Equilibrium analysis
System (1)(3) has three biologically feasible equilibrium points (positive equilibrium) that are described as follows:
Trivial equilibrium point
\(\bar{E}=(0,0,0)\): Existence of the trivial case is obvious as all the three populations are extinct.
Axial equilibrium point
\(\tilde{E}=(\tilde{P},\tilde{M},0)\): \(\tilde{P}=\frac{(K (r  \beta + \gamma ))}{r}\), \(\tilde{M}=\frac{(K\beta (r  \beta + \gamma ))}{r \mu }\). Axial equilibrium exists when \((r+\gamma )>\beta \), i.e., the transition rate from immature to mature stage is less than the sum of intrinsic growth and government efforts. In this case industries no longer exist.
Interior equilibrium point
\(\hat{E}=(\hat{P},\hat{M},\hat{I})\): Existence of interior equilibrium depends upon the industrial density as both immature and mature forestry biomass are defined in terms of industrial density (I). Immature population \(\hat{P}=\frac{K}{r}[(r+ \gamma )(q_{1}+d_{1})\hat{I}\beta ]\), mature population \(\hat{M}=\frac{ \beta \hat{P}}{(q_{2}+d_{2})\hat{I}+\mu }\), both P and M exist when \(\hat{P}>0\), when \((r+\gamma )>(q_{1}+d_{1})\hat{I}\beta \) is satisfied, and I is defined by the positive roots of the quadratic equation
$$ z_{1} \hat{I}^{2}+z_{2}\hat{I}+z_{3}=0. $$
(7)
The quadratic equation (7) has one positive root when \(z_{3}\) is less than zero and is possible when \((r+\gamma )>\beta \). Values of \(z_{1}\), \(z_{2}\), and \(z_{3}\) are given as \(z_{1}=(a_{1} d_{1}^{2} d_{2} K + a_{1} d_{1}^{2} K q_{1} + a_{1} d _{1} d_{2} K q_{1} + a_{1} d_{1} K q_{1}^{2})p_{1} + (a_{1} d_{1} d _{2} K q_{1} + a_{1} d_{1} K q_{1}^{2} + a_{1} d_{2} K q_{1}^{2} + a _{1} K q_{1}^{3}) (p_{1}  \tau_{1})\), \(z_{2}= (a_{1} d_{2} K q _{1} \beta + a_{1} K q_{1}^{2} \beta a_{1} d_{2} K q_{1} r  a_{1} K q_{1}^{2} r  a_{1} d_{2} K q_{1} \gamma  a_{1} K q_{1}^{2} \gamma + a_{1} d_{1} K q_{1} \mu + a_{1} K q_{1}^{2} \mu ) (p_{1}  \tau_{1}) + (a_{2} d_{1} K q_{2} \beta + a_{2} K q_{1} q_{2} \beta )(p_{2}  \tau _{2})+ ( a_{1} d_{1} d_{2} K \beta + a_{1} d_{1} K q_{1} \beta a_{1} d_{1} d_{2} K r  a_{1} d_{1} K q_{1} r  a_{1} d_{1} d_{2} K \gamma  a_{1} d_{1} K q_{1} \gamma + a_{1} d_{1}^{2} K \mu + a_{1} d_{1} K q _{1} \mu )p_{1}+(a_{2} d_{1} K d_{2} \beta + a_{2} K d_{2} q_{1} \beta ) p_{2}+a_{1} c_{1} d_{2} r + a_{2} c_{2} d_{2} r + d_{2} d_{3} r + a_{1} c_{1} q_{1} r + a_{2} c_{2} q_{1} r + d_{3} q_{1} r\), and \(z_{3}= (a_{2} K d_{2} \beta^{2}  a_{2} K d_{2} r \beta  a _{2} K d_{2} \beta \gamma )p_{2} + (a_{1} d_{1} K \beta \mu a_{1} d _{1} K r \mu  a_{1} d_{1} K \gamma \mu )p_{1} + (a_{1} K q_{1} \beta \mu a_{1} K q_{1} r \mu  a_{1} K q_{1} \gamma \mu ) (p_{1}  \tau_{1}) + (a_{2} K q_{2} \beta^{2} a_{2} K q_{2} r \beta  a_{2} K q_{2} \beta \gamma ) (p_{2}  \tau_{2})+a_{1} c_{1} r \mu + a_{2} c _{2} r \mu + d_{3} r \mu \).
In the next section, the local stability criteria are obtained for the coexistence state, as we are interested in the state, when all the populations coexist for the defined system.
Stability analysis
System without maturation delay
In this section, we show the local stability of the system for all the feasible equilibria. A variational matrix corresponding to system (1)(3) is given as follows:
$$ J=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} j_{11} & 0 & j_{13} \\ j_{21} & j_{22} & j_{23} \\ j_{31} & j_{32} & j_{33} \end{array}\displaystyle \right ], $$
where \(j_{11}= r\beta +\gamma \frac{2 r P}{K}(d_{1}+q_{1})I\), \(j _{13}= (d_{1}+q_{1})P\), \(j_{21}= \beta \), \(j_{22}=(d_{2}+q_{2})I \mu \), \(j_{23}= (d_{2}+q_{2})M\), \(j_{31}=a_{1}I(d_{1}p_{1}+q _{1}(p_{1}\tau_{1})) \), \(j_{32}= a_{2}I(d_{2}p_{2}+q_{2}(p_{2}\tau _{2}))\), and \(j_{33}= a_{1}P(d_{1}p_{1}+q_{1}(p_{1}\tau_{1}))+a_{2}M(d _{2}p_{2}+q_{2}(p_{2}\tau_{2}))a_{1}c_{1}a_{2}c_{2}d_{3}\). It is observed that:

1.
Trivial equilibrium Ē is obvious and is unstable.

2.
Axial equilibrium Ẽ with eigenvalues \(\frac{r \tilde{P}}{K}\), −μ, and \(a_{1}c_{1}  a _{2}c_{2} d_{3} +a_{1}d_{1}\tilde{P} p_{1} + a_{2} d_{2} \tilde{M} p _{2} + a_{1} \tilde{P} (p_{1}\tau_{1}) q_{1} + a_{2} \tilde{M} (p _{2}\tau_{2}) q_{2} \) is stable when \(a_{1}c_{1} + a_{2}c_{2}+ d_{3}>a _{1}d_{1}\tilde{P} p_{1} + a_{2} d_{2} \tilde{M} p_{2} + a_{1} \tilde{P} (p_{1}\tau_{1}) q_{1} + a2 \tilde{M} (p_{2}\tau_{2}) q _{2}\).

3.
The stability of interior equilibrium Ê is an interesting case as all the state variables, immature (P) and mature (M) forestry biomass together with industrialization (I) coexist; analysis of them is as follows.
