From the viewpoint of biology, we only study the positive equilibrium of system (1.2). In this section, we shall discuss the local stability of a linearized system at the positive equilibrium and the existence of Hopf bifurcations for system (1.2).
It is easy to show that system (1.2) has a unique positive equilibrium \(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \), where
$$\begin{gathered} x_{1}^{*} = \sqrt{ \frac{r}{a_{2}  rm}}, \\ x_{2}^{*} = \frac{b}{r_{2}}x_{1}^{*}, \\ y^{*} = \frac{(ab  r_{1}r_{2}  br_{2}  cr_{2}x_{1}^{*})(1 + mx_{1}^{*2})}{a_{1}r_{2}x_{1}^{*}} \end{gathered} $$
if the following conditions are satisfied:
$$(\mathrm{H}1)\quad a_{2}  rm > 0, \qquad ab  r_{1}r_{2}  br_{2}  cr_{2}\sqrt{\frac{r}{a_{2}  rm}} > 0. $$
Let \(\bar{x}_{1}(t) = x_{1}(t)  x_{1}^{*} \), \(\bar{x}_{2}(t) = x_{2}(t)  x_{2}^{*} \), \(\bar{y}(t) = y(t)  y^{*} \) and still denote \(\bar{x}_{1}(t) \), \(\bar{x}_{2}(t) \), \(\bar{y}(t) \), respectively. Using Taylor’s expansion to expand system (1.2) at the positive equilibrium \(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \), we have
$$ \begin{gathered} \dot{x}_{1}(t) = a_{11}x_{1}(t) + a_{12}x_{2}(t) + a_{13}y(t) + b_{11}x(t  \tau_{1}) \\ \hphantom{\dot{x}_{1}(t) =}{}+ \sum _{i + j + k + q \ge 2} f_{1}^{(ijkq)} x_{1}^{i}(t)x_{2}^{j}(t)y^{k}x_{1}^{q}(t  \tau_{1}), \\ \dot{x}_{2}(t) = a_{21}x_{1}(t) + a_{22}x_{2}(t) + \sum_{i + j \ge 2} f_{2}^{(ij)} x_{1}^{i}(t)x_{2}^{j}(t), \\ \dot{y}(t) = b_{31}x_{1}(t  \tau_{2}) + a_{33}y(t) + b_{33}y(t  \tau_{2}) + \sum _{i + j + k \ge 2} f_{3}^{(ijk)} y^{i}(t)x_{1}^{j}(t  \tau_{2})y^{k}(t  \tau_{2}), \end{gathered} $$
(2.1)
where
$$\begin{gathered} a_{11} =  r_{1}  cx_{1}^{*}  b  \frac{2a_{1}x_{1}^{*}y^{*}}{[1 + m(x_{1}^{*})^{2}]^{2}},\qquad a_{12} = a,\\ a_{13} = \frac{  a_{1}(x_{1}^{*})^{2}}{1 + m(x_{1}^{*})^{2}},\qquad b_{11} =  cx_{1}^{*}, \\ a_{21} = b,\qquad a_{22} =  r_{2},\qquad b_{31} = \frac{2a_{2}x_{1}^{*}y^{*}}{[1 + m(x_{1}^{*})^{2}]^{2}},\\ b_{33} = \frac{a_{2}(x_{1}^{*})^{2}}{1 + m(x_{1}^{*})^{2}},\qquad a_{33} =  r. \\ f_{1}^{(ijkq)} = \frac{1}{i!j!k!q!}\frac{\partial^{i + j + k + q}f_{1}}{\partial x_{1}^{i}(t)\,\partial x_{2}^{j}(t)\,\partial y^{k}(t)\,\partial x^{q}(x  \tau_{1})}\bigg\vert \bigl(x_{1}^{*},x_{2}^{*},y^{*} \bigr) , \\ f_{2}^{(ij)} = \frac{1}{i!j!}\frac{\partial^{i + j}f_{2}}{\partial x_{1}^{i}(t)\,\partial x_{2}^{j}(t)} \bigg\vert \bigl(x_{1}^{*},x_{2}^{*},y^{*} \bigr) , \\ f_{3}^{(ijk)} = \frac{1}{i!j!k!}\frac{\partial^{i + j + k}f_{3}}{\partial y^{i}(t)\,\partial x_{1}^{j}(t  \tau_{2})\,\partial y^{k}(t  \tau_{2})} \bigg\vert \bigl(x_{1}^{*},x_{2}^{*},y^{*} \bigr) , \\ f_{1} = ax_{2}(t)  bx_{1}(t)  r_{1}x_{1}(t)  cx_{1}(t)x_{1}(t  \tau_{1})  \frac{a_{1}x_{1}^{2}(t)y(t)}{1 + mx_{1}^{2}(t)}, \\ f_{2} = bx_{1}(t)  r_{2}x_{2}(t), \qquad f_{3} = \frac{a_{2}x_{1}^{2}(t  \tau_{2})y(t  \tau_{2})}{1 + mx_{1}^{2}(t  \tau_{2})}  ry(t). \end{gathered} $$
Then we obtain the linearized system of system (2.1) as follows:
