Modeling the effect of budget allocation on the abatement of atmospheric carbon dioxide

Abstract

An increment in carbon dioxide ($CO _2$) results in global warming posed a threat to the mankind. Mitigation of anthropogenic $CO _2$ emission is important for climate change alleviation. In this regard, some significant steps are taken by the government of every country, which requires budget. In this paper, to observe the effect of budget allocation on the abatement of atmospheric concentration of $CO _2$, a non-linear mathematical model is formulated and analyzed. In the modeling process, it is considered that a part of the available budget is used for the control of anthropogenic emission and the remaining part of budget is used for afforestation and reforestation. For the proposed model, feasibility and stability of all the equilibria have been discussed. From the model analysis, we have derived that how much budget one should spend on controlling the anthropogenic emission of $CO _2$ and afforestation/reforestation. Furthermore, numerical simulation has been performed to support analytical findings. It has been shown that the atmospheric level of $CO _2$ can be reduced to an innocuous level if the efficacy of allocated budget to control the anthropogenic emission of $CO _2$ and afforestation/reforestation increases. Moreover, it is found that the growth rate of budget allocation due to an increase in atmospheric level of carbon dioxide may cause stability switch through Hopf-bifurcation.

Appendices

Appendix A

Proof

From the third equation of the model system (1), we have

$$\begin{aligned} \frac{\text {d}F}{\text {d}t} \le uF\left( 1 - \frac{F}{M}\right) + \mu F. \end{aligned}$$

Using the theory of differential inequality, we have

$$\begin{aligned} \limsup _{t\rightarrow \infty } F(t) \le \frac{M(u+\mu )}{u} = F_m \text{(say) }. \end{aligned}$$

From the second equation of model system, we have

$$\begin{aligned} \frac{\text {d}N}{\text {d}t} \le (s + \pi \phi F_m)N - \frac{sN^2}{L}. \end{aligned}$$

Using the same argument as before

$$\begin{aligned} \limsup _{t\rightarrow \infty } N(t) \le \frac{L}{s}(s + \pi \phi F_m) = N_m \text{(say) }. \end{aligned}$$

Now, from the first equation of the model system

$$\begin{aligned} \frac{\text {d}C}{\text {d}t} \le -\alpha (C-C_0) + \lambda N_m. \end{aligned}$$

Then, we have

$$\begin{aligned} \limsup _{t\rightarrow \infty } C(t) \le \frac{\alpha C_0 + \lambda N_m}{\alpha }. \end{aligned}$$

Similarly, we can show that

$$\begin{aligned} \limsup _{t\rightarrow \infty } B(t) \le \frac{K}{r}\left( r + \frac{\eta \lambda N_m}{\alpha }\right) . \end{aligned}$$

$\square$

Appendix B

Proof

For any $N \ge 0$, we have $\frac{\text {d}N}{\text {d}t}\bigg |_{N=0} = 0$ which implies that $N = 0$ is invariant manifold. Due to continuity of system, we can conclude that N would never go below zero if its initial condition is non-negative. Therefore

$$\begin{aligned}&\frac{\text {d}C}{\text {d}t} \ge \alpha C_0 - \alpha C - \frac{\nu k B_m}{q_1 + kB_m}N_m - \lambda _1 CF_m.\\&\quad \Rightarrow \liminf _{t\rightarrow \infty } C(t) \ge \frac{\left( \alpha C_0 - \frac{\nu k B_m}{q_1 + kB_m}N_m\right) }{\alpha + \lambda _1F_m} = C_a \text{(say) }, \end{aligned}$$

provided that $\left( \alpha C_0 - \frac{\nu k B_m}{q_1 + kB_m}N_m\right) > 0$. Moreover

$$\begin{aligned}&\frac{\text {d}B}{\text {d}t} \ge rB - \frac{rB^2}{K} + \eta (C_a - C_0)B, \\&\quad \Rightarrow \liminf _{t\rightarrow \infty } B(t) \ge \frac{K}{r}\{r + \eta (C_a - C_0)\} = B_a \text{(say) } , \end{aligned}$$

provided $r + \eta (C_a - C_0). > 0$. From third equation of the model system (1), we have