A variational matrix for the interior equilibrium Ê is given by
$$ J_{1}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} \frac{r \hat{P}}{K} & 0 & (d_{1}+q_{1})\hat{P} \\ \beta & \mu & (d_{2}+q_{2})\hat{M} \\ a_{1} \hat{I} (d_{1} p_{1} + q_{1} (p_{1}  \tau_{1})) & a_{2} \hat{I} (d_{2} p_{2} + q_{2} (p_{2}  \tau_{2})) & 0 \end{array}\displaystyle \right ], $$
the corresponding characteristic equation for the variational matrix J is
$$ \lambda^{3}+Z_{1} \lambda^{2}+Z_{2} \lambda +Z_{3}=0, $$
where \(Z_{1}=d_{2} \hat{I} + q_{2} \hat{I}+ \frac{r \hat{P}}{K} + \mu\), \(Z_{2}= \frac{r \hat{P}}{K} ((d_{2}+q_{2}) \hat{I}+\mu ) + (a_{1} d _{1}^{2} \hat{I} \hat{P}+a_{1} d_{1} \hat{I} \hat{P} q_{1} ) p_{1} + (a _{2} d_{2}^{2} \hat{I} \hat{M}+a_{2} d_{2} \hat{I} \hat{M} q_{2} ) p _{2} + (a_{1} d_{1} \hat{I} \hat{P} q_{1}+ a_{1} \hat{I} \hat{P} q _{1}^{2} ) (p_{1}\tau_{1}) + (a_{2} d_{2} \hat{I} \hat{M} q_{2} + a _{2} \hat{I} \hat{M} q_{2}^{2} ) (p_{2}\tau_{2})\) and \(Z_{3}= (a_{1} d_{1}^{2}d_{2} I^{2} \hat{P} + a_{1} d_{1} d_{2} \hat{I}^{2} P q_{1} + a_{1} d_{1}^{2} \hat{I}^{2} \hat{P} q_{2} + a_{1} d_{1} \hat{I}^{2} \hat{P} q_{1} q_{2} + a_{1} d_{1}^{2} \hat{I} \hat{P} \mu + a_{1} d _{1} \hat{I} \hat{P} q_{1} \mu ) p_{1}+ (a_{1} d_{1} d_{2} \hat{I} ^{2} P q_{1} + a_{1} d_{2} \hat{I}^{2} \hat{P} q_{1}^{2} + a_{1} d_{1} \hat{I}^{2} \hat{P} q_{1} q_{2} + a_{1} \hat{I}^{2} \hat{P} q _{1}^{2} q_{2} + a_{1} d_{1} \hat{I} \hat{P} q_{1} \mu + a_{1} \hat{I} \hat{P} q_{1}^{2} \mu ) (p_{1}  \tau_{1}) + (\frac{a_{2} d _{2}^{2} \hat{I} \hat{M} \hat{P} r}{K} + \frac{a_{2} d_{2} \hat{I} \hat{M} \hat{P} q_{2} r}{K} +a_{2} d_{1} d_{2} \hat{I} \hat{P} \beta + a_{2} d_{2} \hat{I} \hat{P} q_{1} \beta ) p_{2} + (\frac{a_{2} d_{2} \hat{I} \hat{M} \hat{P} q_{2} r}{K} + \frac{a_{2} \hat{I} \hat{M} \hat{P} q_{2}^{2} r}{K} +a_{2} d_{1} \hat{I} \hat{P} q_{2} \beta + a _{2} \hat{I} \hat{P} q_{1} q_{2} \beta ) (p_{2}  \tau_{2})\).
Now, on applying the RouthHurwitz criteria, the system is locally stable as all the coefficients \(Z_{1}\), \(Z_{2}\), and \(Z_{3}\) are positive and the condition
$$ Z_{1}Z_{2}Z_{3}>0 $$
(8)
is satisfied. Hence, the interior equilibrium is locally asymptotically stable as all the conditions of RouthHurwitz criteria are satisfied.
Remark
We have considered the tax \(\tau_{1}\), \(\tau_{2}\) per unit of forestry biomass is less than the price \(p_{1}\), \(p_{2}\) per unit of forestry biomass. Hence, if the coexistence state of the system exists, it is always stable for the system.
System with maturation delay
For the analysis of delayed system, we refer to [7, 16, 17]. Equilibrium points of the system without maturation delay and of that with maturation delay are the same, as time delay does not change the equilibria of the system. To study the local stability of the interior equilibrium \(\hat{E}=(\hat{P}, \hat{M}, \hat{I})\) for the delay system, first we use linear transformation in the system equations (1)(3) as \(P=P_{1}+\hat{P}\), \(M=M_{1}+\hat{M}\), and \(I=I_{1}+\hat{I}\), where \(P_{1}\ll 1\), \(M_{1}\ll 1\), \(I_{1}\ll 1\). A variational matrix for the delay system about the interior equilibrium is
$$ J_{2}=\left [ \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} \beta e^{\lambda h_{1}}  d_{1} \hat{I}  q_{1} \hat{I} \frac{2r \hat{P}}{K} +r+\gamma & 0 & (d_{1}+q_{1})\hat{P} \\ \beta e^{\lambda h_{1}} & \mu (q_{2}+d_{2})\hat{I} & (d_{2}+q_{2}) \hat{M} \\ a_{1} \hat{I} (d_{1} p_{1} + q_{1} (p_{1}  \tau_{1})) & a_{2} \hat{I} (d_{2} p_{2} + q_{2} (p_{2}  \tau_{2})) & 0 \end{array}\displaystyle \right ], $$
the characteristic equation corresponding to the interior equilibrium is
$$ \lambda^{3}+A_{1}\lambda^{2}+A_{2} \lambda +A_{3}+\beta e^{\lambda h _{1}} \bigl(B_{1} \lambda^{2}+B_{2}\lambda +B_{3} \bigr)=0, $$
(9)
where \(A_{1}=\mu r\gamma + d_{1} \hat{I}+d_{2} \hat{I} + q_{1} \hat{I} + \frac{2 r\hat{P}}{K}+ q_{2} \hat{I}\), \(A_{2}= r \mu  \gamma \mu  d_{2} r \hat{I} q_{2} r \hat{I} d_{2} \gamma \hat{I}  q_{2} \gamma \hat{I} + d_{1} \mu \hat{I} + q_{1} \mu \hat{I} + d_{1} d_{2} \hat{I}^{2} + d_{2} q_{1} \hat{I}^{2} + d 1 q2 \hat{I}^{2} + q_{1} q _{2} \hat{I}^{2} + a_{2} d_{2}^{2} p_{2} \hat{I} \hat{M} +2 a_{2} d _{2} p_{2} q_{2} \hat{I} \hat{M} +a_{2} p_{2} q_{2}^{2} \hat{I} \hat{M} a_{2} d_{2} q_{2} \tau_{2} \hat{I} \hat{M} a_{2} q_{2}^{2} \tau_{2} \hat{I} \hat{M} + \frac{ 2 r \mu \hat{P}}{K} + a_{1} d_{1} ^{2} p_{1} \hat{I} \hat{P} +2 a_{1} d_{1} p_{1} q_{1} \hat{I} \hat{P} +a_{1} p_{1} q_{1}^{2} \hat{I} \hat{P} + \frac{2 d_{2} r \hat{I} \hat{P}}{K} + \frac{2 q_{2} r \hat{I} \hat{P}}{K}  a_{1} d_{1} q_{1} \tau_{1} \hat{I} \hat{P}  a_{1} q_{1}^{2} \tau_{1} \hat{I} \hat{P}\), \(A_{3}=p_{1} (a_{1} d_{1}^{2} \mu \hat{I} \hat{P} + a_{1} d_{1} q_{1} \mu \hat{I} \hat{P} + a_{1} d_{1}^{2} d_{2} \hat{I}^{2} \hat{P} + a _{1} d_{1} d_{2} q_{1} \hat{I}^{2} \hat{P} + a_{1} d_{1}^{2} q_{2} \hat{I}^{2} \hat{P} + a_{1} d_{1} q_{1} q_{2} \hat{I}^{2} \hat{P})+ (p_{1}  \tau_{1}) (a_{1} d_{1} q_{1} \mu \hat{I} \hat{P}) + a_{1} q _{1}^{2} \mu \hat{I} \hat{P} + a_{1} d_{1} d_{2} q_{1} \hat{I}^{2} \hat{P} + a_{1} d_{2} q_{1}^{2} \hat{I}^{2} \hat{P} + a_{1} d_{1} q _{1} q_{2} \hat{I}^{2} \hat{P} + a_{1} q_{1}^{2} q_{2} \hat{I}^{2} \hat{P}) + p_{2} (a_{2} d_{2}^{2} r \hat{I} \hat{M}  a_{2} d_{2} q _{2} r \hat{I} \hat{M}  a_{2} d_{2}^{2} \gamma \hat{I} \hat{M}  a _{2} d_{2} q_{2} \gamma \hat{I} \hat{M} + a_{2} d_{1} d_{2}^{2} \hat{I}^{2} \hat{M} + a_{2} d_{2}^{2} q_{1} \hat{I}^{2} \hat{M} + a _{2} d_{1} d_{2} q_{2} \hat{I}^{2} \hat{M} + a_{2} d_{2} q_{1} q_{2} \hat{I}^{2} \hat{M} +\frac{ 2 a_{2} d_{2}^{2} r \hat{I} \hat{M} \hat{P}}{K} + \frac{ 2 a_{2} d_{2} q_{2} r \hat{I} \hat{M} \hat{P}}{ K} + (p_{2}  \tau_{2}) (a_{2} d_{2} q_{2} r \hat{I} \hat{M} + a_{2} q_{2}^{2} r \hat{I} \hat{M} + a_{2} d_{2} q_{2} \gamma \hat{I} \hat{M} + a_{2} q _{2}^{2} \gamma \hat{I} \hat{M}  a_{2} d_{1} d_{2} q_{2} \hat{I}^{2} \hat{M}  a_{2} d_{2} q_{1} q_{2} \hat{I}^{2} \hat{M}  a_{2} d_{1} q _{2}^{2} \hat{I}^{2} \hat{M}  a_{2} q_{1} q_{2}^{2} \hat{I}^{2} \hat{M}  \frac{ 2 a_{2} d_{2} q_{2} r \hat{I} \hat{M} \hat{P}}{K}  \frac{2 a_{2} q_{2}^{2} r \hat{I} \hat{M} \hat{P}}{K})\), \(B_{1}= \beta \), \(B_{2}=\beta \mu + d_{2} \beta \hat{I} + q_{1} \beta \hat{I}\), and \(B_{3}= \beta d_{2} \hat{I} \hat{P}(d_{1} + q _{1} ) p_{2}+a_{2} q_{2} \hat{I} \hat{P}(d_{1}+ q_{1})\beta (p_{2}  \tau_{2})\).
Previously, it was noticed when there is no delay (\(h_{1}=0\)) the system is locally asymptotically stable. Now, we assume for some \(\lambda =i \omega \) with \(\omega >0 \) be the solution of (9). Then equation (9) can be written as
$$\begin{aligned}& A_{3} + i A_{2} \omega  A_{1} \omega^{2}  i \omega^{3}+ B_{3} \cos( \omega h_{1})+i B_{2} \omega \cos (\omega h_{1}) B_{1} \omega^{2} \cos( \omega h_{1}) \\& \quad {} i B_{3} \sin (\omega h_{1}) + B_{2} \omega \sin (\omega h_{1}) + i B _{1} \omega^{2} \sin (\omega h_{1})=0. \end{aligned}$$
(10)
On separating the real and imaginary parts, we have
$$\begin{aligned}& A_{3}  A_{1} \omega^{2} = B_{1} \omega^{2} \cos (\omega h_{1})B _{3} \cos (\omega h_{1})B_{2}\omega \sin (\omega h_{1}). \end{aligned}$$
(11)
$$\begin{aligned}& A_{2} \omega  \omega^{3} =B_{3} \sin (\omega h_{1})  B_{1} \omega ^{2} \sin (\omega h_{1}) B_{2} \omega \cos (\omega h_{1}). \end{aligned}$$
(12)
Squaring and adding equations (11) and (12) and substituting \(\omega^{2}=\zeta \) leads to a cubic equation
$$ h(\zeta )=\zeta^{3}+k_{1}\zeta^{2}+k_{2} \zeta +k_{3}=0, $$
(13)
where \(k_{1}=A_{2}^{2}B_{2}^{2}2B_{1} B_{3}\), \(k_{2}=A_{1}^{2}2A _{2}B_{1}^{2}\), and \(k_{3}=A_{3}^{2}B_{3}^{2}\). If the condition of the RouthHurwitz criterion satisfies \(h(\zeta )\), then equation (13) will not have any positive real root, and we may not get any positive value of ω, which satisfies equations (11) and (12). In this case the results are given in the form of the following lemma.
Lemma 2
If coefficients of equation (13) are positive, all the conditions of RouthHurwitz criteria are satisfied and the interior equilibrium exists, it is locally asymptotically stable for all
\(h_{1}>0\).
Proof
Now, we consider that the values’ of \(k_{1}\), \(k_{2}\), and \(k_{3}\) do not satisfy the conditions of the RouthHurwitz criterion. In this case a simple assumption
$$ k_{1}>0, \qquad k_{2}>0, \quad \mbox{and} \quad k_{3}< 0 $$
(14)
gives that equation (14) has one positive root. If condition (14) holds good, then equation (9) has a pair of purely imaginary roots \(\pm i \omega_{0}\). Next, using transcendental equations (11)(12), we get
$$ \tan \omega h_{1}=\frac{B_{2} \omega_{0}(A_{3}A_{1} \omega_{0}^{2})(A _{2}\omega_{0}\omega_{0}^{3})(B_{3}B_{1}\omega_{0}^{2})}{B_{2} \omega_{0}(A_{2}\omega_{0}\omega_{0}^{3})+(A_{3}A_{1} \omega_{0} ^{2})(B_{3}B_{1}\omega_{0}^{2})}. $$
For a positive value of \(\omega_{0}\), the value of \(h_{1}\) is defined as
$$ h_{1n}=\frac{n \pi }{\omega_{0}}+\frac{1}{\omega_{0}} \tan^{1} \frac{B _{2} \omega_{0}(A_{3}A_{1} \omega_{0}^{2})(A_{2}\omega_{0}\omega _{0}^{3})(B_{3}B_{1}\omega_{0}^{2})}{B_{2} \omega_{0}(A_{2}\omega _{0}\omega_{0}^{3})+(A_{3}A_{1} \omega_{0}^{2})(B_{3}B_{1}\omega _{0}^{2})} $$
for \(n=0,1,2,\ldots\) .
In the similar logic [17, 18], it can be concluded that the stable interior equilibrium Ê remains stable for \(h_{1}< h_{10}\). Next, we will investigate the possibility of Hopf bifurcation as \(h_{1}\) is greater than \(h_{10}\). □
Lemma 3
The transversality condition
\(\frac{d \operatorname{Re} \lambda_{k}(h_{1k})}{d\tau }>0\), where
\(k=0,1,2,3,\ldots\) , is satisfied, i.e., system (1)(3) undergoes a Hopf bifurcation at the coexisting state of equilibrium
Ê
for
\(h_{1}=h_{1n}\).
