Dynamics of algae blooming: effects of budget allocation and time delay

Abstract

Algal blooms are increasing in coastal waters worldwide. The study on the features of algal pollution in water bodies and the ways to eliminate them is of vital importance. Preventing, treating, and monitoring algal blooms can be an unanticipated cost for a water system. To tame algal bloom in a lake, the government provides funds through budget allocation. In this paper, we propose a mathematical model to investigate the effect of budget allocation on the control of algal bloom in a lake. We assume that the growth of budget follows logistic law and also increases in proportion to the algal density in the lake. A part of the budget is utilized for the control of inflow of nutrients, while the remaining is used in the removal of algae from the lake. Our results show that algal bloom can be mitigated from the lake by reducing the inflow rate of nutrients to a very low value, which can be achieved for very high efficacy of budget allocation for the control of nutrients inflow from outside sources. Also, increasing the efficacy of budget allocation for the removal of algae helps to control the algal bloom. Further, more budget should be used on the control of nutrient’s inflow than on the removal of algae, as the presence of nutrients in high concentration will immediately proliferate the growth of algae. Moreover, the combined effects of controlling the inflow of nutrients and removing algae at high rates will result in nutrients and algae-free aquatic environment. Further, we modify the model by considering a discrete time delay involved in the increment of budget due to increased density of algae in the lake. We observe that chaotic oscillations may arise via equilibrium destabilization on increasing the values of time delay. We apply basic tools of nonlinear dynamics such as Poincaré section and maximum Lyapunov exponent to confirm the chaotic behavior of the system.

Appendix

1.1 Appendix A

Components of the equilibrium $E^*$ are given by

$$\begin{aligned} M^*= & {} \frac{K}{r}(r+\gamma A^*), \end{aligned}$$

(29)

$$\begin{aligned} N^*= & {} \frac{\beta _{12}[\alpha _1qr+K(1-k_1)(\alpha _1+\alpha _2\nu _2)(r+\gamma A^*)]}{(\theta _1\beta _1-\beta _{11}\alpha _1)qr+K(1-k_1)(r+\gamma A^*)\{\theta _1\beta _1-\beta _{11}(\alpha _1+\alpha _2\nu _2)\}}, \end{aligned}$$

(30)

and $A^*$ is a positive root of $G(A)=0$, where

$$\begin{aligned} G(A)= & {} \left[ Q+\left( \pi _1-\frac{1}{\theta _1}\right) \alpha _1A\right] -\frac{Q\nu _1k_1K(r+\gamma A)}{pr+k_1K(r+\gamma A)}-\,\frac{\alpha _2\nu _2K(1-k_1)A(r+\gamma A)}{\theta _1[qr+K(1-k_1)(r+\gamma A)]}\nonumber \\&-\,\frac{\beta _{12}\alpha _0[\alpha _1qr+K(1-k_1)(\alpha _1+\alpha _2\nu _2)(r+\gamma A)]}{(\theta _1\beta _1-\beta _{11}\alpha _1)qr+K(1-k_1)(r+\gamma A)\{\theta _1\beta _1-\beta _{11}(\alpha _1+\alpha _2\nu _2)\}}. \end{aligned}$$

(31)

We note the following properties of G:

$$\begin{aligned}&(\mathrm{i})\, G(0)=Q-\frac{Q\nu _1k_1K}{p+k_1K} -\frac{\beta _{12}\alpha _0[\alpha _1q+(\alpha _1+\alpha _2\nu _2)(1-k_1)K]}{[(\theta _1\beta _1-\beta _{11}\alpha _1)q+\{\theta _1\beta _1-\beta _{11}(\alpha _1+\alpha _2\nu _2)\}(1-k_1)K]}, \end{aligned}$$

which is positive provided condition (9) holds.

(ii):

At $\displaystyle \overline{A}=\frac{Q\theta _1}{\alpha _1(1-\pi _1\theta _1)}>0,$ we have

$$\begin{aligned}&G(\overline{A})=-\frac{Q\nu _1k_1K(r+\gamma \overline{A})}{pr+k_1K(r+\gamma \overline{A})}-\,\frac{\alpha _2\nu _2K(1-k_1)\overline{A}(r+\gamma \overline{A})}{\theta _1[qr+K(1-k_1)(r+\gamma \overline{A})]}\\&-\,\frac{\beta _{12}\alpha _0[\alpha _1qr+K(1-k_1)(\alpha _1+\alpha _2\nu _2)(r+\gamma \overline{A})]}{(\theta _1\beta _1-\beta _{11}\alpha _1)qr+K(1-k_1)\{\theta _1\beta _1-\beta _{11}(\alpha _1+\alpha _2\nu _2)\}(r+\gamma \overline{A})}<0. \end{aligned}$$

(iii):

Also, we have

$$\begin{aligned}&\frac{\mathrm{d}G}{\mathrm{d}A}=-\frac{\alpha _1(1-\pi _1\theta _1)}{\theta _1}-\frac{Q\nu _1k_1Kpr\gamma }{[pr+k_1K(r+\gamma A)]^2}-\,\frac{\alpha _2\nu _2K(1-k_1)[qr(r+2\gamma A)+K(1-k_1)(r+\gamma A)^2]}{\theta _1[qr+K(1-k_1)(r+\gamma A)]^2}\\&-\,\frac{\beta _{12}\alpha _0\theta _1\beta _1\alpha _2\nu _2qrK\gamma (1-k_1)}{[(\theta _1\beta _1-\beta _{11}\alpha _1)qr+K(1-k_1)(r+\gamma A)\{\theta _1\beta _1-\beta _{11}(\alpha _1+\alpha _2\nu _2)\}]^2}<0. \end{aligned}$$

In view of the above three points, Eq. (31) has exactly one positive root (say $A^*$) in the interval $(0,\overline{A})$ if condition (9) holds. For this positive value $A^*$ of A, we get the positive value $M^*$ of M from Eq. (29). Also, we get the positive value $N^*$ of N from Eq. (30) provided condition (10) holds.

1.2 Appendix B

Jacobian of system (5) is given by

$$\begin{aligned} J=\left( \begin{array}{ccc} J_{11} &{} J_{12} &{} J_{13}\\ J_{21} &{} J_{22} &{} J_{23}\\ 0 &{} J_{32} &{} J_{33} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} J_{11}= & {} -\left( \alpha _0+\frac{\beta _1\beta _{12}A}{(\beta _{12}+\beta _{11}N)^2}\right) ,\\ J_{12}= & {} \pi _1\alpha _1-\frac{\beta _1N}{\beta _{12}+\beta _{11}N}, \ J_{13}=-\frac{Qp\nu _1k_1}{(p+k_1M)^2}, \\ J_{21}= & {} \frac{\theta _1\beta _1\beta _{12}A}{(\beta _{12}+\beta _{11}N)^2},\\ J_{22}= & {} \frac{\theta _1\beta _1N}{\beta _{12}+\beta _{11}N}-\alpha _1-\frac{\nu _2(1-k_1)M}{q+(1-k_1)M},\\ J_{23}= & {} -\frac{q\nu _2(1-k_1)A}{(q+(1-k_1)M)^2}, \ J_{32}=\gamma M,\\ J_{33}= & {} r\left( 1-\frac{2M}{K}\right) +\gamma A. \end{aligned}$$

• 1. Evaluating Jacobian J at the equilibrium $E_0$ gives the eigenvalues

  $$\begin{aligned} -\alpha _0, \ \frac{\theta _1\beta _1Q}{\beta _{12}\alpha _0+\beta _{11}Q}-\alpha _1, \ r. \end{aligned}$$

  Since one eigenvalue is always positive, the equilibrium $E_0$ is always unstable in M-direction irrespective of the signs of the other two eigenvalues.