$$ \begin{gathered} \dot{x}_{1}(t) = a_{11}x_{1}(t) + a_{12}x_{2}(t) + a_{13}y(t) + b_{11}x(t  \tau_{1}), \\ \dot{x}_{2}(t) = a_{21}x_{1}(t) + a_{22}x_{2}(t), \\ \dot{y}(t) = b_{31}x_{1}(t  \tau_{2}) + a_{33}y(t) + b_{33}y(t  \tau_{2}). \end{gathered} $$
(2.2)
Therefore, the corresponding characteristic equation of system (2.2) is given by
$$ \begin{aligned}[b] &\lambda^{3} + m_{2} \lambda^{2} + m_{1}\lambda + m_{0} + \bigl(n_{2}\lambda^{2} + n_{1}\lambda + n_{0}\bigr)e^{  \lambda \tau_{1}} \\ &\quad {}+ \bigl(p_{2}\lambda^{2} + p_{1}\lambda + p_{0}\bigr)e^{  \lambda \tau_{2}} + (q_{1}\lambda + q_{0})e^{  \lambda (\tau_{1} + \tau_{2})} = 0, \end{aligned} $$
(2.3)
where
$$\begin{gathered} m_{0} = a_{12}a_{21}a_{33}  a_{11}a_{22}a_{33},\qquad m_{1} = a_{11}a_{22} + a_{22}a_{33} + a_{11}a_{33}  a_{12}a_{21},\\ m_{2} =  (a_{11} + a_{22} + a_{33}), \\ n_{0} =  b_{11}a_{22}a_{33},\qquad n_{1} = b_{11}(a_{22} + a_{33}),\qquad n_{2} =  b_{11}, \\ p_{0} = a_{13}a_{22}b_{31} + a_{12}a_{21}b_{33}  a_{11}a_{22}a_{33}, \\ p_{1} = a_{11}b_{33} + a_{22}b_{33}  a_{13}b_{31},\qquad p_{2} =  b_{33}, \\ q_{0} =  a_{22}b_{11}b_{33},\qquad q_{1} = b_{11}b_{33}. \end{gathered} $$
In order to investigate the root distribution of the transcendental Eq. (2.3), the result of Ruan and Wei [15] is introduced here.
Lemma 2.1
For the transcendental equation
$$\begin{aligned} p\bigl(\lambda,e^{  \lambda \tau_{1}}, \ldots,e^{  \lambda \tau_{m}}\bigr) &= \lambda^{n} + p_{1}^{(0)}\lambda^{n  1} + \cdots + p_{n  1}^{(0)}\lambda + p_{n}^{(0)} \\ &\quad {}+ \bigl[p_{1}^{(1)}\lambda^{n  1} + \cdots + p_{n  1}^{(1)}\lambda + p_{n}^{(1)} \bigr]e^{  \lambda \tau_{1}}+ \cdots \\ &\quad {} + \bigl[p_{1}^{(m)}\lambda^{n  1} + \cdots + p_{n  1}^{(m)}\lambda + p_{n}^{(m)} \bigr]e^{  \lambda \tau_{m}} \\ &= 0, \end{aligned} $$
as
\(( \tau_{1},\tau_{2},\tau_{3}, \ldots,\tau_{m} )\)
vary, the sum of orders of the zeros of
\(p(\lambda,e^{  \lambda \tau_{1}}, \ldots,e^{  \lambda \tau_{m}}) \)
in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.
Next, we will consider the following four cases.
Case 1: \(\tau_{1} = \tau_{2} = 0 \), the characteristic Eq. (2.3) reduces to
$$ \lambda^{3} + m_{12}\lambda^{2} + m_{11}\lambda + m_{10} = 0, $$
(2.4)
where \(m_{10} = m_{0} + n_{0} + p_{0} + q_{0} \), \(m_{11} = m_{1} + n_{1} + p_{1} \), \(m_{12} = m_{2} + n_{2} + p_{2} \).
It is not difficult to verify that \(m_{10} > 0 \), \(m_{12} > 0 \). Thus, all the roots of Eq. (2.4) have negative real parts if the following condition holds:
$$(\mathrm{H}11)\quad m_{11}m_{12} > m_{10}. $$
Namely, the equilibrium point \(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \) is locally asymptotically stable when condition (H11) is satisfied.
Case 2: \(\tau_{1} = 0 \), \(\tau_{2} > 0 \). Equation (2.3) becomes
$$ \lambda^{3} + m_{22}\lambda^{2} + m_{21}\lambda + m_{20} + \bigl(p_{22} \lambda^{2} + p_{21}\lambda + p_{20} \bigr)e^{  \lambda \tau_{2}} = 0, $$
(2.5)
where \(m_{20} = m_{0} + n_{0} \), \(m_{21} = m_{1} + n_{1} \), \(m_{22} = m_{2} + n_{2} \), \(p_{20} = p_{0} + q_{0} \), \(p_{21} = p_{1} + q_{1} \), \(p_{22} = p_{2} \).