Proof
Differentiating (9) with respect to \(h_{1}\) gives
$$ \biggl( \frac{d\lambda }{dh_{1}} \biggr) ^{1}=\frac{(3\lambda^{2}+2A_{1} \lambda +A_{2})+(2B_{1}\lambda +B_{2})e^{\lambda h_{1}}}{\lambda (B _{1}\lambda^{2}+B_{2}\lambda +B_{3})e^{\lambda h_{1}}} \frac{h_{1}}{ \lambda }. $$
(15)
Now,
$$\begin{aligned} \operatorname{sgn} \biggl[ \frac{d (\operatorname{Re} (\lambda ))}{d h_{1}} \biggr] _{h_{1}=h_{10}} =& \operatorname{sgn} \biggl[ \frac{d (\operatorname{Re} (\lambda ))}{d h_{1}} \biggr] _{h_{1}=h_{10}}^{1} \\ =& \operatorname{sgn} \biggl[ \operatorname{Re} \biggl( \frac{d (\lambda )}{d h_{1}} \biggr) ^{1} \biggr] _{\lambda =i \omega_{0}} \\ =& \operatorname{sgn} \biggl[ \frac{3 \omega_{0}^{4}+2k_{1}\omega_{0}^{2}+k_{2}}{B_{2} ^{2}\omega_{0}^{2}+(B_{3}B_{1}\omega_{0}^{2})^{2}} \biggr] \\ =& \operatorname{sgn} \biggl[ \frac{h'(\omega_{0}^{2})}{B_{2}^{2}\omega_{0}^{2}+(B_{3}B _{1}\omega_{0}^{2})^{2}} \biggr]. \end{aligned}$$
The condition defined in equation (15) is satisfied, \(h'(\omega _{0}^{2})>0\). Hence the transversality condition holds, which confirms that the Hopf bifurcation occurs at \(h_{1}=h_{10}\). □
Remark
The condition defined in equation (8) is satisfied when equation (15) holds, the interior equilibrium Ê of system (1)(3) is locally asymptotically stable for \(h_{1}\epsilon [0,h_{10})\) and becomes unstable for \(h_{1}>h_{10}\).
Stability and direction of a Hopf bifurcation point
Previously, we have acquired the condition where the system undergoes bifurcation for the interior equilibrium when \(h_{1}\) crosses the critical threshold. Here, we will derive explicit formulae for determining the direction and bifurcating periodic solutions that emerge through the Hopf bifurcation. We will use the normal form theory and the center manifold theorem as in [19]. Without loss of generality, we denote the critical values of h by \(h_{k}\) at which (9) has a pair of purely imaginary roots and the system undergoes the Hopf bifurcation. Hence, for any root of (9) of the form \(\psi ( h ) =\nu ( h ) +i\omega ( h ), \nu ( h_{k} ) =0, \omega ( h_{k} ) =\omega_{0}\), and \(\arrowvert \frac{d \nu }{d h} \rvert_{h=h_{k}}\neq 0\), let \(h=h_{k}+\mu \), \(\mu \in R \), so that \(\mu =0\) is a Hopf bifurcation value for the system.
Next, taking the space of continuous realvalued functions as \(C=C ( [ 1, 0 ], R^{3} ) \), using the transformation \(u_{1}=P ( t ) \hat{P}, u_{2}=M ( t ) \hat{M}, u _{3}=I ( t ) \hat{I}\), and \(\chi_{i} ( t ) =u_{i} ( h t ) \) for \(i=1, 2, 3\), the delay system (1)(3) then transforms to FDE in C as follows:
$$ \frac{d \chi }{dt}=L_{\mu } \chi_{t}+f ( \mu, \chi_{t} ), $$
(16)
where \(\chi (t)= ( \chi_{1}(t), \chi_{2}(t), \chi_{3}(t) ) ^{T} \in R^{3}, \chi_{t} ( \Theta ) =\chi ( t+\Theta ), \Theta \in [ 1, 0 ] \), \(L_{\mu }: C \rightarrow R^{3}\), and \(f: C\times R \rightarrow R^{3}\) are given by
$$ L_{\mu } \zeta = ( h_{k}+ \mu ) \bigl[ M_{1} \zeta ( 0 ) +M _{2} \zeta ( 1 ) \bigr], $$
(17)
where
$$\begin{aligned}& M_{1}= \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} r+ \gamma \frac{2 r \hat{P}}{K}q_{1} \hat{I}d_{1} \hat{I} & 0 &  ( q_{1}+d_{1} ) \hat{P} \\ 0 &  ( q_{2} \hat{I}+d_{2} \hat{I}+\mu \rbrace &  ( q _{2}+d_{2} ) \hat{M} \\ \{ a_{1}q_{1} ( p_{1}\tau_{1} ) +a_{1} p_{1} d_{1}\} \hat{I} & \{ a_{2}q_{2} ( p_{2}\tau_{2} ) +a_{2} p_{2} d_{2}\} \hat{I} & 0 \end{array}\displaystyle \right ), \end{aligned}$$
(18)
$$\begin{aligned}& M_{2}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} \beta & 0 & 0 \\ \beta & 0 & 0 \\ 0 & 0 & 0 \end{array}\displaystyle \right ) \end{aligned}$$
(19)
and
$$ f ( \mu, \zeta ) = ( h_{k}+\mu ) \left ( \textstyle\begin{array}{@{}c@{}} Z_{1} \\ Z_{2} \\ Z_{3} \end{array}\displaystyle \right ) , $$
(20)
where \(Z_{1}= \frac{r}{K} \zeta_{1}^{2} ( 0 )  ( q _{1}+d_{1} ) \zeta_{1} ( 0 ) \zeta_{3} ( 0 )\), \(Z _{2}= ( q_{2}+d_{2} ) \zeta_{2} ( 0 ) \zeta_{3} ( 0 )\), \(Z_{3}= [ a_{1} \lbrace ( p_{1}\tau_{1} ) q _{1}+p_{1} d_{1} \rbrace ] \zeta_{1} ( 0 ) \zeta _{3} ( 0 ) + [ a_{2} \lbrace ( p_{2}\tau_{2} ) q _{2}+p_{2} d_{2} \rbrace ] \zeta_{2} ( 0 ) \zeta _{3} ( 0 ) \) for \(\zeta = ( \zeta_{1}\zeta_{2},\zeta_{3} ) ^{T} \in C\).
As per the Riesz representation theorem, there exists a function \(\eta ( \Theta, \mu ) \) whose components are of bounded variation for \(\Theta \in [ 1,0 ] \) such that
$$ L_{\mu } \zeta = \int_{0}^{1} d\eta ( \Theta, \mu ) \zeta ( \Theta ) . $$
(21)
In view of equation (17), we can choose
$$ \eta ( \Theta, \mu ) = ( h_{k}+ \mu ) \bigl[ M _{1} \delta ( \Theta ) +M_{2} \delta ( \theta +1 ) \bigr], $$
(22)
where \(\zeta \in C^{1} ( [ 1, 0 ], R^{3} ) \), define
$$ A ( \mu ) \zeta = \textstyle\begin{cases} \frac{d \zeta ( \Theta ) }{d \Theta }, & \Theta \in [ 1,0 ), \\ \int_{1}^{0}d \eta ( \rho, \mu ) \zeta ( \rho ) =L_{\mu } \zeta, &\Theta =0. \end{cases} $$
(23)
Also,
$$ R ( \mu ) \zeta = \textstyle\begin{cases} 0,& \Theta \in [ 1, 0 ), \\ f ( \zeta, \mu ),&\Theta =0. \end{cases} $$
(24)
System (16) is equivalent to
$$ \dot{\chi _{t}}=A(\mu ) \chi_{t}+R (\mu ) \chi_{t}, $$
(25)
where \(\chi_{t} ( \Theta ) =\chi ( t+ \Theta )\) for \(\Theta \in [ 1,0 ] \).