• 2. Evaluation of Jacobian J at the equilibrium $E_1$ leads to the eigenvalues

  $$\begin{aligned} -\alpha _0, \ \frac{\theta _1\beta _1N_1}{\beta _{12}+\beta _{11}N_1}-\alpha _1-\frac{\nu _2(1-k_1)K}{q+(1-k_1)K}, \ -r. \end{aligned}$$

  Note that the first and third eigenvalues are always negative and the second one is negative provided the opposite of condition (9) holds.

• 3. Evaluation of Jacobian J at the equilibrium $E_2$ immediately gives one eigenvalue $r+\eta A_2$ while other two are given by roots of the following quadratic:

  $$\begin{aligned}&\xi ^2+\xi \left( \alpha _0+\frac{\beta _1\beta _{12}A_2}{(\beta _{12}+\beta _{11}N_2)^2}\right) \nonumber \\&\quad +\,\alpha _1(1-\pi _1\theta _1)\frac{\beta _1\beta _{12}A_2}{(\beta _{12}+\beta _{11}N_2)^2}=0. \end{aligned}$$

  (32)

  Note that the feasibility of equilibrium $E_2$ implies that the linear and constant terms in the characteristic equation (32) are always positive. Thus, according to Routh–Hurwitz criterion, roots of Eq. (32) are either negative or have negative real parts. Since one eigenvalue is always positive, the equilibrium $E_2$ is always unstable in M-direction.

• 4. Evaluation of Jacobian J at the equilibrium $E^*$ leads to the following characteristic equation:

  $$\begin{aligned} \xi ^3+C_1\xi ^2+C_2\xi +C_3=0, \end{aligned}$$

  (33)

  where

  $$\begin{aligned} C_1= & {} \alpha _0+\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}+\frac{rM^*}{K},\\ C_2= & {} \frac{rM^*}{K}\left( \alpha _0+\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}\right) \\&+\,\frac{q\alpha _2\nu _2\gamma A^*M^*(1-k_1)}{(q+(1-k_1)M^*)^2}\\&+\,\frac{\theta _1\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}\left[ \frac{\beta _1N^*}{\beta _{12}+\beta _{11}N^*}-\pi _1\alpha _1\right] ,\\ C_3= & {} \frac{q\alpha _2\nu _2\gamma A^*M^*(1-k_1)}{(q+(1-k_1)M^*)^2}\left( \alpha _0+\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}\right) \\&+\,\frac{rM^*}{K}\frac{\theta _1\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}\left[ \frac{\beta _1N^*}{\beta _{12}+\beta _{11}N^*}-\pi _1\alpha _1\right] \\&+\,\frac{Qp\nu _1k_1\gamma M^*}{(p+k_1M^*)^2}\frac{\theta _1\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}. \end{aligned}$$

  Clearly, $C_1>0$ and $C_2>0$. Employing Routh–Hurwitz criterion, the roots of Eq. (33) are either negative or have negative real parts if and only if the condition (12) is satisfied.

1.3 Appendix C

The second additive compound matrix of Jacobian of the system (5) at equilibrium $E^*$ is given by

$$\begin{aligned} J^{[2]}=\left( \begin{array}{ccc} a_{11} &{} a_{23} &{} -a_{13} \\ a_{32} &{} \ \ a_{11}+a_{33} &{} a_{12} \\ 0 &{} a_{21} &{} a_{33} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} a_{11}= & {} -\left( \alpha _0+\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}\right) ,\\ a_{12}= & {} \pi _1\alpha _1-\frac{\beta _1N^*}{\beta _{12}+\beta _{11}N^*}, \ a_{13}=-\frac{Qp\nu _1k_1}{(p+k_1M^*)^2}, \\ a_{21}= & {} \frac{\theta _1\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2},\\ a_{23}= & {} -\frac{q\alpha _2\nu _2(1-k_1)A^*}{(q+(1-k_1)M^*)^2}, \ a_{32}=\gamma M^*, \\ a_{33}= & {} -\frac{rM^*}{K}. \end{aligned}$$

Let $\displaystyle |X|_\infty =\sup _i|X_i|.$ The logarithmic norm $\nu _\infty (J^{[2]})$ of $J^{[2]}$ endowed with the vector norm $|X|_\infty$ is the supremum of $a_{11}+|a_{23}|+|a_{13}|$, $|a_{32}|+a_{11}+a_{33}+|a_{12}|$ and $|a_{21}|+a_{33}$.

Now, $a_{11}+|a_{23}|+|a_{13}|<0$ if

$$\begin{aligned}&\frac{q\alpha _2\nu _2(1-k_1)A^*}{(q+(1-k_1)M^*)^2}+\frac{Qp\nu _1k_1}{(p+k_1M^*)^2}<\alpha _0\\&\quad +\,\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}=L_1; \end{aligned}$$

similarly $a_{11}+a_{33}+|a_{32}|+|a_{12}|<0$ if

$$\begin{aligned}&\frac{\beta _1N^*}{\beta _{12}+\beta _{11}N^*}+\gamma M^*<\alpha _0+\frac{rM^*}{K}+\pi _1\alpha _1\\&\quad +\,\frac{\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}=L_2; \end{aligned}$$

also, $a_{33}+|a_{21}|<0$ if

$$\begin{aligned} \frac{\theta _1\beta _1\beta _{12}A^*}{(\beta _{12}+\beta _{11}N^*)^2}<\frac{rM^*}{K}=L_3. \end{aligned}$$

Following [37], system (5) has no periodic solution around the equilibrium $E^*$ provided condition (15) is satisfied.

1.4 Appendix D

Jacobian of system (16) is given by

$$\begin{aligned} \overline{J}=\left( \begin{array}{ccc} \overline{J}_{11} &{} \overline{J}_{12} &{} 0 \\ \overline{J}_{21} &{} \overline{J}_{22} &{} \overline{J}_{23} \\ 0 &{} \overline{J}_{32} &{} \overline{J}_{33} \end{array} \right) , \end{aligned}$$

where

$$\begin{aligned} \overline{J}_{11}= & {} -\left( \alpha _0+\frac{\beta _1\beta _{12}A}{(\beta _{12}+\beta _{11}N)^2}\right) ,\\ \overline{J}_{12}= & {} \pi _1\alpha _1-\frac{\beta _1N}{\beta _{12}+\beta _{11}N}, \ \overline{J}_{21}=\frac{\theta _1\beta _1\beta _{12}A}{(\beta _{12}+\beta _{11}N)^2},\\ \overline{J}_{22}= & {} \frac{\theta _1\beta _1N}{\beta _{12}+\beta _{11}N}-\alpha _1-\frac{\nu _2(1-k_1)M}{q+(1-k_1)M},\\ \overline{J}_{23}= & {} -\frac{q\nu _2(1-k_1)A}{(q+(1-k_1)M)^2}, \ \overline{J}_{32}=\gamma M, \\ \overline{J}_{33}= & {} r\left( 1-\frac{2M}{K}\right) +\gamma A. \end{aligned}$$

• 1. Evaluating the Jacobian $\overline{J}$ at the equilibrium $\overline{E}_0$ gives the eigenvalues $-\alpha _0$, $-\alpha _1$, and r. Since one eigenvalue is always positive, the equilibrium $\overline{E}_0$ is always unstable.