Let \(i\omega_{2} \) (\(\omega_{2} > 0 \)) be a root of Eq. (2.5), it follows that
$$ \begin{gathered} p_{21}\omega_{2}\sin \omega_{2}\tau_{2} + \bigl(p_{20}  p_{22} \omega_{2}^{2}\bigr)\cos \omega_{2} \tau_{2} = m_{22}\omega_{2}^{2}  m_{20}, \\ p_{21}\omega_{2}\cos \omega_{2}\tau_{2}  \bigl(p_{20}  p_{21}\omega_{2}^{2} \bigr)\sin \omega_{2}\tau_{2} = \omega_{2}^{3}  m_{21}\omega_{2}, \end{gathered} $$
(2.6)
which leads to
$$ \omega_{2}^{6} + e_{22}\omega_{2}^{4} + e_{21}\omega_{2}^{2} + e_{20} = 0, $$
(2.7)
where \(e_{20} = m_{20}^{2}  p_{20}^{2} \), \(e_{21} = m_{21}^{2}  2m_{20}m_{22}  p_{21}^{2} + 2p_{20}p_{22} \), \(e_{22} = m_{22}^{2}  2m_{21}  p_{22}^{2} \).
Let \(\omega_{2}^{2} = v_{2} \), then Eq. (2.7) can be written as
$$ v_{2}^{3} + e_{22}v_{2}^{2} + e_{21}v_{2} + e_{20} = 0. $$
(2.8)
Denote
$$ f_{1}(v_{2}) = v_{2}^{3} + e_{22}v_{2}^{2} + e_{21}v_{2} + e_{20}. $$
(2.9)
Since \(f_{1}(0) = e_{20} \), \(\lim_{v_{2} \to + \infty} f_{1}(v_{2}) = + \infty \), and from Eq. (2.9), we have
$$ f'_{1}(v_{2}) = 3v_{2}^{2} + 2e_{22}v_{2} + e_{21}. $$
(2.10)
After discussion about the roots of Eq. (2.10) similar to that in [16], we have the following lemma.
Lemma 2.2
For the polynomial Eq. (2.8), we have the following results:

(1)
If
$$(\mathrm{H}21)\quad e_{20} \ge 0, \qquad \Delta = e_{22}^{2}  3e_{21} \le 0 $$
holds, then Eq. (2.8) has no positive root;

(2)
If
$$(\mathrm{H}22)\quad e_{20} \ge 0,\qquad \Delta = e_{22}^{2}  3e_{21} > 0, \qquad v_{2}^{*} = \frac{  e_{21} + \sqrt{\Delta}}{3} > 0,\qquad f_{1}\bigl(v_{2}^{*} \bigr) \le 0, $$
or
$$(\mathrm{H}23)\quad e_{20} < 0 $$
holds, then Eq. (2.8) has a positive root.
Suppose that Eq. (2.8) has positive roots. Without loss of generality, we assume that it has three positive roots, which are denoted by \(v_{21} \), \(v_{22} \), and \(v_{23} \). Then Eq. (2.7) has three positive roots \(\omega_{2k} = \sqrt{v_{2k}} \), \(k = 1,2,3 \). The corresponding critical value of time delay \(\tau_{2k}^{(j)} \) is
$$ \tau_{2k}^{(j)} = \frac{1}{\omega_{2k}}\arccos \biggl\{ \frac{A_{24}\omega_{2k}^{4} + A_{22}\omega_{2k}^{2} + A_{20}}{B_{24}\omega_{2k}^{4} + B_{22}\omega_{2k}^{2} + B_{20}} \biggr\} + \frac{2\pi j}{\omega_{2k}},\quad k = 1,2,3; j = 0,1,2, \ldots, $$
(2.11)
where \(A_{20} =  m_{20}p_{20} \), \(A_{22} = m_{22}p_{20} + m_{20}p_{22}  m_{21}p_{21} \), \(A_{24} = p_{21}  m_{22}p_{22} \), \(B_{20} = p_{20}^{2} \), \(B_{22} = p_{21}^{2}  2p_{20}p_{22} \), \(B_{24} = p_{22}^{2} \).
Thus \(\pm \omega_{2k} \) is a pair of purely imaginary roots of Eq. (2.5) with \(\tau_{2} = \tau_{2k}^{(j)} \), and let \(\tau_{20} = \min_{k \in \{1,2,3 \}} \{ \tau_{2k}^{(0)} \} \), \(\omega_{20} = \omega_{2k_{0}} \).
