A delayed prey–predator system with prey subject to the strong Allee effect and disease

Abstract

In this article, an eco-epidemiological model with strong Allee effect in prey population growth is presented by a system of delay differential equations. The time lag in terms of the delay parameter corresponds to the predator gestation period. We inspect elementary mathematical characteristic of the proposed model such as uniform persistence, stability and Hopf bifurcation at the interior equilibrium point of the system. We execute several numerical simulations to illustrate the proposed mathematical model and our analytical findings. We use basic tools of nonlinear dynamic analysis as first return maps, Poincare sections and Lyapunov exponents to identify chaotic behavior of the system. We observe that the system exhibits chaotic oscillation due to the increase of the delay parameter. Such chaotic behavior can be suppressed by the strength of Allee effect.

Appendix

1.1 Proof of the Proposition 1:

Proof

The Jacobian matrix of the model (2) at its any equilibrium point $(S_{*}, I_{*}, P_{*})$ is described as follows

$$\begin{aligned}&J\vert _{\left( S_*,I_*,P_*\right) }\nonumber \\&\quad =\left[ \begin{array}{lll} S_*(1-2S_*-I_*+\theta ) &{}\quad \! (\theta -S_*-\beta )S_* &{}\quad \! 0 \\ \beta I_* &{}\quad \! \beta S_*-aP_*-\mu &{}\quad \! -aI_* \\ 0 &{}\quad \! \alpha P_* &{}\quad \! \alpha I_*-d \end{array} \right] \nonumber \\ \end{aligned}$$

(23)

After substituting $E_i = \left( S^*,I^*,P^*\right) ,~ i~=~0,~\theta ,~1,~2$ into (23), we obtain the eigenvalues for each equilibrium:

• 1. $E_0 = \left( 0, 0, 0\right)$ is always locally asymptotically stable since eigenvalues associated with (23) at $E_0$ can be presented as follows:

  $$\begin{aligned} \lambda _1= & {} -\theta ~~ \left( <0\right) ,\quad \lambda _2 = -\mu ~~ (<0) \quad \hbox {and}\\ \lambda _3= & {} -d~~ (<0). \end{aligned}$$

• 2. $E_\theta = \left( \theta , 0, 0\right)$ is always unstable since eigenvalues associated with (23) at $E_1$ are given by,

  $$\begin{aligned} \lambda _1= & {} \theta (1-\theta ) ~~(>0),\\ \lambda _2= & {} \beta \theta -\mu \biggl \{^{<0\quad \text{ if } \beta \theta <\mu }_{>0\quad \text{ if } \beta \theta >\mu }\quad \text{ and } \lambda _3 = -d \left( <0\right) . \end{aligned}$$

• 3. $E_1 = \left( 1, 0, 0\right)$ is locally asymptotically stable if $\beta < \mu$ since eigenvalues associated with (23) at $E_1$ are given by,

  $$\begin{aligned} \lambda _1= & {} \theta -1 ~~(<0),\\ \lambda _2= & {} \beta -\mu \biggl \{^{<0\quad \text{ if } \beta <\mu }_{>0\quad \text{ if } \beta >\mu }~~~~ \text{ and } \lambda _3 = -d \left( <0\right) . \end{aligned}$$

• 4. $E_{2} = \left( \frac{\mu }{\beta }, \frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta },0\right)$ is locally asymptotically stable if $\alpha \frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta } -d<0$ and $1<\frac{\beta }{\mu }<\min \Bigg \{\frac{1}{\theta },\frac{\beta -\theta +\sqrt{\beta ^2-\beta \theta +\beta }}{\beta +\beta \theta -\theta ^2}\Bigg \}$. Since the eigenvalues associated with (23) at $E_{2}$ are given by

  $$\begin{aligned} \lambda _3= & {} \alpha \frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta }\\&-d\biggl \{^{<0\quad \text{ if } \alpha \frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta } -d<0}_{>0\quad \text{ if } \alpha \frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta } -d>0} \end{aligned}$$

  and the other two eigenvalues are the roots of the equation

  $$\begin{aligned} \lambda ^2 - A\lambda +B = 0. \end{aligned}$$

  Where

  $$\begin{aligned} \begin{array}{lll} A &{}=&{} \frac{\mu }{\beta }\left( 1-2\frac{\mu }{\beta }-\frac{\left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) }{\frac{\mu }{\beta }+\beta -\theta }+\theta \right) \\ &{}=&{} \frac{\left( \beta +\beta \theta -\theta ^2\right) \left( \frac{\beta }{\mu }\right) ^2-2\frac{\beta }{\mu }(\beta -\theta )-1}{\left( \frac{\beta }{\mu }\right) ^2\left( 1+\frac{\beta }{\mu }(\beta -\theta )\right) }\\ B &{}=&{} \mu \left( \frac{\mu }{\beta }-\theta \right) \left( 1-\frac{\mu }{\beta }\right) >0 \end{array} \end{aligned}$$

  (24)

  Thus, we have

  $$\begin{aligned}&A>0\quad \text{ if } \frac{\beta }{\mu }>\frac{\beta -\theta +\sqrt{\beta ^2-\beta \theta +\beta }}{\beta +\beta \theta -\theta ^2} \text{ while } \\&A<0\quad \text{ if } \frac{\beta }{\mu }<\frac{\beta -\theta +\sqrt{\beta ^2-\beta \theta +\beta }}{\beta +\beta \theta -\theta ^2}. \end{aligned}$$

  Therefore, $E_2$ exists and is locally asymptotically stable if

  $$\begin{aligned} 1<\frac{\beta }{\mu }<\min \Bigg \{\frac{1}{\theta },\frac{\beta -\theta +\sqrt{\beta ^2-\beta \theta +\beta }}{\beta +\beta \theta -\theta ^2}\Bigg \} \end{aligned}$$

Again, the Jacobian matrix of the model (2) at its interior equilibrium point $\widetilde{E} = (\widetilde{S}, \widetilde{I}, \widetilde{P})$ can be written as follows

$$\begin{aligned}&J\vert _{\widetilde{E} \,\,=\,\, (\widetilde{S}, \widetilde{I}, \widetilde{P})}\nonumber \\&\quad =\left[ \begin{array}{lll} \widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S}) &{}\quad \widetilde{S}(\theta -\beta -\widetilde{S}) &{}\quad 0 \\ \\ \beta \widetilde{I} &{}\quad 0 &{}\quad -a\widetilde{I} \\ \\ 0&{}\quad \alpha \widetilde{P} &{}\quad 0\\ \end{array} \right] \nonumber \\ \end{aligned}$$

