Asymptotic expansion on 3 nonlinear ordinary differential equations 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
$\dfrac{dc_0}{dt}, \dfrac{ds_0}{dt}, \dfrac{dq_0}{dt}, \dfrac{dc_1}{dt}, \dfrac{ds_1}{dt}$ and $\dfrac{dq_1}{dt}$
but i think $\dfrac{dc_1}{dt}$ will disappear while expanding and equating the up to first power of $\epsilon$, do i need to go further up to $\epsilon{^2}$ because $\dfrac{dc_1}{dt}$ is very important to find and we need 6 approximate differetial equations in total. what can i do? please some one help  me.
 A: This is an example of a singular perturbation problem and so the type of perturbative expansion you are attempting is not likely to give you anything useful.
I would recommend reading about multi-scale analysis, which can be found in most graduate books on nonlinear differential equations (Glenddings book has an introduction).
For a detailed exposition i would refer you to Kervokian and Coles book.
The system you have presented has $3$ time scales; fast time $\epsilon^{-1}t$, 'slow' time $t$, and 'really slow' time $\epsilon t$. 
On the fast time scale, $c$ will rapidly approach the slow manifold, ie the fixed point at $$\dot{c} =0 \implies c(q,s) = \frac{I\alpha}{(K_F+K_D+K_Ns+K_P(1−q))}.$$ 
You can then plug this into the equations for $\dot{s}$ and $\dot{q}$ to produce a two dimensional nonlinear equation for the motion on this slow manifold.
Note that in this reduced system, there is still $\epsilon$ floating around. Since it is not a coefficient of a derivative term, the perturbative expansion attempted in the original post will work.