Lemma 2.3
Suppose that
$$(\mathrm{H}24) \quad f'_{1}\bigl(\omega_{20}^{2} \bigr) \ne 0, $$
then the following transversality condition holds:
$$\biggl\{ \frac{d(\operatorname{Re} \lambda )}{d\tau_{2}} \biggr\} _{\lambda = i\omega_{20}} \ne 0. $$
Proof
Differentiating Eq. (2.5) with respect to \(\tau_{2} \), and noticing that λ is a function of \(\tau_{2} \), we obtain
$$ \biggl(\frac{d\lambda}{d\tau_{2}}\biggr)^{  1} =  \frac{3\lambda^{2} + 2m_{22}\lambda + m_{21}}{\lambda (\lambda^{3} + m_{22}\lambda^{2} + m_{21}\lambda + m_{20})} + \frac{2p_{22}\lambda + p_{21}}{\lambda (p_{22}\lambda^{2} + p_{21}\lambda + n_{20})}  \frac{\tau_{1}}{\lambda}, $$
(2.12)
which leads to
$$\begin{aligned} \operatorname{Re} \biggl(\frac{d\lambda}{d\tau_{2}}\biggr)^{  1} &= \operatorname{Re} \biggl( \frac{3\lambda^{2} + 2m_{22}\lambda + m_{21}}{\lambda (\lambda^{3} + m_{22}\lambda^{2} + m_{21}\lambda + m_{20})}\biggr)_{\lambda = i\omega_{20}}\\ &\quad {} + \operatorname{Re} \biggl(\frac{2p_{22}\lambda + p_{21}}{\lambda (p_{22}\lambda^{2} + p_{21}\lambda + p_{20})}\biggr)_{\lambda = i\omega_{20}} \\ &= \frac{3\omega_{20}^{4} + 2(m_{22}^{2}  2m_{21})\omega_{20}^{2} + m_{21}^{2}  2m_{20}m_{22}}{(\omega_{20}^{3}  m_{21}\omega_{20})^{2} + (m_{20}  m_{22}\omega_{20}^{2})^{2}}  \frac{2p_{22}^{2}\omega_{20}^{2} + p_{21}^{2}  2p_{20}p_{22}}{(p_{22}\omega_{20}^{2}  p_{20})^{2} + p_{21}^{2}\omega_{20}^{2}}. \end{aligned} $$
From Eq. (2.6), we have
$$ \bigl(\omega_{20}^{3}  m_{21} \omega_{20}\bigr)^{2} + \bigl(m_{20}  m_{22}\omega_{20}^{2}\bigr)^{2} = \bigl(p_{22}\omega_{20}^{2}  p_{20} \bigr)^{2} + p_{21}^{2}\omega_{20}^{2}. $$
(2.13)
Noting that \(\{ \frac{d(\operatorname{Re} \lambda )}{d\tau_{1}} \}_{\lambda = i\omega_{20}} \) and \(\{ \operatorname{Re} (\frac{d\lambda}{d\tau_{1}})^{  1} \}_{\lambda = i\omega_{20}} \) have the same sign, then
$$ \begin{aligned}[b] \operatorname{sign} \biggl\{ \frac{d(\operatorname{Re} \lambda )}{d\tau_{1}} \biggr\} _{\lambda = i\omega_{20}} &= \operatorname{sign} \biggl\{ \operatorname{Re} \biggl( \frac{d\lambda}{d\tau_{1}}\biggr)^{  1} \biggr\} _{\lambda = i\omega_{20}} = \frac{3(\omega_{20}^{2})^{2} + 2e_{22}\omega_{20}^{2} + e_{21}}{p_{21}^{2}\omega_{20}^{2} + (p_{20}  p_{22}\omega_{20}^{2})^{2}} \\ &= \frac{f'_{1}(\omega_{20}^{2})}{p_{21}^{2}\omega_{20}^{2} + (p_{20}  p_{22}\omega_{20}^{2})^{2}} \\ &\ne 0. \end{aligned} $$
(2.14)
It follows that \(\{ \frac{d(\operatorname{Re} \lambda )}{d\tau_{2}} \}_{\lambda = i\omega_{20}} \ne 0 \) and the proof is complete. □
By Lemmas 2.1–2.3, and combining the Hopf bifurcation theorem [17–19], we have the following results.
Theorem 2.1
For system (1.2), \(\tau_{1} = 0 \).

(1)
If (H21) holds, then the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
is asymptotically stable for all
\(\tau_{2} \ge 0 \).

(2)
If (H22) or (H23) and (H24) hold, then the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
is asymptotically stable for all
\(\tau_{2} \in [0,\tau_{20}) \)
and unstable for
\(\tau_{2} > \tau_{20} \). Furthermore, system (1.2) undergoes a Hopf bifurcation at the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
when
\(\tau_{2} = \tau_{20} \).
Case 3: \(\tau_{1} = \tau_{2} = \tau \ne 0 \). Equation (2.3) reduces to
$$ \lambda^{3} + m_{32}\lambda^{2} + m_{31}\lambda + m_{30} + \bigl(n_{32} \lambda^{2} + n_{31}\lambda + n_{30} \bigr)e^{  \lambda \tau} + (q_{31}\lambda + q_{30})e^{  2\lambda \tau} = 0, $$
(2.15)
where \(m_{30} = m_{0} \), \(m_{31} = m_{1} \), \(m_{32} = m_{2} \), \(n_{30} = n_{0} + p_{0} \), \(n_{31} = n_{1} + p_{1} \), \(n_{32} = n_{2} + p_{2} \), \(q_{30} = q_{0} \), \(q_{31} = q_{1} \).