(25)

where its characteristic equation reads as follows:

$$\begin{aligned}&\lambda ^3 - \widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S})\lambda ^2\nonumber \\&\qquad +\,\left\{ \beta \widetilde{I} \widetilde{S}(-\theta +\beta +\widetilde{S}) +a\widetilde{I}\alpha \widetilde{P} \right\} \lambda \nonumber \\&\qquad -\,a\alpha \widetilde{I}\widetilde{P}\widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S})\nonumber \\&\quad =\left( \lambda _1-\lambda \right) \left( \lambda _2-\lambda \right) \left( \lambda _3-\lambda \right) = 0. \end{aligned}$$

(26)

with $\lambda _i, i = 1,2,3$ being roots of (26). If all the real parts of $\lambda _i, i=1,2,3$ are negative, then we have

$$\begin{aligned}&\sum \nolimits ^3_{i=1}\lambda _i = \widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S}) < 0 \\&\sum \nolimits ^3_{i,j=1,i\ne j}\lambda _i\lambda _j = \beta \widetilde{I} \widetilde{S}(-\theta +\beta +\widetilde{S}) +a\widetilde{I}\alpha \widetilde{P} > 0 \\&\prod ^3_{i=1}\lambda _i = a\alpha \widetilde{I}\widetilde{P}\widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S}) < 0 \end{aligned}$$

Thus, interior equilibrium is locally asymptotically stable if the following conditions hold

$$\begin{aligned}&\widetilde{S}(1+\theta -\widetilde{I}-2\widetilde{S}) < 0\nonumber \\&\frac{1+\theta -\widetilde{I}}{2}<\widetilde{S} \end{aligned}$$

(27)

So, from (27), we see that $E_2^*$ is stable and $E_1^*$ is unstable.

1.2 Proof of the permanence of the model (1)

Proof

In this section, we shall prove that the boundary planes of ${\mathbb {R}}^{3}_{+}$ repel the positive solutions of system (1) uniformly. Let us define

$$\begin{aligned} C_{1}= & {} \Bigg \{\left( \psi _{1}, \psi _{2}, \psi _{3}\right) \in C\left( \left[ -\tau , 0\right] ,{\mathbb {R}}^{3}_{+}\right) {:}\\&\quad \psi _{1}( \phi ) = 0,~ \phi \in [-\tau , 0]\Bigg \}, \\ C_{2}= & {} \Bigg \{\left( \psi _{1}, \psi _{2}, \psi _{3}\right) \in C\left( \left[ -\tau , 0\right] ,{\mathbb {R}}^{3}_{+} \right) {:}\\&\quad \psi _{2}( \phi ) = 0,~ \psi _{1}( \phi )\ne 0,~ \phi \in [-\tau , 0]\Bigg \}, \\ C_{3}= & {} \Bigg \{\left( \psi _{1}, \psi _{2}, \psi _{3}\right) \in C\left( \left[ -\tau , 0\right] ,{\mathbb {R}}^{3}_{+} \right) {:}\\&\quad \psi _{3}( \phi ) = 0,~ \psi _{1}( \phi ) \psi _{2}( \phi )\ne 0,~ \phi \in [-\tau , 0]\Bigg \}, \end{aligned}$$

where $C([-\tau , 0],~{\mathbb {R}}^{3}_{+})$ denote the space of continuous function mapping $[-\tau , 0]$ in to ${\mathbb {R}}^{3}_{+}$.

If $C_{0} = C_{1}\cup C_{2}\cup C_{3}$ and $C^{0} = int C\left( [-\tau , 0], {\mathbb {R}}^{3}_{+}\right)$, it suffices to show that there exists an $\epsilon _{0} >0$ such that for any solution $u_{t}$ of system (1) initiating from $C_{0}$, $\displaystyle {\lim _{t\rightarrow \infty }\inf \ d(u_{t}, C_{0} )\ge \epsilon _{0}}$.

Now, we verify below that the conditions of Lemma 1 are satisfied. By definition of $C_{0}$ and $C^{0}$ and system (1), it is easy to see that $C_{0}$ and $C^{0}$ are positively invariant. Moreover, it is clear that conditions (i) and (ii) of Lemma 1 are satisfied. Thus, we need to confirm conditions (iii) and (iv).

Three constant solutions in $C_{0}$ corresponding to $\left( S(t) = 0, I(t) = 0, P(t) = 0\right)$, $\left( S(t)=1, I(t)=0, P(t)=0\right)$ and $\left( S(t) = S_{2}, I(t) = I_{2}, P(t) = 0\right)$ are respectively $E_{0}$, $E_{1}$ and $E_{2}$.

If $\left( S(t), I(t), P(t)\right)$ is any solution of system (1) initiating from $C_{1}$ with $\psi _{1}(0) = 0$ then $S(t)\rightarrow 0$, $I(t)\rightarrow 0$, $P(t)\rightarrow 0$ as $t\rightarrow \infty$. If $\left( S(t), I(t), P(t)\right)$ is a solution of system (1) initiating from $C_{2}$ with $\psi _{1}(0) > 0$, it follows that $S(t)\rightarrow 1$, $I(t)\rightarrow 0$, $P(t)\rightarrow 0$ as $t\rightarrow \infty$. If $\left( S(t), I(t), P(t)\right)$ is a solution of system (1) initiating from $C_{3}$ with $\psi _{1}(0) \psi _{2}(0) > 0$, it follows that $S(t)\rightarrow S_{2}$, $I(t)\rightarrow I_{2}$, $P(t)\rightarrow 0$ as $t\rightarrow \infty$.

This shows that invariant sets $E_{0}$, $E_{1}$ and $E_{2}$ are isolated invariant, and then, $E_{0}$, $E_{1}$ and $E_{2}$ are an isolated as well as an acyclic covering, satisfying condition (iii) of Lemma 1.

