It is easy to show that if \(a_{2}r_{2}>md_{4}(r_{2}+d_{3})\) and \(\frac{rr_{1}}{r_{1}+d_{1}}>d_{2}+\frac{ad_{4}(r_{2}+d_{3})}{a_{2}r_{2}-md_{4}(r_{2}+d_{3})}\), then system (4) has a unique positive equilibrium \(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\), where
$$\begin{aligned}& x_{1}^{*}=\frac{rx_{2}^{*}}{r_{1}+d_{1}},\qquad x_{2}^{*}=\frac {d_{4}(r_{2}+d_{3})}{a_{2}r_{2}-md_{4}(r_{2}+d_{3})},\\& y_{1}^{*}=\frac{d_{4}y_{2}^{*}}{r_{2}},\qquad y_{2}^{*}=\frac {(1+mx_{2}^{*})(r_{1}x_{1}^{*}-d_{2}x_{2}^{*}-a(x_{2}^{*})^{2})}{a_{1}x_{2}^{*}}. \end{aligned}$$
Let \(\bar{x}_{1}(t)=x_{1}(t)-x_{1}^{*}\), \(\bar{x}_{2}(t)=x_{2}(t)-x_{2}^{*}\), \(\bar{y}_{1}(t)=y_{1}(t)-y_{1}^{*}\), \(\bar{y}_{2}(t)=y_{2}(t)-y_{2}^{*}\). Dropping the bars for convenience, system (4) gets the following form:
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{dx_{1}(t)}{dt}=a_{11}x_{1}(t)+a_{12}x_{2}(t), \\ \frac{dx_{2}(t)}{dt}=a_{21}x_{1}(t)+a_{22}x_{2}(t)+a_{24}y_{2}(t)+b_{22}x_{2}(t-\tau _{1})+f_{2}, \\ \frac{dy_{1}(t)}{dt}=a_{33}y_{1}(t)+c_{32}x_{2}(t-\tau_{2})+c_{34}y_{2}(t-\tau _{2})+f_{3}, \\ \frac{dy_{2}(t)}{dt}=a_{43}y_{1}(t)+a_{44}y_{2}(t), \end{array}\displaystyle \right . $$
(5)
where
$$\begin{aligned}& a_{11}=-(d_{1}+r_{1}),\qquad a_{12}=r,\qquad a_{21}=r_{1},\qquad a_{22}=-d_{2}-ax_{2}^{*}- \frac{a_{1}y_{2}^{*}}{(1+mx_{2}^{*})^{2}},\\& a_{24}=-\frac{a_{1}x_{2}^{*}}{1+mx_{2}^{*}},\qquad a_{33}=-(d_{3}+r_{2}),\qquad a_{43}=r_{2},\qquad a_{44}=-d_{4},\\& b_{22}=-ax_{2}^{*},\qquad c_{32}=\frac{a_{2}y_{2}^{*}}{(1+mx_{2}^{*})^{2}},\qquad c_{34}=\frac{a_{2}x_{2}^{*}}{1+mx_{2}^{*}}, \end{aligned}$$
and
$$\begin{aligned}& f_{2}=a_{25}x_{2}^{2}(t)+a_{26}x_{2}(t)y_{2}(t)+a_{27}x_{2}(t)x_{2}(t- \tau_{1})\\& \hphantom{f_{2}=}{}+a_{28}x_{2}^{2}(t)y_{2}(t)+a_{29}x_{2}^{3}(t)+ \cdots,\\& f_{3}=a_{34}x_{2}^{2}(t- \tau_{2})+a_{35}x_{2}(t-\tau_{2})y_{2}(t- \tau_{2})\\& \hphantom{f_{3}=}{}+a_{36}x_{2}^{2}(t-\tau_{2})y_{2}(t- \tau_{2})+a_{37}x_{2}^{3}(t- \tau_{2})+\cdots, \end{aligned}$$
with
$$\begin{aligned}& a_{25}=\frac{ma_{1}y_{2}^{*}}{(1+mx_{2}^{*})^{3}},\qquad a_{26}=-\frac {a_{1}}{(1+mx_{2}^{*})^{2}},\qquad a_{27}=-a,\\& a_{28}=\frac{ma_{1}}{(1+mx_{2}^{*})^{3}},\qquad a_{29}=-\frac {m^{2}a_{1}y_{2}^{*}}{(1+mx_{2}^{*})^{4}},\\& a_{34}=-\frac{ma_{2}y_{2}^{*}}{(1+mx_{2}^{*})^{3}},\qquad a_{35}=\frac {a_{2}}{(1+mx_{2}^{*})^{2}},\\& a_{36}=-\frac{ma_{2}}{(1+mx_{2}^{*})^{3}},\qquad a_{37}=\frac {m^{2}a_{2}y_{2}^{*}}{(1+mx_{2}^{*})^{4}}. \end{aligned}$$
The linearized system of (5) is
$$ \left \{ \textstyle\begin{array}{@{}l} \frac{dx_{1}(t)}{dt}=a_{11}x_{1}(t)+a_{12}x_{2}(t), \\ \frac {dx_{2}(t)}{dt}=a_{21}x_{1}(t)+a_{22}x_{2}(t)+a_{24}y_{2}(t)+b_{22}x_{2}(t-\tau _{1}), \\ \frac{dy_{1}(t)}{dt}=a_{33}y_{1}(t)+c_{32}x_{2}(t-\tau_{2})+c_{34}y_{2}(t-\tau _{2}), \\ \frac{dy_{2}(t)}{dt}=a_{43}y_{1}(t)+a_{44}y_{2}(t). \end{array}\displaystyle \right . $$
(6)
The characteristic equation of system (6) at the positive equilibrium \(E^{*}\) is of the form
$$\begin{aligned} &\lambda^{4}+A_{3}\lambda^{3}+A_{2} \lambda^{2}+A_{1}\lambda+A_{0}+ \bigl(B_{3}\lambda^{3}+B_{2}\lambda^{2}+B_{1} \lambda+B_{0}\bigr)e^{-\lambda\tau_{1}} \\ &\quad{}+\bigl(C_{2}\lambda^{2}+C_{1} \lambda+C_{0}\bigr)e^{-\lambda\tau_{2}}+(D_{1}\lambda +D_{0})e^{-\lambda(\tau_{1}+\tau_{2})}=0, \end{aligned}$$
(7)
where