• 2. Evaluation of Jacobian $\overline{J}$ at the equilibrium $\overline{E}_1$ leads to the eigenvalues

  $$\begin{aligned} -\alpha _0, \ -\left( \alpha _1+\frac{\nu _2(1-k_1)K}{q+(1-k_1)K}\right) , \ -r. \end{aligned}$$

  Since the eigenvalues are negative, the equilibrium $\overline{E}_1$ is unconditionally stable.

• 3. At the equilibrium $\overline{E}_2$, one eigenvalue of $\overline{J}$ is immediately obtained as $r+\gamma \overline{A}_2$, whereas two can be obtain as roots of the equation

  $$\begin{aligned}&\xi ^2+\xi \left( \alpha _0+\frac{\beta _1\beta _{12}\overline{A}_2}{(\beta _{12}+\beta _{11}\overline{A}_2)^2}\right) \nonumber \\&\quad +\,\frac{\theta _1\beta _1\beta _{12}\overline{A}_2}{(\beta _{12}+\beta _{11}\overline{N}_2)^2}\left[ \frac{\beta _1\overline{N}_2}{\beta _{12}+\beta _{11}\overline{N}_2}-\pi _1\alpha _1\right] =0.\nonumber \\ \end{aligned}$$

  (34)

  Since one eigenvalue is positive, the equilibrium is unstable irrespective of the signs of roots of Eq. (34).

• 4. Evaluation of Jacobian $\overline{J}$ at the equilibrium $\overline{E}^*$ leads to the following characteristic equation:

  $$\begin{aligned} \xi ^3+D_1\xi ^2+D_2\xi +D_3=0, \end{aligned}$$

  (35)

  where

  $$\begin{aligned} D_1= & {} \alpha _0+\frac{\beta _1\beta _{12}\overline{A}^*}{(\beta _{12}+\beta _{11}\overline{N}^*)^2}+\frac{r\overline{M}^*}{K},\\ D_2= & {} \frac{r\overline{M}^*}{K}\left( \alpha _0+\frac{\beta _1\beta _{12}\overline{A}^*}{(\beta _{12}+\beta _{11}\overline{N}^*)^2}\right) \\&+\,\frac{q\alpha _2\nu _2\gamma \overline{A}^*\overline{M}^*(1-k_1)}{(q+(1-k_1)\overline{M}^*)^2}\\&+\,\frac{\theta _1\beta _1\beta _{12}\overline{A}^*}{(\beta _{12}+\beta _{11}\overline{N}^*)^2}\left[ \frac{\beta _1\overline{N}^*}{\beta _{12}+\beta _{11}\overline{N}^*}-\pi _1\alpha _1\right] ,\\ D_3= & {} \frac{q\alpha _2\nu _2\gamma \overline{A}^*\overline{M}^*(1-k_1)}{(q+(1-k_1)\overline{M}^*)^2}\left( \alpha _0+\frac{\beta _1\beta _{12}\overline{A}^*}{(\beta _{12}+\beta _{11}\overline{N}^*)^2}\right) \\&+\,\frac{r\overline{M}^*}{K}\frac{\theta _1\beta _1\beta _{12}\overline{A}^*}{(\beta _{12}+\beta _{11}\overline{N}^*)^2} \left[ \frac{\beta _1\overline{N}^*}{\beta _{12}+\beta _{11}\overline{N}^*}-\pi _1\alpha _1\right] . \end{aligned}$$

  Clearly, $D_1>0$. Hence, according to Routh–Hurwitz criterion, roots of Eq. (35) are either negative or have negative real parts if and only if conditions in (19) are satisfied.

1.5 Appendix E

Differentiating (23) with respect to $\tau$, we get

$$\begin{aligned} \frac{\mathrm{d}\xi }{\mathrm{d}\tau }=\frac{\xi (q_1\xi +q_0)e^{-\xi \tau }}{[3\xi ^2+2p_2\xi +p_1+q_1e^{-\xi \tau }-(q_1\xi +q_0)\tau e^{-\xi \tau }]}. \end{aligned}$$

This gives

$$\begin{aligned} \left( \frac{\mathrm{d}\xi }{\mathrm{d}\tau }\right) ^{-1}=\frac{3\xi ^2+2p_2\xi +p_1+q_1e^{-\xi \tau }}{\xi (q_1\xi +q_0)e^{-\xi \tau }}-\frac{\tau }{\xi }. \end{aligned}$$

Now,

$$\begin{aligned} {\mathrm{sgn}\left[ \frac{\mathrm{d}(Re(\xi ))}{\mathrm{d}\tau }\right] }_{\tau =\tau ^0_{\pm }}= & {} {\mathrm{sgn}\left[ \frac{\mathrm{d}(Re(\xi ))}{\mathrm{d}\tau }\right] }^{-1}_{\tau =\tau ^0_{\pm }}\\= & {} {\mathrm{sgn}\left[ Re\left( \frac{\mathrm{d}\xi }{\mathrm{d}\tau }\right) ^{-1}\right] }_{\xi =i\omega _{\pm }}\\= & {} {\mathrm{sgn}\left[ \frac{3\omega ^4_{\pm }+2B_2\omega ^2_{\pm }+B_1}{q^2_0+q^2_1\omega ^2_{\pm }}\right] }\\= & {} {\mathrm{sgn}\left[ \frac{\varPsi '(\omega ^2_{\pm })}{q^2_0+q^2_1\omega ^2_{\pm }}\right] }. \end{aligned}$$

Hence, it follows from the hypothesis ($h_4$) that $\varPsi '(\omega ^2_{+})>0$ and $\varPsi '(\omega ^2_{-})<0$. Thus, the transversality conditions are satisfied.

1.6 Appendix F

Without loss of generality, we denote anyone of the critical values $\tau =\tau ^k$ ($k=0,1,2,\cdots$) by $\widetilde{\tau }$ at which the characteristic equation (23) corresponding to the delay differential equation (20) has a pair of purely imaginary roots $\pm i\omega _0$ and the system goes under Hopf bifurcation. Hence, for any root of characteristic equation of the form $\xi (\tau )=v(\tau )+i\omega (\tau )$, $v(\widetilde{\tau })=0$, $\omega (\widetilde{\tau })=\omega _0$ and $\displaystyle \frac{\mathrm{d}v}{\mathrm{d}\tau }\Big \vert _{\tau =\widetilde{\tau }}\ne 0.$

We denote $\tau$ as $\tau =\widetilde{\tau }+\mu$, $\mu \in \mathbb {R}$; so that $\mu =0$ is Hopf bifurcation value for the system. Denote the space of continuous real-valued functions as $C=C([-1,0],\mathbb {R}^3)$. Using the transformation $u_1(t)=N(t)-N^*$, $u_2(t)=A(t)-A^*$, $u_3(t)=M(t)-M^*$, and $x_i(t)=u_i(\tau t)$ ($i=1,2,3$); the delay system (20) transforms to the following differential equation

$$\begin{aligned} \frac{\mathrm{d}x(t)}{\mathrm{d}t}=L_\mu x_t+F(\mu ,x_t), \end{aligned}$$

(36)

where $x(t)=(x_1(t),x_2(t),x_3(t))^T\in C$, $x_t(\varTheta )=x(t+\varTheta )$, $\varTheta \in [-1,0]$ and $L_\mu : \ C\rightarrow \mathbb {R}^3$ is given by

$$\begin{aligned} L_\mu \xi =(\widetilde{\tau }+\mu )[R_1\xi (0)+R_2\xi (-1)], \end{aligned}$$

(37)

where $R_1$ and $R_2$ are matrices corresponding to linearize delay differential equation (22).

