Regular perturbation nonlinear problem

[Carla1985]

Hi all, I have this (nondimensionalised) system of ODEs that I am trying to analyse:
\[
\begin{align}
\frac{dr}{dt}= &\ - \left(\alpha+\frac{\epsilon}{2}\right)r + \left(1-\frac{\epsilon}{2}\right)\alpha p - \alpha^2\beta r p + \frac{\epsilon}{2} \\
\frac{dp}{dt}= &\ \left(1-\frac{\epsilon}{2\alpha}\right)r - \left(1+\frac{\epsilon}{2}\right)p - \alpha\beta r p + \frac{\epsilon}{2\alpha}
\end{align}
\]
with initial conditions $r(0)=1, p(0)=0$ and $\epsilon$ is a small parameter. After substituting in the asymptotic approximations $r\approx r_0+\epsilon r_1+...$, $p\approx p_0+\epsilon p_1+...$ I get a leading order problem as
\[
\begin{align}
\frac{dr_0}{dt}= &\ - \alpha r_0 + \alpha p_0 -\alpha^2\beta r_0 p_0 \\
\frac{dp_0}{dt}= &\ r_0 - p_0 -\alpha\beta r_0 p_0
\end{align}
\]

Clearly, this is still of no help as it's still nonlinear. So my question is, is there any way of getting past this?

The only other piece of information I have is that at equilibrium the solution for both will be $O(\epsilon^{1/2})$. I did attempt to rescale for this but I then get a singular perturbation problem in which the limit of the inner solution is infinity so I could not do the matching (I will explain in more detail what I did for this if needed). If anyone could suggest a way in which I can solve this problem I would be very grateful.

Thank you!
Carla

 

[Theia]

What are \(\displaystyle \alpha\) and \(\displaystyle \beta\)? What we know about them?

 

[HOI]

The non-linearity has nothing to do with $$\epsilon$$ but rather with $$\alpha$$ and $$\beta$$.

 

[Carla1985]

$\alpha$ and $\beta$ are grouped parameters from how I did the nondimensionalisation. Basically the system is a type of drug-receptor binding system. After nondimensionalising I have 3 parameters: $\alpha$, $\beta$ and $\gamma$. We know that $\alpha$ is $O(1)$ but have limit information about $beta$ and $gamma$. So we are trying to use asymptotic to explore these. To do this we fix one of the parameters to also be $O(1)$ and set the other to be either small ($\epsilon$) or large ($1/\epsilon$). I have done the cases of $\beta=\epsilon$, $\beta=1/\epsilon$ and $\gamma=1/\epsilon$ without too many issues. This one that I'm having trouble with is $\gamma=\epsilon$.

I understand the answer may well be that in this situation there is nothing I can do further but I thought I would ask in case there was something else I could do with it that I wasn't aware of. Either way I appreciate the help.

Thanks
Carla

 

[Theia]

In that case, I'd simply use either numerical methods to find out the behaviour of the solution, or use some sort of series solution to approximate the solution.

 

[Alone]

$\alpha$ and $\beta$ are grouped parameters from how I did the nondimensionalisation. Basically the system is a type of drug-receptor binding system. After nondimensionalising I have 3 parameters: $\alpha$, $\beta$ and $\gamma$. We know that $\alpha$ is $O(1)$ but have limit information about $beta$ and $gamma$. So we are trying to use asymptotic to explore these. To do this we fix one of the parameters to also be $O(1)$ and set the other to be either small ($\epsilon$) or large ($1/\epsilon$). I have done the cases of $\beta=\epsilon$, $\beta=1/\epsilon$ and $\gamma=1/\epsilon$ without too many issues. This one that I'm having trouble with is $\gamma=\epsilon$.

I understand the answer may well be that in this situation there is nothing I can do further but I thought I would ask in case there was something else I could do with it that I wasn't aware of. Either way I appreciate the help.

Thanks
Carla

In your ODEs in your OP I don't see any mention of a $\gamma$ there, where should it appear?