$$\begin{aligned}&\frac{\text {d}F}{\text {d}t} \ge uF - \frac{uF^2}{M} - \phi N_m F + \frac{\mu (1-k)B_a}{q_2+(1-k)B_a}F, \\&\quad \Rightarrow \liminf _{t\rightarrow \infty } F(t) \ge \frac{M}{u}\left( u - \phi N_m + \frac{\mu (1-k)B_a}{q_2+(1-k)B_a}\right) = F_a \text{(say) }, \end{aligned}$$

provided that $\left( u - \phi N_m + \frac{\mu (1-k)B_a}{q_2+(1-k)B_a}\right) > 0$. From the second equation of the model (1), we have

$$\begin{aligned}&\frac{\text {d}N}{\text {d}t} \ge sN - \frac{sN^2}{L} - \theta (C_m - C_0)N + \pi \phi F_a N, \\&\quad \Rightarrow \liminf _{t\rightarrow \infty } N(t) \ge \frac{L}{s}(s - \theta (C_m - C_0) + \pi \phi F_a) = N_a \text{(say) }, \end{aligned}$$

provided that $(s - \theta (C_m - C_0) + \pi \phi F_a) > 0$. Taking $M_1 = \min (C_a, N_a, F_a, B_a)$, we have

$$\begin{aligned} M_1 \le \liminf _{t\rightarrow \infty }(C(t), N(t), F(t), B(t)), \end{aligned}$$

, and from Lemma 1, the system is uniformly bounded. Hence, the system is persistence. $\square$

Appendix C

Proof

In this part, we will proof the result for direction of bifurcating periodic solutions. For this, we translate the origin to the interior equilibrium $E^*$, by substituting $C = C^* + x_1$, $N = N^* + x_2$, $F = F^* + x_3$ and $B = B^* + x_4$, where $x = (x_1, x_2, x_3, x_4)^T$ are small perturbation. Now, we have following system:

$$\begin{aligned} \left( \begin{array}{c} \frac{\text {d}x_1}{\text {d}t} \\ \frac{\text {d}x_2}{\text {d}t} \\ \frac{\text {d}x_3}{\text {d}t} \\ \frac{\text {d}x_4}{\text {d}t} \end{array}\right) = \left( \begin{array}{c} g_1(x_1, x_2, x_3, x_4) \\ g_2(x_1, x_2, x_3, x_4) \\ g_3(x_1, x_2, x_3, x_4) \\ g_4(x_1, x_2, x_3, x_4) \end{array}\right) + O(|x|^3), \end{aligned}$$

where

$$\begin{aligned} g_1(x_1, x_2, x_3, x_4)&= -(\alpha + \lambda _1F^*)x_1 + \left( \lambda - \frac{\nu k B^*}{q_1 + kB^*}\right) x_2 - \lambda _1C^*x_3 - \frac{q_1k\nu N^*}{(q_1 + kB^*)^2}x_4\\&\quad - \lambda _1x_1x_3 - \frac{\nu kq_1x_2x_4}{(q_1+kB^*)^2} + \frac{\nu q_1k^2N^*x_4^2}{(q_1+kB^*)^3}, \\ g_2(x_1, x_2, x_3, x_4)&= -\theta N^*x_1 - \frac{sN^*}{L}x_2 + \pi \phi N^*x_3 - \frac{sx_2^2}{L} - \theta x_1x_2 + \pi \phi x_2x_3,\\ g_3(x_1, x_2, x_3, x_4)&= -\phi F^*x_2 - \frac{uF^*}{M} x_3 + \frac{q_2\mu (1-k)F^*}{(q_2 + (1-k)B^*)^2}x_4 -\frac{ux_3^2}{M} - \phi x_2x_3\\&\quad + \frac{\mu q_2(1-k)x_3x_4}{(q_2+(1-k)B^*)^2} - \frac{\mu (1-k)^2q_2F^*x_4^2}{(q_2+(1-k)B^*)^3}, \\ g_4(x_1, x_2, x_3, x_4)&= \eta B^*x_1 -\frac{rB^*}{K}x_4 - \frac{rx_4^2}{K} + \eta x_1x_4. \end{aligned}$$