We now show that $W^{s}(E_{0}) \cap C^{0} = {\varPhi }$, $W^{s}(E_{1}) \cap C^{0} = {\varPhi }$ and $W^{s}(E_{2}) \cap C^{0} = {\varPhi }$. The proof for the first part is simple, so we ignore it. We shall prove the second part through contradiction. Let us assume that $W^{s}(E_{1}) \cap C^{0} \ne {\varPhi }$, then there exists a positive solution $\left( S(t), I(t), P(t)\right)$ of system (1) such that $\left( S(t), I(t), P(t)\right) \rightarrow (1, 0, 0)$ as $t\rightarrow +\infty$. Let us choose $\epsilon _{1} > 0$ small enough such that

$$\begin{aligned}&\left( 1-\frac{\mu + a\epsilon _1}{\beta }\right) \left( \frac{\mu + a\epsilon _1}{\beta }-\theta \right) >0 \text{ and } \\&\quad -\epsilon _{1} < P(t) < \epsilon _{1} \end{aligned}$$

for some $t>t_{1}$, where $t_{1}$ be sufficiently large. Then, from first and second equations of the system (1), we have for $t>t_{1}$

$$\begin{aligned} \frac{\hbox {d}S(t)}{\hbox {d}t}\ge & {} S\{(1- S - I)(S- \theta )-\beta I\}, \nonumber \\ \frac{\hbox {d}I(t)}{\hbox {d}t}\ge & {} I\{\beta S -\mu -a\epsilon _1\}. \end{aligned}$$

(28)

Now let us consider

$$\begin{aligned} \frac{\hbox {d}y_1(t)}{\hbox {d}t}\ge & {} y_1\left\{ (1-y_1 -y_2)(y_1-\theta )-\beta y_2\right\} , \nonumber \\ \frac{\hbox {d}y_2(t)}{\hbox {d}t}\ge & {} y_2\left\{ \beta y_1 -\mu -a\epsilon _1\right\} . \end{aligned}$$

(29)

Let $V = (v_1,v_2)$ and $\zeta >0$ be small enough such that $\zeta v_1<S(t_1),\zeta v_2<I(t_1)$. If $\left( y_1(t),y_2(t)\right)$ is a solution of system (29) satisfying $y_i(t_1) = \zeta v_i, i = 1, 2$. We know from comparison theorem that $S(t) > y_1(t),~ I(t) > y_2(t)$ for all $t>t_1$. We can check easily that the system (29) has a unique positive equilibrium

$$\begin{aligned} \left( y^*_1,y^*_2\right) = \left( \frac{\mu +a\epsilon _1}{\beta }, \frac{\left( 1-\frac{\mu + a\epsilon _1}{\beta }\right) \left( \frac{\mu + a\epsilon _1}{\beta }-\theta \right) }{\frac{\mu + a\epsilon _1}{\beta }-\theta +\beta }\right) . \end{aligned}$$

Now $S(t) > y_1(t), I(t) > y_2(t)$ for all $t>t_1$ and $\displaystyle {\lim _{t\rightarrow \infty } y_2(t)}=y^*_2$. This is a contradiction. Hence, $W^{s}(E_{2} ) \cap C^{0} = {\varPhi }$.

Let $W^{s}(E_{2}) \cap C^{0} \ne {\varPhi }$. Then, there exists a positive solution $\left( S(t), I(t), P(t)\right)$ of the system such that $\left( S(t), I(t), P(t)\right) \rightarrow (S_{2}, I_{2}, 0)$ as $t\rightarrow \infty$. Let us choose $\epsilon _2>0$ small enough such that $I_2-\epsilon _2 < I(t) < I_2+\epsilon _2$ for $t > t_2-\tau$.

Then, from third equation of the system (1), we have for $t > t_{2}-\tau$

$$\begin{aligned} \frac{\hbox {d} P(t)}{\hbox {d}t} \ge \left( \alpha P(t-\tau )(I_2-\epsilon _2)-dP\right) . \end{aligned}$$

(30)

Now, let us consider

$$\begin{aligned} \frac{\hbox {d} z(t)}{\hbox {d}t} \ge \left( \alpha z(t-\tau )(I_2-\epsilon _2)-dz\right) . \end{aligned}$$

(31)

Let $u_{1}$ and $v>0$ be small enough such that $vu_{1}<P(t_{2})$. If $z_{1}$ is a solution of system (31) satisfying $z_{1}(t_{2})=wu_{1}$, we know from comparison theorem, $P(t)\ge z_{1}(t)$ for all $t >t_{2}-\tau$. We also observe that the solution $z_{1}$ of Eq. (31) satisfies $\displaystyle {\lim _{t\rightarrow \infty } z_{1}(t)} \rightarrow +\infty$ (from condition (ii)).

Since $P(t)\ge z_{1}(t)$ for all $t >t_{2}$, so $\displaystyle {\lim _{t\rightarrow \infty } P(t)}\nrightarrow 0$. This contradicts that $W^{s}(E_{2} ) \cap C^{0} = {\varPhi }$. From Lemma 1, we conclude that $C_{0}$ repels the positive solutions of (1) uniformly. Hence, the system (1) is permanent. This proves the theorem.

1.3 Direction and stability of Hopf bifurcation of model (1)

We consider the transformation $z_{1}(t) = S(\tau t)-S_{*}$, $z_{2}(t) = I(\tau t)-I_{*}$, $z_{3}(t) = P(\tau t)-P_{*}$.

Let $\tau =\tau ^{*}+\mu$, $\mu \in \mathbf {R}$. Then, $\mu =0$ is the Hopf bifurcation value of the system (1). The Eq. (1) can be written in the form

$$\begin{aligned} \dot{z}(t)=L_{\mu }(z_{t})+F(\mu , z_{t}), \end{aligned}$$

(32)

where $z(t)=(z_{1}(t), z_{2}(t), z_{3}(t))^{T}\in \mathbf {R^{3}}$. For $\psi =(\psi _{1}, \psi _{2}, \psi _{3})^{T}\in \mathbf {C}([-1, 0], \mathbf {R^{3}_{+}})$; $L_{\mu }: \mathbf {C}\rightarrow \mathbf {R}$ and $F: \mathbf {R}\times \mathbf {C}\rightarrow \mathbf {R}$ are given by

$$\begin{aligned} L_{\mu }(\psi )= & {} (\tau ^{*}+\mu )A_3 \left( \begin{array}{c} \psi _{1}(0) \\ \psi _{2}(0) \\ \psi _{3}(0) \\ \end{array} \right) \nonumber \\&+\,(\tau ^{*}+\mu )A_4 \left( \begin{array}{c} \psi _{1}(-1) \\ \psi _{2}(-1) \\ \psi _{3}(-1) \\ \end{array} \right) , \end{aligned}$$

(33)

and

$$\begin{aligned} F(\mu , \psi )=(\tau ^{*}+\mu ) A_5, \end{aligned}$$

(34)

where

$$\begin{aligned} A_3= & {} \left( \begin{array}{lll} 2S_* (1+\theta )-\theta -3S_{*}^2+I_* (\theta -\beta -2I_*) &{}\quad -S_* (S_* -\theta )-\beta S_* &{}\quad 0 \\ \beta I_* &{}\quad \beta S_* -a P_* -\mu &{}\quad -aI_* \\ 0 &{}\quad 0 &{}\quad -d\\ \end{array} \right) ,\\ A_4= & {} \left( \begin{array}{ccc} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 \\ 0 &{} \alpha P_* &{} \alpha I_* \\ \end{array} \right) ,\\ A_5= & {} \left( \begin{array}{c} (1+\theta -3S_*)\psi _{1}^{2}(0)+(\theta -\beta - 2S_* ) \psi _{1}(0)\psi _{2}(0) \\ \beta \psi _{1}(0)\psi _{2}(0)- a \psi _{2}(0)\psi _{3}(0) \\ \alpha \psi _{2}(-1)\psi _{3}(-1) \\ \end{array} \right) . \end{aligned}$$

