MHB Regular perturbation nonlinear problem

Carla1985
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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
 
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What are $$\alpha$$ and $$\beta$$? What we know about them?
 
The non-linearity has nothing to do with \epsilon but rather with \alpha and \beta.
 
$\alpha$ 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 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.
 
Carla1985 said:
$\alpha$ 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?
 
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