In the above expression, we are not interested in coefficient of third or higher degree. Now, the system takes the form

$$\begin{aligned} \dot{x} = J^*x + G(x), \end{aligned}$$

(.1)

where

$$\begin{aligned} x = \left( \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ x_4 \end{array}\right) , J^* = \left( \begin{array}{cccc} -(\alpha + \lambda _1F^*) &{} \left( \lambda - \frac{\nu k B^*}{q_1 + kB^*}\right) &{} -\lambda _1C^* &{} \frac{-q_1k\nu N^*}{(q_1 + kB^*)^2} \\ -\theta N^* &{} -\frac{sN^*}{L} &{} \pi \phi N^* &{} 0 \\ 0 &{} -\phi F^* &{} -\frac{uF^*}{M} &{} \frac{q_2\mu (1-k)F^*}{(q_2 + (1-k)B^*)^2} \\ \eta B^* &{} 0 &{} 0 &{} -\frac{rB^*}{K} \end{array}\right) \end{aligned}$$

and

$$\begin{aligned} G(x) = \left( \begin{array}{c} g_1 \\ g_2 \\ g_3 \\ g_4 \end{array}\right) = \left( \begin{array}{c} -\lambda _1x_1x_3 - \frac{\nu kq_1x_2x_4}{(q_1+kB^*)^2} - \frac{\nu q_1k^2N^*x_4^2}{(q_1+kB^*)^3}\\ -\frac{sx_2^2}{L} - \theta x_1x_2 + \pi \phi x_2x_3 \\ -\frac{ux_3^2}{M} - \phi x_2x_3 + \frac{\mu q_2(1-k)x_3x_4}{(q_2+(1-k)B^*)^2} + \frac{\mu (1-k)^2q_2F^*x_4^2}{(q_2+(1-k)B^*)^3}\\ -\frac{rx_4^2}{K} + \eta x_1x_4 \end{array}\right) . \end{aligned}$$

The eigenvectors $u_1$, $u_2$, and $u_3$ of the Jacobian matrix $J^*$ corresponding to the eigenvalues $i\omega _0$, $\rho _3$, and $\rho _4$ are obtained as follows:

$$\begin{aligned}&u_1 = \left( \begin{array}{c} u_{11} - iu_{12}\\ u_{21} - iu_{22}\\ u_{31} - iu_{32}\\ u_{41} - iu_{42} \end{array} \right) , u_2 = \left( \begin{array}{c} u_{13}\\ u_{23}\\ u_{33}\\ u_{43} \end{array}\right) , \\&u_3 = \left( \begin{array}{c} u_{14}\\ u_{24}\\ u_{34}\\ u_{44} \end{array}\right) . \end{aligned}$$