By the Riesz representation theorem [87], there exists a function $\eta (\theta , \mu )$ of bounded variation for $\theta \in [-1, 0]$, such that

$$\begin{aligned} L_{\mu }\psi =\int _{-1}^{0} \hbox {d}\eta (\theta , \mu )\psi (\theta ), \ \ \ \ \text{ for } \ \ \psi \in \mathbf {C}. \end{aligned}$$

(35)

In fact, we can choose

$$\begin{aligned} \eta (\theta , \mu )= & {} (\tau ^{*}+\mu )\left( \begin{array}{lll} 2S_* (1+\theta )-\theta -3S_{*}^2+I_* (\theta -\beta -2I_*) &{}\quad -S_* (S_* -\theta )-\beta S_* &{}\quad 0 \\ \beta I_* &{}\quad \beta S_* -a P_* -\mu &{}\quad -aI_* \\ 0 &{}\quad 0 &{}\quad -d\\ \end{array} \right) \delta (\theta ) \nonumber \\&-(\tau ^{*}+\mu ) \left( \begin{array}{lll} 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad 0 &{}\quad 0 \\ 0 &{}\quad \alpha P_* &{}\quad \alpha I_* \\ \end{array} \right) \delta (\theta +1), \end{aligned}$$

(36)

where $\delta$ is defined by $\delta (\theta )=\Big \{^{1, \ \ \theta =0,}_{0, \ \ \theta \ne 0.}$

For $\psi \in \mathbf {C}^{1}\left( [-1, 0], \mathbf {R^{3}_{+}}\right)$, define

$$\begin{aligned} A(\mu )\psi =\left\{ \begin{array}{ll} \frac{\hbox {d}\psi (\theta )}{\hbox {d}\theta } &{} \quad \theta \in [-1, 0) \\ \displaystyle \int _{-1}^{0} \hbox {d}\eta (\mu , s)\psi (s)&{}\quad \theta =0 \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} R(\mu )\psi =\left\{ \begin{array}{c} 0, \quad \theta \in [-1, 0), \\ F(\mu , \psi ), \quad \theta =0. \end{array} \right. \end{aligned}$$

Then, the system (32) is of the form

$$\begin{aligned} \dot{z_{t}}=A(\mu )z_{t}+R(\mu )z_{t}, \end{aligned}$$

(37)

where $z_{t}(\theta )=z_{t}(t+\theta )$ for $\theta \in [-1, 0]$.

For $\phi \in \mathbf {C}^{1}([0, 1], (\mathbf {R^{3}_{+}})^{*})$, define

$$\begin{aligned} A^{*}\phi (s)=\left\{ \begin{array}{ll} -\frac{\hbox {d}\phi (s)}{\hbox {d}s}, &{}\quad s \in (0, 1], \\ \\ {\displaystyle \int _{-1}^{0}}\hbox {d}\eta ^{T}(t, 0)\phi (-t), &{}\quad s=0, \end{array} \right. \end{aligned}$$

and a bilinear inner product

$$\begin{aligned}&\langle \phi (s), \psi (\theta )\rangle =\overline{\phi }(0)\psi (0)\nonumber \\&\quad -\,\int _{-1}^{0}\int _{\alpha =0}^{\theta }\overline{\phi }(\alpha -\theta )\hbox {d}\eta (\theta )\psi (\alpha )\hbox {d}\alpha , \end{aligned}$$

(38)

where $\eta (\theta )=\eta (\theta , 0)$. Clearly, A(0) and $A^{*}$ are adjoint operators. We know that $\pm i\rho _{0}\tau ^{*}$ are eigenvalues of A(0). So, they are also eigenvalues of $A^{*}$. Now we search for the eigenvector of A(0) and $A^{*}$ corresponding to $i\rho _{0}\tau ^{*}$ and $-i\rho _{0}\tau ^{*}$ respectively.

We assume that $q(\theta )=(1, u, w)^{T} e^{i\rho _{0}\tau ^{*}\theta }$ and $q^{*}(s)$ are the eigenvectors of A(0) and $A^{*}$ corresponding to $i\rho _{0}\tau ^{*}$ and $-i\rho _{0}\tau ^{*}$, respectively. Then, we have $A(0)q(\theta )=i\rho _{0}\tau ^{*}q(\theta )$. By the definition of A(0) and from (35) and (36), it follows that

$$\begin{aligned}&\tau ^{*}\left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) +i\rho _{0} &{}\quad +S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu +i\rho _{0} &{}\quad aI_* \\ 0 &{}\quad -\alpha P_* e^{-i\rho _{0}\tau ^{*}} &{}\quad -\alpha I_* e^{-i\rho _{0}\tau ^{*}} +d +i\rho _{0}\\ \end{array} \right) q(0)\\&\quad =\left( \begin{array}{c} 0 \\ 0 \\ 0 \\ \end{array} \right) . \end{aligned}$$

Then, we can get $q(0)=(1, u, w)^{T}$,

where

$$\begin{aligned} u= & {} -\frac{-2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) \,+\,i\rho _{0}}{S_* (S_* -\theta )+\beta S_*}, \nonumber \\ w= & {} -\frac{-2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) \,+\,i\rho _{0}}{S_* (S_* -\theta )+\beta S_*}\nonumber \\&\times \frac{\alpha P_* e^{-i\rho _{0}\tau ^{*}}}{-\alpha I_* e^{-i\rho _{0}\tau ^{*}} \,+\,d \,+\,i\rho _{0}}. \end{aligned}$$

(39)

Similarly, we can obtain

$$\begin{aligned}&q^{*}(s) = D\left( 1, u^{*}, w^{*}\right) ^{T} e^{i\rho _{0}\tau ^{*}s}, \\&\quad = D e^{i\rho _{0}\tau ^{*}s}\left( 1,\frac{-2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) -i\rho _{0}}{\beta I_*}, \right. \\&\qquad \left. -\,\frac{aI_*}{-\alpha I_* e^{-i\rho _{0}\tau ^{*}} +d -i\rho _{0}}\right. \\&\qquad \times \left. \frac{-2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) -i\rho _{0}}{\beta I_*}\right) . \end{aligned}$$

We choose D in such a way that $\langle q^{*}(s), q(\theta ) \rangle =1$, $\langle q^{*}(s), \overline{q}(\theta ) \rangle =0$.