For \(\psi \in C^{1} ( [ 1,0 ],(R^{3})^{*} ) \), define
$$ A^{*} \xi ( \rho ) = \textstyle\begin{cases} \frac{d \xi ( \rho ) }{ds},& \rho \in ( 0,1 ], \\ \int_{1}^{0}d \eta^{T} ( t,0 ) \xi ( t ),&\rho =0, \end{cases} $$
(26)
and a bilinear inner product
$$ \langle \xi, \zeta \rangle =\bar{\xi } ( 0 ). \zeta ( 0 )  \int_{\Theta =1}^{0} \int_{\nu =0}^{\Theta } \bar{ \xi }^{T} ( \nu  \Theta )\,d\eta ( \Theta ) \zeta ( \nu )\,d\nu. $$
(27)
Here, \(\eta ( \Theta ) =\eta ( \Theta, 0 )\), then \(A( 0 )\) (from here onwards we use \(A ( 0 ) \) as A) and \(A^{*}\) are adjoint operators. Since \(\pm i \omega_{0} h_{k}\) are the eigenvalues of A, they are also the eigenvalues of \(A^{*}\). Next, we compute the eigenvectors of A and \(A^{*}\) corresponding to \(+i\omega_{0} h_{k}\) and \(i\omega_{0} h_{k}\), respectively.
Suppose \(q ( \Theta ) = ( 1,b_{1},b_{2} ) ^{T}e^{i \omega_{0} h_{k} \Theta }\) is the eigenvector of A corresponding to eigenvalues \(i \omega_{0} h_{k}\), then
$$ A q ( \Theta ) = i \omega_{0} h_{k} q ( \Theta ), $$
(28)
which for \(\Theta =0\) gives
$$ h_{k}\left ( \textstyle\begin{array}{cccc} m_{11} & 0 & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{array}\displaystyle \right ) q ( 0 ) =\left ( \textstyle\begin{array}{c} 0 \\ 0 \\ 0 \end{array}\displaystyle \right ), $$
(29)
where \(m_{11}=i \omega_{0} + \beta e^{i \omega_{0} h_{k}} ( r+ \gamma ) + ( \frac{2 r \hat{P}}{K}+q_{1} \hat{I}+d_{1} \hat{I} )\), \(m_{13}= ( q_{1}+d_{1} ) \hat{P}\), \(m_{21}=\beta e^{ i \omega_{0} h_{k}}\), \(m_{22}=i \omega_{0}+q_{2} \hat{I}+d_{2} \hat{I}+\mu \), \(m_{23}= ( q_{2}+d_{2} ) \hat{M}, m_{31}= [ a_{1}\{ ( p_{1}\tau_{1} ) q_{1}+p _{1} d_{1}\} ] \hat{I}\), \(m_{32}=  [ a_{2}\{ ( p_{2}\tau _{2} ) q_{2}+p_{2} d_{2} \} ] \hat{I}\), \(m_{33}=i \omega_{0}\).
Solving the system of equation (29), we get \(b_{1}= \frac{i \omega_{0} a_{2} [ a_{1} \lbrace ( p_{1} \tau_{1} ) q_{1}+p_{1} d_{1} \rbrace ] \hat{I}}{ [ a _{2} \lbrace ( p_{2}\tau_{2} ) q_{2}+p_{2} d_{2} \rbrace ] \hat{I}}\), \(b_{2}= \frac{i \omega_{0}+\beta e ^{i \omega_{0} h_{k}}+ ( \frac{2 r \hat{P}}{K}+q_{1} \hat{I}+d _{1} \hat{I} ) r\gamma }{ ( q_{1}+d_{1} ) \hat{P}}\).
Similarly, we calculate \(q^{*} ( \rho ) = ( 1,b_{1}^{*},b _{2}^{*} ) ^{T}e^{i \omega_{0} h_{k} \rho }\) such that
$$ A^{*} q^{*} ( \rho ) =i \omega_{0} h_{k}q^{*} ( \rho ), $$
(30)
where \(b_{1}^{*}=\frac{C_{2} b_{2}^{*}}{C_{1}}\), \(b_{2}^{*}=\frac{C _{1} ( q_{1}+d_{1} ) \hat{P}}{C_{1} i \omega_{0}C_{2} ( q _{2}+d_{2} ) \hat{M} }\), \(C_{1}=i \omega_{0}+ \lbrace ( q _{2}+d_{2} ) \hat{I}+\mu \rbrace \), \(C_{2}= [ a_{2} \lbrace ( p_{2}\tau_{2} ) q_{2}+p_{2} d_{2} \rbrace ] \hat{I}\).
The normalization condition gives \(\langle q^{*} ( \rho ).q ( \Theta ) \rangle =1\), \(\bar{q}^{*} ( 0 ).q ( 0 ) \bar{D}\int_{\Theta =1}^{0}\int_{\nu =0}^{\Theta }\bar{q}^{*} ( 0 ) e^{i \omega_{0}h_{k} ( \nu \Theta ) }\,d\eta (\Theta ) q ( 0 ) e^{i \omega_{0} h_{k} \nu }\,d\nu =1\), \(\bar{D} [ 1+b_{1} \bar{b}_{1}^{*}+b_{2} \bar{b}_{2} ^{*}e^{i \omega_{0} h_{k}} h_{k} \phi \bar{b}_{2}^{*} ] =1\).
Thus, D̄ is chosen such that
$$ \bar{D}=\frac{1}{1+b_{1} \bar{b}_{1}^{*}+b_{2} \bar{b}_{2}^{*}e^{i \omega_{0} h_{k}} h_{k} \phi \bar{b}_{2}^{*}}. $$
(31)
Proceeding the same as in [19] and using the same notation, we compute the coordinates to describe the center manifold \(C_{0}\) at \(\mu =0\). Let \(\chi_{t}\) be a solution of equation (25) when \(\mu =0\). Define
$$ Z ( t ) = \bigl\langle q^{*}, \chi_{t} \bigr\rangle ,\qquad W ( t, \Theta ) =\chi_{t} ( \Theta ) 2 \operatorname{Re} \bigl\lbrace Z ( t ) q ( \Theta ) \bigr\rbrace . $$
(32)
On the center manifold \(C_{0}\), we have
$$\begin{aligned}& W ( t,\theta ) =W ( Z,\bar{Z},\Theta ), \end{aligned}$$
(33)
$$\begin{aligned}& W ( z,\bar{z},\Theta ) =W_{20} ( \Theta ) \frac{z^{2}}{2}+W_{11} ( \Theta ) z \bar{z}+W_{02} ( \Theta ) \frac{\bar{z}^{2}}{2}+\cdots \end{aligned}$$
(34)
z and z̄ are local coordinates for the center manifold \(C_{0}\) in the direction of \(q^{*}\) and \(\bar{q}^{*}\). Note that W is real if \(\chi_{t}\) is real. Here, we consider only real solutions. For solution \(\chi_{t} \in C_{0}\) of equation (25), since \(\mu =0\), we have
$$\begin{aligned} \dot{z}&=i \omega_{0} h_{k} z+\bar{q}^{*} ( 0 ) .f \bigl( 0,W ( z,\bar{z},0 ) +2\operatorname{Re} \bigl\lbrace zq ( 0 ) \bigr\rbrace \bigr) \\ &=i \omega_{0}h_{k}z+\bar{q}^{*} ( 0 ).f _{0} ( z,\bar{z} ). \end{aligned}$$