Multiplying by \(e^{\lambda \tau} \), Eq. (2.15) becomes
$$ \bigl(\lambda^{3} + m_{32}\lambda^{2} + m_{31}\lambda + m_{30}\bigr)e^{\lambda \tau} + \bigl(n_{32}\lambda^{2} + n_{31}\lambda + n_{30}\bigr) + (q_{41}\lambda + q_{40})e^{  \lambda \tau} = 0. $$
(2.16)
Let iω (\(\omega > 0 \)) be the root of Eq. (2.16), and separate the real and imaginary parts, we have
$$ \begin{gathered} E_{31}\sin \omega \tau + E_{32} \cos \omega \tau = E_{35}, \\ E_{33}\cos \omega \tau + E_{34}\sin \omega \tau = E_{36}, \end{gathered} $$
(2.17)
where \(E_{31} =  m_{31}\omega + \omega^{3} + q_{31}\omega \), \(E_{32} = m_{30}  m_{32}\omega^{2} + q_{30} \),
$$\begin{gathered} E_{33} = m_{31}\omega  \omega^{3} + q_{31}\omega,\qquad E_{34} = m_{30}  m_{32}\omega^{2}  q_{30}, \\ E_{35} = n_{32}\omega^{2}  n_{30}, \qquad E_{36} =  n_{31}\omega. \end{gathered} $$
It follows that
$$ \sin \omega \tau = \frac{A_{35}\omega^{5} + A_{33}\omega^{3} + A_{31}\omega}{\omega^{6} + B_{34}\omega^{4} + B_{32}\omega^{2} + B_{30}},\qquad \cos \omega \tau = \frac{A_{34}\omega^{4} + A_{32}\omega^{4} + A_{30}}{\omega^{6} + B_{34}\omega^{4} + B_{32}\omega^{2} + B_{30}}, $$
(2.18)
where
$$\begin{gathered} A_{30} = (q_{30}  m_{30})n_{30},\qquad A_{31} = (m_{31} + q_{31})n_{30}  (m_{30} + q_{30})n_{31}, \\ A_{32} = (m_{30}  q_{30})n_{32}  m_{31}n_{31} + m_{32}n_{30} + q_{31}n_{31},\\\ A_{33} = m_{32}n_{31}  n_{30}  m_{31}n_{32}  q_{31}n_{32}, \\ A_{34} = n_{31}  m_{32}n_{32},\qquad A_{35} = n_{32},\qquad B_{30} = m_{30}^{2}  q_{30}^{2}, \\ B_{32} = m_{31}^{2}  q_{31}^{2}  2m_{30}m_{32},\qquad B_{34} = m_{32}^{2}  2m_{31}. \end{gathered} $$
From Eq. (2.18), we get
$$ \omega^{12} + e_{35}\omega^{10} + e_{34} \omega^{8} + e_{33}\omega^{6} + e_{32} \omega^{4} + e_{31}\omega^{2} + e_{30} = 0, $$
(2.19)
where
$$\begin{gathered} e_{30} = B_{30}^{2}  A_{30}^{2}, \qquad e_{31} = 2B_{30}B_{32}  A_{31}^{2}  2A_{30}A_{32}, \\ e_{32} = B_{32}^{2} + 2B_{30}B_{34}  A_{32}^{2}  2A_{30}A_{34}  2A_{31}A_{33}, \\ e_{33} = 2B_{30} + 2B_{32}B_{34}  A_{33}^{2}  2A_{31}A_{35}  2A_{32}A_{34}, \\ e_{34} = B_{34}^{2} + 2B_{32}  A_{34}^{2}  2A_{33}A_{35},\qquad e_{35} = 2B_{34}  A_{35}^{2}. \end{gathered} $$
Let \(\omega^{2} = v_{3} \), then Eq. (2.19) can be written as
$$ v_{3}^{6} + e_{35}v_{3}^{5} + e_{34}v_{3}^{4} + e_{33}v_{3}^{3} + e_{32}v_{3}^{2} + e_{31}v_{3} + e_{30} = 0. $$
(2.20)
Suppose that Eq. (2.20) has at least one positive root, and without loss of generality, we assume that it has six positive roots which are denoted by \(v_{31} \), \(v_{32} \), \(v_{33} \), \(v_{34} \), \(v_{35} \), \(v_{36} \), then Eq. (2.19) has six positive roots \(\omega_{k} = \sqrt{v_{3k}} \), \(k = 1,2,3,4,5,6 \). The corresponding critical value of time delay \(\tau_{k}^{(j)} \) is
$$ \begin{aligned}[b] &\tau_{k}^{(j)} = \frac{1}{\omega_{k}}\arccos \biggl\{ \frac{A_{34}\omega_{k}^{4} + A_{32}\omega_{k}^{2} + A_{30}}{\omega_{k}^{6} + B_{34}\omega_{k}^{4} + B_{32}\omega_{k}^{2} + B_{30}} \biggr\} + \frac{2\pi j}{\omega_{k}},\\ &\quad k = 1,2,3,4,5,6, j = 0,1,2, \ldots. \end{aligned} $$
(2.21)
Then \(\pm \omega_{k} \) is a pair of purely imaginary roots of Eq. (2.16) with \(\tau = \tau_{k}^{(j)} \), and let \(\tau_{0} = \min_{k \in \{ 1  6 \}} \{ \tau_{k}^{(0)} \} \), \(\omega_{0} = \omega_{k_{0}} \).