Hence

$$\begin{aligned}&\langle q^{*}(s), q(\theta ) \rangle \\&\quad = \overline{D}(1, \overline{u^{*}}, \overline{w^{*}})(1, u, w)^{T} \\&\qquad - \,\int _{-1}^{0}\int _{\zeta =0}^{\theta }\overline{D}(1, \overline{u^{*}},\overline{w^{*}})e^{-i\rho _{0}\tau ^{*}(\zeta -\theta )}\hbox {d}\eta (\theta ) \\&\qquad (1, u, w)^{T}e^{i\rho _{0}\tau ^{*}\zeta } \hbox {d}\zeta \\&\quad = \overline{D}\left[ 1+\overline{u^{*}}u+\overline{w^{*}}w \right. \\&\qquad \left. -\,\int _{-1}^{0}(1, \overline{u^{*}} ,\overline{w^{*}})\theta e^{i\rho _{0}\tau ^{*}\theta } \hbox {d}\eta (\theta )(1, u, w)^{T} \right] \\&\quad = \overline{D}\left[ 1+\overline{u^{*}}u \right. \\&\qquad \left. +\,\overline{w^{*}}w+ \tau ^{*} w \alpha (\overline{u^{*}}P_*+\overline{w^{*}}I_*) e^{-i\rho _{0}\tau ^{*}} \right] . \end{aligned}$$

Thus, we can choose D as $D=\frac{1}{1+\overline{u}u^{*}+\overline{w}w^{*}+ \tau ^{*} \overline{w} \alpha (u^{*}P_*+ w^{*}I_*) e^{i\rho _{0}\tau ^{*}}}$.

To describe the center manifold $\mathbf {C}_{0}$ at $\mu =0$, we compute the coordinates by using the same notations and procedures as proposed by Hassard et al. [36].

Let $z_{t}$ be the solution of of Eq. (32) when $\mu =0$.

Define

$$\begin{aligned} \text {z}(t)=\langle q^{*}, z_{t}\rangle , \quad W(t,\theta )=z_{t}(\theta )-2\hbox {Re}\{\text {z}(t)q(\theta ) \}. \nonumber \\ \end{aligned}$$

(40)

On the center manifold $\mathbf {C}_{0}$, we have

$$\begin{aligned} \ \ \ \ W(t, \theta )=W\Bigl (\text {z}(t), \overline{\text {z}}(t), \theta \Bigr ), \end{aligned}$$

where

$$\begin{aligned}&W(\text {z}, \overline{\text {z}}, \theta ) =W_{20}(\theta )\frac{\text {z}^{2}}{2}+ W_{11}(\theta )\text {z}\overline{\text {z}}+W_{02}(\theta )\frac{\overline{\text {z}}^{2}}{2}\\&\quad +\, W_{30}(\theta ) \frac{\text {z}^{3}}{6}+\cdots , \end{aligned}$$

$\text {z}$ and $\overline{\text {z}}$ are local coordinates for center manifold $\mathbf {C}_{0}$ in the direction of $q^{*}$ and $\overline{q}^{*}$. Here W is real when $z_{t}$ is real. Now, we consider only real solutions. For solution $z_{t} \in \mathbf {C}_{0}$ of Eq. (32), since $\mu =0$, we can obtain

$$\begin{aligned} \dot{\text {z}}(t)= & {} i\rho _{0}\tau ^{*}\text {z} \\&+\, \Bigl \langle \overline{q}^{*}(\theta ), F\Bigl (0, W(\text {z}, \overline{\text {z}},\theta )+2\hbox {Re}\{\text {z}q(\theta )\} \Bigr ) \Bigr \rangle \\= & {} i\rho _{0}\tau ^{*}\text {z}+\overline{q}^{*}(0)F\Bigl (0, W(\text {z}, \overline{\text {z}},0) \\&+\,2\hbox {Re}\left\{ \text {z}q(0)\right\} \Bigr ) \mathop {=}\limits ^{\text {def}} i\rho _{0}\tau ^{*}\text {z}+\overline{q}^{*}(0)F_{0}(\text {z}, \overline{\text {z}}); \end{aligned}$$

we rewrite this equation as $\dot{\text {z}}=i\rho _{0}\tau ^{*}\text {z}+g(\text {z}, \overline{\text {z}})$ with

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

(41)

Then, from Eq. (40), we have

$$\begin{aligned} z_{t}(\theta )= & {} (z_{1t}(\theta ), z_{2t}(\theta ), z_{3t}(\theta ))\nonumber \\= & {} W(t,\theta )+2\hbox {Re}\left\{ \text {z}(t)q(\theta ) \right\} \nonumber \\= & {} W_{20}(\theta )\frac{\text {z}^{2}}{2}+ W_{11}(\theta )\text {z}\overline{\text {z}} +W_{02}(\theta )\frac{\overline{\text {z}}^{2}}{2}\nonumber \\&+\,(1, u, w)^{T} e^{i\rho _{0}\tau ^{*}\theta } \text {z}+ (1, \overline{u}, \overline{w})^{T} e^{-i\rho _{0}\tau ^{*}\theta } \overline{\text {z}} \nonumber \\&+\,O\left( {\mid }(\text {z}, \overline{\text {z}}){\mid }^{3}\right) . \end{aligned}$$

(42)