Further, $\xi : \ [-1,0]\rightarrow \mathbb {R}^3$, $\xi (t)=(\xi _1(t),\xi _2(t),\xi _3(t))^T$ and

$$\begin{aligned} F(\mu ,\xi )=(\widetilde{\tau }+\mu )\left[ \begin{array}{c} V_1\\ V_2\\ V_3 \end{array}\right] , \ \xi =(\xi _1,\xi _2,\xi _3)^T\in C, \end{aligned}$$

where

$$\begin{aligned} V_1= & {} \sum _{i+j+k\ge 2}\frac{f^{(1)}_{ijk}}{i!j!k!}\xi ^i_1(0)\xi ^j_2(0)\xi ^k_3(0), \\ V_2= & {} \sum _{i+j+k\ge 2}\frac{f^{(2)}_{ijk}}{i!j!k!}\xi ^i_1(0)\xi ^j_2(0)\xi ^k_3(0), \\ V_3= & {} \sum _{i+j+k\ge 2}\frac{f^{(3)}_{ijk}}{i!j!k!}\xi ^i_1(0)\xi ^j_2(-1)\xi ^k_3(0) \end{aligned}$$

with

$$\begin{aligned} f^{(1)}(N,A,M)\equiv & {} Q\left( 1-\frac{\nu _1k_1M}{p+k_1M}\right) -\alpha _0N\\&-\,\frac{\beta _1NA}{\beta _{12}+\beta _{11}N}+\pi _1\alpha _1A,\\ f^{(2)}(N,A,M)\equiv & {} \frac{\theta _1\beta _1NA}{\beta _{12}+\beta _{11}N}-\alpha _1A\\&-\,\alpha _2\frac{\nu _2(1-k_1)AM}{q+(1-k_1)M},\\ f^{(3)}(N,A_1,M)\equiv & {} rM\left( 1-\frac{M}{K}\right) +\gamma A_1M, \end{aligned}$$

where $A_1=A(t-\tau )$ and

$$\begin{aligned} f^{(1)}_{ijk}= & {} \frac{\partial ^{i+j+k}f^{(1)}}{\partial N^i\partial A^j\partial M^k}\bigg \vert _{E^*}, \ f^{(2)}_{ijk}=\frac{\partial ^{i+j+k}f^{(2)}}{\partial N^i\partial A^j\partial M^k}\bigg \vert _{E^*}, \\ f^{(3)}_{ijk}= & {} \frac{\partial ^{i+j+k}f^{(3)}}{\partial N^i\partial A^j_1\partial M^k}\bigg \vert _{E^*}. \end{aligned}$$

Then, $L_\mu$ is a continuous linear function mapping $C([-1,0],\mathbb {R}^3)$ into $\mathbb {R}^3$.

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

$$\begin{aligned} L_\mu \xi =\int ^0_{-1}\mathrm{d}\eta (\varTheta ,\mu )\xi (\varTheta ). \end{aligned}$$

(38)

In view of Eq. (37), we can choose

$$\begin{aligned} \eta (\varTheta ,\mu )=(\widetilde{\tau }+\mu )[R_1\delta (\varTheta )-R_2\delta (\varTheta +1)], \end{aligned}$$

(39)

where $\delta (\varTheta )$ is the Dirac delta function.

For $\xi \in C^1([-1,0],\mathbb {R}^3)$, we define

$$\begin{aligned} B(\mu )\xi ={\left\{ \begin{array}{ll} {\displaystyle \frac{\mathrm{d}\xi (\varTheta )}{\mathrm{d}\varTheta },} &{} \varTheta \in [-1,0),\\ {\displaystyle \int ^0_{-1}\mathrm{d}\eta (y,\mu )\xi (y)=L_\mu \xi ,} &{} \varTheta =0 \end{array}\right. }\nonumber \\ \end{aligned}$$

(40)

and

$$\begin{aligned} R(\mu )\xi ={\left\{ \begin{array}{ll} 0, &{} \varTheta \in [-1,0)\\ F(\mu ,\xi ), &{} \varTheta =0. \end{array}\right. } \end{aligned}$$

(41)

Then, the system (36) is equivalent to

$$\begin{aligned} \dot{x}_t=B(\mu )x_t+R(\mu )x_t, \end{aligned}$$

(42)

where $x_t(\varTheta )=x(t+\varTheta )$ for $\varTheta \in [-1,0]$.

The adjoint operator $B^*$ of B is defined by

$$\begin{aligned} B^*(\mu )\varphi ={\left\{ \begin{array}{ll} {\displaystyle -\frac{\mathrm{d}\varphi (s)}{\mathrm{d}s},} &{} 0<s\le 1\\ \int ^0_{-1}\varphi (-t)\mathrm{d}\eta (t,0), &{} s=0, \end{array}\right. } \end{aligned}$$

(43)

associated with a bilinear form

$$\begin{aligned} \langle {\varphi ,\xi }\rangle= & {} \overline{\varphi }(0)\cdot \xi (0)\nonumber \\&-\,\int ^0_{\varTheta =-1}\int ^\varTheta _{\nu =0}\overline{\varphi }(\nu -\varTheta )\mathrm{d}\eta (\varTheta )\xi (\nu )\mathrm{d}\nu ,\nonumber \\ \end{aligned}$$

(44)

where $\eta (\varTheta )=\eta (\varTheta ,0)$. Then, B(0) (from here onward we shall refer B(0) by B) and $B^*$ are adjoint operators. Since $\pm i\omega _0\widetilde{\tau }$ are the eigenvalues of B, they are also eigenvalues of $B^*$. We need to compute eigenvectors of B and $B^*$ corresponding to $+i\omega _0\widetilde{\tau }$ and $-i\omega _0\widetilde{\tau }$, respectively.

Suppose $q(\varTheta )=(d_1,1,d_2)^Te^{i\omega _0\widetilde{\tau }\varTheta }$ be the eigenvector of B corresponding to eigenvalue $i\omega _0\widetilde{\tau }$, then

$$\begin{aligned} B(0)q(\varTheta )=i\omega _0\tau q(\varTheta ). \end{aligned}$$

(45)

For $\varTheta =0$, this gives

$$\begin{aligned} \widetilde{\tau }\left[ \begin{array}{ccc} r_{11} &{} r_{12} &{} r_{13}\\ r_{21} &{} 0 &{} r_{23}\\ 0 &{} r_{32}e^{-i\omega _0\widetilde{\tau }} &{} r_{33} \end{array}\right] \left[ \begin{array}{c} d_1\\ 1\\ d_2 \end{array}\right] =i\omega _0\widetilde{\tau }\left[ \begin{array}{c} d_1\\ 1\\ d_2 \end{array}\right] . \end{aligned}$$

(46)

Solving the system of Eq. (46), we get

$$\begin{aligned} d_1=\frac{r_{12}+r_{13}d_2}{i\omega _0-r_{11}}, \ d_2=\frac{r_{32}e^{-i\omega _0\widetilde{\tau }}}{i\omega _0-r_{33}}. \end{aligned}$$

Similarly, we can calculate $q^*(y)=D(d^*_1,1,d^*_2)e^{i\omega _0\widetilde{\tau }y}$ such that

$$\begin{aligned} B^*{q^*}^T(y)=-i\omega _0\widetilde{\tau }{q^*}^T(y). \end{aligned}$$

(47)

We get,

$$\begin{aligned} d^*_1=-\frac{r_{21}}{i\omega _0+r_{11}}, \ d^*_2=-\frac{r_{23}+r_{13}d^*_1}{r_{33}+i\omega _0}. \end{aligned}$$