(35)
We can rewrite this equation as
$$\begin{aligned}& \dot{z} = i \omega_{0}h_{k}z+g ( z,\bar{z} ),\quad\mbox{where } g ( z,\bar{z} ) =\bar{q}^{*} ( 0 ). f_{0} ( z, \bar{z} ) \\& \mbox{or} = g_{20} \frac{z^{2}}{2}+g_{11}z \bar{z}+g_{02} \frac{\bar{z}^{2}}{2} +g_{21}\frac{z^{2}\bar{z}}{2}+ \cdots . \end{aligned}$$
(36)
It follows from (33) and (35) that
$$\begin{aligned}& \chi_{t} ( \Theta ) = W ( z,\bar{z},\Theta ) +2 \operatorname{Re} \bigl\lbrace z q ( \Theta ) \bigr\rbrace , \\& \mbox{or} = W_{20} ( \Theta ) \frac{z^{2}}{2}+W_{11} ( \Theta ) z \bar{z}+W_{02} ( \Theta ) \frac{\bar{z}^{2}}{2}+z ( 1,b _{1},b_{2} ) ^{T} \\& \hphantom{\mbox{or} ={}}{}\times e^{i \omega_{0}h_{k}\Theta }+\bar{z} ( 1,\bar{b_{1}}, \bar{b_{2}} ) ^{T}e^{i \omega_{0}h_{k}\Theta }+\cdots . \end{aligned}$$
(37)
Also, we have
$$\begin{aligned}& g ( z, \bar{z} ) = \bar{q}^{*} ( 0 ). f_{0} ( 0, \chi_{t} ) =h_{k}\bar{D} \bigl( 1, \bar{b_{1}}^{*}, \bar{b_{2}}^{*} \bigr) ^{T}\left ( \textstyle\begin{array}{c} V_{1} \\ V_{2} \\ V_{3} \end{array}\displaystyle \right ) , \end{aligned}$$
where \(V_{1}=\frac{r}{K} \chi_{1t}^{2} ( 0 )  ( q_{1}+d_{1} ) \chi_{1t} ( 0 ) \chi_{3t} ( 0 )\), \(V_{2}= ( q _{2}+d_{2} ) \chi_{2t} ( 0 ) \chi_{3t} ( 0 )\), \(V_{3}= [ a_{1} \lbrace ( p_{1}\tau_{1} ) q_{1}+p _{1} d_{1} \rbrace ] \chi_{1t} ( 0 ) \chi_{3t} ( 0 ) + [ a_{2} \lbrace ( p_{2}\tau_{2} ) q _{2}+p_{2} d_{2} \rbrace ] \chi_{2t} ( 0 ) \chi _{3t} ( 0 ) \), so that
$$\begin{aligned}& \chi_{1t} ( \Theta ) =W_{20}^{1} ( \Theta ) \frac{z ^{2}}{2}+W_{11}^{1} ( \Theta ) z \bar{z}+W_{02}^{1} ( \Theta ) \frac{ \bar{z}^{2}}{2}+ze^{i \omega_{0}h_{k}\Theta }+ \bar{z}e^{i \omega_{0}h _{k} \Theta }+\cdots, \\& \chi_{2t} ( \Theta ) =W_{20}^{2} ( \Theta ) \frac{z ^{2}}{2}+W_{11}^{2} ( \Theta ) z \bar{z}+W_{02}^{2} ( \Theta ) \frac{ \bar{z}^{2}}{2}+b_{1}ze^{i \omega_{0}h_{k}\Theta }+ \bar{b_{1}}\bar{z}e ^{i \omega_{0}h_{k} \Theta }+\cdots, \\& \chi_{3t} ( \Theta ) =W_{20}^{3} ( \Theta ) \frac{z ^{2}}{2}+W_{11}^{3} ( \Theta ) z \bar{z}+W_{02}^{3} ( \Theta ) \frac{ \bar{z}^{2}}{2}+b_{2}ze^{i \omega_{0}h_{k}\Theta }+ \bar{z}\bar{b_{2}}e ^{i \omega_{0}h_{k} \Theta }+\cdots. \end{aligned}$$
Therefore,
$$\begin{aligned}& \chi_{1t} ( 0 ) =z+\bar{z}+W_{20}^{1} ( 0 ) \frac{z ^{2}}{2}+W_{11}^{1} ( 0 ) z \bar{z}+W_{02}^{1} ( 0 ) \frac{ \bar{z}^{2}}{2}+\cdots, \\& \chi_{2t} ( 0 ) =b_{1} z+\bar{b_{1}} \bar{z}+W_{20}^{2} ( 0 ) \frac{z^{2}}{2}+W_{11}^{2} ( 0 ) z \bar{z}+W_{02}^{1} ( 0 ) \frac{\bar{z}^{2}}{2}+\cdots, \\& \chi_{3t} ( 0 ) =b_{2} z+\bar{b_{2}} \bar{z}+W_{20}^{3} ( 0 ) \frac{z^{2}}{2}+W_{11}^{3} ( 0 ) z \bar{z}+W_{02}^{3} ( 0 ) \frac{\bar{z}^{2}}{2}+\cdots, \\& g ( z,\bar{z} ) =h_{k}\bar{D} \bigl( 1,\bar{b_{1}}^{*}, \bar{b _{2}}^{*} \bigr) ^{T}.\left ( \textstyle\begin{array}{c} V_{1} \\ V_{2} \\ V_{3} \end{array}\displaystyle \right ) \end{aligned}$$
(38)
$$\begin{aligned}& \hphantom{g ( z,\bar{z} )}=h_{k}\bar{D} \biggl[ \frac{r}{K} \chi_{1t}^{2} ( 0 ) + \bigl\lbrace  ( q_{1}+d_{1} ) +a_{1} \bar{b_{2}}^{*} \bigl( ( p_{1}\tau _{1} ) q_{1}+p_{1}d_{1} \bigr) \bigr\rbrace \chi_{1t} ( 0 ) \chi_{3t} ( 0 ) \biggr] \\& \hphantom{g ( z,\bar{z} )=}{}+h_{k}\bar{D} \bigl[ \bigl\lbrace \bar{b_{1}}^{*} ( q_{2}+d_{2} ) + a_{2}\bar{b_{2}}^{*} \bigl\lbrace ( p_{2}\tau_{2} ) q _{2} +p_{2}d_{2} \bigr\rbrace \bigr\rbrace \chi_{2t} ( 0 ) \chi_{3t} ( 0 ) \bigr]. \end{aligned}$$
(39)
Comparing the coefficients in (37) with those in (41), we get
$$\begin{aligned}& g_{20} = 2 h_{k} \bar{D} \biggl[\frac{r}{K}b_{2} ( q_{1}+d_{1} ) +b _{2}\bar{b}_{2}^{*}+a_{1} \bigl\{ ( p_{1}\tau_{1} ) q_{1}+p _{1} d_{1} \bigr\} \\& \hphantom{g_{20} =}{}  b_{1}b_{2}\bar{b_{1}}^{*} ( q_{2}+d_{2} ) +b_{1} b_{2} \bar{b}_{2}^{*}a_{2} \bigl\{ ( p_{2} \tau_{2} ) q_{2}+p_{2}d_{2} \bigr\} \biggr], \end{aligned}$$
(40)
$$\begin{aligned}& g_{11} = 2 h_{k} \bar{D} \biggl[\frac{r}{K}+ \operatorname{Re} ( b_{2} ) \bigl\{  ( q_{1}+d_{1} ) + \bar{b}_{2}^{*}a_{1} \bigl( ( p_{1} \tau _{1} ) q_{1}+p_{1} d_{1} \bigr) \bigr\} \biggr] \\& \hphantom{g_{11} =}{}+ 2 h_{k} \bar{D} \bigl[ ( \bar{b_{1}}b_{2}+b_{1} \bar{b}_{2} ) \bigl\{ \bar{b_{1}}^{*} ( q_{2}+d_{2} ) +\bar{b}_{2}^{*}a_{2} \bigl( ( p _{2}\tau_{2} ) q_{2}+p_{2}d_{2} \bigr) \bigr\} \bigr], \end{aligned}$$