Thus, from Eq. (41), we can get

$$\begin{aligned}&g\left( \text {z}, \overline{\text {z}}\right) =\overline{q}^{*}\left( 0\right) F_{0}\left( \text {z}, \overline{\text {z}}\right) \nonumber \\&\quad =\overline{D}\left( 1, \overline{u^{*}}, \overline{w^{*}}\right) \tau ^{*}\nonumber \\&\qquad \left( \begin{array}{c} -\left( 3S_*-1-\theta \right) z_{1t}^{2}\left( 0\right) -\left( \beta +2S_*-\theta \right) z_{1t}\left( 0\right) z_{2t}\left( 0\right) \\ \beta z_{1t}\left( 0\right) z_{2t}\left( 0\right) -a z_{2t}\left( 0\right) z_{3t}\left( 0\right) \\ \alpha z_{2t}\left( -1\right) z_{3t}\left( -1\right) \\ \end{array} \right) \nonumber \\&\quad = -\overline{D}\tau ^{*} \left[ \left( 3S_*-1-\theta \right) \left\{ z+\overline{z} \right. \right. \nonumber \\&\qquad +\,W_{20}^{1}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{1}\left( 0\right) \text {z}\overline{\text {z}}\nonumber \\&\qquad \left. +\,W_{02}^{1}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} ^{2} \nonumber \\&\qquad + \,\left( \beta +2S_*-\theta \right) \left\{ z+\overline{z}+W_{20}^{1}\left( 0\right) \frac{\text {z}^{2}}{2} \right. \nonumber \\&\qquad \left. +\, W_{11}^{1}\left( 0\right) \text {z}\overline{\text {z}} +W_{02}^{1}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \nonumber \\&\qquad \times \,\left\{ uz+ \overline{u} \ \overline{z} +W_{20}^{2}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{2}\left( 0\right) \text {z}\overline{\text {z}} \right. \nonumber \\&\qquad \left. \left. +\,W_{02}^{2}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \right] \nonumber \\&\qquad +\, \overline{D}\tau ^{*}\overline{u^{*}} \left[ \left\{ \beta \left( z+\overline{z}+W_{20}^{1}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{1}\left( 0\right) \text {z}\overline{\text {z}} \right. \right. \right. \nonumber \\&\qquad \left. \left. +\,W_{02}^{1}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right) \right\} \nonumber \\&\qquad \times \,\left\{ uz+ \overline{u} \ \overline{z}+W_{20}^{2}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{2}\left( 0\right) \text {z}\overline{\text {z}} \right. \nonumber \\&\qquad \left. +\,W_{02}^{2}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \nonumber \\&\qquad -\,a\left\{ uz+ \overline{u} \ \overline{z}+W_{20}^{2}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{2}\left( 0\right) \text {z}\overline{\text {z}} \right. \nonumber \\&\qquad \left. +\,W_{02}^{2}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \nonumber \\&\qquad \times \,\left\{ wz+ \overline{w}\ \overline{z}+W_{20}^{3}\left( 0\right) \frac{\text {z}^{2}}{2}+ W_{11}^{3}\left( 0\right) \text {z}\overline{\text {z}} \right. \nonumber \\&\qquad \left. \left. +\,W_{02}^{3}\left( 0\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \right] \nonumber \\&\qquad +\,\overline{D}\tau ^{*} \overline{w^{*}}\alpha \left\{ wz e^{-i\rho _{0}\tau ^{*}}+ \overline{w} \ \overline{z} e^{i\rho _{0}\tau ^{*}}+W_{20}^{3}\left( -1\right) \frac{\text {z}^{2}}{2} \right. \nonumber \\&\qquad \left. +\, W_{11}^{3}\left( -1\right) \text {z}\overline{\text {z}} +W_{02}^{3}\left( -1\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \nonumber \\&\qquad \times \, \left\{ uz e^{-i\rho _{0}\tau ^{*}}+ \overline{u} \ \overline{z} e^{i\rho _{0}\tau ^{*}}+W_{20}^{2}\left( -1\right) \frac{\text {z}^{2}}{2}\right. \nonumber \\&\qquad \left. +\, W_{11}^{2}\left( -1\right) \text {z}\overline{\text {z}} +W_{02}^{2}\left( -1\right) \frac{\overline{\text {z}}^{2}}{2}+O\left( {\mid }\left( \text {z}, \overline{\text {z}}\right) {\mid }^{3}\right) \right\} \end{aligned}$$

(43)

Comparing with the coefficients with (41), we can obtain

$$\begin{aligned} g_{20}= & {} 2\overline{D}\tau ^{*}\left[ -\{(3S_*-1-\theta )+u(2S_*+\beta -\theta )\}\right. \nonumber \\&\left. +\,u\overline{u^{*}}(\beta -aw)+\alpha uw\overline{w^{*}}e^{-2i\rho _{0}\tau ^{*}}\right] ,\nonumber \\ g_{11}= & {} 2\overline{D}\tau ^{*}\left[ -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}\right. \nonumber \\&\left. +\,\overline{u^{*}}(\beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\})+\alpha \hbox {Re}\{u\overline{w}\}\overline{w^{*}}\right] ,\nonumber \\ g_{02}= & {} 2\overline{D}\tau ^{*}\left[ -\{(3S_*-1-\theta )+\overline{u}(2S_*+\beta -\theta )\}\right. \nonumber \\&\left. +\,\overline{u}\overline{u^{*}}(\beta -a\overline{w})+\alpha \overline{uw}\overline{w^{*}}e^{2i\rho _{0}\tau ^{*}}\right] ,\nonumber \\ g_{21}= & {} \overline{D}\tau ^{*}\left[ -(2S_*+\beta -\theta )( W_{20}^{2}(0)+ 2W_{11}^{2}(0)\right. \nonumber \\&+\,\overline{u} W_{20}^{1}(0)+ 2uW_{11}^{1}(0))\nonumber \\&-\,2(3S_*-1-\theta ) ( W_{20}^{1}(0)+ 2W_{11}^{1}(0))\nonumber \\&+\,\overline{u^{*}}\{\beta ( 2W_{11}^{2}(0)+W_{20}^{2}(0)+ \overline{u} W_{20}^{1}(0)\nonumber \\&+\, 2uW_{11}^{1}(0)) - a ( 2uW_{11}^{3}(0)\nonumber \\&+\,\overline{u} W_{20}^{3}(0)+\overline{w} W_{20}^{2}(0)+ 2wW_{11}^{2}(0)\} \nonumber \\&+\,\alpha \overline{w^{*}}( 2uW_{11}^{2}(-1)e^{-i\rho _{0}\tau ^{*}}\nonumber \\&+\,\overline{u}W_{20}^{3}(-1)e^{i\rho _{0}\tau ^{*}} + \overline{w}W_{20}^{2}(-1)e^{i\rho _{0}\tau ^{*}}\nonumber \\&\left. +\, 2wW_{11}^{2}(-1)e^{-i\rho _{0}\tau ^{*}})\right] . \end{aligned}$$

(44)

To calculate the value of $g_{21}$, we need to compute the values of $W_{20}(\theta )$ and $W_{11}(\theta )$. From Eqs. (37) and (40), we have

$$\begin{aligned}&\dot{W}=\dot{z}_{t} -\dot{\text {z}}q- \dot{\overline{\text {z}}}\ \overline{q} \nonumber \\&\quad =\left\{ \begin{array}{c} A W- 2\hbox {Re}\{\overline{q}^{*}(0)F_{0}q(\theta ) \}, \quad \theta \in [-1, 0),\\ A W-2\hbox {Re}\{\overline{q}^{*}(0)F_{0}q(\theta )\}+F_{0}, \quad \theta = 0, \end{array} \right. \nonumber \\&\quad \mathop {=}\limits ^{\text {def}} AW+H(\text {z}, \overline{\text {z}}, \theta ), \end{aligned}$$

