In this section, we give the following results about the existence and stability of the positive equilibrium of system (1.3).
Proposition 1
For system (1.3), assume that
\(a_{2}\), \(b_{1}\), \(d_{1}\), \(d_{2}\)
are positive constants such that

\((\mathrm{H}_{1})\)
:

\(a_{2}^{2}d_{1}>b_{1}d_{2}^{2}\)
holds, then the system has a unique positive equilibrium
\(E^{\ast }=(y_{1}^{\ast},y_{2}^{\ast})\). Furthermore, when system (1.3) has no delay, i.e. \(\tau_{i}\ (i=1,2,3)=0\), then
\(E^{\ast}\)
is globally asymptotically stable.
Proof
For system (1.3), assumption \((\mathrm{H}_{1})\) is the parameter condition which ensures the existence of the positive equilibrium \(E^{\ast}\). The proof for the existence of \(E^{\ast}\) is similar to that in [23], we omit it here.
We now prove the global asymptotic stability. When \(\tau_{i}\ (i=1,2,3)=0\), system (1.3) is reduced to the ODE system (1.1). Defining Dulac function as \(B(x_{1},x_{2})=\frac{1}{x_{1}x_{2}}\), and
$$ \begin{aligned}D&=\frac{\partial \{\frac{1}{x_{1}x_{2}} [r_{1}x_{1}(t)(1\frac {{x_{1}(t)}}{K_{1}}\frac{\alpha(x_{2}(t)c_{2})^{2}}{K_{2}}) ] \}}{\partial x_{1}}+\frac{\partial \{\frac {1}{x_{1}x_{2}} [r_{2}x_{2}(t)(1\frac{{x_{2}(t)}}{K_{2}}+\frac{\beta (x_{1}(t)c_{1})^{2}}{K_{1}}) ] \}}{\partial x_{2}} \\ &={\frac{r_{{1}}}{K_{{1}}x_{2}}}{\frac{r_{{2}}}{x_{1}K_{{2}}}},\end{aligned} $$
we easily get \(D<0\) in the \(\operatorname{int} \mathbb{R}_{+}^{2}=\{ (x_{1},x_{2}):x_{1}>0,x_{2}>0\}\) space. By [29] (Theorem 4.1.2, Chap. 4, p. 72), it follows from Dulac’s principle that the system has no closed path curve. So \(E^{\ast}\) is globally asymptotically stable when system (1.3) has no delay and also when the nonnegative delays are sufficiently small. □
If a pair of complex roots with negative real parts and nonzero imaginary parts cross the imaginary axis as τ increases, this potentially results in Hopf bifurcation and the positive equilibrium \(E^{\ast}\) loses stability. Now we discuss the existence of a local Hopf bifurcation occurring at \(E^{\ast}\). Let \(u_{1}(t)=y_{1}(t)y_{1}^{\ast}\), \(u_{2}(t)=y_{2}(t)y_{2}^{\ast}\), then system (1.3) becomes
$$ \textstyle\begin{cases} \dot{{u}}_{1}(t)= (u_{1} (t)+y_{1}^{\ast}+c_{1} )[a_{1}u_{1}(t\tau_{1}) b_{1}{u_{2}}^{2}(t\tau_{2})2b_{1}y_{2}^{\ast}u_{2}(t\tau_{2})],\\ \dot{{u}_{2}}(t)= (u_{2} (t)+y_{2}^{\ast}+c_{1} )[a_{2}u_{2}(t\tau_{1}) +b_{2}{u_{1}}^{2}(t\tau_{3})+2b_{2}y_{2}^{\ast}u_{1}(t\tau_{3})], \end{cases} $$
(2.1)
the linearization of system (2.1) at \(E^{\ast}\) is
$$ \textstyle\begin{cases} \dot{u}_{1}(t)=a_{1}(y_{1}^{\ast}+c_{1})u_{1}(t\tau _{1})2b_{1}y_{2}^{\ast}(y_{1}^{\ast}+c_{1})u_{2}(t\tau_{2}),\\ \dot{u}_{2}(t)=2b_{2}y_{2}^{\ast}(y_{2}^{\ast}+c_{1})u_{1}(t\tau _{3})a_{2}(y_{2}^{\ast}+c_{1})u_{2}(t\tau_{1}), \end{cases} $$
(2.2)
and the associated characteristic equation of (2.2) is
$$ \begin{vmatrix} \lambda+a_{1} \bigl(y_{1}^{\ast}+c_{1}\bigr)e^{\lambda\tau_{1}} &2b_{1}y_{2}^{\ast}\bigl(y_{1}^{\ast}+c_{1} \bigr)e^{\lambda\tau_{2}} \\ 2b_{2}y_{2}^{\ast}\bigl(y_{2}^{\ast}+c_{1} \bigr)e^{\lambda\tau_{3}} &\lambda+a_{2}\bigl(y_{2}^{\ast}+c_{1} \bigr)e^{\lambda\tau_{1}} \end{vmatrix} =0. $$
For the above characteristic equation, it is hard to do the complete analysis for the distribution of the roots, so we assume that

\((\mathrm{H}_{2})\)
:

\(\tau_{2}+\tau_{3}= \tau_{1}\)
holds, and \(\tau_{1}\triangleq\tau\).