(41)
$$\begin{aligned}& g_{02} = 2 h_{k}\bar{D} \biggl[\frac{r}{K} \bar{b}_{2} ( q_{1}+d_{1} ) +\bar{b _{2}}\bar{b}_{2}^{*}a_{1} \bigl\{ ( p_{1}\tau_{1} ) q_{1}+p_{1}d _{1} \bigr\} \biggr] \\& \hphantom{g_{02} =}{}+2 h_{k}\bar{D} \bigl[\bar{b_{1}}\bar{b}_{2} \bigl\{  \bar{b_{1}}^{*} ( q_{2}+d_{2} ) +a _{2}\bar{b}_{2}^{*} \bigl( ( p_{2} \tau_{2} ) q_{2}+p_{2}d_{2} \bigr) \bigr\} \bigr]. \end{aligned}$$
(42)
$$\begin{aligned}& g_{21} = 2 h_{k} \bar{D} \biggl[ \frac{r}{K} \bigl( W_{20}^{ ( 1 ) }+2 W_{11}^{ ( 1 ) } \bigr) + \bigl\{  ( q_{1}+d_{1} ) +a_{1}\bar{b _{2}}^{*} \bigl(( p_{1}\tau_{1})q_{1} \\& \hphantom{g_{21} =}{}+ p_{1}d_{1} \bigr) \bigr\} \biggl(b_{2}W_{11}^{ ( 1 ) }+ \frac{\bar{b}_{2}}{2}W_{20}^{ ( 1 ) }+ \frac{W_{20}^{ ( 3 ) }}{2}+W_{11}^{(3 )} \biggr) \biggr]+2 h_{k} \bar{D} \biggl[ \bar{b_{1}}^{*}(q _{2}+d_{2}) \\& \hphantom{g_{21} =}{} + a_{2}\bar{b}_{2}^{*} \bigl\{ ( p_{2}\tau_{2})q_{2}+p_{2}d_{2} \bigr\} \biggl(b_{2}W _{11}^{(2)}+ \bar{b}_{2}W_{20}^{(2 )}+ \frac{\bar{b_{1}}W_{20}^{( 3) }}{2}+b_{1}W_{11}^{(3)} \biggr) \biggr]. \end{aligned}$$
(43)
In order to compute (43), we need \(W_{20} ( \Theta )\) and \(W_{11} ( \Theta ) \). From equations (33) and (37), we have
$$\begin{aligned} \dot{W}&=\dot{\chi _{t}}\dot{z}q\dot{z}\bar{q} \\ &= \textstyle\begin{cases} AW2 \operatorname{Re} \{ \bar{q^{*}}{(0)} ( 0 ). f_{0}q ( \Theta )\}, & \Theta \in [ 1,0 ), \\ AW2 \operatorname{Re} \{ \bar{q^{*}}{(0)} ( 0 ). f_{0}q ( 0 ) \}+f_{0}, & \Theta =0 \end{cases}\displaystyle \end{aligned}$$
(44)
$$\begin{aligned} &=AW+H ( z,\bar{z},\Theta ) \end{aligned}$$
(45)
with
$$ H ( z,\bar{z},\Theta ) =H_{20} ( \Theta ) \frac{z ^{2}}{2}+H_{11} ( \Theta ) z\bar{z}+H_{02} ( \Theta ) \frac{ \bar{z}^{2}}{2}+\cdots. $$
(46)
Also, on \(C_{0}\), using the chain rule, we get
$$ \dot{W}=W_{z}\dot{z}+W_{\bar{z}}\dot{\bar{z}}. $$
(47)
It follows from (54), (44), and (46) that
$$\begin{aligned}& ( A2i\omega_{0}h_{k} ) W_{20}=H_{20}, \end{aligned}$$
(48)
$$\begin{aligned}& AW_{11}=H_{11}. \end{aligned}$$
(49)
Now, for \(\Theta \in [ 1,0 ) \), we have
$$\begin{aligned} H ( z,\bar{z},\Theta ) =&\bar{q}^{*} ( 0 ).f_{0}q ( \Theta ) \bar{q}^{*} ( 0 ).\bar{f}_{0}\bar{q} ( \Theta ) \\ =&g ( z,\bar{z} ) q ( \Theta ) \bar{g} ( z, \bar{z} ) \bar{q} ( \Theta ) \\ =& \bigl( g_{20}q ( \Theta ) +\bar{g_{02}} \bar{q} ( \Theta ) \bigr) \frac{z^{2}}{2} \bigl( g_{11}q ( \Theta ) + \bar{g}_{11}\bar{q} ( \Theta ) \bigr) z\bar{z}+\cdots, \end{aligned}$$
(50)
which on computing the coefficients with (45) gives
$$\begin{aligned}& H_{20} ( \Theta ) =g_{20}q ( \Theta )  \bar{g_{02}}\bar{q} ( \Theta ), \end{aligned}$$
(51)
$$\begin{aligned}& H_{11} ( \Theta ) =g_{11} q ( \Theta )  \bar{g_{11}}\bar{q} ( \Theta ). \end{aligned}$$
(52)
From (48), (51), and the definition of A, we have
$$ W_{20}^{'} ( \Theta ) =2i \omega_{0}h_{k}W_{20} ( \Theta ) +g _{20}q ( \Theta ) +\bar{g_{02}}\bar{q} ( \Theta ). $$
(53)
Note that \(q ( \Theta ) =q ( 0 ) e^{i \omega_{0}h_{k} \Theta }\), hence
$$ W_{20} ( \Theta ) =\frac{i g_{20}}{\omega_{0} h_{k}}q ( \Theta ) +\frac{i \bar{g_{20}}}{3 \omega_{0} h_{k}} \bar{q} ( \Theta ) +F_{1}e^{2 i \omega_{0}h_{k}\Theta }. $$
(54)
Similarly, from (48), (51), and the definition of A, we get
$$\begin{aligned}& W_{11}^{'} ( \Theta ) =g_{11}q ( \Theta ) + \bar{g_{11}}\bar{q} ( \Theta ), \end{aligned}$$
(55)
$$\begin{aligned}& W_{11} ( \Theta ) =\frac{i g_{11}}{\omega_{0}h_{k}}q ( \Theta ) +\frac{i \bar{g_{11}}}{ \omega_{0} h_{k}} \bar{q} ( \Theta ) +F_{2}, \end{aligned}$$
(56)
where \(F_{1}= ( F_{1}^{ ( 1 ) }, F_{1}^{ ( 2 ) }, F_{1}^{ ( 3 ) } )\) and \(F_{2}= ( F_{2}^{ ( 1 ) }, F_{2}^{ ( 2 ) }, F_{2}^{ ( 3 ) } ) \in R^{3}\) are constant vectors to be determined. It follows from the definition of A and (52) that
$$\begin{aligned}& \int_{1}^{0}d\eta ( \Theta ) W_{20} ( \Theta ) =2 i \omega_{0}h_{k} W_{20} ( 0 ) H_{20} ( 0 ), \end{aligned}$$
(57)
$$\begin{aligned}& \int_{1}^{0}d\eta ( \Theta ) W_{11} ( \Theta ) =H _{11} ( 0 ). \end{aligned}$$
(58)
From equations (53) and (55), we get
$$\begin{aligned} H_{20}(0)&=g_{20}q ( 0 ) \bar{g_{02}} \bar{q ( 0 ) } \\ &\quad {}+2 h_{k}\left ( \textstyle\begin{array}{c} {\frac{r}{K}+b_{2} ( q_{1}+d_{1} ) } \\ b_{1}b_{2} ( q_{2}+d_{2} ) \\ b_{2}a_{1}{ ( p_{1}\tau_{1} ) q_{1}+p_{1}d_{1}} +b_{1}b_{2}a _{2}{ ( p_{2}\tau_{2} ) q_{2}+p_{2}d_{2}} \end{array}\displaystyle \right ) \end{aligned}$$