$$\begin{aligned}& A_{0}=(a_{11}a_{22}-a_{12}a_{21})a_{33}a_{44},\\& A_{1}=(a_{12}a_{21}-a_{11}a_{22}) (a_{33}+a_{44})-a_{33}a_{44}(a_{11}+a_{22}),\\& A_{2}=a_{11}a_{22}+a_{33}a_{44}-a_{12}a_{21}+(a_{11}+a_{22}) (a_{33}+a_{44}),\\& A_{3}=-(a_{11}+a_{22}+a_{33}+a_{44}),\\& B_{0}=a_{11}a_{33}a_{44}b_{22},\qquad B_{1}=-(a_{11}a_{33}+a_{11}a_{44}+a_{33}a_{44})b_{22},\\& B_{2}=(a_{11}+a_{33}+a_{44})b_{22},\qquad B_{3}=-b_{22},\\& C_{0}=(a_{12}a_{21}-a_{11}a_{22})a_{43}c_{34}+a_{11}a_{24}a_{43}c_{32},\\& C_{1}=a_{43}c_{34}(a_{11}+a_{22})-a_{24}a_{43}c_{32},\qquad C_{2}=-a_{43}c_{34},\\& D_{0}=-a_{11}a_{43}b_{22}c_{34},\qquad D_{1}=a_{43}b_{22}c_{34}. \end{aligned}$$
Case 1. \(\tau_{1}=\tau_{2}=0\). Equation (7) becomes
$$ \lambda^{4}+A_{13}\lambda^{3}+A_{12} \lambda^{2}+A_{11}\lambda+A_{10}=0, $$
(8)
where
$$\begin{aligned}& A_{10}=A_{0}+B_{0}+C_{0}+D_{0},\\& A_{11}=A_{1}+B_{1}+C_{1}+D_{1},\\& A_{12}=A_{2}+B_{2}+C_{2},\qquad A_{13}=A_{3}+B_{3}. \end{aligned}$$
Obviously, \(\operatorname{det}_{1}=A_{13}=d_{1}+d_{2}+d_{3}+d_{4}+r_{1}+r_{2}+2ax_{2}^{*}+\frac {a_{1}y_{2}^{*}}{(1+mx_{2}^{*})^{2}}>0\). Thus, all roots of (8) have negative real parts if the condition (H1): (9) is satisfied. We have
$$ \begin{aligned} &\operatorname{det}_{2}= \begin{vmatrix} A_{13}&1\\ A_{11}&A_{12} \end{vmatrix}>0,\qquad \operatorname{det}_{3}= \begin{vmatrix} A_{13}&1&0\\ A_{11}&A_{12}&A_{13}\\ 0&A_{10}&A_{11} \end{vmatrix}>0,\\ &\operatorname{det}_{4}= \begin{vmatrix} A_{13}&1&0&0\\ A_{11}&A_{12}&A_{13}&1\\ 0&A_{10}&A_{11}&A_{12}\\ 0&0&0&A_{10} \end{vmatrix}>0. \end{aligned} $$
(9)
Thus, the positive equilibrium of system (4) without delay is locally asymptotically stable under the condition (H1): (9) holds.
Case 2. \(\tau_{1}>0\), \(\tau_{2}=0\).
When \(\tau_{2}=0\), (7) becomes
$$ \lambda^{4}+A_{23}\lambda^{3}+A_{22} \lambda^{2}+A_{21}\lambda+A_{20}+ \bigl(B_{23}\lambda^{3}+B_{22}\lambda^{2}+B_{21} \lambda+B_{20}\bigr)e^{-\lambda\tau_{1}}=0, $$
(10)
where
$$\begin{aligned}& A_{20}=A_{0}+C_{0}, \qquad A_{21}=A_{1}+C_{1},\qquad A_{22}=A_{2}+C_{2},\qquad A_{23}=A_{3},\\& B_{23}=B_{3},\qquad B_{22}=B_{2},\qquad B_{21}=B_{1}+D_{1},\qquad B_{20}=B_{0}+D_{0}. \end{aligned}$$
Let \(\lambda=i\omega_{1}\) (\(\omega_{1}>0\)) be a root of (10). Then
$$\left \{ \textstyle\begin{array}{@{}l} (B_{21}\omega_{1}-B_{23}\omega_{1}^{3})\sin\omega_{1}\tau _{1}+(B_{20}-B_{22}\omega_{1}^{2})\cos\omega_{1}\tau_{1}=A_{22}\omega _{1}^{2}-\omega _{1}^{4}-A_{20}, \\ (B_{21}\omega_{1}-B_{23}\omega_{1}^{3})\cos\omega_{1}\tau _{1}-(B_{20}-B_{22}\omega_{1}^{2})\sin\omega_{1}\tau_{1}=A_{23}\omega _{1}^{3}-A_{21}\omega_{1}, \end{array}\displaystyle \right . $$
from which it follows that
$$ \omega_{1}^{8}+e_{23} \omega_{1}^{6}+e_{22}\omega_{1}^{4}+e_{21} \omega_{1}^{2}+e_{20}=0, $$
(11)
where
$$\begin{aligned}& e_{20}=A_{20}^{2}-B_{20}^{2},\qquad e_{21}=A_{21}^{2}-B_{21}^{2}-2A_{20}A_{22}+2B_{20}B_{22},\\& e_{22}=A_{22}^{2}-B_{22}^{2}+2A_{20}-2A_{21}A_{23}+2B_{21}B_{23},\qquad e_{23}=A_{23}^{2}-B_{23}^{2}-2A_{22}. \end{aligned}$$
Let \(\omega_{1}^{2}=v_{1}\), then (11) becomes
$$ v_{1}^{4}+e_{23}v_{1}^{3}+e_{22}v_{1}^{2}+e_{21}v_{1}+e_{20}=0. $$
(12)
Discussion of the roots of (12) is similar to that in [22]. Denote
$$ f_{1}(v_{1})=v_{1}^{4}+e_{23}v_{1}^{3}+e_{22}v_{1}^{2}+e_{21}v_{1}+e_{20}. $$
(13)
Clearly, if \(e_{20}<0\), then (12) has at least one positive root. From (13), one can get
$$f_{1}^{\prime}(v_{1})=4v_{1}^{3}+3e_{23}v_{1}^{2}+2e_{22}v_{1}+e_{21}. $$
Set
$$ 4v_{1}^{3}+3e_{23}v_{1}^{2}+2e_{22}v_{1}+e_{21}=0. $$
(14)
Let \(y_{1}=v_{1}+\frac{3e_{23}}{4}\). Then (14) becomes
$$y_{1}^{3}+p_{1}y_{1}+q_{1}=0, $$
where
