The 3 nonlinear differential equations are as follows

\begin{equation}

\epsilon \frac{dc}{dt}=\alpha I + \ c (-K_F - K_D-K_N s-K_P(1-q)), \nonumber

\end{equation}

\begin{equation}

\frac{ds}{dt}= \lambda_b P_C \ \epsilon \ c (1-s)- \lambda_r (1-q) \ s, \nonumber

\end{equation}

\begin{equation}

\frac{dq}{dt}= K_P (1-q) \frac{P_C}{P_Q} \ \ c - \gamma \ q, \nonumber

\end{equation}

I want to use asymptotic expansion on $c, s$ and $q$.

And values of parameters are:

$K_F = 6.7 \times 10^{-2},$

$K_N = 6.03 \times 10^{-1}$

$K_P = 2.92 \times 10^{-2}$,

$K_D = 4.94 \times 10^{-2}$,

$\lambda_b= 0.0087$,

$I=1200$

$P_C = 3 \times 10^{11}$

$P_Q = 2.304 \times 10^{9}$

$\gamma=2.74 $

$\lambda_{b}=0.0087 $

$\lambda_{r}= 835$

$\alpha=1.14437 \times 10^{-3}$

For initial conditions:

\begin{equation}

c_0(0)= c(0) = 0.25 \nonumber

\end{equation}

\begin{equation}

s_0(0)= cs(0) = 0.02 \nonumber \nonumber

\end{equation}

\begin{equation}

q_0(0)=q(0) = 0.98 \nonumber \nonumber

\end{equation}

and

\begin{equation}

c_i(0)= 0, \ i>0\nonumber

\end{equation}

\begin{equation}

s_i(0)= 0, \ i>0 \nonumber \nonumber

\end{equation}

\begin{equation}

q_i(0)=0, i>0. \nonumber \nonumber

\end{equation}

=> i started with the expansions :

\begin{equation}

c= c_0+ \epsilon c_1 + \epsilon^2 c_2+......... \nonumber

\end{equation}

\begin{equation}

s= s_0+ \epsilon s_1 + \epsilon^2 s_2+......... \nonumber

\end{equation}

\begin{equation}

q= q_0+ \epsilon q_1 + \epsilon^2 q_2+......... \nonumber

\end{equation}

we are only interseted in up to fisrt power of $\epsilon$.

so, we should get total 6 approximate differential equations to get answer for

$\frac{dc_0}{dt}, \frac{ds_0}{dt}, \frac{dq_0}{dt}, \frac{dc_1}{dt}, \frac{ds_1}{dt}$ and $\frac{dq_1}{dt}$

but i think $\frac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\frac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help me.