Lemma 2.4
Suppose that
$$(\mathrm{H}31)\quad AC + BD \ne 0 $$
holds, then the following transversality condition is satisfied:
$$\biggl\{ \frac{d(\operatorname{Re} \lambda )}{d\tau} \biggr\} _{\lambda = i\omega_{0}} \ne 0. $$
Proof
Differentiating Eq. (2.16) with respect to τ, we obtain
$$ \biggl(\frac{d\lambda}{d\tau} \biggr)^{  1} = \frac{2n_{32}\lambda + n_{31} + (3\lambda^{2} + 2m_{32}\lambda + m_{31})e^{\lambda \tau}}{  \lambda (\lambda^{3} + m_{32}\lambda^{2} + m_{31}\lambda + m_{30})e^{\lambda \tau} + (q_{31}\lambda^{2} + q_{30})\lambda e^{  \lambda \tau}}  \frac{\tau}{ \lambda}, $$
(2.22)
substituting \(\lambda = i\omega_{0} \) into Eq. (2.22), we get
$$ \operatorname{Re} \biggl(\frac{d\lambda}{d\tau} \biggr)_{\lambda = i\omega_{0}}^{  1} = \operatorname{Re} \biggl(\frac{A + Bi}{C + Di}\biggr) = \frac{AC + BD}{C^{2} + D^{2}}, $$
(2.23)
where
$$\begin{gathered} A = \bigl(m_{31}  3\omega_{0}^{2} \bigr)\cos \omega_{0}\tau_{0}  2m_{32} \omega_{0}\sin \omega_{0}\tau_{0} + q_{31}\cos \omega_{0}\tau_{0} + n_{31}, \\ B = \bigl(m_{31}  3\omega_{0}^{2}\bigr)\sin \omega_{0}\tau_{0} + 2m_{32}\omega_{0} \cos \omega_{0}\tau_{0}  q_{31}\sin \omega_{0}\tau_{0} + 2n_{32}\omega_{0}, \\ C = \bigl(m_{31}  q_{31}  \omega_{0}^{2} \bigr)\omega_{0}^{2}\cos \omega_{0} \tau_{0} + \bigl(q_{30} + m_{30}  m_{32} \omega_{0}^{2}\bigr)\omega_{0}\sin \omega_{0}\tau_{0}, \\ D = \bigl(m_{31} + q_{31}  \omega_{0}^{2} \bigr)\omega_{0}^{2}\sin \omega_{0} \tau_{0} + \bigl(q_{30}  m_{30} + m_{32} \omega_{0}^{2}\bigr)\omega_{0}\cos \omega_{0}\tau_{0}. \end{gathered} $$
Noting that \(\{ \frac{d(\operatorname{Re} \lambda )}{d\tau} \}_{\lambda = i\omega_{0}} \) and \(\{ \operatorname{Re} (\frac{d\lambda}{d\tau} )^{  1} \}_{\lambda = i\omega_{0}} \) have the same sign, if condition (H31) holds, we obtain \(\{\frac{d(\operatorname{Re} \lambda )}{d\tau} \}_{\lambda = i\omega_{0}} \ne 0 \).
This completes the proof. □
By applying Lemma 2.4 to Eq. (2.16), we obtain the existence of a Hopf bifurcation as stated in the following theorem.
Theorem 2.2
For system (1.2), \(\tau_{1} = \tau_{2} = \tau \ne 0 \). Suppose that condition (H31) holds, then the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
is asymptotically stable for all
\(\tau \in [0,\tau_{0}) \)
and unstable for
\(\tau > \tau_{0} \). Furthermore, system (1.2) undergoes a Hopf bifurcation at the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
when
\(\tau = \tau_{0} \).
Case 4: \(\tau_{1} > 0 \), \(\tau_{2} \in [0,\tau_{20}) \), and \(\tau_{1} \ne \tau_{2} \).