Hence, the characteristic equation is equivalent to
$$ \lambda^{2}+e^{\lambda\tau}(p\lambda+q)+re^{2\lambda\tau}=0, $$
(2.3)
where \(p=a_{1}(y_{1}^{\ast}+c_{1})+a_{2}(y_{2}^{\ast}+c_{1})\), \(q=4b_{1}b_{2}(y_{2}^{\ast})^{2}(y_{1}^{\ast}+c_{1})(y_{2}^{\ast}+c_{1})\), \(r=a_{1}a_{2}(y_{1}^{\ast}+c_{1})(y_{2}^{\ast}+c_{1})\).
Since the characteristic equation (2.3) has the same form as equation (2.4) in [30], so by Theorem 2.5 in [30], we can get the following result, which presents the conditions for a Hopf bifurcation to occur in system (1.3).
Proposition 2
Suppose that
\((\mathrm{H}_{1})\)
and
\((\mathrm{H}_{2})\)
hold. Then
$$ \tau^{j}_{k}=1/{\omega_{k}}\bigl[\arccos q/ \bigl({\omega^{2}_{k}r}\bigr)+2j\pi \bigr],\quad k=1,2,3,4,j=0,1,2, \ldots $$
are Hopf bifurcation values at
\(E^{\ast}\), where
\(i\omega _{k}\) (\(k=1,2,3,4\)) are the roots of (2.3). And
\(E^{\ast}\)
is locally asymptotically stable for
\(\tau\in[0,\tau _{1}^{0}]\)
and unstable where
\(\tau>\tau_{1}^{0}\).
Remark 1
The characteristic equation (2.3) has some pairs of purely imaginary roots denoted by \(\lambda=\pm i\omega_{k}\) with \(\tau=\tau^{j}_{k}\) under the condition of \((\mathrm{H}_{1})\), \((\mathrm{H}_{2})\). Define \(\tau^{0}=\tau^{0}_{k_{0}}=\min_{1\leq k\leq4}\{\tau^{0}_{k}\} \), \(\omega_{0}=\omega_{k_{0}}\), where \(k_{0}\in\{1,2,3,4\}\). Then \(\tau^{0}\) is the first value of τ such that (2.3) has purely imaginary roots. For convenience, we denote \(\tau^{j}_{k}\) by \(\tau^{j}\) (\(j=0,1,2,\ldots\)) for fixed \(k\in\{1,2,3,4\}\).
Remark 2
Let \(\lambda(\tau)=\alpha(\tau)\pm i\omega(\tau)\) be the roots of (2.3) near \(\tau=\tau^{j}\) satisfying \(\alpha(\tau^{j})=0\), \(\omega(\tau^{j})=\omega_{0}\) (\(j=0,1,2,\ldots\)). By the theory of DDEs, for \(\forall\tau^{j}_{k}\), \(\exists\varepsilon>0\) s.t. \(\lambda(\tau )\) in \(\tau\tau^{j}_{k}<\varepsilon\) about τ is continuous and differentiable. The transversality condition \(\frac{{d}\operatorname{Re}\lambda(\tau)}{{d}\tau } _{\tau=\tau_{j}}>0\) is satisfied (more details are provided in [30]).
In the previous part, it was shown that system (2.1) undergoes a Hopf bifurcation under certain conditions. Here we will derive explicit formulae determining the direction of the Hopf bifurcation and the stability of the periodic solutions bifurcating from \(E^{\ast}\) at \(\tau^{j}\) (\(j=0,1,2,\ldots\)), by employing centre manifold theory and the normal form method. For convenience, denote \(\tau^{j}\) by τ̃ and \(\tau =\widetilde{\tau}+\mu\), \(\mu\in\mathbb{R}\), then \(\mu=0\) is the Hopf bifurcation value for system (1.3), where \(\widetilde{\tau}=\widetilde{\tau_{2}}+\widetilde{\tau_{3}}\), \(\tau=\widetilde{\tau_{2}}+\widetilde{\tau_{3}}+\mu\). Without loss of generality, assume \(\widetilde{\tau_{2}}<\widetilde{\tau_{3}}\).
The discussion will be divided into five steps as follows.
Step 1. Transform system (
2.1
) into the abstract ODE.
System (2.1) can locally be represented as the following DDE in \(C=C([\widetilde{\tau},0],R^{2})\):
$$ \dot{u}(t)=L_{\mu}(u_{t})+F( \mu,u_{t}), $$
(2.4)
where \(u(t)= (u_{1}(t),u_{2}(t) )^{T}\), \(u_{t}(\theta)=u(t+\theta )\), \(L_{\mu}:C\rightarrow R\) is a bounded linear operator and \(F:R\times C\rightarrow R\) is continuous and differentiable with
$$ \begin{aligned} L_{\mu}\phi&= (\widetilde{ \tau}+\mu) \begin{pmatrix}a_{1}(y_{1}^{\ast}+c_{1})\phi_{1}(\tau _{1})+2b_{1}y_{2}^{\ast}(y_{1}^{\ast}+c_{1})\phi_{2}(\tau_{2})\\ 2b_{2}y_{2}^{\ast}(y_{2}^{\ast}+c_{1})\phi_{1}(\tau _{3})a_{2}(y_{2}^{\ast}+c_{1})\phi_{2}(\tau_{1}) \end{pmatrix} , \end{aligned} $$
and
$$ \begin{aligned} F(\mu,\phi)={}&(\widetilde{\tau}+\mu)\\ &\times\begin{pmatrix}a_{1}\phi_{1}(0)\phi_{1}(\tau_{1})b_{1}\phi_{1}(0)\phi ^{2}_{2}(\tau_{2}) 2b_{1}y^{\ast}_{2}\phi_{1}(0)\phi_{2}(\tau_{2})b_{1}(y^{\ast }_{1}+c_{1})\phi^{2}_{2}(\tau_{2}) \\a_{2}\phi_{2}(0)\phi_{2}(\tau_{1})+b_{2}\phi_{2}(0)\phi ^{1}_{2}(\tau_{3}) +2b_{2}y^{\ast}_{2}\phi_{2}(0)\phi_{1}(\tau_{3})+b_{2}(y^{\ast }_{2}+c_{1})\phi^{2}_{1}(\tau_{3}) \end{pmatrix} ,\end{aligned} $$
where \(\phi=(\phi_{1}(\theta),\phi_{2}(\theta))\in C\).