$$p_{1}=\frac{e_{22}}{2}-\frac{3}{16}e_{23}^{2},\qquad q_{1}=\frac{e_{23}^{3}}{32}-\frac {e_{22}e_{23}}{8}+e_{21}. $$
Define
$$\begin{aligned}& \alpha_{1}=\biggl(\frac{q_{1}}{2}\biggr)^{2}+\biggl( \frac{p_{1}}{3}\biggr)^{3},\qquad \beta_{1}=\frac{-1+\sqrt {3}i}{2},\\& y_{11}=\sqrt[3]{-\frac{q_{1}}{2}+\sqrt{\alpha_{1}}}+ \sqrt[3]{-\frac{q_{1}}{2}-\sqrt{\alpha_{1}}},\\& y_{12}=\sqrt[3]{-\frac{q_{1}}{2}+\sqrt{\alpha_{1}} \beta_{1}}+\sqrt[3]{-\frac {q_{1}}{2}-\sqrt{\alpha_{1}} \beta_{1}^{2}},\\& y_{13}=\sqrt[3]{-\frac{q_{1}}{2}+\sqrt{\alpha_{1}} \beta_{1}^{2}}+\sqrt[3]{-\frac{q_{1}}{2}-\sqrt{ \alpha_{1}}\beta_{1}},\\& v_{1i}=y_{1i}-\frac{3e_{23}}{4}, \quad i=1, 2, 3. \end{aligned}$$
Then we have the following results according to the Lemma 2.2 in [22].
Lemma 1
For (12),
-
(i)
if
\(e_{20}\geq0\)
and
\(\alpha_{1}\geq0\), then (12) has positive roots if and only if
\(v_{11}>0\)
and
\(f_{1}(v_{11})<0\);
-
(ii)
if
\(e_{20}\geq0\)
and
\(\alpha_{1}<0\), then (12) has positive roots if and only if there exists at least one
\(v_{1*}\in\{ v_{11}, v_{12}, v_{13}\}\), such that
\(v_{1*}>0\)
and
\(f_{1}(v_{1*})\leq0\).
In what follows, we assume that we have (H21): the coefficients in \(f_{1}(v_{1})\) satisfy one of the following conditions in (α)-(γ): (α) \(e_{20}<0\); (β) \(e_{20}\geq0\), \(\alpha _{1}\geq0\), \(v_{11}>0\), and \(f_{1}(v_{11})<0\); (γ) \(e_{20}\geq0\), \(\alpha_{1}<0\), and there exists at least one \(v_{1*}\in\{v_{11}, v_{12}, v_{13}\}\), such that \(v_{1*}>0\) and \(f_{1}(v_{1*})\leq0\).
If the condition (H21) holds, (11) has at least one positive root \(\omega_{10}\) such that (10) has a pair of purely imaginary roots \(\pm i\omega_{10}\) and the corresponding critical value of the delay is
$$\begin{aligned} \tau_{1k} =&\frac{1}{\omega_{10}}\arccos\biggl\{ \frac {(B_{22}-A_{23}B_{23})\omega _{10}^{6}+(A_{21}B_{23}+A_{23}B_{21}-A_{22}B_{22}-B_{20})\omega _{10}^{4}}{(B_{20}-B_{22}\omega_{10}^{2})^{2}+(B_{21}\omega _{10}-B_{23}\omega _{10}^{3})^{2}}\\ &{}+\frac{(A_{20}B_{22}+A_{22}B_{20}-A_{21}B_{21})\omega _{10}^{2}-A_{20}B_{20}}{(B_{20}-B_{22}\omega_{10}^{2})^{2}+(B_{21}\omega _{10}-B_{23}\omega_{10}^{3})^{2}}\biggr\} +\frac{2k\pi}{\omega_{10}},\quad k=0, 1, 2, \ldots. \end{aligned}$$
Differentiating the two sides of (10), we can get
$$\biggl[\frac{d\lambda}{d\tau_{1}}\biggr]^{-1}=-\frac{4\lambda ^{3}+3A_{23}\lambda ^{2}+2A_{22}\lambda+A_{21}}{\lambda(\lambda^{4}+A_{23}\lambda ^{3}+A_{22}\lambda^{2}+A_{21}\lambda+A_{20})}+ \frac{3B_{23}\lambda ^{2}+2B_{22}\lambda+B_{21}}{\lambda(B_{23}\lambda^{3}+B_{22}\lambda ^{2}+B_{21}\lambda+B_{20})}-\frac{\tau_{1}}{\lambda}. $$
Thus,
$$\operatorname{Re}\biggl[\frac{d\lambda}{d\tau_{1}}\biggr]^{-1}_{\tau _{1}=\tau_{10}}= \frac {f_{1}^{\prime}(v_{1}^{*})}{(B_{20}-B_{22}\omega_{10}^{2})^{2}+(B_{21}\omega _{10}-B_{23}\omega_{10}^{3})^{2}}, $$
where \(v_{1}^{*}=\omega_{10}^{2}\). Obviously, if the condition (H22): \(f_{1}^{\prime}(v_{1}^{*})\neq0\) holds, then \(\operatorname{Re}[\frac{d\lambda}{d\tau_{1}}]^{-1}_{\tau_{1}=\tau_{10}}\neq 0\). In conclusion, we have the following results according to the Hopf bifurcation theorem in [23].
Theorem 1
Suppose that the conditions (H21)-(H22) hold. The positive equilibrium
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
of system (4) is asymptotically stable for
\(\tau_{1}\in[0,\tau_{10})\)
and system (4) undergoes a Hopf bifurcation at
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
when
\(\tau_{1}=\tau_{10}\).
Case 3.
\(\tau_{2}>0\), \(\tau_{1}=0\).