Now, we need to determine the value of D such that $\langle {q^*(p),q(\varTheta )}\rangle =1$. Using (44), we have

$$\begin{aligned}&\overline{q}^*(0)q(0)-\overline{D}\int ^0_{\varTheta =-1}\int ^\varTheta _{\nu =0}\overline{q}^*(0)e^{-i\omega _0\widetilde{\tau }(\nu -\varTheta )}\\&\quad \mathrm{d}\eta (\varTheta ).q(0)e^{i\omega _0\widetilde{\tau }\nu }\mathrm{d}\nu =1. \end{aligned}$$

After some calculations, we get

$$\begin{aligned} \overline{D}[1+d_1\overline{d}^*_1+d_2\overline{d}^*_2+e^{-i\omega _0\widetilde{\tau }}\widetilde{\tau }r_{32}\overline{d}^*_2]=1. \end{aligned}$$

Thus, D is chosen such that

$$\begin{aligned} \overline{D}=\frac{1}{1+d_1\overline{d}^*_1+d_2\overline{d}^*_2+e^{-i\omega _0\widetilde{\tau }}\widetilde{\tau }r_{32}\overline{d}^*_2}. \end{aligned}$$

(48)

Moreover, we can verify that $\langle {q^*(p),\overline{q(\varTheta )}}\rangle =0$.

Proceeding as in Hassard et al. [29] and using the same notation, we compute the coordinates to describe the center manifold $C_0$ at $\mu =0$. Let $x_t$ be solution of Eq. (42) when $\mu =0$. Define,

$$\begin{aligned} z(t)=\langle {q^*,x_t}\rangle , \ W(t,\varTheta )=x_t(\varTheta )-2Re\{z(t)q(\varTheta )\}.\nonumber \\ \end{aligned}$$

(49)

On the center manifold $C_0$, we have

$$\begin{aligned} W(t,\varTheta )=W(z,\overline{z},\varTheta ), \end{aligned}$$

(50)

where

$$\begin{aligned} W(z,\overline{z},\varTheta )= & {} W_{20}(\varTheta )\frac{z^2}{2}+W_{11}(\varTheta )z\overline{z}\nonumber \\&+\,W_{02}(\varTheta )\frac{\overline{z}^2}{2}+\cdots , \end{aligned}$$

(51)

z and $\overline{z}$ are local coordinates for center manifold $C_0$ in the direction of $q^*$ and $\overline{q}^*$, respectively. Note that W is real if $x_t$ is real; we consider only real solutions. Since $\mu =0$, for solution $x_t\in C_0$ of Eq. (42), we have

$$\begin{aligned} \dot{z}= & {} i\omega _0\widetilde{\tau }z+\overline{q}^*(0)F(0,W(z,\overline{z},0)+2Re\{zq(0)\})\end{aligned}$$

(52)

$$\begin{aligned}= & {} i\omega _0\widetilde{\tau }z+\overline{q}^*(0)F_0(z,\overline{z}). \end{aligned}$$

(53)

We rewrite this equation as

$$\begin{aligned} \dot{z}=i\omega _0\widetilde{\tau }z+g(z,\overline{z}), \end{aligned}$$

(54)

where

$$\begin{aligned} g(z,\overline{z})= & {} \overline{q}^*(0).F_0(z,\overline{z})=g_{20}\frac{z^2}{2}+g_{11}z\overline{z}+g_{12}\frac{\overline{z}^2}{2}\nonumber \\&+\,g_{21}\frac{z^2\overline{z}}{2}+\cdots . \end{aligned}$$

(55)

From (49) and (51), it follows that

$$\begin{aligned} x_t(\varTheta )= & {} W(z,\overline{z},\varTheta )+2Re\{zq(\varTheta )\}, \end{aligned}$$

(56)

$$\begin{aligned}= & {} W_{20}(\varTheta )\frac{z^2}{2}+W_{11}(\varTheta )z\overline{z}+W_{02}(\varTheta )\frac{\overline{z}^2}{2}\nonumber \\&+\,z(d_1,1,d_2)^Te^{i\omega _0\widetilde{\tau }\varTheta }\nonumber \\&+\,\overline{z}(\overline{d}_1,1,\overline{d}_2)^Te^{-i\omega _0\widetilde{\tau }\varTheta }+\cdots , \end{aligned}$$

(57)

so that

$$\begin{aligned} x_{1t}(\varTheta )= & {} W^{(1)}_{20}(\varTheta )\frac{z^2}{2}+W^{(1)}_{11}(\varTheta )z\overline{z}+W^{(1)}_{02}\frac{\overline{z}^2}{2}\\&+\,d_1ze^{i\omega _0\widetilde{\tau }\varTheta } +\overline{d}_1\overline{z}e^{-i\omega _0\widetilde{\tau }\varTheta }+\cdots ,\\ x_{2t}(\varTheta )= & {} W^{(2)}_{20}(\varTheta )\frac{z^2}{2}+W^{(2)}_{11}(\varTheta )z\overline{z}+W^{(2)}_{02}(\varTheta )\frac{\overline{z}^2}{2}\\&+\,ze^{i\omega _0\widetilde{\tau }\varTheta } +\overline{z}e^{-i\omega _0\widetilde{\tau }\varTheta }+\cdots ,\\ x_{3t}(\varTheta )= & {} W^{(3)}_{20}(\varTheta )\frac{z^2}{2}+W^{(3)}_{11}(\varTheta )z\overline{z}+W^{(3)}_{02}\frac{\overline{z}^2}{2}\\&+\,d_2ze^{i\omega _0\widetilde{\tau }\varTheta } +\overline{d}_2\overline{z}e^{-i\omega _0\widetilde{\tau }\varTheta }+\cdots . \end{aligned}$$

Thus, we have

$$\begin{aligned} x_{1t}(0)= & {} W^{(1)}_{20}(0)\frac{z^2}{2}+W^{(1)}_{11}(0)z\overline{z}+W^{(1)}_{02}(0)\frac{\overline{z}^2}{2}\nonumber \\&+\,d_1z+\overline{d}_1\overline{z}+\cdots ,\nonumber \\ x_{2t}(0)= & {} z+\overline{z}+W^{(2)}_{20}(0)\frac{z^2}{2}+W^{(2)}_{11}(0)z\overline{z}+W^{(2)}_{02}\frac{\overline{z}^2}{2}+\cdots ,\nonumber \\ x_{3t}(0)= & {} d_2z+\overline{d}_2\overline{z}+W^{(3)}_{20}(0)\frac{z^2}{2}+W^{(3)}_{11}(0)z\overline{z}\nonumber \\&+\,W^{(3)}_{02}\frac{\overline{z}^2}{2}+\cdots ,\nonumber \\ x_{2t}(-1)= & {} ze^{-i\omega _0\widetilde{\tau }}+\overline{z}e^{i\omega _0\widetilde{\tau }}+W^{(2)}_{20}(-1)\frac{z^2}{2}\nonumber \\&+\,W^{(2)}_{11}(-1)z\overline{z} +W^{(2)}_{02}(-1)\frac{\overline{z}^2}{2}. \end{aligned}$$

(58)

From the definition of $F(\mu ,\phi )$, we have

$$\begin{aligned} g(z,\overline{z})= & {} \overline{q}^*(0)\cdot F(0,x_t)=\overline{D}(\overline{d}^*_1,1,\overline{d}^*_2)\cdot \widetilde{\tau }\left[ \begin{array}{c} V_1\\ V_2\\ V_3 \end{array} \right] \nonumber \\= & {} \widetilde{\tau }\overline{D}[\overline{d}^*_1V_1+V_2+\overline{d}^*_2V_3], \end{aligned}$$