Here

$$\begin{aligned}&u_{11} = -\frac{rB^*}{K}\left( \frac{sN^*}{L}\frac{uF^*}{M} - \omega ^2 + \pi \phi ^2N^*F^*\right) + \omega ^2\left( \frac{sN^*}{L} + \frac{uF^*}{M}\right) ,\\&u_{12} = -\left\{ \frac{rB^*}{K}\omega \left( \frac{sN^*}{L}+\frac{uF^*}{M}\right) - \omega \left( \frac{sN^*}{L}\frac{uF^*}{M} - \omega ^2 + \pi \phi ^2N^*F^*\right) \right\} ,\\&u_{21} = \frac{\theta N^*}{\eta B^*}\left( \frac{rB^*}{K}\frac{uF^*}{M} - \omega ^2\right) - \frac{\pi \phi N^*q_2\mu (1-k)F^*}{(q_2+(1-k)F^*)^2},\\&u_{22} = \left\{ \frac{\omega \theta N^*}{\eta B^*}\left( \frac{rB^*}{K}+\frac{uF^*}{M}\right) \right\} ,\\&u_{31} = -\frac{sN^*}{L}\frac{q_2\mu (1-k)F^*}{(q_2+(1-k)B^*)^2)} - \phi F^*\frac{\theta N^*}{\eta B^*}\frac{rB^*}{K},\\&u_{32} = -\left\{ \frac{\omega q_2\mu (1-k)F^*}{(q_2+(1-k)B^*)^2} + \frac{\omega \phi F^*\theta N^*}{\eta B^*}\right\} ,\\&u_{41} = -\frac{sN^*}{L}\frac{uF^*}{M} + \omega ^2 - \pi \phi ^2N^*F^*,\\&u_{42} = -\omega \left( \frac{sN^*}{L} + \frac{uF^*}{M}\right) ,\\&u_{13} = -\left( \frac{rB^*}{K} + \lambda _3\right) \left[ \left( \frac{sN^*}{L}+\lambda _3\right) \left( \frac{uF^*}{M}+\lambda _3\right) + \pi \phi ^2N^*F^*\right] ,\\&u_{23} = \frac{\theta N^*}{\eta B^*}\left( \frac{rB^*}{K}+\lambda _3\right) \left( \frac{uF^*}{M}+\lambda _3\right) - \frac{\pi \phi N^*q_2\mu (1-k)F^*}{(q_2+(1-k)B^*)^2},\\&u_{33} = -\left( \frac{sN^*}{L} + \lambda _3\right) \left( \frac{q_2\mu (1-k)F^*}{q_2+(1-k)B^*)^2}\right) - \frac{\phi F^*\theta N^*}{\eta B^*}\left( \frac{rB^*}{K}+\lambda _3\right) ,\\&u_{43} = -\left( \frac{sN^*}{L} + \lambda _3\right) \left( \frac{uF^*}{M} + \lambda _3\right) - \pi \phi ^2N^*F^*,\\&u_{14} = -\left( \frac{rB^*}{K} + \lambda _4\right) \left[ \left( \frac{sN^*}{L}+\lambda _4\right) \left( \frac{uF^*}{M}+\lambda _4\right) +\pi \phi ^2N^*F^*\right] ,\\&u_{24} = \frac{\theta N^*}{\eta B^*}\left( \frac{rB^*}{K}+\lambda _3\right) \left( \frac{uF^*}{M}+\lambda _3\right) -\frac{\pi \phi N^*q_2\mu (1-k)F^*}{(q_2+(1-k)B^*)^2},\\&u_{34} = -\left( \frac{sN^*}{L} + \lambda _4\right) \left( \frac{q_2\mu (1-k)F^*}{q_2+(1-k)B^*)^2}\right) - \frac{\phi F^*\theta N^*}{\eta B^*}\left( \frac{rB^*}{K}+\lambda _4\right) ,\\&u_{44} = -\left( \frac{sN^*}{L} + \lambda _4\right) \left( \frac{uF^*}{M} + \lambda _4\right) - \pi \phi ^2N^*F^*.\\ \end{aligned}$$

Define $U = (Re(u_1), -Im(u_1), u_2, u_3)$, that is

$$\begin{aligned} U = \left( \begin{array}{cccc} u_{11} &{} u_{12} &{} u_{13} &{} u_{14} \\ u_{21} &{} u_{22} &{} u_{23} &{} u_{24} \\ u_{31} &{} u_{32} &{} u_{33} &{} u_{34} \\ u_{41} &{} u_{42} &{} u_{43} &{} u_{44} \end{array}\right) . \end{aligned}$$

The matrix U is non-singular, such that

$$\begin{aligned} U^{-1}PU = \left( \begin{array}{cccc} 0 &{} -\omega _0 &{} 0 &{} 0 \\ \omega _0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} \rho _3 &{} 0 \\ 0 &{} 0 &{} 0 &{} \rho _4 \end{array}\right) . \end{aligned}$$