Substitute \(\tau_{1}=0\) into (7) and we have
$$ \lambda^{4}+A_{33}\lambda^{3}+A_{32} \lambda^{2}+A_{31}\lambda+A_{30}+ \bigl(B_{32}\lambda^{2}+B_{31}\lambda+B_{30} \bigr)e^{-\lambda\tau_{2}}=0, $$
(15)
where
$$\begin{aligned}& A_{30}=A_{0}+B_{0},\qquad A_{31}=A_{1}+B_{1},\qquad A_{32}=A_{2}+B_{2},\\& A_{33}=A_{3}+B_{3},\qquad B_{32}=C_{2},\qquad B_{31}=C_{1}+D_{1},\qquad B_{30}=C_{0}+D_{0}. \end{aligned}$$
Let \(\lambda=i\omega_{2}\) (\(\omega_{2}>0\)) be the root of (15). Then
$$\left \{ \textstyle\begin{array}{@{}l} B_{31}\omega_{2}\sin\omega_{2}\tau_{2}+(B_{30}-B_{32}\omega_{2}^{2})\cos \omega _{2}\tau_{2}=A_{32}\omega_{2}^{2}-\omega_{2}^{4}-A_{30}, \\ B_{31}\omega_{2}\cos\omega_{2}\tau_{2}-(B_{30}-B_{32}\omega_{2}^{2})\sin \omega _{2}\tau_{2}=A_{33}\omega_{2}^{3}-A_{31}\omega_{2}, \end{array}\displaystyle \right . $$
from which it follows that
$$ \omega_{2}^{8}+e_{33} \omega_{2}^{6}+e_{32}\omega_{2}^{4}+e_{31} \omega_{2}+e_{30}=0, $$
(16)
where
$$\begin{aligned}& e_{30}=A_{30}^{2}-B_{30}^{2},\qquad e_{31}=A_{31}^{2}-B_{31}^{2}-2A_{30}A_{32}+2B_{30}B_{32},\\& e_{32}=A_{32}^{2}-B_{32}^{2}+2A_{30}-2A_{31}A_{33},\qquad e_{33}=A_{33}^{2}-2A_{32}. \end{aligned}$$
Let \(\omega_{2}^{2}=v_{2}\), then (16) becomes
$$ v_{2}^{4}+e_{33}v_{2}^{3}+e_{32}v_{2}^{2}+e_{31}v_{2}+e_{30}=0. $$
(17)
Define
$$f_{2}(v_{2})=v_{2}^{4}+e_{33}v_{2}^{3}+e_{32}v_{2}^{2}+e_{31}v_{2}+e_{30}. $$
Then
$$f_{2}^{\prime}(v_{2})=4v_{2}^{3}+3e_{33}v_{2}^{2}+2e_{32}v_{2}+e_{31}. $$
Set
$$ 4v_{2}^{3}+3e_{33}v_{2}^{2}+2e_{32}v_{2}+e_{31}=0. $$
(18)
Let \(y_{2}=v_{2}+\frac{3e_{33}}{4}\). Then (18) becomes
$$y_{2}^{3}+p_{2}y_{2}+q_{2}=0, $$
where
$$p_{2}=\frac{e_{32}}{2}-\frac{3}{16}e_{33}^{2},\qquad q_{2}=\frac{e_{33}^{3}}{32}-\frac {e_{32}e_{33}}{8}+e_{31}. $$
Define
$$\begin{aligned}& \alpha_{2}=\biggl(\frac{q_{2}}{2}\biggr)^{2}+\biggl( \frac{p_{2}}{3}\biggr)^{3}, \qquad\beta_{2}=\frac{-1+\sqrt {3}i}{2},\\& y_{21}=\sqrt[3]{-\frac{q_{2}}{2}+\sqrt{\alpha_{2}}}+ \sqrt[3]{-\frac {q_{2}}{2}-\sqrt{\alpha_{2}}},\\& y_{22}=\sqrt[3]{-\frac{q_{2}}{2}+\sqrt{\alpha_{2}} \beta_{2}}+\sqrt[3]{-\frac {q_{2}}{2}-\sqrt{\alpha_{2}} \beta_{2}^{2}},\\& y_{23}=\sqrt[3]{-\frac{q_{2}}{2}+\sqrt{\alpha_{2}} \beta_{2}^{2}}+\sqrt[3]{-\frac{q_{2}}{2}-\sqrt{ \alpha_{2}}\beta_{2}},\\& v_{2i}=y_{2i}-\frac{3e_{33}}{4},\quad i=1, 2, 3. \end{aligned}$$
According to Lemma 1, we can conclude that if we may consider the condition (H31): the coefficients in \(f_{2}(v_{2})\) satisfy one of the following conditions in (\(\alpha^{\prime}\))-(\(\gamma^{\prime}\)): (\(\alpha^{\prime}\)) \(e_{30}<0\); (\(\beta^{\prime}\)) \(e_{30}\geq0\), \(\alpha _{2}\geq\), \(v_{21}>0\), and \(f_{2}(v_{21})<0\); (\(\gamma^{\prime}\)) \(e_{30}\geq 0\), \(\alpha_{2}<0\), and there exists at least one \(v_{2*}\in\{v_{21}, v_{22}, v_{23}\}\), such that \(v_{2*}>0\) and \(f_{2}(v_{2*})\leq0\).
If the condition (H31) holds, (16) has at least one positive root \(\omega_{20}\) such that (15) has a pair of purely imaginary roots \(\pm i\omega_{20}\) and the corresponding critical value of the delay is
$$\begin{aligned} \tau_{2k} =&\frac{1}{\omega_{20}}\arccos\biggl\{ \frac {B_{32}\omega _{20}^{6}+(A_{33}B_{31}-A_{32}B_{32}-B_{30})\omega _{20}^{4}}{B_{31}^{2}\omega _{20}^{2}+(B_{30}-B_{32}\omega_{20}^{2})^{2}}\\ &{}+\frac{(A_{30}B_{32}+A_{32}B_{30}-A_{31}B_{31})\omega _{20}^{2}-A_{30}B_{30}}{B_{31}^{2}\omega_{20}^{2}+(B_{30}-B_{32}\omega _{20}^{2})^{2}}\biggr\} +\frac{2k\pi}{\omega_{20}},\quad k=0, 1, 2, \ldots. \end{aligned}$$
Similar as in Case 2, if the condition (H32): \(f_{2}^{\prime}(v_{2}^{*})\neq0\) holds, where \(v_{2}^{*}=\omega_{20}^{2}\), then \(\operatorname{Re}[\frac {d\lambda}{d\tau_{2}}]^{-1}_{\tau_{2}=\tau_{20}}\neq0\). In conclusion, we have the following results according to the Hopf bifurcation theorem in [23].