(59)

where

$$\begin{aligned} V_1= & {} \sum _{i+j+k\ge 2}\frac{f^{(1)}_{ijk}}{i!j!k!}x^i_{1t}(0)x^j_{2t}(0)x^k_{3t}(0), \\ V_2= & {} \sum _{i+j+k\ge 2}\frac{f^{(2)}_{ijk}}{i!j!k!}x^i_{1t}(0)x^j_{2t}(0)x^k_{3t}(0), \\ V_3= & {} \sum _{i+j+k\ge 2}\frac{f^{(3)}_{ijk}}{i!j!k!}x^i_{1t}(0)x^j_{2t}(-1)x^k_{3t}(0). \end{aligned}$$

Using the expressions for $x_{1t}(0)$, $x_{2t}(0)$, and $x_{3t}(0)$ from (58) in (59) and comparing the coefficients of $z^2$, $z\overline{z}$, $\overline{z}^2$, and $z^2\overline{z}$ of the resulting expression with those in (55), we get

$$\begin{aligned} g_{20}= & {} 2\widetilde{\tau }\overline{D}\left[ \overline{d}^*_1\left\{ \frac{f^{(1)}_{200}}{2!}d^2_1+f^{(1)}_{110}d_1+\frac{f^{(1)}_{002}}{2!}d^2_2\right\} \right. \\&\quad +\,\left\{ \frac{f^{(2)}_{200}}{2!}d^2_1+f^{(2)}_{110}d_1+f^{(2)}_{011}d_2+\frac{f^{(2)}_{002}}{2!}d^2_2\right\} \\&\quad \left. +\,\overline{d}^*_2\left\{ \gamma d_2e^{-i\omega _0\widetilde{\tau }}-\frac{r}{K}d^2_2\right\} \right] ,\\ g_{11}= & {} \widetilde{\tau }\overline{D}\left[ \overline{d}^*_1\left\{ \frac{f^{(1)}_{200}}{2!}2d_1\overline{d}_1+f^{(1)}_{110}(d_1+\overline{d}_1)\right. \right. \\&\left. \quad +\,\frac{f^{(1)}_{002}}{2!}2d_2\overline{d_2}\right\} \\&\quad +\,\left\{ \frac{f^{(2)}_{200}}{2!}2d_1\overline{d}_1+f^{(2)}_{110}(d_1+\overline{d}_1)\right. \\&\quad \left. +\,f^{(2)}_{011}(d_2+\overline{d}_2)+\frac{f^{(2)}_{002}}{2!}2d_2\overline{d}_2\right\} \\&\quad \left. +\,\overline{d}^*_2\left\{ \gamma (d_2e^{i\omega _0\widetilde{\tau }}+\overline{d}_2e^{-i\omega _0\widetilde{\tau }})-\frac{r}{K}2d_2\overline{d}_2\right\} \right] ,\\ g_{02}= & {} 2\widetilde{\tau }\overline{D}\left[ \overline{d}^*_1\left\{ \frac{f^{(1)}_{200}}{2!}\overline{d}^2_1+f^{(1)}_{110}\overline{d}_1+\frac{f^{(1)}_{002}}{2!}\overline{d}^2_2\right\} \right. \\&\quad +\,\left\{ \frac{f^{(2)}_{200}}{2!}\overline{d}^2_1+f^{(2)}_{110}\overline{d}_1+f^{(2)}_{011}\overline{d}_2+\frac{f^{(2)}_{002}}{2!}\overline{d}^2_2\right\} \\&\quad \left. +\,\overline{d}^*_2\left\{ \gamma \overline{d}_2e^{i\omega _0\widetilde{\tau }}-\frac{r}{K}\overline{d}^2_2\right\} \right] \end{aligned}$$

and

$$\begin{aligned}&g_{21}=2\widetilde{\tau }\overline{D}\left[ \overline{d}^*_1\left\{ \frac{f^{(1)}_{200}}{2!}(2d_1W^{(1)}_{11}(0)+\overline{d}_1W^{(1)}_{20}(0))\right. \right. \\&\quad +\,f^{(1)}_{110}\left( d_1W^{(2)}_{11}(0)+W^{(1)}_{11}(0)+\frac{1}{2}\overline{d}_1W^{(2)}_{20}(0)+\frac{1}{2}W^{(1)}_{20}(0)\right) \\&\quad \left. +\,\frac{f^{(1)}_{002}}{2!}\left( 2d_2W^{(3)}_{11}(0)+\overline{d}_2W^{(3)}_{20}(0)\right) \right\} \\&\quad +\,\left\{ \frac{f^{(2)}_{200}}{2!}\left( 2d_1W^{(1)}_{11}(0)+\overline{d}_1W^{(1)}_{20}(0)\right) \right. \\&\quad +\,f^{(2)}_{110}\left( d_1W^{(2)}_{11}(0)+W^{(1)}_{11}(0)+\frac{1}{2}\overline{d}_1W^{(2)}_{20}(0)+\frac{1}{2}W^{(1)}_{20}(0)\right) \\&\quad +\,f^{(2)}_{011}\left( W^{(3)}_{11}(0)+d_2W^{(2)}_{11}(0)+\frac{1}{2}\overline{d}_2W^{(2)}_{20}(0)+\frac{1}{2}W^{(3)}_{20}(0)\right) \\&\quad \left. +\,\frac{f^{(2)}_{002}}{2!}\left( 2d_2W^{(3)}_{11}(0)+\overline{d}_2W^{(3)}_{20}(0)\right) \right\} \\&\quad +\,\overline{d}^*_2\left\{ \eta \left( W^{(3)}_{11}(0)e^{-i\omega _0\widetilde{\tau }}+d_2W^{(2)}_{11}(-1)+\frac{1}{2}W^{(3)}_{20}(0)e^{i\omega _0\overline{\tau }}\right. \right. \\&\quad \left. \left. +\,\frac{1}{2}\overline{d}_2W^{(2)}_{20}(-1)\right) -\frac{r}{K}\left( 2d_2W^{(3)}_{11}(0)+\overline{d}_2W^{(3)}_{20}(0)\right) \right\} \\&\quad +\,\overline{d}^*_1\left\{ \frac{f^{(1)}_{300}}{3!}3d^2_1\overline{d}_1+\frac{f^{(1)}_{210}}{2!}(d^2_1+2d_1\overline{d}_1)+f^{(1)}_{003}3d^2_2\overline{d}_2\right\} \\&\quad +\,\left\{ \frac{f^{(2)}_{300}}{3!}3d^2_1\overline{d}_1+\frac{f^{(2)}_{210}}{2!}(d^2_1+2d_1\overline{d}_1)\right. \\&\quad \left. \left. +\,\frac{f^{(2)}_{012}}{2!}(d^2_2+2d_2\overline{d}_2)+\frac{f^{(2)}_{003}}{3!}3d^2_2\overline{d}_2\right\} \right] . \end{aligned}$$

In order to compute $g_{21}$, we need to compute $W_{20}(\varTheta )$ and $W_{11}(\varTheta )$. From Eqs. (49) and (52), we have

$$\begin{aligned} \dot{W}= & {} \dot{x}_t-\dot{z}q-\dot{\overline{z}} \overline{q}\nonumber \\= & {} \left\{ \begin{array}{ll} B(0)W-2Re\{\overline{q}^*(0).F_0q(\varTheta )\}, &{} \varTheta \in [-1,0)\\ B(0)W-2Re\{\overline{q}^*(0).F_0q(0)\}+F_0, &{} \varTheta =0 \end{array}\right. \end{aligned}$$