Inverse of matrix U is given by

$$\begin{aligned}&U^{-1} = \left( \begin{array}{cccc} y_{11} &{} y_{12} &{} y_{13} &{} y_{14} \\ y_{21} &{} y_{22} &{} y_{23} &{} y_{24} \\ y_{31} &{} y_{32} &{} y_{33} &{} y_{34} \\ y_{41} &{} y_{42} &{} y_{43} &{} y_{44} \end{array}\right) .\\&y_{11} = \frac{1}{\Delta }[u_{22}(u_{33}u_{44} - u_{34}u_{43}) + u_{23}(u_{42}u_{34} - u_{44}u_{32}) + u_{24}(u_{43}u_{32} - u_{42}u_{33})],\\&y_{12} = \frac{1}{\Delta }[u_{12}(u_{43}u_{34} - u_{44}u_{33}) + u_{13}(u_{32}u_{44} - u_{34}u_{42}) + u_{14}(u_{33}u_{42} - u_{43}u_{32})],\\&y_{13} = \frac{1}{\Delta }[u_{12}(u_{44}u_{23} - u_{43}u_{24}) + u_{13}(u_{42}u_{24} - u_{44}u_{22}) + u_{14}(u_{43}u_{22} - u_{23}u_{42})],\\&y_{14} = \frac{1}{\Delta }[u_{12}(u_{24}u_{33} - u_{23}u_{34}) + u_{13}(u_{22}u_{34} - u_{32}u_{24}) + u_{14}(u_{23}u_{32} - u_{33}u_{22})],\\&y_{21} = \frac{1}{\Delta }[u_{21}(u_{34}u_{43} - u_{44}u_{33}) + u_{23}(u_{44}u_{31} - u_{34}u_{41}) + u_{24}(u_{41}u_{33} - u_{31}u_{43})],\\&y_{22} = \frac{1}{\Delta }[u_{11}(u_{33}u_{44} - u_{34}u_{43}) + u_{13}(u_{34}u_{41} - u_{44}u_{31}) + u_{14}(u_{31}u_{43} - u_{33}u_{41})],\\&y_{23} = \frac{1}{\Delta }[u_{11}(u_{43}u_{24} - u_{44}u_{23}) + u_{13}(u_{21}u_{44} - u_{24}u_{41}) + u_{14}(u_{41}u_{23} - u_{43}u_{21})],\\&y_{24} = \frac{1}{\Delta }[u_{11}(u_{23}u_{34} - u_{24}u_{33}) + u_{13}(u_{31}u_{24} - u_{34}u_{21}) + u_{14}(u_{33}u_{21} - u_{23}u_{31})],\\&y_{31} = \frac{1}{\Delta }[u_{21}(u_{32}u_{44} - u_{42}u_{34}) + u_{22}(u_{34}u_{41} - u_{31}u_{44}) + u_{24}(u_{31}u_{42} - u_{32}u_{41})],\\&y_{32} = \frac{1}{\Delta }[u_{11}(u_{42}u_{34} - u_{44}u_{32}) + u_{12}(u_{44}u_{31} - u_{34}u_{41}) + u_{14}(u_{41}u_{32} - u_{42}u_{31})],\\&y_{33} = \frac{1}{\Delta }[u_{11}(u_{44}u_{22} - u_{42}u_{24}) + u_{12}(u_{41}u_{24} - u_{21}u_{44}) + u_{14}(u_{41}u_{32} - u_{42}u_{31})],\\&y_{34} = \frac{1}{\Delta }[u_{11}(u_{32}u_{24} - u_{34}u_{22}) + u_{12}(u_{21}u_{34} - u_{24}u_{31}) + u_{14}(u_{22}u_{31} - u_{21}u_{32})],\\&y_{41} = \frac{1}{\Delta }[u_{21}(u_{42}u_{33} - u_{43}u_{32}) + u_{22}(u_{43}u_{31} - u_{33}u_{41}) + u_{23}(u_{41}u_{32} - u_{31}u_{42})],\\&y_{42} = \frac{1}{\Delta }[u_{11}(u_{43}u_{32} - u_{42}u_{33}) + u_{12}(u_{33}u_{41} - u_{31}u_{43}) + u_{13}(u_{42}u_{31} - u_{32}u_{41})],\\&y_{43} = \frac{1}{\Delta }[u_{11}(u_{42}u_{23} - u_{43}u_{22}) + u_{12}(u_{43}u_{21} - u_{23}u_{41}) + u_{13}(u_{41}u_{22} - u_{21}u_{42})],\\&y_{44} = \frac{1}{\Delta }[u_{11}(u_{33}u_{22} - u_{32}u_{23}) + y_{12}(u_{31}u_{23} - u_{21}u_{33}) + u_{13}(u_{32}u_{21} - u_{31}u_{22})]. \end{aligned}$$

Consider the transformation $x = Uw$, i.e., $w = U^{-1}x$, where $w = (w_1, w_2, w_3, w_4)$. Under the linear transformation, the system takes the form

$$\begin{aligned} \dot{w} = (U^{-1}J^*U)w + g(w), \end{aligned}$$

(.2)

where $g(w) = U^{-1}G(Uw)$. This can be written as

$$\begin{aligned} \dot{w_1}&= -\omega _0w_2 + g^1(w_1, w_2, w_3, w_4)\\ \dot{w_2}&= -\omega _0w_1 + g^2(w_1, w_2, w_3, w_4)\\ \dot{w_3}&= \rho _3w_3 + g^3(w_1, w_2, w_3, w_4)\\ \dot{w_3}&= \rho _4w_4 + g^4(w_1, w_2, w_3, w_4),\\ \end{aligned}$$