By the Riesz representation theorem, there exists a \(2\times2\) matrix whose elements are a bounded variation function \(\eta(\theta,\mu)\) in \(\theta\in[\widetilde{\tau},0]\) such that
$$ L_{\mu}\phi= \int_{\widetilde{\tau}}^{0} d\eta(\theta,\mu)\phi(\theta), \quad\phi\in C, $$
where \(\eta(\theta,\mu)\) can be chosen as
$$\eta(\theta,\mu)= \textstyle\begin{cases} (\widetilde{\tau}+\mu)(M+N+P),&\theta\in[\widetilde{\tau }_{2},0),\\ (\widetilde{\tau}+\mu)(M+N),&\theta\in(\widetilde{\tau }_{3},\widetilde{\tau}_{2}),\\ (\widetilde{\tau}+\mu)M,&\theta\in(\widetilde{\tau},\widetilde {\tau}_{3}],\\ 0,&\theta=\widetilde{\tau}, \end{cases} $$
with
$$ \begin{gathered} M= \begin{pmatrix}a_{1}(y_{1}^{\ast}+c_{1})&0\\0&a_{2}(y_{2}^{\ast}+c_{1}) \end{pmatrix} , \qquad N= \begin{pmatrix}0&2b_{1}y_{2}^{\ast}(y_{1}^{\ast}+c_{1})\\0&0 \end{pmatrix} , \\ P= \begin{pmatrix}0&0\\2b_{2}y_{2}^{\ast}(y_{2}^{\ast}+c_{1})&0 \end{pmatrix} . \end{gathered} $$
For \(\phi\in C\), let
$$\begin{gathered} A(\mu)\phi(\theta)= \textstyle\begin{cases} d\phi(\theta)/{d\theta},&\theta\in[\widetilde{\tau},0),\\ \int_{\widetilde{\tau}}^{0}d\eta(\mu,\theta)\phi(\theta),&\theta=0, \end{cases}\displaystyle \\ R(\mu)\phi(\theta)= \textstyle\begin{cases} 0,&\theta\in[\widetilde{\tau},0),\\ F(\mu,\phi),& \theta=0, \end{cases}\displaystyle \end{gathered}$$
then system (2.4) is equivalent to the following abstract operator equation:
$$ \dot{u}(t)=A({\mu})u_{t}+R(\mu)u_{t}. $$
(2.5)
Step 2. Calculate the eigenfunctions of
\(A=A(0)\)
and the adjoint operator
\(A^{\ast}\)
corresponding to
\(i\omega_{0}\widetilde{\tau}\)
and
\(i\omega _{0}\widetilde{\tau}\)
.
For \(\psi\in C([0,\widetilde{\tau}],\mathbb{(}{C}^{2})^{\ast})\), where \(\mathbb{(}{C}^{2})^{\ast}\) is the twodimensional complex space of row vectors, we define the adjoint operator \(A^{\ast}\) of A
$$A^{\ast}\psi(s)= \textstyle\begin{cases} d\psi(s)/{ds},&s\in(0,\widetilde{\tau}],\\\int_{\widetilde {\tau}}^{0}d\eta^{T}(\mu,t)\psi(t),& s=0, \end{cases} $$
and the bilinear form is given by
$$ \bigl\langle \psi(s),\phi(\theta)\bigr\rangle =\overline{\psi}(0)\phi(0)  \int_{\widetilde{\tau}}^{0} \int_{\xi=0}^{\theta}\overline{\psi }^{T}(\xi \theta)\,d\eta(\theta)\phi(\xi)\,d\xi, $$
where \(\eta(\theta)=\eta(\theta,0)\). Then \(A=A(0)\) and \(A^{\ast}(0)\) are adjoint operators.