(60)

$$\begin{aligned}\equiv & {} B(0)W+H(z,\overline{z},\varTheta ), \end{aligned}$$

(61)

where

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

(62)

Also, on the center manifold $C_0$ near the origin,

$$\begin{aligned} \dot{W}=W_z\dot{z}+W_{\overline{z}}\dot{\overline{z}}. \end{aligned}$$

(63)

From Eqs. (54), (61) and (63), it follows that

$$\begin{aligned} (B(0)-2i\omega _0\widetilde{\tau })W_{20}= & {} -H_{20}, \end{aligned}$$

(64)

$$\begin{aligned} B(0)W_{11}= & {} -H_{11}. \end{aligned}$$

(65)

Now, for $\varTheta \in [-1,0)$, we have

$$\begin{aligned} H(z,\overline{z},\varTheta )= & {} -\overline{q}^*(0).F_0q(\varTheta )-q^*(0).\overline{F}_0\overline{q}(\varTheta )\nonumber \\= & {} -g(z,\overline{z})q(\varTheta )-\overline{g}(z,\overline{z})\overline{q}(\varTheta )\nonumber \\= & {} -(g_{20}q(\varTheta )+\overline{g}_{02}q(\varTheta ))\frac{z^2}{2}\nonumber \\&-\,(g_{11}q(\varTheta )+\overline{g}_{11}\overline{q}(\varTheta ))z\overline{z}+\cdots ,\nonumber \\ \end{aligned}$$

(66)

which on comparing the coefficients with (62) gives

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

(67)

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

(68)

From Eqs. (64) and (67), and the definition of B(0), we have

$$\begin{aligned} W'_{20}(\varTheta )=2i\omega _0\widetilde{\tau }W_{20}(\varTheta )+g_{20}q(\varTheta )+\overline{g}_{02}\overline{q}(\varTheta ). \end{aligned}$$

(69)

Note that $q(\varTheta )=q(0)e^{i\omega _0\widetilde{\tau }\varTheta }$, hence

$$\begin{aligned} W_{20}(\varTheta )=\frac{ig_{20}}{\omega _0\widetilde{\tau }}q(\varTheta )+\frac{i\overline{g}_{02}}{3\omega _0\widetilde{\tau }}\overline{q}(\varTheta )+E_1e^{2i\omega _0\widetilde{\tau }\varTheta }. \end{aligned}$$

(70)

Similarly, from Eqs. (65) and (68), and the definition of B(0), we have

$$\begin{aligned} W'_{11}(\varTheta )= & {} g_{11}q(\varTheta )+\overline{g}_{11}\overline{q}(\varTheta ), \end{aligned}$$

(71)

which gives

$$\begin{aligned} W_{11}(\varTheta )=-\frac{ig_{11}}{\omega _0\widetilde{\tau }}q(\varTheta )+\frac{i\overline{g}_{11}}{\omega _0\widetilde{\tau }}\overline{q}(\varTheta )+E_2, \end{aligned}$$

(72)

where $E_1=(E^{(1)}_1,E^{(2)}_1,E^{(3)}_1)$ and $E_2=(E^{(1)}_2,E^{(2)}_2,E^{(3)}_2)\in \mathbb {R}^3$ are constant vectors to be determined. It follows from Eq. (64) and the definition of B(0) that

$$\begin{aligned} \int ^0_{-1}\mathrm{d}\eta (\varTheta )W_{20}(\varTheta )= & {} 2i\omega _0\widetilde{\tau }W_{20}(0)-H_{20}(0), \end{aligned}$$

(73)

$$\begin{aligned} \int ^0_{-1}\mathrm{d}\eta (\varTheta )W_{11}(\varTheta )= & {} -H_{11}(0). \end{aligned}$$

(74)

From Eqs. (60) and (62), we get

$$\begin{aligned} H_{20}= & {} -(g_{20}q(0)+\overline{g}_{02}\overline{q}(0))\nonumber \\&+\,2\widetilde{\tau }\left[ \begin{array}{c} \displaystyle \frac{1}{2}f^{(1)}_{200}d^2_1+f^{(1)}_{110}d_1+\frac{1}{2}f^{(1)}_{002}d^2_2\\ \displaystyle \frac{1}{2}f^{(2)}_{200}d^2_1+f^{(2)}_{110}d_1+f^{(2)}_{011}d_2+\frac{1}{2}f^{(2)}_{002}d^2_2\\ \displaystyle \gamma d_2e^{-i\omega _0\widetilde{\tau }}-\frac{r}{K}d^2_2 \end{array}\right] \nonumber \\ \end{aligned}$$

(75)

and

$$\begin{aligned}&H_{11}(0)=-g_{11}q(0)-\overline{g}_{11}\overline{q}(0)\nonumber \\&\quad +\,2\widetilde{\tau }\left[ \begin{array}{c} \displaystyle \frac{1}{2}f^{(1)}_{200}|d_1|^2+f^{(1)}_{110}Re(d_1)+\frac{1}{2}f^{(1)}_{002}|d_2|^2\\ \displaystyle \frac{1}{2}f^{(2)}_{200}|d_1|^2+f^{(2)}_{110}Re(d_1)+f^{(2)}_{011}Re(d_2)+\frac{1}{2}f^{(2)}_{002}|d_2|^2\\ \displaystyle \gamma (Re(d_2e^{i\omega _0\widetilde{\tau }}))-\frac{r}{K}|d_2|^2 \end{array}\right] .\nonumber \\ \end{aligned}$$

(76)

Using Eqs. (70) and (75) in Eq. (73), and noting that $q(\varTheta )$ is eigenvector of B(0), we have

$$\begin{aligned}&(2i\omega _0\widetilde{\tau }I-\int ^0_{-1}e^{2i\omega _0\widetilde{\tau }\varTheta }\mathrm{d}\eta (\varTheta ))E_1\nonumber \\&\quad =2\widetilde{\tau }\left[ \begin{array}{c} \displaystyle \frac{1}{2}f^{(1)}_{200}d^2_1+f^{(1)}_{110}d_1+\frac{1}{2}f^{(1)}_{002}d^2_2\\ \displaystyle \frac{1}{2}f^{(2)}_{200}d^2_1+f^{(2)}_{110}d_1+f^{(2)}_{011}d_2+\frac{1}{2}f^{(2)}_{002}d^2_2\\ \displaystyle \gamma d_2e^{-i\omega _0\widetilde{\tau }}-\frac{r}{K}d^2_2, \end{array}\right] \nonumber \\ \end{aligned}$$

(77)

i.e.,

$$\begin{aligned}&\left[ \begin{array}{ccc} 2i\omega _0-r_{11} &{} -r_{12} &{} -r_{13}\\ -r_{21} &{} 2i\omega _0 &{} -r_{23}\\ 0 &{} -r_{32}e^{-2i\omega _0\widetilde{\tau }} &{} \ \ 2i\omega _0-r_{33} \end{array}\right] \left[ \begin{array}{c} E^{(1)}_1\\ E^{(2)}_1\\ E^{(3)}_1\end{array}\right] \nonumber \\&\quad =2\left[ \begin{array}{c } \displaystyle \frac{1}{2}f^{(1)}_{200}d^2_1+f^{(1)}_{110}d_1+\frac{1}{2}f^{(1)}_{002}d^2_2\\ \displaystyle \frac{1}{2}f^{(2)}_{200}d^2_1+f^{(2)}_{110}d_1+f^{(2)}_{011}d_2+\frac{1}{2}f^{(2)}_{002}d^2_2\\ \displaystyle \gamma d_2e^{-i\omega _0\widetilde{\tau }}-\frac{r}{K}d^2_2 \end{array}\right] .\nonumber \\ \end{aligned}$$

