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.