By [27], \(\pm i\omega_{0}\widetilde{\tau}\) are eigenvalues of \(A(0)\), so they are also eigenvalues of \(A^{\ast}(0)\). Suppose that \(q(\theta)=(1,\alpha)^{T}e^{i\omega_{0}\theta}\) is the eigenfunction of \(A(0)\) corresponding to the eigenvalue \(i\omega _{0}\widetilde{\tau}\) and \(q^{\ast}(s)=G(\beta,1)e^{i\omega_{0}s}\) is the eigenfunction of \(A^{\ast}\) corresponding to the eigenvalue \(i\omega_{0}\widetilde{\tau}\), where
$$\begin{gathered} \alpha= \bigl[i\omega_{0}+a_{2}\bigl(y^{\ast}_{2}+c_{1} \bigr)i\omega _{0}e^{i\omega_{0}\widetilde{\tau}} \bigr]/{2b_{1}y^{\ast}_{2} \bigl(y^{\ast }_{1}+c_{1}\bigr)e^{i\omega_{0}\widetilde{\tau}}}, \\ \beta= \bigl[i\omega_{0}+a_{1}\bigl(y^{\ast}_{1}+c_{1} \bigr)i\omega_{0}e^{i\omega _{0}\widetilde{\tau}} \bigr]/{2b_{1}y^{\ast}_{2} \bigl(y^{\ast }_{1}+c_{1}\bigr)e^{i\omega_{0}\widetilde{\tau}}}, \\ \begin{aligned}G&= \bigl\{ \overline{\beta}+\alpha2\alpha\overline{\beta}b_{1}y^{\ast }_{2} \bigl(y^{\ast}_{1}+c_{1}\bigr)\widetilde{ \tau}_{2}e^{i\omega_{0}\widetilde {\tau}_{2}} +2\alpha\overline{\beta}b_{2}y^{\ast}_{2} \bigl(y^{\ast}_{2}+c_{1}\bigr)\widetilde { \tau}_{3}e^{i\omega_{0}\widetilde{\tau}_{3}} \\ &\quad{}+ \bigl[\overline{\beta}a_{1}\bigl(y^{\ast}_{1}+c_{1} \bigr)\alpha _{2}\bigl(y^{\ast}_{2}+c_{1} \bigr) \bigr]\widetilde{\tau}e^{i\omega_{0}\widetilde {\tau}} \bigr\} ^{1},\end{aligned}\end{gathered} $$
which assures that \(\langle{q}^{\ast}(s),q(\theta)\rangle=1\), \(\langle{q}^{\ast }(s),\overline{q}(\theta)\rangle=0\).
Step 3. Obtain the reduced system on the centre manifold.
In this part, we will use the same notations as in [28] and compute the coordinates to describe the centre manifold \(\mathbf {C}_{0}\) at \(\mu=0\) (a local centre manifold is in general not unique, and the dimension of local centre manifold is 2). Let \(u_{t}\in C\) be the solution of system (2.5) when \(\mu=0\), and define
$$ z(t)=\bigl\langle q^{\ast},u_{t}\bigr\rangle ,\qquad W(t,\theta)=u_{t}( \theta)z(t)q(\theta)\overline {z}(t)\overline{q}(\theta), $$
(2.6)
where z and z̅ are local coordinates for the centre manifold \(\mathbf{C}_{0}\) in the direction of \(q^{\ast}\) and \(\overline{q}^{\ast}\). On the centre manifold \(\mathbf{C}_{0}\), we have \(W(t,\theta )=W(z(t),\overline{z}(t),\theta)\), where
$$\begin{aligned} W(z,\overline{z},\theta)=W_{20}(\theta)z^{2}/{2}+W_{11}( \theta )z\overline{z}+W_{02}(\theta)\overline{z}^{2}/{2}+ \cdots. \end{aligned}$$
(2.7)
The existence of a centre manifold enables us to reduce (2.5) to an ODE on \(\mathbf{C}_{0}\). Note that W is real if \(u_{t}\) is real, we consider only real solutions. For solution \(u_{t}\in\mathbf{C}_{0}\) of system (2.5) at \(\mu=0\),
$$ \begin{aligned}[b] \dot{z}(t)&=\bigl\langle q^{\ast},\dot{u}_{t}\bigr\rangle =\bigl\langle q^{\ast},A(u_{t})+R(u_{t})\bigr\rangle =\bigl\langle A^{\ast}\bigl(q^{\ast}\bigr),u_{t}\bigr\rangle +\bigl\langle q^{\ast},R(u_{t})\bigr\rangle \\ &=i\omega_{0}\widetilde{\tau}z(t)+\overline{q}^{\ast}(0)\cdot f\bigl(0,W(z,\overline{z},\theta)+2\operatorname{Re}\bigl\{ z(t)q(\theta)\bigr\} \bigr) \\ &=i\omega_{0}\widetilde{\tau}z(t)+\overline{q}^{\ast}(0)\cdot f(0,u_{t}),\end{aligned} $$
(2.8)
with
$$ \overline{q}^{\ast}(0)\cdot f(0,u_{t})\triangleq g(z,\overline {z}). $$
Rewriting (2.8), we obtain that the reduced system on \(\mathbf {C}_{0}\) is described by
$$ \dot{z}(t)=i\omega_{0}\widetilde{\tau}z(t)+g(z, \overline{z}), $$
(2.9)
where
$$ g(z,\overline{z})=g_{20}(\theta)z^{2}/{2}+g_{11}( \theta)z\overline{z} +g_{02}(\theta)\overline{z}^{2}/{2}+g_{21}( \theta)z^{2}\overline {z}/{2}+\cdots. $$
(2.10)
We will mainly discuss equation (2.9) in the following part.
Step 4. Obtain the values of
\(g_{20}\)
,
\(g_{11}\)
,
\(g_{02}\)
,
\(g_{21}\)
in (
2.10
).