(78)

Similarly, using Eqs. (72) and (76) in Eq. (74), we get

$$\begin{aligned}&\left[ \begin{array}{ccc} -r_{11} &{} -r_{12} &{} -r_{13}\\ -r_{21} &{} 0 &{} -r_{23}\\ 0 &{} -r_{32} &{} -r_{33} \end{array}\right] \left[ \begin{array}{c} E^{(1)}_2\\ E^{(2)}_2\\ E^{(3)}_2 \end{array}\right] \nonumber \\&\quad =2\left[ \begin{array}{c} \displaystyle \frac{1}{2}f^{(1)}_{200}|d_1|^2+f^{(1)}_{110}Re(d_1)+\frac{1}{2}f^{(1)}_{002}|d_2|^2\\ \displaystyle \frac{1}{2}f^{(2)}_{200}|d_1|^2+f^{(2)}_{110}Re(d_1)+f^{(2)}_{011}Re(d_2)+\frac{1}{2}f^{(2)}_{002}|d_2|^2\\ \displaystyle \gamma (Re(d_2e^{i\omega _0\widetilde{\tau }}))-\frac{r}{K}|d_2|^2 \end{array}\right] .\nonumber \\ \end{aligned}$$

(79)

Now, we solve systems (78) and (79) for $E_1$ and $E_2$, as follows.

Let

$$\begin{aligned} \varDelta _1= & {} \det \left[ \begin{array}{ccc} 2i\omega _0-r_{11} &{} -r_{12} &{} -r_{13}\\ -r_{21} &{} 2i\omega _0 &{} -r_{23}\\ 0 &{} -r_{32}e^{-2i\omega _0\widetilde{\tau }} &{} \ \ 2i\omega _0-r_{33} \end{array} \right] , \\ B_1= & {} \left[ \begin{array}{c} B^{(1)}_1\\ B^{(2)}_1\\ B^{(3)}_1 \end{array}\right] \\= & {} \left[ \begin{array}{c} \frac{1}{2}f^{(1)}_{200}d^2_1+f^{(1)}_{110}d_1+\frac{1}{2}f^{(1)}_{002}d^2_2\\ \frac{1}{2}f^{(2)}_{200}d^2_1+f^{(2)}_{110}d_1+f^{(2)}_{011}d_2+\frac{1}{2}f^{(2)}_{002}d^2_2\\ \gamma d_2e^{-i\omega _0\widetilde{\tau }}-\frac{r}{K}d^2_2 \end{array}\right] ,\\ \varDelta _{11}= & {} 2\det \left[ \begin{array}{ccc} B^{(1)}_1 &{} -r_{12} &{} -r_{13}\\ B^{(2)}_1 &{} 2i\omega _0 &{} -r_{23}\\ B^{(3)}_1 &{} \ \ -r_{32}e^{-2i\omega _0\widetilde{\tau }} &{} \ \ 2i\omega _0-r_{33} \end{array}\right] , \\ \varDelta _{12}= & {} 2\det \left[ \begin{array}{ccc} 2i\omega _0-r_{11} &{} \ \ B^{(1)}_1 &{} -r_{13}\\ -r_{21} &{} B^{(2)}_1 &{} -r_{23}\\ 0 &{} B^{(3)}_1 &{} \ \ 2i\omega _0-r_{33} \end{array}\right] ,\\ \varDelta _{13}= & {} 2\det \left[ \begin{array}{ccc} 2i\omega _0-r_{11} &{} -r_{12} &{} B^{(1)}_1\\ -r_{21} &{} 2i\omega _0 &{} B^{(2)}_1\\ 0 &{} -r_{32}e^{-2i\omega _0\widetilde{\tau }} &{} \ \ B^{(3)}_1 \end{array}\right] , \end{aligned}$$

then

$$\begin{aligned} E^{(1)}_1=\frac{\varDelta _{11}}{\varDelta _1}, \ E^{(2)}_1=\frac{\varDelta _{12}}{\varDelta _1}, \ E^{(3)}_1=\frac{\varDelta _{13}}{\varDelta _1}. \end{aligned}$$

Similarly, let

$$\begin{aligned} \varDelta _2= & {} \det \left[ \begin{array}{ccc} -r_{11} &{} -r_{12} &{} -r_{13}\\ -r_{21} &{} 0 &{} -r_{23}\\ 0 &{} -r_{32} &{} -r_{33} \end{array} \right] , \ B_2=\left[ \begin{array}{c} B^{(1)}_2\\ B^{(2)}_2\\ B^{(3)}_2 \end{array}\right] \\= & {} \left[ \begin{array}{c} \frac{1}{2}f^{(1)}_{200}|d_1|^2+f^{(1)}_{110}Re(d_1)+\frac{1}{2}f^{(1)}_{002}|d_2|^2\\ \frac{1}{2}f^{(2)}_{200}|d_1|^2+f^{(2)}_{110}Re(d_1)+f^{(2)}_{011}Re(d_2)+\frac{1}{2}f^{(2)}_{002}|d_2|^2\\ \gamma (Re(d_2e^{i\omega _0\widetilde{\tau }}))-\frac{r}{K}|d_2|^2 \end{array}\right] ,\\ \varDelta _{21}= & {} 2\det \left[ \begin{array}{ccc} B^{(1)}_2 &{} -r_{12} &{} -r_{13}\\ B^{(2)}_2 &{} 0 &{} -r_{23}\\ B^{(3)}_2 &{} -r_{32} &{} -r_{33} \end{array}\right] , \\ \varDelta _{22}= & {} 2\det \left[ \begin{array}{ccc} -r_{11} &{} B^{(1)}_2 &{} -r_{13}\\ -r_{21} &{} B^{(2)}_2 &{} -r_{23}\\ 0 &{} B^{(3)}_2 &{} -r_{33} \end{array}\right] ,\\ \varDelta _{23}= & {} 2\det \left[ \begin{array}{ccc} -r_{11} &{} -r_{12} &{} B^{(1)}_2\\ -r_{21} &{} 0 &{} B^{(2)}_2\\ 0 &{} -r_{32} &{} B^{(3)}_2 \end{array}\right] , \end{aligned}$$

then

$$\begin{aligned} E^{(1)}_2=\frac{\varDelta _{21}}{\varDelta _2}, \ E^{(2)}_2=\frac{\varDelta _{22}}{\varDelta _2}, \ E^{(3)}_2=\frac{\varDelta _{23}}{\varDelta _2}. \end{aligned}$$

Using these values, we determine $W_{20}$ and $W_{11}$ and hence $g_{21}$. Now, to determine the direction, stability, and period of bifurcating periodic solutions of system (20) at the critical value $\tau =\widetilde{\tau }$, we compute the following quantities:

$$\begin{aligned} C_1(0)= & {} \frac{i}{2\omega _0\widetilde{\tau }}\left( g_{20}g_{11}-2|g_{11}|^2-\frac{|g_{02}|^2}{3}\right) +\frac{g_{21}}{3}, \end{aligned}$$

(80)

$$\begin{aligned} \mu _2= & {} -\frac{Re\{C_1(0)\}}{Re\{\varPsi '(\widetilde{\tau })\}},\end{aligned}$$

(81)

$$\begin{aligned} \beta _2= & {} 2Re\{C_1(0)\},\end{aligned}$$

(82)

$$\begin{aligned} T_2= & {} -\frac{Im\{C_1(0)\}+\mu _2Im\{\varPsi '(\widetilde{\tau })\}}{\omega _0\widetilde{\tau }}. \end{aligned}$$

(83)