We consider Eq. (2.3) with \(\tau_{2} \) in its stable interval, and \(\tau_{1} \) is regarded as the parameter. Let \(i\omega_{1_{*}}\) (\(\omega_{1_{*}} > 0\)) be the root of Eq. (2.3), then we obtain
$$ \begin{gathered} E_{41}\sin \omega_{1_{*}} \tau_{1} + E_{42}\cos \omega_{1_{*}} \tau_{1} = E_{43}, \\ E_{41}\cos \omega_{1_{*}}\tau_{1}  E_{42}\sin \omega_{1_{*}}\tau_{1} = E_{44}, \end{gathered} $$
(2.24)
where
$$\begin{gathered} E_{41} = n_{1} \omega_{1_{*}}  q_{0}\sin \omega_{1_{*}} \tau_{2} + q_{1}\omega_{1_{*}}\cos \omega_{1_{*}}\tau_{2}, \\ E_{42} = n_{0}  n_{2}\omega_{1_{*}}^{2} + q_{0}\cos \omega_{1_{*}}\tau_{2} + q_{1}\omega_{1_{*}}\sin \omega_{1_{*}}\tau, \\ E_{43} = m_{2}\omega_{1_{*}}^{2}  m_{0} + \bigl(p_{2}\omega_{1_{*}}^{2}  p_{0}\bigr)\cos \omega_{1_{*}}\tau_{2}  p_{1}\omega_{1_{*}}\sin \omega_{1_{*}} \tau_{2}, \\ E_{44} = \omega_{1_{*}}^{3}  m_{1} \omega_{1_{*}}  \bigl(p_{2}\omega_{1_{*}}^{2}  p_{0}\bigr)\sin \omega_{1_{*}}\tau_{2}  p_{1}\omega_{1_{*}}\cos \omega_{1_{*}} \tau_{2}. \end{gathered} $$
From Eq. (2.24), we have
$$ \begin{aligned}[b] &\omega_{1_{*}}^{6} + e_{42} \omega_{1_{*}}^{4} + e_{41}\omega_{1_{*}}^{2} + e_{40} + \bigl(c_{44}\omega_{1_{*}}^{4} + c_{42}\omega_{1_{*}}^{2} + c_{40}\bigr) \cos \omega_{1_{*}}\tau_{2} \\ &\quad {}+\bigl(c_{45}\omega_{1_{*}}^{5} + c_{43}\omega_{1_{*}}^{3} + c_{41} \omega_{1_{*}}\bigr)\sin \omega_{1_{*}}\tau_{2} = 0, \end{aligned} $$
(2.25)
where
$$\begin{gathered} e_{40} = m_{0}^{2} + p_{0}^{2}  n_{0}^{2}  q_{0}^{2}, \qquad e_{41} = m_{1}^{2} + p_{1}^{2}  n_{1}^{2}  q_{1}^{2} + 2n_{0}n_{2}  2m_{0}m_{2}  2p_{0}p_{2}, \\ e_{42} = m_{2}^{2}  n_{2}^{2}  2m_{1} + p_{2}^{2},\qquad c_{40} = 2m_{0}p_{0}  2n_{0}q_{0}, \\ c_{41} = 2p_{1}m_{0}  2p_{0}m_{1} + 2n_{1}q_{0}  2n_{0}q_{1}, \\ c_{42} = 2p_{1}m_{1}  2p_{0}m_{2} + 2n_{2}q_{0}  2p_{2}m_{0}  2n_{1}q_{1},\\ c_{43} = 2p_{0}  2p_{1}m_{2} + 2p_{2}m_{1} + 2n_{2}q_{1},\qquad c_{44} =  2p_{1} + 2p_{2}m_{2},\qquad c_{45} =  2p_{2}. \end{gathered} $$
In order to give the main results, we provide the following assumption.
$$(\mathrm{H}41)\quad \mbox{Eq. (2.25) has at least a finite positive root}. $$
We denote the positive roots of Eq. (2.25) by \(\omega_{1_{*}}^{(1)} \), \(\omega_{1_{*}}^{(2)} \), \(\omega_{1_{*}}^{(3)} \), \(\omega_{1_{*}}^{(4)} \), \(\omega_{1_{*}}^{(5)} \), and \(\omega_{1_{*}}^{(6)} \). For every \(\omega_{1_{*}}^{(i)} \) (\(i = 1,2,3,4,5,6 \)), the corresponding critical value of time delay \(\tau_{1i}^{(j)} \), \(j = 1,2,3 \ldots \) , is
$$ \begin{aligned}[b] &\tau_{1i}^{(j)} = \frac{1}{\omega_{1_{*}}}\arccos \biggl\{ \frac{E_{41}E_{44} + E_{42}E_{43}}{E_{41}^{2} + E_{42}^{2}} + 2\pi j \biggr\} _{\omega_{1_{*}} = \omega_{1_{*}}^{i}},\\ &\quad i = 1,2,3,4,5,6;j = 0,1,2 \ldots. \end{aligned} $$
(2.26)
Let \(\tau '_{10} = \min \{ \tau_{1i}^{(0)} \vert i = 1,2, \ldots 6;j = 0,1,2 \ldots \} \), \(\omega '_{10} \) is the corresponding root of Eq. (2.25) with \(\tau '_{10} \).