where $g = (g^1, g^2, g^3, g^4)^T$

$$\begin{aligned} g^1 = y_{11}h_1 + y_{12}h_2 + y_{13}h_3 + y_{14}h_4,\\ g^2 = y_{21}h_1 + y_{22}h_2 + y_{23}h_3 + y_{24}h_4,\\ g^3 = y_{31}h_1 + y_{32}h_2 + y_{33}h_3 + y_{34}h_4,\\ g^4 = y_{41}h_1 + y_{42}h_2 + y_{43}h_3 + y_{44}h_4; \end{aligned}$$

here,

$$\begin{aligned} h_1 =&-\lambda _1(u_{11}w_1 + u_{12}w_2 + u_{13}w_3 + u_{14}w_4)(u_{31}w_1 + u_{32}w_2 + u_{33}w_3 + u_{34}w_4)\\&- \frac{\nu kq_1(u_{21}w_1 + u_{22}w_2 + u_{23}w_3 + u_{24}w_4)(u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4)}{(q_1+kB^*)^2} \\&- \frac{\nu q_1^2k^2N^*}{(q_1+kB^*)^2}(u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4)^2, \\ h_2 =&-\frac{s}{L}(u_{21}w_1 + u_{22}w_2 + u_{23}w_3 + u_{24}w_4)^2 - \theta (u_{11}w_1 + u_{12}w_2 + u_{13}w_3 + u_{14}w_4)\\&(u_{21}w_1 + u_{22}w_2 + u_{23}w_3 + u_{24}w_4) \\&+ \pi \phi (u_{21}w_1 + u_{22}w_2 + u_{23}w_3 + u_{24}w_4)(u_{31}w_1 + u_{32}w_2 + u_{33}w_3 + u_{34}w_4), \\ h_3 =&-\frac{u}{M}(u_{31}w_1 + u_{32}w_2 + u_{33}w_3 + u_{34}w_4)^2 - \phi (u_{21}w_1 + u_{22}w_2 + u_{23}w_3 + u_{24}w_4)\\&(u_{31}w_1 + u_{32}w_2 + u_{33}w_3 + u_{34}w_4)\\&+ \frac{\mu (1-k)^2q_2^2F^*}{(q_2+(1-k)B^*)^3}(u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4)^2\\&+ \frac{\mu q_2(1-k)}{(q_2+(1-k)B^*)^2}(u_{31}w_1 + u_{32}w_2 + u_{33}w_3 + u_{34}w_4)\\&\times (u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4),\\ h_4 =&-\frac{r}{K}(u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4)^2 + \eta (u_{11}w_1 + u_{12}w_2 + u_{13}w_3 + u_{14}w_4)\\&(u_{41}w_1 + u_{42}w_2 + u_{43}w_3 + u_{44}w_4). \end{aligned}$$

Furthermore, we can calculate $h_{11}$, $h_{02}$, $h_{20}$, $H_{21}$, $H_{110}^1$, $H_{110}^2$, $H_{101}^1$, $H_{101}^2$, $\sigma _{11}^1$, $\sigma _{11}^2$, $\sigma _{20}^1$, $\sigma _{20}^2$ following the procedure given in Hassard et al (Hassard et al. 1981). Using the above, we can find the following quantities:

$$\begin{aligned} h_{21}&= H_{21}+ 2(H_{110}^1\sigma _{11}^1 + H_{110}^2\sigma _{11}^2) + H_{101}^1\sigma _{20}^1 + H_{101}^2\sigma _{20}^2,\\ c_1(0)&= \frac{i}{2\omega _0}\left( h_{11}h_{20}-2|h_{11}|^2 - \frac{|h_{02}|^2}{3}\right) + \frac{h_{21}}{2},\\ \mu _2&= -\frac{Re(c_1(0))}{\phi '(0)},\\ \tau _2&= -\frac{(Im(c_1(0))+\mu _2\sigma '(0))}{\omega _0},\\ \beta _2&= -2\mu _2\phi '(0), \end{aligned}$$

where, $\phi '(0) = \frac{d}{d\eta }(Re(\rho (\eta )))|_{\eta =\eta _c}$ and $\sigma '(0) = \frac{d}{d\eta }(Im(\rho (\eta )))|_{\eta =\eta _c}.$ $\square$