In this part, we calculate the coefficients \(W_{20}(\theta)\), \(W_{11}(\theta)\), \(W_{02}(\theta)\), … and substitute them in (2.8) to get the reduced system (2.9) on \(\mathbf{C}_{0}\).
It follows from (2.6) that
$$ \begin{aligned}[b] u_{t}(\theta)&=u(t+ \theta)=W(t,\theta)+2\operatorname{Re}\bigl\{ z(t),q(\theta)\bigr\} \\ &=W_{20}(\theta)z^{2}/{2}+W_{11}(\theta)z \overline{z}+W_{02}(\theta )\overline{z}^{2}/{2} +(1, \alpha)^{T}e^{i\omega_{0}\widetilde{\tau}\theta}z \\ &\quad{}+(1,\overline{\alpha})^{T}e^{i\omega_{0}\widetilde{\tau}\theta }\overline{z}+ \cdots. \end{aligned} $$
And we have
$$ \begin{gathered} u_{1}(t)=z+ \overline{z}+W^{(1)}(t,0),\qquad u_{2}(t)=z\alpha+\overline{z} \overline{\alpha}+W^{(2)}(t,0), \\ u_{1}(t\widetilde{\tau})=ze^{i\omega_{0}\widetilde{\tau}}+\overline {z}e^{i\omega_{0}\widetilde{\tau}}+W^{(1)}(\widetilde{\tau}), \\ u_{2}(t\widetilde{\tau})=z\alpha e^{i\omega_{0}\widetilde{\tau }}+\overline{z} \overline{\alpha}e^{i\omega_{0}\widetilde{\tau }}+W^{(2)}(\widetilde{\tau}), \\ u_{2}(t\widetilde{\tau}_{2})=z\alpha e^{i\omega_{0}\widetilde{\tau}_{2}}+ \overline{z} \overline{\alpha}e^{i\omega_{0}\widetilde{\tau }_{2}}+W^{(2)}(\widetilde{ \tau}_{2}), \\ u_{1}(t\widetilde{\tau}_{3})=ze^{i\omega_{0}\widetilde{\tau }_{3}}+ \overline{z}e^{i\omega_{0}\widetilde{\tau }_{3}}+W^{(1)}(\widetilde{\tau}_{3}), \\ u_{2}(t\widetilde{\tau}_{3})=z\alpha e^{i\omega_{0}\widetilde{\tau}_{3}}+ \overline{z} \overline{\alpha}e^{i\omega_{0}\widetilde{\tau }_{3}}+W^{(2)}(\widetilde{ \tau}_{3}). \end{gathered} $$
(2.11)
It follows that together with \(F(\mu,\phi)\) we get
$$ \begin{gathered} f(0,u_{t}) =\widetilde{\tau} \begin{pmatrix}a_{1}u_{1}(t)u_{1}(t\widetilde{\tau })b_{1}u_{1}(t)u^{2}_{2}(t\widetilde{\tau}_{2})2b_{1}y^{\ast }_{2}u_{1}(t)u_{2}(t\widetilde{\tau}_{2})b_{1}(y^{\ast }_{1}+c_{1})u^{2}_{2}(t\widetilde{\tau}_{2}) \\a_{2}u_{2}(t)u_{2}(t\widetilde{\tau })+b_{2}u_{2}(t)u^{1}_{2}(t\widetilde{\tau}_{3})+2b_{2}y^{\ast }_{2}u_{2}(t)u_{1}(t\widetilde{\tau}_{3})+b_{2}(y^{\ast }_{2}+c_{1})u^{2}_{1}(t\widetilde{\tau}_{3}) \end{pmatrix} .\end{gathered} $$
(2.12)
Substituting (2.11) into (2.12), then this substitution into (2.8), and comparing the coefficients with (2.10), we obtain
$$ \begin{aligned} g_{20} &=2\widetilde{\tau} \overline{G} \bigl\{ \overline{\beta} \bigl[a_{1}e^{i\omega_{0}\widetilde{\tau}}2b_{1}y^{\ast}_{2}{ \alpha }e^{i\omega_{0}\widetilde{\tau_{2}}}b_{1}\bigl(y^{\ast}_{1}+c_{1} \bigr){\alpha }^{2}e^{2i\omega_{0}\widetilde{\tau_{2}}} \bigr] \\ &\quad{}a_{2}{\alpha}^{2}e^{i\omega_{0}\widetilde{\tau}}+2b_{2}y^{\ast }_{2}{ \alpha}e^{i\omega_{0}\widetilde{\tau_{3}}}+b_{2}\bigl(y^{\ast }_{2}+c_{1} \bigr)e^{2i\omega_{0}\widetilde{\tau}} \bigr\} , \\ g_{11} &=\widetilde{\tau}\overline{G} \bigl\{ \overline{\beta} \bigl[a_{1}\bigl(e^{i\omega _{0}\widetilde{\tau}} +e^{i\omega_{0}\widetilde{\tau}} \bigr)2b_{1}y^{\ast}_{2}\bigl(\overline{\alpha} e^{i\omega_{0}\widetilde{\tau}}+\alpha e^{i\omega_{0}\widetilde{\tau }}\bigr)2b_{1} \bigl(y^{\ast}_{1}+c_{1}\bigr)\alpha\overline{\alpha} \bigr] \\ &\quad{}a_{2}\alpha\overline{\alpha}\bigl(e^{i\omega_{0}\widetilde{\tau }}+e^{i\omega_{0}\widetilde{\tau}} \bigr) +2b_{2}y^{\ast}_{2}\bigl(\alpha e^{i\omega_{0}\widetilde{\tau_{3}}}+\overline {\alpha} e^{i\omega_{0}\widetilde{\tau_{3}}}\bigr)+2b_{2} \bigl(y^{\ast }_{2}+c_{1}\bigr) \bigr\} , \\ g_{02} &=2\widetilde{\tau}\overline{G} \bigl\{ \overline{\beta} \bigl[a_{1}e^{i\omega_{0}\widetilde{\tau}}2b_{1}y^{\ast}_{2} \overline {\alpha}e^{i\omega_{0}\widetilde{\tau}} b_{1}\bigl(y^{\ast}_{1}+c_{1} \bigr)\overline{\alpha}^{2}e^{2i\omega_{0}\widetilde {\tau}} \bigr] a_{2} \overline{\alpha}^{2}e^{i\omega_{0}\widetilde{\tau}} \\ &\quad{}+2b_{2}y^{\ast}_{2}\overline{ \alpha}e^{i\omega_{0}\widetilde {\tau}}+b_{2}\bigl(y^{\ast}_{2}+c_{1} \bigr) \bigr\} , \\ g_{21} &=\widetilde{\tau}\overline{G} \bigl\{ \overline{\beta} \bigl[a_{1} \bigl(W^{(1)}_{11}(\widetilde{ \tau})+W^{(1)}_{20}(\widetilde{\tau})/{2} +e^{i\omega_{0}\widetilde{\tau}}W^{(1)}_{11}(0) +e^{i\omega_{0}\widetilde{\tau}}W^{(1)}_{20}(0)/{2} \bigr) \\ &\quad{}b_{1}\bigl(2\alpha\overline{\alpha}+\overline{ \alpha}^{2}e^{2i\omega _{0}\widetilde{\tau_{{2}}}}\bigr) 2b_{1}y^{\ast}_{2} \bigl(W^{(2)}_{11}(\widetilde{\tau })+W^{(2)}_{20}( \widetilde{\tau})/{2}+\alpha e^{i\omega_{0}\widetilde {\tau}} W^{(1)}_{11}(0) \\ &\quad{}+\overline{\alpha}e^{i\omega_{0}\widetilde{\tau }}W^{(1)}_{20}(0)/{2} \bigr) b_{1}\bigl(y^{\ast}_{1}+c_{1}\bigr) \bigl(\alpha e^{i\omega_{0}\widetilde{\tau }}W^{(2)}_{11}(\widetilde{\tau}) + \overline{\alpha}e^{i\omega_{0}\widetilde{\tau }}W^{(2)}_{20}(\widetilde{ \tau})/{2} \bigr) \bigr] \\ &\quad{}a_{2} \bigl(\alpha W^{(2)}_{11}( \widetilde{\tau})+\overline {\alpha}W^{(2)}_{20}(\widetilde{ \tau})/{2}+\alpha e^{i\omega _{0}\widetilde{\tau}} W^{(2)}_{11}(0)+\overline{ \alpha}e^{i\omega _{0}\widetilde{\tau}}W^{(2)}_{20}(0)/{2} \bigr) \\ &\quad{}+b_{2}\bigl(2\alpha+e^{2i\omega_{0}\widetilde{\tau}}\bigr)+2b_{2}y^{\ast }_{2} \bigl[\alpha W^{(1)}_{11}(\widetilde{\tau })+W^{(1)}_{20}(\widetilde{\tau})/2+e^{i\omega_{0}\widetilde{\tau }}W^{(1)}_{11}(0) \\ &\quad{}+e^{i\omega_{0}\widetilde{\tau}}W^{(1)}_{20}(0) \bigr] +b_{2}\bigl(y^{\ast}_{2}+c_{1}\bigr) \bigl[2e^{i\omega_{0}\widetilde{\tau }}W^{(1)}_{11}(\widetilde{ \tau_{3}}) +e^{i\omega_{0}\widetilde{\tau}}W^{(1)}_{20}( \widetilde{\tau_{3}}) \bigr] \bigr\} . \end{aligned} $$
(2.13)
Since there are \(W_{20}(\theta)\) and \(W_{11}(\theta)\) in \(g_{21}\), we still need to compute them.