Theorem 2
Suppose that the conditions (H31)-(H32) hold. The positive equilibrium
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
of system (4) is asymptotically stable for
\(\tau_{2}\in[0,\tau_{20})\)
and system (4) undergoes a Hopf bifurcation at
\(E^{*}E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
when
\(\tau_{2}=\tau_{20}\).
Case 4.
\(\tau_{1}=\tau_{2}=\tau>0\).
Substitute \(\tau_{1}=\tau_{2}=\tau\) into (7); then (7) becomes
$$\begin{aligned} &\lambda^{4}+A_{43}\lambda^{3}+A_{42} \lambda^{2}+A_{41}\lambda+A_{40}+ \bigl(B_{43}\lambda^{3}+B_{42}\lambda^{2}+B_{41} \lambda+B_{40}\bigr)e^{-\lambda\tau} \\ &\quad{}+(C_{41} \lambda+C_{40})e^{-2\lambda\tau}=0, \end{aligned}$$
(19)
where
$$\begin{aligned}& A_{40}=A_{0},\qquad A_{41}=A_{1},\qquad A_{42}=A_{2},\qquad A_{43}=A_{3},\\& B_{40}=B_{0}+C_{0},\qquad B_{41}=B_{1}+C_{1},\qquad B_{42}=B_{2}+C_{2},\\& B_{43}=B_{3},\qquad C_{40}=D_{0},\qquad C_{41}=D_{1}. \end{aligned}$$
Multiplying (19) by \(e^{\lambda\tau}\), then (19) becomes
$$\begin{aligned} &B_{43}\lambda^{3}+B_{42} \lambda^{2}+B_{41}\lambda+B_{40}+\bigl(\lambda ^{4}+A_{43}\lambda^{3}+A_{42} \lambda^{2}+A_{41}\lambda+A_{40} \bigr)e^{\lambda\tau } \\ &\quad{}+(C_{41}\lambda+C_{40})e^{-\lambda\tau}=0. \end{aligned}$$
(20)
Let \(\lambda=i\omega\) (\(\omega>0\)) be the root of (20), then
$$\left \{ \textstyle\begin{array}{@{}l} (\omega^{4}-A_{42}\omega^{2}+A_{40}+C_{40})\cos\tau\omega +(A_{43}\omega ^{3}-A_{41}\omega+C_{41}\omega)\sin\tau\omega=B_{42}\omega ^{2}-B_{40}, \\ (\omega^{4}-A_{42}\omega^{2}+A_{40}-C_{40})\sin\tau\omega -(A_{43}\omega ^{3}-A_{41}\omega-C_{41}\omega)\cos\tau\omega=B_{43}\omega ^{3}-B_{41}\omega, \end{array}\displaystyle \right . $$
from which it follows that
$$\begin{aligned}& \sin(\tau\omega)=\frac{g_{7}\omega^{7}+g_{5}\omega^{5}+g_{3}\omega ^{3}+g_{1}\omega }{\omega^{8}+h_{6}\omega^{6}+h_{4}\omega^{4}+h_{2}\omega^{2}+h_{0}}, \\& \cos(\tau \omega )=\frac{g_{6}\omega^{6}+g_{4}\omega^{4}+g_{2}\omega^{2}+g_{0}}{\omega ^{8}+h_{6}\omega^{6}+h_{4}\omega^{4}+h_{2}\omega^{2}+h_{0}}, \end{aligned}$$
where
$$\begin{aligned}& g_{0}=(C_{40}-A_{40})B_{40},\qquad g_{1}=(A_{41}+C_{41})B_{40}-(A_{40}+C_{40}),\\& g_{2}=A_{40}B_{42}+A_{42}B_{40}+B_{41}C_{41}-A_{41}B_{41}-B_{42}C_{40},\\& g_{3}=A_{40}B_{43}+A_{42}B_{41}+B_{43}C_{40}-A_{41}B_{42}-A_{43}B_{40}-B_{42}C_{41},\\& g_{4}=A_{41}B_{43}+A_{43}B_{41}-A_{42}B_{42}-B_{43}C_{41}-B_{40},\\& g_{5}=A_{43}B_{42}-A_{42}B_{43}-B_{41},\qquad g_{6}=B_{42}-A_{43}B_{43},\qquad g_{7}=B_{43},\\& h_{0}=A_{40}^{2}-C_{40}^{2},\qquad h_{2}=A_{41}^{2}-C_{41}^{2}-2A_{40}A_{42},\\& h_{4}=A_{42}^{2}+2A_{40}-2A_{41}A_{43},\qquad h_{6}=A_{43}^{2}-2A_{42}. \end{aligned}$$
Then we can obtain
$$ \omega^{16}+e_{47}\omega^{14}+e_{46} \omega^{12}+e_{45}\omega^{10}+e_{44} \omega^{8}+e_{43}\omega^{6}+e_{42} \omega^{4}+e_{41}\omega^{2}+e_{40}=0, $$
(21)
where
$$\begin{aligned}& e_{40}=h_{0}^{2}-g_{0}^{2},\qquad e_{41}=2h_{0}h_{2}-2g_{0}g_{2}-g_{1}^{2},\\& e_{42}=h_{2}^{2}-g_{2}^{2}+2h_{0}h_{4}-2g_{1}g_{3}-2g_{0}g_{4},\\& e_{43}=2h_{0}h_{6}+2h_{2}h_{4}-g_{3}^{2}-2g_{0}g_{6}-2g_{1}g_{5}-2g_{2}g_{4},\\& e_{44}=h_{4}^{2}+2h_{0}+2h_{2}h_{6}-g_{4}^{2}-2g_{1}g_{7}-2g_{2}g_{6}-2g_{3}g_{5},\\& e_{45}=2h_{2}+2h_{4}h_{6}-g_{5}^{2}-2g_{3}g_{5}-2g_{4}g_{6},\\& e_{46}=h_{6}^{2}-g_{6}^{2}+2h_{4}-2g_{5}g_{7},\qquad e_{47}=2h_{6}-g_{7}^{2}. \end{aligned}$$
Let \(\omega^{2}=v\), then (21) becomes
$$ v^{8}+e_{47}v^{7}+e_{46}v^{6}+e_{45}v^{5}+e_{44}v^{4}+e_{43}v^{3}+e_{42}v^{2}+e_{41}v+e_{40}=0. $$
(22)
If the coefficients of system (4) are given, the roots of (22) can be obtained by the Matlab software package. Therefore, we make the following assumption in order to get the main results in this paper.