(45)

where

$$\begin{aligned} H(\text {z}, \overline{\text {z}}, \theta )= & {} H_{20}(\theta )\frac{\text {z}^{2}}{2}+ H_{11}(\theta )\text {z}\overline{\text {z}}\nonumber \\&+H_{02}(\theta )\frac{\overline{\text {z}}^{2}}{2}+ \cdots \ . \end{aligned}$$

(46)

Expanding the above series and comparing the corresponding coefficients, we obtain

$$\begin{aligned} (A-i2\rho _{0}\tau ^{*}I)W_{20}(\theta )= & {} -H_{20}(\theta ),\nonumber \\ AW_{11}(\theta )= & {} -H_{11}(\theta ). \end{aligned}$$

(47)

From Eq. (45), we know that for $\theta \in [-1, 0),$

$$\begin{aligned} H(\text {z}, \overline{\text {z}}, \theta )= & {} -\overline{q}^{*}(0)F_{0}q(\theta )-q^{*}(0)\overline{F}_{0}\overline{q}(\theta )\nonumber \\= & {} -g q(\theta )-\overline{g}\ \overline{q}(\theta ). \end{aligned}$$

(48)

Comparing the coefficients with (46) gives that

$$\begin{aligned} H_{20}( \theta )=-g_{20}q(\theta )-\overline{g}_{02} \overline{q}(\theta ) \end{aligned}$$

(49)

and

$$\begin{aligned} H_{11}( \theta )=-g_{11}q(\theta )-\overline{g}_{11} \overline{q}(\theta ). \end{aligned}$$

(50)

From (47) and (49), we have

$$\begin{aligned} \dot{W}_{20}(\theta )=i2\rho _{0}\tau ^{*}W_{20}(\theta )+g_{20}q(\theta )+\overline{g}_{02} \overline{q}(\theta ). \end{aligned}$$

Since $q(\theta )=(1, u, w)^{T} e^{i\rho _{0}\tau ^{*}\theta }$, hence

$$\begin{aligned} W_{20}(\theta )= & {} \frac{ig^{}_{20}}{\rho _{0}\tau ^{*}} q(0)e^{i\rho _{0}\tau ^{*}\theta }+\frac{i\overline{g}^{}_{20}}{3\rho _{0}\tau ^{*}} \overline{q}(0)e^{-i\rho _{0}\tau ^{*}\theta }\nonumber \\&+\,E_{1}e^{i2\rho _{0}\tau ^{*}\theta }, \end{aligned}$$

(51)

where $E_{1}=(E_{1}^{(1)}, E_{1}^{(2)}, E_{1}^{(3)}) \in \mathbf {R}^{3}$ is a constant vector.

Similarly, from Eqs. (47) and (50), we get

$$\begin{aligned} W_{11}(\theta )= & {} -\frac{ig^{}_{11}}{\rho _{0}\tau ^{*}} q(0)e^{i\rho _{0}\tau ^{*}\theta }\nonumber \\&+\,\frac{i\overline{g}^{}_{11}}{\rho _{0}\tau ^{*}} \overline{q}(0)e^{-i\rho _{0}\tau ^{*}\theta }+E_{2}, \end{aligned}$$

(52)

where $E_{2}=(E_{2}^{(1)}, E_{2}^{(2)}, E_{2}^{(3)}) \in \mathbf {R}^{3}$ is a constant vector.

In what follows, we shall seek appropriate $E_{1}$ and $E_{2}$ in (51) and (52) respectively. From the definition of A and (47), we obtain

$$\begin{aligned} \int _{-1}^{0}\hbox {d}\eta (\theta )W_{20}(\theta )=i2\rho _{0}\tau ^{*} W_{20}(0)-H_{20}(0) \end{aligned}$$

(53)

and

$$\begin{aligned} \int _{-1}^{0}\hbox {d}\eta (\theta )W_{11}(\theta )=-H_{11}(0), \end{aligned}$$

(54)

where $\eta (\theta )=\eta (0, \theta )$. From (47), we have

$$\begin{aligned} H_{20}(0)= & {} -g_{02}q(0)-\overline{g}_{02}\overline{q}(0)+2 \tau ^{*}\nonumber \\&\left( \begin{array}{c} -\left\{ (3S_*-1-\theta )+u(2S_*+\beta -\theta )\right\} \\ u(\beta -a w)\\ \alpha uwe^{-2i\rho _{0}\tau ^{*}} \\ \end{array} \right) \nonumber \\ \end{aligned}$$

(55)

and

$$\begin{aligned}&H_{11}(0)=-g_{11}q(0)-\overline{g}_{11}\overline{q}(0)\nonumber \\&\quad +\,2 \tau ^{*}\left( \begin{array}{c} -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}\\ \beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}\\ \alpha \hbox {Re}\{u\overline{w}\}\\ \end{array} \right) . \nonumber \\ \end{aligned}$$

(56)

Noting that

$$\begin{aligned} \left( i\rho _{0}\tau ^{*}I-\int _{-1}^{0}e^{i\rho _{0}\tau ^{*}\theta }\hbox {d}\eta \left( \theta \right) \right) q\left( 0\right) =0, \end{aligned}$$

and

$$\begin{aligned} \left( -i\rho _{0}\tau ^{*}I-\int _{-1}^{0}e^{-i\rho _{0}\tau ^{*}\theta }\hbox {d}\eta \left( \theta \right) \right) \overline{q}\left( 0\right) =0, \end{aligned}$$

and putting (51) and (55) into (53), we get

$$\begin{aligned}&\left( i2\rho _{0}\tau ^{*}I-\int _{-1}^{0}e^{i2\rho _{0}\tau ^{*}\theta }\hbox {d}\eta (\theta )\right) E_{1}\nonumber \\&\quad =2\tau ^{*} \left( \begin{array}{c} -\left\{ (3S_*-1-\theta )+u(2S_*+\beta -\theta )\right\} \\ u(\beta -a w)\\ \alpha uwe^{-2 i\rho _{0}\tau ^{*}} \\ \end{array} \right) , \end{aligned}$$

which implies that

$$\begin{aligned} \begin{array}{lcl} \left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) +2i\rho _{0} &{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu +2i\rho _{0} &{}\quad aI_* \\ 0 &{}\quad -\alpha P_* e^{-2i\rho _{0}\tau ^{*}} &{}\quad -\alpha I_* e^{-2i\rho _{0}\tau ^{*}} +d +2i\rho _{0}\\ \end{array} \right) E_{1}\\ \quad =2 \left( \begin{array}{c} -\left\{ (3S_*-1-\theta )+u(2S_*+\beta -\theta )\right\} \\ u(\beta -a w)\\ \alpha uwe^{-2i\rho _{0}\tau ^{*}} \\ \end{array} \right) , \end{array} \end{aligned}$$