From (2.5) and (2.6), we have
$$ \begin{aligned} \dot{W}=\dot{u}_{t} \dot{z}q\dot{\overline{z}} \overline{q}&= \textstyle\begin{cases} AW2\operatorname{Re}\{gq(\theta)\},&\theta\in[1,0),\\ AW2\operatorname {Re}\{gq(0)\}+f_{0},&\theta=0, \end{cases}\displaystyle \end{aligned} $$
(2.14)
where
$$ \begin{aligned} f_{0}=f_{z^{2}}z^{2}/{2}+f_{z \overline{z}}z \overline{z}+f_{\overline {z}^{2}}\overline{z}^{2}/{2}+f_{z^{2}\overline{z}}z^{2} \overline {z}/{2}\cdots. \end{aligned} $$
On the other hand, near the origin, on the centre manifold \(\mathbf {C}_{0}\), according to (2.7), we obtain
$$ \begin{aligned}[b] \dot{W}&=W_{z}\dot{z}+W_{\overline{z}}\dot{ \overline{z}} = \bigl[W_{20}(\theta)z+W_{11}(\theta) \overline{z} \bigr]\dot{z}+ \bigl[W_{11}(\theta)z+W_{02}( \theta)\overline{z} \bigr]\dot{\overline {z}} \\ &= \bigl[W_{20}(\theta)z+W_{11}(\theta)\overline{z} \bigr] \bigl(i\omega _{0}z+g(z,\overline{z}) \bigr) \\ &\quad{}+ \bigl[W_{11}(\theta)z+W_{02}(\theta)\overline{z} \bigr] \bigl(\overline{g}(z,\overline{z})i\omega_{0}\overline{z} \bigr)+\cdots .\end{aligned} $$
(2.15)
Substituting (2.7) into the righthand side of (2.14), equating terms of \(\frac{z^{2}}{2}\) and zz̅ of (2.14) with (2.15), we obtain
$$\begin{aligned}& (2i\omega_{0}IA)W_{20}(\theta)= \textstyle\begin{cases} g_{20}q(\theta)\overline{g}_{02}\overline{q}(\theta),&\theta\in [\widetilde{\tau},0),\\ g_{20}q(0)\overline{g}_{02}\overline {q}(0)+f_{z^{2}},&\theta=0, \end{cases}\displaystyle \end{aligned}$$
(2.16)
$$\begin{aligned}& AW_{11}(\theta)= \textstyle\begin{cases} g_{11}q(\theta)\overline{g}_{11}\overline{q}(\theta),&\theta\in [\widetilde{\tau},0),\\ g_{11}q(0)\overline{g}_{11}\overline{q}(0)+f_{z \overline{z}},& \theta=0. \end{cases}\displaystyle \end{aligned}$$
(2.17)
According to the definition of A and from (2.16), (2.17) for \(\theta\in[\widetilde{\tau},0)\), we get
$$\begin{gathered} \dot{W}_{20}(\theta)=2i\omega_{0} W_{20}( \theta)+g_{20}q(\theta )+\overline{g}_{02}\overline{q}( \theta), \\ \dot{W}_{11}(\theta)=g_{11}q(\theta)+\overline{g}_{11} \overline {q}(\theta).\end{gathered} $$
Solving for \({W}_{20}(\theta)\) and \({W}_{11}(\theta)\), we obtain
$$\begin{aligned}& {W}_{20}(\theta)=ig_{20}/{\omega_{0}}\cdot q(0)e^{i\omega_{0}\theta} +i\overline{g}_{02}/{3\omega_{0}}\cdot \overline{q}(0)e^{i\omega _{0}\theta} +E_{1}e^{2i\omega_{0}\theta}, \end{aligned}$$
(2.18)
$$\begin{aligned}& {W}_{20}(\theta)=ig_{11}/{\omega_{0}}\cdot q(0)e^{i\omega_{0}\theta} +i\overline{g}_{11}/{\omega_{0}}\cdot \overline{q}(0)e^{i\omega _{0}\theta} +E_{2}, \end{aligned}$$
(2.19)
where \(E_{1}=(E^{(1)}_{1},E^{(2)}_{1})^{T}\in R^{2}\) and \(E_{2}=(E^{(1)}_{2},E^{(2)}_{2})^{T}\in R^{2}\) are constant vectors.
In what follows we shall seek appropriate \(E_{1}\) and \(E_{2}\) in (2.18) and (2.19), respectively. According to the definition of A and (2.16), (2.17) for \(\theta=0\), we have
$$\begin{aligned}& \int_{\widetilde{\tau}}^{0}d\eta(\theta){W}_{20}( \theta)=2i\omega_{0} W_{20}(0)+g_{20}q(0)+ \overline{g}_{02}\overline{q}(0)f_{z^{2}}, \end{aligned}$$
(2.20)
$$\begin{aligned}& \int_{\widetilde{\tau}}^{0}d\eta(\theta){W}_{11}(\theta )=g_{11}q(0)+\overline{g}_{11}\overline{q}(0)f_{z \overline{z}}, \end{aligned}$$
(2.21)
where \(\eta(\theta)=\eta(0,\theta)\) and