Suppose that (H41): (22) has at least one positive root.
If the condition (H41) holds, without loss of generality, we assume that (22) has eight positive roots which are denoted by \(v_{1}, v_{2}, \ldots, v_{8}\), respectively. Then (21) has eight positive roots \(\omega_{k}=\sqrt{v_{k}}\), \(k =1, 2, \ldots, 8\). For every \(\omega_{k}\), the corresponding critical value of the time delay is
$$\begin{aligned}& \tau_{k}^{(j)}=\frac{1}{\omega_{k}}\arccos\frac{g_{6}\omega _{k}^{6}+g_{4}\omega _{k}^{4}+g_{2}\omega_{k}^{2}+g_{0}}{\omega_{k}^{8}+h_{6}\omega_{k}^{6}+h_{4}\omega _{k}^{4}+h_{2}\omega _{k}^{2}+h_{0}}+ \frac{2j\pi}{\omega_{k}}, \quad k=1, 2, 3,\ldots, 8; j=0, 1, 2, \ldots. \end{aligned}$$
Let
$$\tau_{0}=\min\bigl\{ \tau_{k}^{(0)}\bigr\} ,\quad k=1, 2, \ldots, 8, \omega_{0}=\omega_{k}|_{\tau=\tau_{0}}. $$
Thus, when \(\tau=\tau_{0}\), (20) has a pair of purely imaginary roots \(\pm i\omega_{0}\).
Differentiating both sides of (20) with respect to τ, we get
$$\biggl[\frac{d\lambda}{d\tau}\biggr]^{-1}=-\frac{(4\lambda ^{3}+3A_{43}\lambda ^{2}+2A_{42}\lambda+A_{41})e^{\lambda\tau}+C_{41}e^{-\lambda\tau }+3B_{43}\lambda^{2}+2B_{42}\lambda+B_{41}}{\lambda[(\lambda ^{4}+A_{43}\lambda^{3}+A_{42}\lambda^{2}+A_{41}\lambda+A_{40})e^{\lambda \tau }-(C_{41}\lambda+C_{40})e^{-\lambda\tau}]}- \frac{\tau}{\lambda}. $$
Then we have
$$\operatorname{Re}\biggl[\frac{d\lambda}{d\tau}\biggr]^{-1}_{\tau=\tau_{0}}= \frac {P_{41}Q_{41}+P_{42}Q_{42}}{Q_{41}^{2}+Q_{42}^{2}}, $$
where
$$\begin{aligned}& P_{41}=\bigl(A_{41}+C_{41}-3A_{43} \omega_{0}^{2}\bigr)\cos\tau_{0}\omega _{0}-\bigl(2A_{42}\omega_{0}-4\omega_{0}^{3} \bigr)\sin\tau_{0}\omega_{0}-3B_{43}\omega _{0}^{2}+B_{41},\\& P_{42}=\bigl(A_{41}-C_{41}-3A_{43} \omega_{0}^{2}\bigr)\sin\tau_{0}\omega _{0}+\bigl(2A_{42}\omega_{0}-4\omega_{0}^{3} \bigr)\cos\tau_{0}\omega_{0}+2B_{42} \omega_{0}, \\& Q_{41}=\bigl(A_{43}\omega_{0}^{4}-A_{41} \omega_{0}^{2}-C_{41}\omega_{0}^{2} \bigr)\cos\tau_{0}\omega_{0}-\bigl(\omega_{0}^{5}-A_{42} \omega_{0}^{3}+A_{40}\omega_{0}+C_{40} \omega_{0}\bigr)\sin\tau_{0}\omega_{0}, \\& Q_{42}=\bigl(A_{43}\omega_{0}^{4}-A_{41} \omega_{0}^{2}+C_{41}\omega_{0}^{2} \bigr)\sin\tau_{0}\omega_{0}+\bigl(\omega_{0}^{5}-A_{42} \omega_{0}^{3}+A_{40}\omega_{0}-C_{40} \omega_{0}\bigr)\cos\tau_{0}\omega_{0}. \end{aligned}$$
Obviously, if the condition (H42): \(P_{41}Q_{41}+P_{42}Q_{42}\neq 0\) holds, then \(\operatorname{Re}[\frac{d\lambda}{d\tau}]^{-1}_{\tau=\tau_{0}}\neq0\). Thus, according to the Hopf bifurcation theorem in [23], we have the following results.
Theorem 3
Suppose that the conditions (H41)-(H42) hold. The positive equilibrium
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
of system (4) is asymptotically stable for
\(\tau_{1}\in[0,\tau_{0})\)
and system (4) undergoes a Hopf bifurcation at
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
when
\(\tau=\tau_{0}\).
Case 5.
\(\tau_{2}>0\) and \(\tau_{1}\in(0, \tau_{10})\). We consider (7) with \(\tau_{1}\) in its stable interval and \(\tau_{2}\) is considered as a parameter.