(59)
and
$$\begin{aligned} H_{11}(0)&=g_{11}q(0)\bar{g_{11}} \bar{q(0)} \\ &\quad {}+2 h_{k}\left ( \textstyle\begin{array}{c} { \frac{r}{K}+\operatorname{Re} ( b_{2})( q_{1}+d_{1})} \\ (\bar{b_{1}}b_{2}+b_{1}\bar{b}_{2})(q_{2}+d_{2}) \\ \operatorname{Re} (b_{2})a_{1}{(p_{1}\tau_{1})q_{1}+p_{1}d_{1}}+(\bar{b_{1}}b_{2}+b _{1}\bar{b}_{2})a_{2}\{ ( p_{2}\tau_{2} ) q_{2}+p_{2}d_{2}\} \end{array}\displaystyle \right ). \end{aligned}$$
(60)
Using (53) and (58) in (56) and noting that \(q ( \Theta ) \) is an eigenvector of A, we have
$$\begin{aligned}& \biggl( 2 i \omega_{0}h_{k} I \int_{1}^{0}e^{2 i \omega_{0}h_{k} \Theta }\,d\eta ( \Theta ) \biggr) F_{1} \\& \quad =2 h_{k}\left ( \textstyle\begin{array}{c} \{\frac{r}{K}+b_{2} ( q_{1}+d_{1} ) \} \\ b_{1}b_{2} ( q_{2}+d_{2} ) \\ b_{2}a_{1} \lbrace ( p_{1}\tau_{1} ) q_{1}+p_{1}d_{1} \rbrace +b _{1}b_{2}a_{2} \lbrace ( p_{2}\tau_{2} ) q_{2}+p_{2}d _{2} \rbrace \end{array}\displaystyle \right ), \end{aligned}$$
(61)
$$\begin{aligned}& \small{ \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 2 i\omega_{0}+\beta e^{i \omega_{0}h_{k}}(r+\gamma )+\frac{2 r \hat{P}}{K}+q_{1} \hat{I}+d_{1} \hat{I} & 0 & ( q_{1}+d_{1}) \hat{I} \\ \beta e^{i \omega_{0}h_{k}} & 2 i \omega_{0}+q_{2} \hat{I}+d_{2} \hat{I}+\mu & ( q_{2}+d_{2} ) \hat{M} \\ \{a_{1}q_{1} ( p_{1}\tau_{1} ) +a_{1} p_{1} d_{1}\}\hat{I} & \{a_{2}q_{2} ( p_{2}\tau_{2} ) +a_{2} p_{2} d_{2}\} \hat{I} & 2 i \omega_{0} \end{array}\displaystyle \right )} \\& \quad {}\times \left ( \textstyle\begin{array}{c} F_{1}^{ ( 1 ) } \\ F_{1}^{ ( 2 ) } \\ F_{1}^{ ( 3 ) } \end{array}\displaystyle \right ) =2\left ( \textstyle\begin{array}{c}  \lbrace \frac{r}{K}+b_{2} ( q_{1}+d_{1} ) \rbrace \\ b_{1}b_{2} ( q_{2}+d_{2} ) \\ b_{2}a_{1} \lbrace ( p_{1}\tau_{1} ) q_{1}+p_{1}d_{1} \rbrace +b _{1}b_{2}a_{2} \lbrace ( p_{2}\tau_{2} ) q_{2}+p_{2}d _{2} \rbrace \end{array}\displaystyle \right ). \end{aligned}$$
(62)
Similarly, using (55) and (56) in (59), we get
$$\begin{aligned}& \left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} \beta  ( r+\gamma ) +\frac{2 r \hat{P}}{K}+q_{1} \hat{I}+d _{1} \hat{I} & 0 & ( q_{1}+d_{1} ) \hat{P} \\ \beta & q_{2} \hat{I}+d_{2} \hat{I}+\mu & ( q_{2}+d_{2} ) \hat{M} \\  \lbrace a_{1}q_{1} ( p_{1}\tau_{1} ) +a_{1} p_{1} d _{1} \rbrace \hat{I} &  \lbrace a_{2}q_{2} ( p_{2}\tau _{2} ) +a_{2} p_{2} d_{2} \rbrace \hat{I} & 0 \end{array}\displaystyle \right ) \left ( \textstyle\begin{array}{c} F_{2}^{ ( 1 ) } \\ F_{2}^{ ( 2 ) } \\ F_{2}^{ ( 3 ) } \end{array}\displaystyle \right ) \\& \quad =2\left ( \textstyle\begin{array}{c}  \lbrace \frac{r}{K}+\operatorname{Re} ( b_{2} ) ( q_{1}+d_{1} ) \rbrace \\  ( \bar{b_{1}}b_{2}+b_{1}\bar{b}_{2} ) ( q_{2}+d_{2} ) \\ \operatorname{Re} ( b_{2} ) a_{1} \lbrace ( p_{1}\tau_{1} ) q _{1}+p_{1}d_{1} \rbrace + ( \bar{b_{1}}b_{2}+b_{1}\bar{b _{2}} ) a_{2} \lbrace ( p_{2}\tau_{2} ) q_{2}+p_{2}d _{2} \rbrace \end{array}\displaystyle \right ). \end{aligned}$$
(63)
Next, solving (60) for \(F_{1}\) and (61) for \(F_{2}\) and using these values, we can determine \(W_{20}\) and \(W_{11}\), and hence \(g_{21}\). Afterwards, to determine the direction, stability, and the period of bifurcating solutions from a critical point at the critical threshold \(h=h_{k}\), we can come to the following necessary quantities as given by [19]:
$$\begin{aligned}& b_{1} ( 0 ) =\frac{i}{2 \omega_{0} h_{k}} \biggl( g_{11}g_{20}2 \lvert g_{11}\rvert^{2}\frac{\lvert g_{02}\rvert^{2}}{3} \biggr) + \frac{g _{21}}{2}, \end{aligned}$$
(64)
$$\begin{aligned}& \mu_{2}=\frac{\operatorname{Re} \lbrace b_{1} ( 0 ) \rbrace }{\operatorname{Re} \lbrace \psi^{'} ( h_{k} ) \rbrace }, \end{aligned}$$
(65)
$$\begin{aligned}& m_{2}=2\operatorname{Re} \bigl\lbrace b_{1} ( 0 ) \bigr\rbrace , \end{aligned}$$
(66)
$$\begin{aligned}& T_{2}=\frac{I_{m}b_{1} ( 0 ) +\mu I_{m} \lbrace \psi^{'} ( h_{k} ) \rbrace }{\omega_{0} h_{k}}. \end{aligned}$$
(67)
Hence, using the results of [19], we have the following theorem.
Theorem 1
If
\(\mu_{2}>0 \) (\(\mu_{2}<0\)), then the Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist for
\(h>h_{k}\) (\(h< h_{k}\)). The bifurcating periodic solution is stable (unstable) if
\(m_{2}<0\) (\(m_{2} >0\)), and the period increases (decreases) if
\(T_{2}>0\) (\(T_{2}<0\)).