$$\begin{gathered} f_{z^{2}}= \begin{pmatrix}a_{1}e^{i\omega_{0}\widetilde{\tau}}2b_{1}y^{\ast }_{2}{\alpha}e^{i\omega_{0}\widetilde{\tau_{2}}}b_{1}(y^{\ast }_{1}+c_{1}){\alpha}^{2}e^{2i\omega_{0}\widetilde{\tau_{2}}}\\ a_{2}{\alpha}^{2}e^{i\omega_{0}\widetilde{\tau}}+2b_{2}y^{\ast }_{2}{\alpha}e^{i\omega_{0}\widetilde{\tau_{3}}}+b_{2}(y^{\ast }_{2}+c_{1})e^{2i\omega_{0}\widetilde{\tau}} \end{pmatrix} , \\ f_{z\overline{z}}= \begin{pmatrix} a_{1}(e^{i\omega_{0}\widetilde{\tau}} +e^{i\omega_{0}\widetilde{\tau}})2b_{1}y^{\ast}_{2}(\overline{\alpha} e^{i\omega_{0}\widetilde{\tau}}+\alpha e^{i\omega_{0}\widetilde{\tau }})2b_{1}(y^{\ast}_{1}+c_{1})\alpha\overline{\alpha}\\ a_{2}\alpha\overline{\alpha}(e^{i\omega_{0}\widetilde{\tau }}+e^{i\omega_{0}\widetilde{\tau}}) +2b_{2}y^{\ast}_{2}(\alpha e^{i\omega_{0}\widetilde{\tau_{3}}}+\overline {\alpha} e^{i\omega_{0}\widetilde{\tau_{3}}})+2b_{2}(y^{\ast}_{2}+c_{1}) \end{pmatrix} . \end{gathered} $$
Substituting (2.18) into (2.20), we obtain
$$ \biggl(2i\omega_{0} I \int_{\widetilde{\tau}}^{0}e^{2i\omega_{0}\theta}\, d\eta(\theta) \biggr) E_{1} =f_{z^{2}}, $$
that is
$$ \begin{pmatrix}2i\omega_{0}+a_{1}(y^{\ast}_{1}+c_{1})e^{i\omega _{0}\widetilde{\tau}} &2b_{1}y^{\ast}_{2}(y^{\ast}_{1}+c_{1})e^{i\omega_{0}\widetilde{\tau _{2}}}\\ 2b_{2}y^{\ast}_{2}(y^{\ast}_{2}+c_{1})e^{i\omega_{0}\widetilde{\tau_{3}}} &2i\omega_{0}+a_{2}(y^{\ast}_{2}+c_{1})e^{i\omega_{0}\widetilde{\tau}} \end{pmatrix} E_{1}=f_{z^{2}}. $$
(2.22)
Similarly, substituting (2.19) into (2.21), we get
$$ \int_{\widetilde{\tau}}^{0}d\eta(\theta)E_{2}=f_{z\overline {z}}, $$
that is
$$ \begin{pmatrix}a_{1}(y^{\ast}_{1}+c_{1})e^{i\omega_{0}\widetilde{\tau}} &2b_{1}y^{\ast}_{2}(y^{\ast}_{1}+c_{1})e^{i\omega_{0}\widetilde{\tau _{2}}}\\ 2b_{2}y^{\ast}_{2}(y^{\ast}_{2}+c_{1})e^{i\omega_{0}\widetilde{\tau_{3}}} &a_{2}(y^{\ast}_{2}+c_{1})e^{i\omega_{0}\widetilde{\tau}} \end{pmatrix} E_{2}=f_{z\overline{z}}. $$
(2.23)
We have obtained the values of \(E_{1}\) and \(E_{2}\) as (2.22) and (2.23) and, ultimately, the reduced system (2.9).
Step 5. Obtain the key values
\(\mu_{2}\)
,
\(\beta_{2}\)
,
\(T_{2}\)
to determine the property of the Hopf bifurcation.
As with the calculation of the ODE Hopf bifurcation parameter and as in [28], according to the analysis above and the expressions of \(g_{20}\), \(g_{11}\), \(g_{02}\) and \(g_{21}\), we can compute the following values:
$$ \begin{gathered}c_{1}(0)=i/{2\omega_{0}\widetilde{\tau}} \bigl(g_{11}g_{20}2 \vert g_{11} \vert ^{2} \vert g_{02} \vert ^{2}/{3} \bigr)+g_{21}/{2}, \\ \mu_{2}=\operatorname{Re}\bigl\{ c_{1}(0)\bigr\} /{ \alpha'(\widetilde{\tau})}, \\ \beta_{2}=2\operatorname{Re}\bigl\{ c_{1}(0)\bigr\} , \\ T_{2}= \bigl[\operatorname{Im}\bigl\{ c_{1}(0)\bigr\} + \mu_{2}\omega'(\widetilde{\tau }) \bigr]/{ \omega_{0}},\end{gathered} $$
(2.24)
where \(\lambda(\tau)=\alpha(\tau)\pm i\omega(\tau)\) is the characteristic root of (2.3), which is a continuous differentiable family. \(\alpha'(\widetilde{\tau })\) and \(\omega'(\widetilde{\tau})\) can be obtained by taking the derivative of the two sides of (2.3) and taking values at τ̃.
These formulae give a description of the Hopf bifurcation periodic solution of system (1.3) at \(\tau=\tau^{j}\) (\(j=0,1,2,\ldots\)) on the centre manifold. Thus, we can obtain the following results according to the discussion about properties of Hopf bifurcating periodic solutions of dynamical system in [30].
Proposition 3
Assume that
\((\mathrm{H}_{1})\)
and
\((\mathrm{H}_{2})\)
hold. Then

(i)
\(\mu_{2}\)
determines the direction of the Hopf bifurcation. If
\(\mu_{2}>0\) (\(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (subcritical);

(ii)
\(\beta_{2}\)
determines the stability of the bifurcating periodic solutions. If
\(\beta_{2}<0\) (\(\beta_{2}>0\)), then bifurcating periodic solution is stable (unstable);

(iii)
\(T_{2}\)
determines the period of the bifurcating periodic solutions. If
\(T_{2}>0\) (\(T_{2}<0\)), then periods of the periodic solutions increase (decrease).