Let \(\lambda=i\omega_{2}^{\prime}\) (\(\omega_{2}^{\prime}>0\)) be the root of (7). Then we get
$$\left \{ \textstyle\begin{array}{@{}l} \Delta_{51}\sin\tau_{2}\omega_{2}^{\prime}+\Delta_{52}\cos\tau _{2}\omega _{2}^{\prime}=\Delta_{53}, \\ \Delta_{51}\cos\tau_{2}\omega_{2}^{\prime}-\Delta_{52}\sin\tau _{2}\omega _{2}^{\prime}=\Delta_{54}, \end{array}\displaystyle \right . $$
where
$$\begin{aligned}& \Delta_{51}=C_{1}\omega_{2}^{\prime}-D_{0} \sin\tau_{1}\omega_{2}^{\prime}+D_{1}\omega _{2}^{\prime}\cos\tau_{1}\omega_{2}^{\prime}, \\& \Delta_{52}=C_{0}-C_{2}\bigl( \omega_{2}^{\prime}\bigr)^{2}+D_{0}\cos \tau_{1}\omega_{2}^{\prime}+D_{1} \omega_{2}^{\prime}\sin\tau_{1}\omega_{2}^{\prime}, \\& \Delta_{53}=\bigl(B_{2}\bigl(\omega_{2}^{\prime}\bigr)^{2}-B_{0}\bigr)\cos\tau_{1} \omega_{2}^{\prime}+\bigl(\bigl(\omega_{2}^{\prime}\bigr)^{3}-B_{1}\omega_{2}^{\prime}\bigr)\sin \tau_{1}\omega_{2}^{\prime}-\bigl(\omega_{2}^{\prime}\bigr)^{4}+A_{2} \bigl(\omega_{2}^{\prime}\bigr)^{2}-A_{0}, \\& \Delta_{54}=\bigl(B_{0}-B_{2}\bigl( \omega_{2}^{\prime}\bigr)^{2}\bigr)\sin \tau_{1}\omega_{2}^{\prime}+\bigl(\bigl( \omega_{2}^{\prime}\bigr)^{3}-B_{1} \omega_{2}^{\prime}\bigr)\cos\tau_{1} \omega_{2}^{\prime}+A_{3}\bigl(\omega_{2}^{\prime}\bigr)^{3}-A_{1} \omega_{2}^{\prime}. \end{aligned}$$
It follows that
$$ e_{50}\bigl(\omega_{2}^{\prime}\bigr)+e_{51}\bigl(\omega_{2}^{\prime}\bigr)\cos \tau_{1}\omega_{2}^{\prime}+e_{52}\bigl( \omega_{2}^{\prime}\bigr)\sin\tau_{1} \omega_{2}^{\prime}=0, $$
(23)
where
$$\begin{aligned}& e_{50}\bigl(\omega_{2}^{\prime}\bigr)=\bigl( \omega_{2}^{\prime}\bigr)^{8}+\bigl(A_{3}^{2}+B_{3}^{2}-2A_{2} \bigr) \bigl(\omega_{2}^{\prime}\bigr)^{6}\\& \hphantom{e_{50}\bigl(\omega_{2}^{\prime}\bigr)=}{}+\bigl(A_{2}^{2}+B_{2}^{2}-C_{2}^{2}+2A_{0}-2A_{1}A_{3}-2B_{1}B_{3} \bigr) \bigl(\omega_{2}^{\prime}\bigr)^{4}\\& \hphantom{e_{50}\bigl(\omega_{2}^{\prime}\bigr)=}{}+\bigl(A_{1}^{2}+B_{1}^{2}-C_{1}^{2}-D_{1}^{2}-2A_{0}A_{2}-2B_{0}B_{2}+2C_{0}C_{2} \bigr) \bigl(\omega_{2}^{\prime}\bigr)^{2}\\& \hphantom{e_{50}\bigl(\omega_{2}^{\prime}\bigr)=}{}+A_{0}^{2}+B_{0}^{2}-C_{0}^{2}-D_{0}^{2}, \\& e_{51}\bigl(\omega_{2}^{\prime}\bigr)=2(A_{3}B_{3}-B_{2}) \bigl(\omega_{2}^{\prime}\bigr)^{6}+2(A_{2}B_{2}+B_{0}-A_{3}B_{1}-A_{1}B_{3}) \bigl(\omega_{2}^{\prime}\bigr)^{4}\\& \hphantom{e_{51}\bigl(\omega_{2}^{\prime}\bigr)=}{}+2(A_{1}B_{1}-A_{0}B_{2}-A_{2}B_{0}-C_{1}D_{1}+C_{2}D_{0}) \bigl(\omega_{2}^{\prime}\bigr)^{2}+2(A_{0}B_{0}-C_{0}D_{0}),\\& e_{52}\bigl(\omega_{2}^{\prime}\bigr)=-2B_{3} \bigl(\omega_{2}^{\prime}\bigr)^{7}+2(A_{2}B_{3}-A_{3}B_{2}+B_{1}) \bigl(\omega_{2}^{\prime}\bigr)^{5}\\& \hphantom{e_{52}\bigl(\omega_{2}^{\prime}\bigr)=}{}+2(A_{3}B_{0}+A_{1}B_{2}-A_{0}B_{3}-A_{2}B_{1}+C_{2}D_{1}) \bigl(\omega_{2}^{\prime}\bigr)^{3}\\& \hphantom{e_{52}\bigl(\omega_{2}^{\prime}\bigr)=}{}+2(A_{0}B_{1}-A_{1}B_{0}+C_{1}q_{D}-C_{0}D_{1}) \omega_{2}^{*}. \end{aligned}$$
Suppose that we have (H51): (23) has at least finite positive roots. We denote the positive roots of (23) as \(\omega_{21}^{\prime}, \omega_{22}^{\prime}, \ldots, \omega_{2k}^{\prime}\). Then, for every fixed \(\omega_{2i}^{\prime}\) (\(i=1, 2, \ldots, k\)), the corresponding critical value of time delay is
$$\tau_{2i}^{(j)\prime}=\frac{1}{\omega_{2i}^{\prime}}\arccos\biggl\{ \frac{\Delta _{51}\Delta_{54}+\Delta_{52}\Delta_{53}}{\Delta_{51}^{2}+\Delta _{52}^{2}}\Big|_{\omega_{2}^{\prime}=\omega_{2i}^{\prime}}\biggr\} +\frac {2j\pi }{\omega _{2i}^{\prime}}, $$
with \(i=1, 2, \ldots, k\); \(j=0, 1, 2,\ldots\) .
