_{7}+2m_{41}w_{11}^{(7)}(-\tau _{4}))] \nonumber \\&+\,2 e^{-i\omega _{c}\tau _{1}}[ (3m_{14}q_{3}^{2}+q_{8}(2m_{15}q_{3}+m_{16}q_{8}))\overline{q}_{3}\overline{q}_{1}^{*}\\&+\,(m_{15}q_{3}^{2}+q_{8}(2m_{16}q_{3}+3m_{17}q_{8}))\overline{q}_{8}\overline{q}_{1}^{*} \nonumber \\&+\,(2m_{11}q_{3}+m_{12}q_{8})\overline{q}_{1}^{*}w_{11}^{(3)}(-\tau _{1})\\&+\,\overline{q}_{5}^{*}((3m_{17}q_{4}^{2} +\,q_{7}(2m_{16}q_{4}+m_{15}q_{7}))\overline{q}_{4} \nonumber \\&+\,(m_{16}q_{4}^{2}+2m_{15}q_{4}q_{7}+3m_{14}q_{7}^{2})\overline{q}_{7}+(2m_{13}q_{4}+m_{12}q_{7})w_{11}^{(4)}(-\tau _{1}) \nonumber \\&+\,(m_{12}q_{4}+2m_{11}q_{7})w_{11}^{(7)}(-\tau _{1}))+(m_{12}q_{3}+2m_{13}q_{8})\overline{q}_{1}^{*}w_{11}^{(8)}(-\tau _{1})]\nonumber \\&+\,2m_{18}(\overline{q}_{1}\overline{q}_{1}^{*}w_{20}^{(1)}(0)+\overline{q}_{5}\overline{q}_{5}^{*}w_{20}^{(5)}(0))\\&+2m_{21}(\overline{q}_{2}\overline{q}_{2}^{*}w_{20}^{(2)}(0)+\overline{q}_{6}\overline{q}_{6}^{*}w_{20}^{(6)}(0)) \nonumber \\&+\, 2m_{31}(\overline{q}_{3}\overline{q}_{3}^{*}w_{20}^{(3)}(0)+\overline{q}_{7}\overline{q}_{7}^{*}w_{20}^{(7)}(0))\\&+\,2m_{43}(\overline{q}_{4}\overline{q}_{4}^{*}w_{20}^{(4)}(0)+\overline{q}_{8}\overline{q}_{8}^{*}w_{20}^{(8)}(0)) \nonumber \\&+\,2 e^{i\omega _{c}\tau _{4}} m_{41}(\overline{q}_{3}\overline{q}_{4}^{*} w_{20}^{(3)}(-\tau _{4})+\overline{q}_{7}\overline{q}_{8}^{*}w_{20}^{(7)}(-\tau _{4})) \nonumber \\&+\,e^{i\omega _{c}\tau _{1}}[ \overline{q}_{5}^{*}((2m_{13}\overline{q}_{4}+m_{12}\overline{q}_{7})w_{20}^{(4)}(-\tau _{1})\\&+\,m_{12}\overline{q}_{4}+2m_{11}\overline{q}_{7})w_{20}^{(7)}(-\tau _{1})) \nonumber \\&+\,\overline{q}_{3}\overline{q}_{1}^{*}(2m_{11}w_{20}^{(3)}(-\tau _{1})+m_{12}w_{20}^{(8)}(-\tau _{1}))\\&+\,\overline{q}_{8}\overline{q}_{1}^{*}(m_{12}w_{20}^{(3)}(-\tau _{1})+2m_{13}w_{20}^{(8)}(-\tau _{1}))]\}, \end{aligned}$$

where $w_{20}^{(k)}(\theta )$ and $w_{11}^{(k)}(\theta )$ are the kth components of $w_{20}(\theta )$ and $w_{11}(\theta )$ respectively. And for $-\tau _{M}\le \theta <0$, it can be computed that

$$\begin{aligned} \left\{ \begin{array}{lll} w_{20}(\theta )&{}=&{}\frac{ig_{20}}{\omega _{c}}q(0)e^{i\omega _{c}\theta }-\frac{\bar{g}_{02}}{3i\omega _{c}}\bar{q}(0) e^{-i\omega _{c}\theta }+E_{1}e^{2i\omega _{c}\theta }\\ w_{11}(\theta )&{}=&{}\frac{g_{11}}{i\omega _{c}}q(0)e^{i\omega _{c}\theta }- \frac{\bar{g}_{11}}{i\omega _{c}}\bar{q}(0)e^{-i\omega _{c}\theta }+E_{2}, \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} E_{1}= & {} \left( 2i\omega _{c}I-\int _{-\tau _{M}}^{0}e^{2i\omega _{c}\theta }d\eta (\theta ,0)\right) ^{-1}\\&\left( \begin{array}{c} 2[m_{18}q_{1}^{2}+e^{-2i\omega _{c}\tau _{1}}(m_{11}q_{3}^{2}+q_{8}(m_{12}q_{3}+m_{13}q_{8}))]\\ 2m_{21}q_{2}^{2}\\ 2m_{31}q_{3}^{2}\\ 2(e^{-2i\omega _{c}\tau _{4}}m_{41}q_{3}^{2}+m_{43}q_{4}^{2})\\ 2[m_{18}q_{5}^{2}+e^{-2i\omega _{c}\tau _{1}}(m_{13}q_{4}^{2}+q_{7}(m_{12}q_{4}+m_{11}q_{7}))]\\ 2m_{21}q_{6}^{2}\\ 2m_{31}q_{7}^{2}\\ 2(e^{-2i\omega _{c}\tau _{4}}m_{41}q_{7}^{2}+m_{43}q_{8}^{2}) \end{array}\right) ,\\ E_{2}= & {} \left[ -\int _{-\tau _{M}}^{0}d\eta (\theta ,0)\right] ^{-1} \\&\left( \begin{array}{c} 2m_{18}q_{1}\overline{q}_{1}+2m_{1 1}q_{3}\overline{q}_{3}+m_{12}(q_{8}\overline{q}_{3}+q_{3}\overline{q}_{8})+2m_{13}q_{8}\overline{q}_{8}\\ 2m_{21}q_{2}\overline{q}_{2}\\ 2m_{31}q_{3}\overline{q}_{3}\\ 2m_{41}q_{3}\overline{q}_{3}+2m_{43}q_{4}\overline{q}_{4}\\ 2m_{13}q_{4}\overline{q}_{4}+m_{12}(q_{7}\overline{q}_{4}+q_{4}\overline{q}_{7})+2m_{18}q_{5}\overline{q}_{5}+2m_{11}q_{7}\overline{q}_{7}\\ 2m_{21}q_{6}\overline{q}_{6}\\ 2m_{31}q_{7}\overline{q}_{7}\\ 2m_{41}q_{7}\overline{q}_{7}+2m_{43}q_{8}\overline{q}_{8} \end{array}\right) . \end{aligned}$$

With $g_{20},~g_{11},~g_{02}$, and $g_{21}$, we can compute $C_{1}(r_c),~p_{2},~\zeta _{2}$, and $T_{2}$ for the stability and the other properties for the bifurcating periodic solution.