(57)

It follows that

$$\begin{aligned} E_{1}^{(1)}=\frac{|{\varDelta }_{11}|}{|{\varDelta }_{1}|}, \ \ E_{1}^{(2)}=\frac{|{\varDelta }_{12}|}{|{\varDelta }_{1}|}, \ \ E_{1}^{(3)}=\frac{|{\varDelta }_{13}|}{|{\varDelta }_{1}|}, \nonumber \\ \end{aligned}$$

(58)

where

$$\begin{aligned} {\varDelta }_{11}= & {} 2\left( \begin{array}{lll} -\{(3S_*-1-\theta )+u(2S_*+\beta -\theta )\}&{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0\\ u(\beta -a w)&{}\quad -\beta S_* +a P_* +\mu +2i\rho _{0} &{}\quad aI_* \\ \alpha uwe^{-2i\rho _{0}\tau ^{*}} &{}\quad -\alpha P_* e^{-2i\rho _{0}\tau ^{*}} &{}\quad -\alpha I_* e^{-2i\rho _{0}\tau ^{*}} +d +2i\rho _{0}\\ \end{array} \right) ,\\ {\varDelta }_{12}= & {} 2\left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) +2i\rho _{0} &{}\quad -\{(3S_*-1-\theta )+u(2S_*+\beta -\theta )\}&{}\quad 0\\ -\beta I_* &{} u(\beta -a w) &{} aI_* \\ 0 &{}\quad \alpha uwe^{-2i\rho _{0}\tau ^{*}} &{}\quad -\alpha I_* e^{-2i\rho _{0}\tau ^{*}} +d +2i\rho _{0}\\ \end{array} \right) ,\\ {\varDelta }_{13}= & {} 2\left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) +2i\rho _{0} &{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad -\{(3S_*-1-\theta )+u(2S_*+\beta -\theta )\} \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu +2i\rho _{0} &{}\quad u(\beta -a w) \\ 0 &{}\quad -\alpha P_* e^{-i\rho _{0}\tau ^{*}} &{}\quad \alpha uwe^{-2i\rho _{0}\tau ^{*}}\\ \end{array} \right) ,\\ {\varDelta }_{1}= & {} \left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) +2i\rho _{0} &{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu +2i\rho _{0} &{}\quad aI_* \\ 0 &{}\quad -\alpha P_* e^{-2i\rho _{0}\tau ^{*}} &{}\quad -\alpha I_* e^{-2i\rho _{0}\tau ^{*}} +d +2i\rho _{0}\\ \end{array}\right) . \end{aligned}$$

Similarly putting (52) and (56) into (54), we have

$$\begin{aligned} \left( \int _{-1}^{0}\hbox {d}\eta (\theta ) \right) E_{2}=2\tau ^{*} \left( \begin{array}{c} -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}\\ \beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}\\ \alpha \hbox {Re}\{u\overline{w}\}\\ \end{array} \right) , \end{aligned}$$

which implies that

$$\begin{aligned}&\left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) &{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu &{}\quad aI_* \\ 0 &{}\quad -\alpha P_* &{}\quad -\alpha I_* +d \\ \end{array} \right) E_{2}\\&=2 \left( \begin{array}{c} -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}\\ \beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}\\ \alpha \hbox {Re}\{u\overline{w}\}\\ \end{array} \right) , \end{aligned}$$

and hence,

$$\begin{aligned} \begin{array}{ll} E_{2}^{(1)}=\frac{|{\varDelta }_{21}|}{|{\varDelta }_{2}|}, \ \ E_{2}^{(2)}=\frac{|{\varDelta }_{22}|}{|{\varDelta }_{2}|}, \ \ E_{2}^{(3)}=\frac{|{\varDelta }_{23}|}{|{\varDelta }_{2}|}, \end{array} \end{aligned}$$

(59)

where

$$\begin{aligned} {\varDelta }_{21}= & {} \left( \begin{array}{lll} -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}&{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ \beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}&{}\quad -\beta S_* +a P_* +\mu &{}\quad aI_* \\ \alpha \hbox {Re}\{u\overline{w}\}&{}\quad -\alpha P_* &{}\quad -\alpha I_* +d \\ \end{array} \right) ,\\ {\varDelta }_{22}= & {} \left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) &{}\quad -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}&{}\quad 0\\ -\beta I_* &{}\quad \beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}&{}aI_*\\ 0 &{}\quad \alpha \hbox {Re}\{u\overline{w}\}&{}\quad -\alpha I_* +d \\ \end{array} \right) ,\\ {\varDelta }_{23}= & {} \left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*)&{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad -\{(3S_*-1-\theta )+\hbox {Re}\{u\}(2S_*+\beta -\theta )\}\\ -\beta I_*&{} -\beta S_* +a P_* +\mu &{}\beta \hbox {Re}\{u\}-a \hbox {Re}\{u\overline{w}\}\\ 0&{}\quad -\alpha P_* &{}\quad \alpha \hbox {Re}\{u\overline{w}\} \\ \end{array} \right) ,\\ {\varDelta }_{2}= & {} \left( \begin{array}{lll} -2S_* (1+\theta )+\theta +3S_{*}^2-I_* (\theta -\beta -2I_*) &{}\quad S_* (S_* -\theta )+\beta S_* &{}\quad 0 \\ -\beta I_* &{}\quad -\beta S_* +a P_* +\mu &{} aI_* \\ 0 &{}\quad -\alpha P_* &{}\quad -\alpha I_* +d \\ \end{array} \right) . \end{aligned}$$

From the above analysis, we can compute the following quantities:

$$\begin{aligned} C_{1}(0)= & {} \frac{i}{2\rho _{0}\tau ^{*}}\left( g_{20}g_{11}-2|g_{11}|^2-\frac{1}{3}|g_{02}|^2\right) \\&+\,\frac{1}{2}g_{21}, \\ \mu _{2}= & {} -\frac{\hbox {Re}\{C_{1}(0)\}}{\hbox {Re}\{\acute{\lambda }(\tau ^{*})\}}, \\ \beta _{2}= & {} 2\hbox {Re}\{C_{1}(0)\}, \\ \tau _{2}= & {} -\frac{\hbox {Im}\left\{ C_{1}(0)\right\} +\mu _{2}\hbox {Im}left\{\acute{\lambda }(\tau ^{*})right\}}{\rho _{0}\tau ^{*}}. \end{aligned}$$

The above three quantities $\mu _2$, $\beta _2$ and $\tau _2$ will determine the direction, stability and the periods of the bifurcating periodic solutions.