Lemma 2.5
Suppose that
$$(\mathrm{H}42)\quad A'C' + B'D' \ne 0 $$
holds, then the following transversality condition holds:
$$\biggl\{ \frac{d(\operatorname{Re} \lambda )}{d\tau_{1}} \biggr\} _{\lambda = i\omega '_{10}} \ne 0. $$
Proof
Taking the derivative of λ with respect to \(\tau_{1} \) in Eq. (2.3) and substituting \(\lambda = i\omega '_{10} \), we get
$$ \operatorname{Re} \biggl(\frac{d\lambda}{d\tau_{1}}\biggr)_{\lambda = i\omega _{10}}^{\prime  1} = \operatorname{Re} \biggl(\frac{A' + B'i}{C' + D'i}\biggr) = \frac{A'C' + B'D'}{C^{\prime2} + D^{\prime2}}, $$
(2.27)
where
$$\begin{aligned}& \begin{aligned} A' &= m_{1}  3\omega _{10}^{\prime2} + 2n_{2}\omega '_{10} \sin \omega '_{10}\tau '_{10} + n_{1}\cos \omega '_{10}\tau '_{10} \\ &\quad {}+\sin \omega '_{10}\tau_{2}\bigl( p_{1}\omega '_{10}\tau_{2} + 2p_{2}\omega '_{10} + q_{1}\sin \omega '_{10}\tau '_{10}\bigr) \\ &\quad {}+\cos \omega '_{10}\tau_{2} \bigl(p_{2}\tau_{2}\omega _{10}^{\prime2} + p_{1}  p_{0}\tau_{2} + q_{1}\cos \omega '_{10}\tau '_{10}\bigr), \end{aligned} \\& \begin{aligned} B' &= 2m_{2}\omega '_{10}  n_{1}\sin \omega '_{10}\tau '_{10} + 2n_{2}\omega '_{10} \cos \omega '_{10}\tau '_{10} \\ &\quad {}+\sin \omega '_{10}\tau_{2}\bigl( p_{1} + p_{0}\tau_{2}  p_{2} \tau_{2}\omega _{10}^{\prime2}  q_{1}\cos \omega '_{10}\tau '_{10}\bigr) \\ &\quad {}+\cos \omega '_{10}\tau_{2} \bigl(2p_{2}\omega '_{10}  p_{1} \omega '_{10}\tau_{2}  q_{1}\sin \omega '_{10}\tau '_{10}\bigr), \end{aligned} \\& \begin{aligned} C' &= \bigl(n_{0}\omega '_{10}  n_{2}\omega _{10}^{\prime3} \bigr)\sin \omega '_{10}\tau '_{10}  n_{1}\omega _{10}^{\prime2}\cos \omega '_{10}\tau '_{10} \\ &\quad {}+\bigl(q_{0}\omega '_{10}\cos \omega '_{10}\tau '_{10} + q_{1}\omega _{10}^{\prime2}\sin \omega '_{10}\tau '_{10}\bigr)\sin \omega '_{10}\tau_{2} \\ &\quad {}+ \bigl(q_{0}\omega '_{10}\sin \omega '_{10}\tau '_{10}  q_{1}\omega _{10}^{\prime2}\cos \omega '_{10}\tau '_{10}\bigr)\cos \omega '_{10}\tau_{2}, \end{aligned} \\& \begin{aligned} D' &= \bigl(n_{0}\omega '_{10}  n_{2}\omega _{10}^{\prime3} \bigr)\cos \omega '_{10}\tau '_{10} + n_{1}\omega _{10}^{\prime2}\sin \omega '_{10}\tau '_{10} \\ &\quad {}+\bigl( q_{0}\omega '_{10}\sin \omega '_{10}\tau '_{10} + q_{1}\omega _{10}^{\prime2}\cos \omega '_{10}\tau '_{10}\bigr)\sin \omega '_{10}\tau_{2} \\ &\quad {}+\bigl(q_{0}\omega '_{10}\cos \omega '_{10}\tau '_{10} + q_{1}\omega _{10}^{\prime2}\sin \omega '_{10}\tau '_{10}\bigr)\cos \omega '_{10}\tau_{2}. \end{aligned} \end{aligned}$$
Obviously, if condition (H42) holds, then we have \(\{\frac{d(\operatorname{Re} \lambda )}{d\tau_{1}} \}_{\lambda = i\omega '_{10}} \ne 0 \). This completes the proof of the lemma. □
By the above analysis, we have the following theorem.
Theorem 2.3
For system (1.2), \(\tau_{1} > 0 \), \(\tau_{2} \in [0,\tau_{20}) \), and
\(\tau_{1} \ne \tau_{2} \). Suppose that conditions (H41) and (H42) hold, then the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
is asymptotically stable for all
\(\tau_{1} \in [0,\tau '_{10}) \)
and unstable for
\(\tau_{1} > \tau '_{10} \). Furthermore, system (1.2) undergoes a Hopf bifurcation at the positive equilibrium
\(E^{*}(x_{1}^{*},x_{2}^{*},y^{*}) \)
when
\(\tau_{1} = \tau '_{10} \).