Let \(\tau_{20}^{*}=\min\{\tau_{2i}^{(0)\prime}| i=1, 2, \ldots, k\} \). When \(\tau_{2}=\tau_{20}^{*}\), (7) has a pair of purely imaginary roots \(\pm i\omega_{2}^{*}\) for \(\tau_{1}\in(0, \tau_{10})\). Differentiating (7) with respect to \(\tau_{2}\), one can obtain
$$\biggl[\frac{d\lambda}{d\tau_{2}}\biggr]^{-1}=\frac{p_{0}(\lambda )+p_{1}(\lambda )e^{-\lambda\tau_{1}}+p_{2}(\lambda)e^{-\lambda\tau_{2}}+p_{3}(\lambda )e^{-\lambda(\tau_{1}+\tau_{2})}}{q_{1}(\lambda)e^{-\lambda\tau _{2}}+q_{2}(\lambda )e^{-(\lambda_{1}+\lambda_{2})}}- \frac{\tau_{2}}{\lambda}, $$
with
$$\begin{aligned}& p_{0}(\lambda)=4\lambda^{3}+3A_{3} \lambda^{2}+2A_{2}\lambda+A_{1},\\& p_{1}(\lambda)=-\tau_{1}B_{3} \lambda^{3}+(3B_{3}-\tau_{1}B_{2}) \lambda^{2}+(2B_{2}-\tau_{1}B_{1}) \lambda+B_{1}-\tau_{1}B_{0},\\& p_{2}(\lambda)=2C_{2}\lambda+C_{1},\qquad p_{3}(\lambda)=D_{1},\\& q_{1}(\lambda)=C_{2}\lambda^{3}+C_{1} \lambda^{2}+C_{0}\lambda,\qquad q_{2}(\lambda )=D_{1}\lambda^{2}+D_{0}\lambda. \end{aligned}$$
Hence,
$$\operatorname{Re}\biggl[\frac{d\lambda}{d\tau_{2}}\biggr]^{-1}_{\tau=\tau _{20}^{*}}=- \frac{P_{51}Q_{51}-P_{52}Q_{52}}{Q_{51}^{2}+Q_{52}^{2}}, $$
where
$$\begin{aligned}& P_{51}=\bigl(2C_{2}\omega_{2}^{*}-D_{1} \sin\tau_{1}\omega_{2}^{*}\bigr)\sin\tau_{20}^{*}\omega _{2}^{*}+\bigl(C_{1}+D_{1}\cos\tau_{1} \omega_{2}^{*}\bigr)\cos\tau_{20}^{*}\omega_{2}^{*} \\& \hphantom{P_{51}=}{}+\bigl(\tau_{1}B_{3}\bigl(\omega_{2}^{*} \bigr)^{3}+(2B_{2}-\tau_{1}B_{1}) \omega_{2}^{*}\bigr)\sin\tau_{1}\omega_{2}^{*} \\& \hphantom{P_{51}=}{}+\bigl((\tau_{1}B_{2}-3B_{3}) \bigl( \omega_{2}^{*}\bigr)^{2}+B_{1}-\tau_{1}B_{0} \bigr)\cos\tau_{1}\omega_{2}^{*}-3A_{3}\bigl( \omega_{2}^{*}\bigr)^{2}+A_{1}, \\& P_{52}=\bigl(2C_{2}\omega_{2}^{*}-D_{1} \sin\tau_{1}\omega_{2}^{*}\bigr)\cos\tau_{20}^{*}\omega _{2}^{*}-\bigl(C_{1}+D_{1}\cos\tau_{1} \omega_{2}^{*}\bigr)\sin\tau_{20}^{*}\omega_{2}^{*}\\& \hphantom{P_{52}=}{}+\bigl(\tau_{1}B_{3}\bigl(\omega_{2}^{*} \bigr)^{3}+(2B_{2}-\tau_{1}B_{1}) \omega_{2}^{*}\bigr)\cos\tau_{1}\omega_{2}^{*}\\& \hphantom{P_{52}=}{}-\bigl((\tau_{1}B_{2}-3B_{3}) \bigl( \omega_{2}^{*}\bigr)^{2}+B_{1}-\tau_{1}B_{0} \bigr)\sin\tau_{1}\omega_{2}^{*}-4\bigl(\omega_{2}^{*} \bigr)^{3}+2A_{2}\omega_{2}^{*},\\& Q_{51}=\bigl(C_{2}\bigl(\omega_{2}^{*} \bigr)^{3}-C_{0}\omega_{2}^{*}-D_{1}\bigl( \omega_{2}^{*}\bigr)^{2}\sin\tau_{1} \omega_{2}^{*}-D_{0}\omega_{2}^{*}\cos \tau_{1}\omega_{2}^{*}\bigr)\sin\tau_{20}^{*}\omega _{2}^{*}\\& \hphantom{Q_{51}=}{}+\bigl(C_{1}\bigl(\omega_{2}^{*}\bigr)^{2}+D_{1} \bigl(\omega_{2}^{*}\bigr)^{2}\cos\tau_{1} \omega_{2}^{*}-D_{0}\omega_{2}^{*}\sin \tau_{1}\omega_{2}^{*}\bigr)\cos\tau_{20}^{*} \omega_{2}^{*},\\& \begin{aligned}[b] Q_{52}={}&\bigl(C_{2}\bigl(\omega_{2}^{*} \bigr)^{3}-C_{0}\omega_{2}^{*}-D_{1}\bigl( \omega_{2}^{*}\bigr)^{2}\sin\tau_{1} \omega_{2}^{*}-D_{0}\omega_{2}^{*}\cos \tau_{1}\omega_{2}^{*}\bigr)\cos\tau_{20}^{*}\omega _{2}^{*}\\ &{}-\bigl(C_{1}\bigl(\omega_{2}^{*}\bigr)^{2}+D_{1} \bigl(\omega_{2}^{*}\bigr)^{2}\cos\tau_{1} \omega_{2}^{*}-D_{0}\omega_{2}^{*}\sin \tau_{1}\omega_{2}^{*}\bigr)\sin\tau_{20}^{*} \omega_{2}^{*}. \end{aligned} \end{aligned}$$
Obviously, if the condition (H52): \(P_{51}Q_{51}\neq P_{52}Q_{52}\) holds, then \(\operatorname{Re}[\frac{d\lambda}{d\tau_{2}}]^{-1}_{\tau=\tau _{20}^{*}}\neq0\). Namely, if the condition (H52) holds, the transversality condition is satisfied. Thus, according to the Hopf bifurcation theorem in [23], we have the following results.
Theorem 4
If the conditions (H51)-(H52) hold and
\(\tau_{1}\in(0, \tau_{10})\), then the positive equilibrium
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
of system (4) is asymptotically stable for
\(\tau_{2}\in[0,\tau_{20}^{*})\)
and system (4) undergoes a Hopf bifurcation at
\(E^{*}(x_{1}^{*}, x_{2}^{*}, y_{1}^{*}, y_{2}^{*})\)
when
\(\tau_{2}=\tau_{20}^{*}\).